Description
Credit risk refers to the risk that a borrower will default on any type of debt by failing to make payments which it is obligated to do
ABSTRACT
Title of Dissertation: THREE ESSAYS ON MORTGAGE BACKED
SECURITIES: HEDGING INTEREST RATE AND
CREDIT RISKS
Jian Chen, Doctor of Philosophy, 2003
Dissertation directed by: Professor Michael C. Fu
The Robert H. Smith School of Business
This dissertation includes three essays on hedging the interest rate and credit risks
of Mortgage-Backed Securities (MBS).
Essay one addresses the problem of how to efficiently estimate interest rate
sensitivity parameters of MBS. To do this in Monte Carlo simulation, we derive
perturbation analysis (PA) gradient estimators in a general setting. Then we apply the
Hull-White interest rate model and a common prepayment model to derive the
corresponding specific PA estimators, assuming the shock of interest rate term structure
takes the form of a trigonometric polynomial series. Numerical experiments comparing
finite difference (FD) estimators with our PA estimators indicate that the PA estimators
can provide better accuracy than FD estimators, while using much lower computational
cost. Using the estimators, we analyze the impact of term structure shifts on various
mortgage products. Based these analysis, we propose a new product to mitigate interest
rate risk.
Essay two addresses the problem of how to measure interest rate yield curve shift
more realistically, and how to use these risk measures to hedge the interest rate risk of
MBS. We use a Principal Components Analysis (PCA) approach to analyze historical
interest rate data, and acquire the volatility factors we need in Heath-Jarrow-Morton
interest rate model simulation. Then we propose a hedging algorithm to hedge MBS,
based on PA gradient estimators derived upon these PCA factors. Our results show that
the new hedging method can achieve much better hedging efficiency than traditional
duration and convexity hedging.
Essay three addresses the application a new regression method on credit spread
data. Previous research has shown that variables in traditional structural model have
limited explanatory power in credit spread regression. We argue that this is partially due
to the non-constancy of the credit spread gradients to state variables. We use a Random
Coefficient Regression (RCR) model to accommodate this problem. The explanatory
power increases dramatically with the new RCR model, without adding new independent
variables. This is the first work to address the dependence between credit spread
sensitivities and state variables of structural in a systematic way. Also our estimates are
consistent with prediction from Merton’s structural model.
THREE ESSAYS ON MORTGAGE BACKED SECURITIES:
HEDGING INTEREST RATE AND CREDIT RISKS
By
Jian Chen
Dissertation submitted to the Faculty of the Graduate School
of the University of Maryland in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
November 2003
Advisory Committee:
Professor Michael Fu, Chair
Professor Dilip Madan
Professor Haluk Unal
Professor Nengjiu Ju
Professor Eric Slud
©Copyright by
Jian Chen
2003
ii
Dedicated to
Huixian Jackie Xu, my lovely wife
Li Xueqing and Chen Zhixuan, my parents
Chuan Chen, my son
iii
ACKNOWLEDGMENTS
I would like to sincerely thank my advisor, Dr. Fu, who admitted me six years ago, and
has arduously helped me since then, especially during the lengthy process of my
dissertation writing. I also would like to thank Dr. Unal, for providing me most valuable
advice for my last essay. I also would like to thank Dr. Madan, Dr. Ju, and Dr. Slud for
being my committee members and providing a lot of insightful comments on the
dissertation proposal.
A significant portion of this dissertation is completed during my tenure at Fannie Mae. I
benefited a lot from conversations with my colleagues there. I would like to thank Dr.
Yigao Liang for his inputs on Essay I. I would like to thank Dr. Alex Philipov and Dr.
Arash Sotoodehnia for discussion on HJM model. I want to thank Ms. CJ Zhao on credit
spread discussion. I also want to thank Dr. Jay Guo, who helped me a lot for the last
essay. Also I would like to thank Dr. Levant Guntay of Indiana University, who kindly
provided some data for the last essay, and has given me many research ideas during our
discussions. Needless to say, all errors are mine own.
All opinions expressed in this dissertation are not Fannie Mae’s but mine own.
iv
TABLE OF CONTENTS
List of Figures vii
List of Tables x
List of Abbreviations xi
1. Introduction 1
1.1 Efficient Sensitivity Analysis of Mortgage Backed Securities 3
1.2 Hedging MBS in HJM Framework 9
1.3 Hedging Credit Risk of MBS: A Random Coefficient Approach 10
2. Efficient Sensitivity Analysis of Mortgage Backed Securities 12
2.1 Problem Setting 9
2.2 Derivation of General PA Estimators 14
2.2.1 Gradient Estimator for Cash Flow 15
2.2.2 Gradient Estimator for Discounting Factor 19
2.3 Applying the Gradients 20
2.3.1 Interest Model Setup 20
2.3.2 Trigonometric Polynomial Shocks 23
2.3.3 Derivation of Gradients with respect to Modified Fourier Series 27
2.3.4 Derivation of Gradients with respect to Volatility: Vega 31
2.3.5 Derivation of Second Order Gradients: Gamma 31
2.3.6 Derivation of ARM estimators 35
2.4 Numerical Example 41
2.4.1 Specification of Numerical Example 41
v
2.4.2 Comparison of PA and FD estimators 42
2.4.3 Result Analysis 53
2.5 Interpretation of the Results 60
2.5.1 Overview of the Results 60
2.5.2 Modified Fourier Shock Impact 62
2.5.3 Potential New Product 70
2.6 Conclusion 75
3. Hedging MBS in HJM Framework 77
3.1 Motivation 77
3.2 Estimation of Volatility Factors via PCA 80
3.3 Simulation in HJM Framework 84
3.4 Deriving PA Estimators in HJM Framework 86
3.5 Hedging MBS in HJM Framework 89
3.6 Hedging Performance Analysis 92
3.7 Conclusion 94
4. Hedging Credit Risk of MBS: A Random Coefficient Approach 95
4.1 Motivation 95
4.2 Literature Review 100
4.3 Introduction to Random Coefficient Model 104
4.4 Random Coefficient Model for Credit Spread Changes 107
4.5 Data Description 112
4.6 Results Analysis 115
4.6.1 Dependence of Credit Spread Sensitivities to State Variables 115
vi
4.6.2 Results by Rating and Maturity 123
4.6.3 Non-Constancy of Credit Spread Sensitivities 130
4.7 Conclusion and Future Work 135
Bibliography 136
vii
LIST OF FIGURES
2.1 ?R(0,t) with Original Fourier series 25
2.2 ?R(0,t) with T
0
=10 modified Fourier series 25
2.3 Coefficients Estimation for Modified Fourier series 27
2.4 WAC as a function of Index 37
2.5 Gradient Estimator Comparison for ?d(t)/ ??
n
44
2.6 Gradient Estimator Comparison for ?c(t)/ ??
n
44
2.7 Gradient Estimator Comparison for ?PV(t)/ ??
n
45
2.8 95% Confidence Interval for dPV(t)/d?
n
45
2.9 Gradient Estimator Comparison for ?d(t)/?? 46
2.10 Gradient Estimator Comparison for ?c(t)/?? 47
2.11 Gradient Estimator Comparison for ?PV(t)/?? 47
2.12 95% Confidence Interval for dPV(t)/d? 48
2.13 gamma estimators for
2
2
) (
i
t d
? ?
?
, i=1, 2, 3, 4 49
2.14 gamma estimators for
2
2
) (
i
t CPR
? ?
?
, i=1, 2, 3, 4 49
2.15 gamma estimators for
2
2
) (
i
t CF
? ?
?
, i=1, 2, 3, 4 50
2.16 gamma estimators for
2
2
) (
i
t PV
? ?
?
, i=1, 2, 3, 4 50
2.17 Gradient Estimator Comparison for
i
t WAC
? ?
? ) (
, i=1, 2, 3, 4 51
viii
2.18 Gradient Estimator Comparison for
i
t PV
? ?
? ) (
, i=1, 2, 3, 4 52
2.19 Gradient Estimator Comparison for
? ?
? ) (t PV
52
2.20 Difference of FD/PA ?f(0, t)/ ??
n
estimators 57
2.21 Fuction of xe
-x
58
2.22 Duration vs. Products 62
2.23 The Impact of Modified Fourier Order 0 on FRM30 63
2.24 The Impact of Modified Fourier Order 1 on FRM30 64
2.25 The Impact of Modified Fourier Order 2 on FRM30 65
2.26 The Impact of Modified Fourier Order 3 on FRM30 66
2.27 The Impact of Modified Fourier Order 0 on ARM TSY 1 67
2.28 The Impact of Modified Fourier Order 1 on ARM TSY 1 68
2.29 The Impact of Modified Fourier Order 2 on ARM TSY 1 69
2.30 The Impact of Modified Fourier Order 3 on ARM TSY 1 70
2.31 10-Year T Rate, 1-Year T Rate, and mortgage rate 73
2.32 New ARM TSY 10 Durations 74
3.1 The first four principal components 82
3.2 Match monthly yield curve shift 83
3.3 Match annual yield curve shift 83
3.4 Mean Hedging Error of PCA vs. D&C 93
3.5 STD of Hedging Error: PCA vs. D&C 93
4.1 Credit Spread vs. Risk-free Rate 118
4.2 Credit Spread vs. Volatility 119
ix
4.3 Sensitivity to Volatility at different Leverage 120
4.4 Sensitivity to Volatility vs. Interest Rate 121
4.5 Sensitivity to Volatility vs. Maturity 122
4.6 Coefficient for ?r in RCR vs. Linear Model 130
4.7 Coefficient for ?vol in RCR vs. Linear Model 131
4.8 Coefficient for ?lev in RCR vs. Linear Model 131
4.9 Three-Month Treasury Rate from 1990 to 1997 133
4.10 VIX index from 1990 to 1997 133
x
LIST OF TABLES
2.1 Product Specification for Mortgage Pricing 41
2.2 Comparison of PA/FD Duration 53
2.3 Comparison of Convexity Estimators 55
2.4 Comparison of Computing Costs 55
2.5 Durations of Different Products 60
3.1 Statistics for Principal Components 81
4.1 Comparison of three papers on credit spread regression 102
4.2 Comparison of RCR vs. linear model 116
4.3 Relationship between state variables and credit spread sensitivities 117
4.4 RCR coefficients for AA-AAA group 123
4.5 RCR coefficients for A group 124
4.6 RCR coefficients for BBB group 125
4.7 RCR coefficients for BB group 126
4.8 RCR coefficients for B and other group 127
4.9 Summary of RCR coefficients 128
xi
LIST OF ABBREVIATIONS
A(t) Amortization Factor at time t
AGE(t) Aging Multiplier, a parameter to capture the aging effect in prepayment
rate
ARM Adjustable Rate Mortgage
B(t) Balance at time t
BM(t) Burnout Multiplier, a parameter to capture the burnout effect in
prepayment rate
C(t) Cash flow at time t
CMO Collateralized Mortgage Obligation, a special type of MBS
CPR Conditional Prepayment Rate, annualized prepayment rate
CS(t) Credit Spread at maturity t
D(t) Discounting Factor at time t
f(0,t) Instantaneous forward rate starting from t observed at now
FNMA Federal National Mortgage Association, also known as Fannie Mae
FD Finite Difference
FHLMC Federal Home Loan Mortgage Corporation, also known as Freddie Mac
FRM Fixed Rate Mortgage
GNMA Government National Mortgage Association, also known as Ginnie Mae
GLS Generalized Least Square
GSE Government-Sponsored Enterprise, mainly refers to Fannie Mae,
Freddie Mac.
xii
H(t) Haircut at maturity t
HJM Heath-Jarrow-Morton interest rate model, defined in Heath et al. [1992]
IP(t) Interest Payment at time t
lev
t
Leverage at time t
LTV Loan to Value ratio, an 80 LTV loan means the loan amounts for 80% of
the property value
MBS Mortgage-Backed Securities
MM(t) Monthly Multiplier, a parameter to capture the seasonal effect in
prepayment rate
MP(t) Mortgage Monthly Payment at time t
OFHEO Office of Federal Housing Enterprise Oversight, a government agency
under Department of Housing and Urban Development, regulator of
Fannie Mae and Freddie Mac.
OLS Ordinary Least Square
PA Perturbation Analysis
PCA Principal Components Analysis
PDE Partial Differential Equation
PMI Primary Mortgage Insurance
PP(t) Principal Prepayment at time t
PV(t) Present Value of Cash flow at time t
R(0,t) Spot rate for maturity t observed at now
r(t) Short rate at time t
r
10
(t) 10-year rate at time t
xiii
RCR Random Coefficient Regression
REO Real Estate Owned by the GSEs, in case borrower defaults
RI(t) Refinance Incentive, a parameter to capture the refinance incentive
effect in prepayment rate
SMM Simple Monthly Mortality, monthly prepayment rate
SP(t) Scheduled Principal Payment at time t
TPP(t) Total Principal Payment at time t
Vega The security price sensitivity to volatility
WAC Weighted Average Coupon rate for MBS
WAM Weighted Average Maturity for MBS
w.r.t. with respect to
1
Chapter 1
Introduction
Mortgage-backed securities (MBS) have become increasingly important fixed
income instruments, both because of their volume and the role they play in fund
investment and portfolio management. However, there has not been a very
comprehensive set of risk indicators to measure and manage the risks involved with
MBS. Hedging the interest rate and credit risk of MBS remains a complicated problem in
the fixed income industry. This dissertation develops a set of risk measures for interest
rate risk and credit risk, and then attempts to hedge the risks effectively using such risk
measures. Specifically, the dissertation consists of three essays addressing the following
problems: efficiently estimating these new measures of interest rate risk of MBS, hedging
MBS with these new measures, and hedging the credit risk of MBS with advanced
models for credit spread regression.
The first essay is mainly positioned to answer the following research questions:
• How to measure the interest rate risk in a more comprehensive approach, rather than
simply using the traditional duration
1
and convexity
2
?
1
Duration is the first order derivative of the price of a fixed income security to interest rate, expressed as a
percentage change, see Fabozzi [2001] for more details.
2
Convexity is the second order derivative of the price of a fixed income security to interest rate, expressed
as a percentage change, see Fabozzi [2001] for more details.
2
• How to efficiently estimate the risk measures if more factors are introduced into the
measurement problem?
In answering these two questions, based on the results over a broad spectrum of mortgage
products, we propose a new mortgage product, which could be attractive to MBS
investors and mortgage borrowers.
The second essay tries to answer the following questions:
• What would be a realistic method to measure the term structure shift?
• How can we hedge MBS effectively with these measures, in a general interest rate
model framework?
We use Principal Components Analysis (PCA) method to extract the empirical volatility
factors of term structure, which provides some justification for the form of possible term
structure shifts proposed in the first essay. Then we use the Heath-Jarrow-Morton model
to incorporate the factors in developing new risk measures and show that the hedging
effectiveness is far better than traditional duration/convexity hedging.
The third essay is related to credit risks MBS issuers incur when they purchase
mortgage pool insurance from a third party, and attempts to answer the following
questions:
• How to estimate the sensitivity of credit spread in a regression framework more
effectively than a simple linear regression model?
• What implication does the model have on traditional structural models for credit
spread?
3
We use a Random Coefficient Regression (RCR) model to build our regression model for
credit spread changes. This model has explicit sensitivity measures dependent on state
variables. We acquire much better explanatory power with this new model, without
adding new independent state variables. Also our model supports the dependence of
sensitivity of credit spread on state variables predicted by Merton’s structural model
(Merton [1974]).
1.1 Efficient Sensitivity Analysis of Mortgage Backed Securities
A mortgage-backed security (MBS) is a security collateralized by residential or
commercial mortgage loans. An MBS is generally securitized, guaranteed and issued by
three major MBS originating agencies: Ginnie Mae, Fannie Mae, and Freddie Mac. The
cash flow of an MBS is generally the collected payment from the mortgage borrower,
after the deduction of servicing and guaranty fees. However, the cash flows of an MBS
are not as stable as that of a government or corporate coupon bond. Because the mortgage
borrower has the prepayment option, mainly exercised when moving or refinancing, an
MBS investor is actually writing a call option. Furthermore, the mortgage borrower also
has the default option, which is likely to be exercised when the property value drops
below the mortgage balance, and continuing mortgage payments would not be
economically reasonable. In this case the guarantor is writing the borrower a put option,
and the guarantor absorbs the cost. However, the borrower does not always exercise the
options whenever it is financially optimal to do so, because there are always non-
monetary factors associated with the home, like shelter, sense of stability, etc. And it is
also very hard for the borrower to tell whether it is financially optimal to exercise these
4
options because of lack of complete and unbiased information, e.g., they may not be able
to obtain an accurate home price, unless they are selling it. And there are also some other
fixed/variable costs associated with these options, such as the commission paid to the real
estate agent, the cost to initialize another loan, and the negative credit rating impact when
the borrower defaults on a mortgage. All these factors contribute to the complexity of
MBS cash flows. In practice, the cash flows are generally projected by complicated
prepayment models, which are based on statistical estimation on large historical data sets.
Because of the complicated behaviors of the MBS cash flow, due to the complex
relationships with the underlying interest rate term structures, and path dependencies in
prepayment behaviors, Monte Carlo simulation is generally the only applicable method to
price MBS.
Associated with the uncertainty of cash flows are different kinds of risks.
Treasury bonds only bear interest rate risk, whereas non-callable corporate bonds carry
interest rate and credit risk. MBS are further complicated by prepayment risk (resulting
from both voluntary prepayment and default). Thus risk management is especially critical
for portfolios with large holdings in MBS. Duration and convexity are the main risk
measurements for fixed income portfolio mangers. Many practitioners use either the
Macaulay duration, or modified duration (Kopprasch [1987]) to capture the MBS price
sensitivity with respect to interest rate changes, but these duration measures assume a
constant yield and known deterministic prepayment pattern, which is rarely the case in
practice. So these two approaches to calculate duration can lead to serious errors when
used in hedging. Golub [2001] proposed four different approaches to estimate the
5
duration: Percent of Price (POP), Option-Adjusted Duration (OAD), Implied Duration,
and Coupon Curve Duration (CCD). The first two approaches apply parallel shifts in the
yield curve, which is not a very realistic assumption. The latter two approaches require
large numbers of previous or current accurate MBS prices that are comparable to the
MBS whose duration is to be measured. This might not be practical for on-the-fly pricing
and sensitivity analysis. Another drawback of these approaches is that they handle only
duration and convexity, but not sensitivity to interest rate volatility. OAD method can
estimate the vega (the price sensitivity to volatility) using a finite difference approach,
which requires 3 simulations to estimate one gradient: the base, up and down cases. And
non-parallel yield curve shifts require more parameters to characterize the shift. Thus, in
the setting we consider in Chapter 2, to estimate the duration with respect to yield curve
shift of 4 summed harmonic functions would require 9 (2n+1, n=4) simulations. To
estimate vega requires 2 additional simulations. So estimating the duration and vega
roughly increases the computational cost by a factor of 10. Calculating convexity would
require 75 duration estimators to calculate 25 convexity estimators, increasing the
simulation factor to 225. In other words, if one were to use 10,000 replications to
estimate the MBS price, over 2.25 million simulations would be required to estimate the
various sensitivities. Our work aims to decrease this computational burden dramatically.
Most literature on MBS has concentrated on prepayment model estimations,
although some of the recent work has focused on computational efficiency, e.g.,
dimensionality reduction via Brownian bridge ( Caflisch et al. [1997] ), and quasi-Monte
Carlo (Åkesson and Lehoczky [2000]). However, there is no work that we are aware of
6
that addresses efficient sensitivity analysis of MBS pricing and hedging. Related work in
equities includes Fu and Hu [1995], Broadie and Glasserman [1996], Fu et al. [2000],
[2001], and Wu and Fu [2001]. Perhaps the most relevant paper to our work is
Glasserman [1999], which applied perturbation analysis (PA) method for caplet price
sensitivity analysis. Yet most of these models involve only a single exercise decision with
a one-time payoff, whereas an MBS is a pool of homogenous mortgages rather than an
individual mortgage loan. So the cash flows exist until the maturity of the collateral, and
they are highly path dependent, which makes sensitivity analysis of MBS more
complicated.
The other relevant body of research literature analyzes the duration of different
mortgage products. We know that adjustable rate mortgage (ARM) products will have a
different response from fixed rate mortgage (FRM) products, due to ARMs’ coupon-reset
plan and different prepayment function. In a series of papers, Kau et al.[1990,1992,1993]
priced the ARMs and performed some sensitivity analysis. Chiang [1997] applied a
simple simulation scheme to estimate the modified duration of ARMs. Stanton [1999]
calculated the duration of different indexed ARMs via a scheme like Kau’s. However,
most of these papers are based on solving models based on partial differential equation
(PDE), using simplified assumptions that often miss essential features that can be
captured by Monte Carlo simulation. The three major drawbacks of these models that
make them impractical in the mortgage industry are the following:
• They assume borrowers exercise the prepayment option only when it is financially
optimal to do so. This ignores the fact that people routinely prepay even in financially
7
adverse environments, e.g., house sales. Also seasoning and burnout effects are not
considered.
• By solving the PDE, one can only obtain a set of present values of the MBS along the
interest rate axis. By applying the finite difference method, duration of the MBS
could be acquired. However, it provides no information about the discounting factor
and cash flows along the time horizon. So you will have no information about how
the interest shift affects different components of the present value.
• The PDE method generally uses one-factor interest rate model, which applies the
same interest rate both for discounting and for the prepayment model, which ignores
the difference between short-term and long-term interest rates.
In the first essay, we apply perturbation analysis (PA) to estimate the sensitivities
of MBS. Our work makes the following contribution to MBS literature:
• We decompose any interest rate term structure change into four Fourier-like series,
which is based on Fourier cosine series, and can better measure the yield curve shift;
• We derive PA estimators for these Fourier-like factors, as well as interest rate
volatility, which can largely save computation effort. In our example, we calculate 5
duration estimators and 16 convexity estimators in our simulation, which would
require 155 simulations using a conventional simulation scheme.
• Based on our comprehensive analysis of the sensitivity measures we calculated for a
full spectrum of mortgage product, we propose a new mortgage product, which can
potentially benefit both the MBS investor and mortgage borrower.
8
This essay is organized in the following manner. Section 1 describes the problem
setting. We then derive the framework for PA in a general setting in section 2, without
restrictions to any specific interest rate model or prepayment model. Then we consider
the well-known Hull-White interest rate model (Hull and White [1993]) and a common
prepayment model to derive the corresponding PA sensitivities for FRM and ARM
products in section 3, assuming the shock of interest rate term structure takes the form of
a series of trigonometric polynomial functions. Section 4 presents numerical examples, in
which we compare the performance of FD and PA estimators, indicating that the PA
estimator is at least as good as the FD estimator, while the computation cost is reduced
dramatically. Section 5 gives the insights from our simulation results. Section 6 gives
conclusions.
9
1.2 Hedging MBS in HJM Framework
There is a large body of literature on hedging with different interest risk measures,
like first-order hedging with duration (Ilmanen [1992]), second order hedging with
convexity (Kahn and Lochoff [1990], Lacey and Nawalkha [1993]), principal
components hedging (Golub and Tilman [1997]), key rates hedging (Ho [1992]),
level/slope/curvature hedging (Willner [1996]), etc. Yet there has not been a unifying
effort in combining hedging the term structure together with hedging volatility factors.
This essay tries to extract the empirical volatility factors from historical term
structure data, via principal components analysis (PCA), and apply these factors in a
HJM framework for pricing MBS, while deriving the risk measures for hedging MBS. It
makes the following contribution in the MBS literature:
• The first paper to hedge MBS with PCA factors empirically extracted from historical
interest rate data;
• Hedging efficiency is proved to increase significantly, compared with traditional
duration/convexity hedging.
This essay is organized in the following way. Section 1 gives the motivation for
this research question. Section 2 describes the interest rate data set and PCA method we
used to extract the volatility factors. Section 3 applies these factors in interest rate
simulation within a HJM framework. Section 4 derives the PA estimators in the HJM
framework. Section 5 gives the hedging algorithm for MBS, and Section 6 gives the
performance analysis of this hedging method. Section 7 concludes the essay.
10
1.3 Hedging Credit Risk of MBS: A Random Coefficient
Approach
In order to hedge the credit risk of MBS, the MBS issuer sometimes needs to
purchase pool insurance from a third party, beyond the protection of mortgage collateral,
and primary mortgage insurance. In this case, it is important to model the credit risk of
the third party. Recently there has been increased interest in some research papers to use
regression method to determine what factors affect credit spread. Most of the papers,
which use simple linear regression, found that variables in structural models lack
explanatory power in such regression. We argue that the problem partially results from
non-constant credit spread sensitivities to state variables.
We try to overcome the problem by proposing a Random Coefficient Regression
(RCR) model. We collected data from multiple database, and constructed our data set.
Our regression results show that our assumption of non-constancy of credit spread
sensitivities is correct. As a result of improved regression, we improved adjusted R
2
to
28%, compared with 8% adjusted R
2
for a simple linear regression approach, using the
same set of independent variables. Another important result of our RCR model is that it
validated the relationship between credit spread sensitivities and state variables, which
has been predicted by Merton’s model.
This essay makes the following research contributions to the finance literature:
• The first paper to use the RCR method on credit spread data;
11
• The first paper to explicitly build a dependence relationship between credit spread
sensitivity and state variables;
• The first paper to empirically validate the dependence relationship between credit
spread sensitivity and state variables predicted by a structural model, such as
Merton’s model.
This essay is organized in the following way. Section 1 gives the motivation for
this research question. Section 2 describes several previous papers on this topic. Section 3
gives a brief introduction to the Random Coefficient Regression model. Section 4 applies
this model to changes of credit spread. Section 5 gives the data description used in the
regression, and Section 6 gives the results analysis of this regression method. Section 7
concludes the essay.
12
Chapter 2
Efficient Sensitivity Analysis of MBS
2.1 Problem Setting
Generally the price of any security can be written as the net present value (NPV)
of its discounted cash flows. Specifying the price of an MBS (here we consider only the
pass-through MBS
1
) is as follows:
(
¸
(
¸
=
(
¸
(
¸
= =
? ?
= =
M
t
M
t
t c t d E t PV E V E P
0 0
) ( ) ( ) ( ] [ , (2.1)
where P is the price of the MBS,
V is the value of the MBS, which is a random variable, dependent on the
realization of the economic scenario,
PV(t) is the present value for cash flow at time t,
d(t) is the discounting factor at time t,
c(t) is the cash flow at time t,
M is the maturity of the MBS.
1
A pass-through MBS is an MBS that passes through the principal and interest payments collected from a
mortgage pool, minus the guaranty fee and servicing fee, to the MBS investor directly. This is in contrast to
Collaterized Mortgage Obligations (CMOs), which have multiple tranches and pay the principal payments
according to the seniorities of tranches. In this essay, we assume that mortgages in the MBS pool are
homogenous.
13
Monte Carlo simulation is used to generate cash flows on many paths. By the
strong law of large numbers, we have the following:
| |
?
=
? ?
=
N
i
i N
V
N
V E
1
1
lim , (2.2)
where V
i
is the value calculated out in path i.
The calculation of d(t) is found from the short-term (risk-free) interest rate
process,
? ?
?
=
?
=
? ? = ? ? = ? =
1
0
1
0
} ] ) ( [ exp{ ) ) ( exp( ) , 1 ( ) 2 , 1 ( ) 1 , 0 ( ) (
t
i
t
i
t i r t i r t t d d d t d , (2.3)
where d(i, i+1) is the discounting factor for the end of period i+1 at the end of period i;
r(i) is the short term rate used to generate d(i, i+1), observed at the end of period
i;
?t is the time step in simulation, generally monthly, i.e. ?t= 1 month.
An interest rate model is used to generate the short term-rate r(i); then d(t) is instantly
available when the short-term rate path is generated.
The difficult part is to generate c(t), the path dependent cash flow of MBS for
month t, which is observed at the end of month t. From chapter 19 of Fabozzi [1993], we
have the following formula for c(t):
); ( ) ( ) (
); ( ) ( ) (
); ( ) ( ) ( ) ( ) (
t PP t SP t TPP
t IP t SP t MP
t IP t TPP t PP t MP t c
+ =
+ =
+ = + =
(2.4)
where MP(t): Scheduled Mortgage Payment for month t;
TPP(t): Total Principal Payment for month t;
14
IP(t): Interest Payment for month t;
SP(t): Scheduled Principal Payment for month t;
PP(t): Principal Prepayment for month t.
These quantities are calculated as follows:
; ) ( 1 1 ) (
); ( ) 1 ( ) (
)); ( ) 1 ( )( ( ) (
;
12
) 1 ( ) (
;
) 12 / 1 ( 1
12 /
) 1 ( ) (
12
t CPR t SMM
t TPP t B t B
t SP t B t SMM t PP
WAC
t B t IP
WAC
WAC
t B t MP
t WAM
? ? =
? ? =
? ? =
? =
|
|
.
|
\
|
+ ?
? =
+ ?
(2.5)
B(t): The principal balance of MBS at end of month t;
WAC
2
: Weighted Average Coupon rate for MBS;
WAM
3
: Weighted Average Maturity for MBS;
SMM(t): Single Monthly Mortality for month t, observed at the end of
month t;
CPR(t): Conditional Prepayment Rate for month t, observed at the end of
month t.
In Monte Carlo simulation, along the sample path, CPR(t) is the primary variable
to be simulated. Everything else can be calculated out once CPR(t) is known. Different
prepayment models offer different CPR(t), and it is not our goal to derive a new
2
WAC is the weighted average mortgage rate for a mortgage pool, weighted by the balance of each
mortgage.
3
WAM is the weighted average maturity in month for a mortgage pool, weighted by the balance of each
mortgage.
15
prepayment model or compare existing prepayment models. Instead, our concern is,
given a prepayment model, how can we efficiently estimate the price sensitivities of MBS
against parameters of interest? Generally different prepayment models will lead to
different sensitivity estimates, so it is at the user’s discretion to choose an appropriate
prepayment function, as our method for calculating the “Greeks” is universally
applicable.
2.2 Derivation of General PA Estimators
If P, the price of the MBS, is a continuous function of the parameter of interest,
say ?, and assuming the interchange of expectation and differentiation is permissible
4
, we
have the following PA estimator by differentiating both sides of (2.1):
). , (
) , (
) , (
) , ( )) , ( (
,
) , (
) , (
)] ( [ ) (
1
1
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
t d
t c
t c
t d
d
t PV d
d
t dPV
E
d
t PV d
E
d
V dE
d
dP
M
t
M
t
?
?
+
?
?
=
(
¸
(
¸
=
(
(
(
(
¸
(
¸
= =
?
?
=
=
(2.6)
Now we have reduced the original problem from estimating the gradient of a sum
function to estimating the sum of a bunch of gradients. Actually now we only need to
estimate two gradient estimators,
?
?
?
? ) , (t c
and
?
?
?
? ) , (t d
, at each time step.
4
Strictly speaking, sample pathwise continuity of price function with respect to ? will result in the
interchange being valid.
16
2.2.1 Gradient Estimator for Cash Flow
We first derive
?
?
?
? ) , (t c
. To simplify notation, we write c(t) for c(t, ?).
A simplified expression for c(t) is derived from (2.4) and (2.5) as follows:
{ }, ) ( )] ( 1 )[ ( ) 1 (
) ( )
12
1 )( 1 ( )] ( 1 [
) 12 / 1 ( 1
12 /
) 1 (
) ( )
12
1 )( 1 ( )) ( 1 )( (
) ( )]} ( ) ( [ ) 1 ( { ) (
) ( )] ( ) 1 ( [ ) ( ) ( ) ( ) (
t SMM g t SMM t A t B
t SMM
WAC
t B t SMM
WAC
WAC
t B
t SMM
WAC
t B t SMM t MP
t SMM t IP t MP t B t MP
t SMM t SP t B t MP t PP t MP t c
t WAM
+ ? ? =
+ ? + ?
+ ?
? =
+ ? + ? =
? ? ? + =
? ? + = + =
+ ?
(2.7)
where
).
12
1 (
,
) 12 / 1 ( 1
12 /
) (
WAC
g
WAC
WAC
t A
t WAM
+ =
+ ?
=
+ ?
(2.8)
Then we can derive the gradient for c(t), if WAC and t are independent
5
of ?:
{ } ] ) ( )[ 1 (
) (
) ( )] ( 1 )[ (
) 1 ( ) (
g t A t B
t SMM
t SMM g t SMM t A
t B t c
+ ? ?
?
?
+ + ?
?
? ?
=
?
?
? ? ?
. (2.9)
This leads to recursive equations for calculation of the above gradient estimator
from (2.5) and (3.2):
.
) ( ) 1 ( ) (
); ( ) 1 ( ) ( )
12
1 )( 1 ( ) ( ) 1 ( ) (
); 1 (
12
) ( ) ( ) ( ) (
? ? ? ?
?
?
?
? ?
=
?
?
? ? = ? + ? = ? ? =
? ? = ? =
t c
g
t B t B
t c g t B t c
WAC
t B t TPP t B t B
t B
WAC
t c t IP t c t TPP
(2.10)
5
A fixed Rate Mortgage (FRM) would satisfy this assumption; however an Adjustable Rate Mortgage
(ARM) will not, so we derive the gradient estimator for ARMs later in section 4.
17
Assuming we know that the initial balance is not dependent as ?; we have the
initial conditions:
). ) 1 ( )( 0 (
) 1 ( ) 1 (
; 0
) 0 (
g A B
SMM c
B
+ ?
?
?
=
?
?
=
?
?
? ?
?
(2.11)
This leads to the following
,
) 1 ( ) 0 ( ) 1 (
? ? ? ?
?
?
?
?
=
?
? c
g
B B
(2.12)
{ } ), ) 2 ( )( 1 (
) 2 (
) 2 ( ) 2 ( 1 )( 2 (
) 1 ( ) 2 (
g A B
SMM
SMM g SMM A
B c
+ ?
?
?
+ + ?
?
?
=
?
?
? ? ?
(2.13)
M t
t c
,..., 2 , 1 ,
) (
=
?
?
?
.
Thus the problem of calculating the gradient estimator of cash flow c(t) is reduced
to calculating:
. ,..., 1 ,
) (
M t
t SMM
=
?
?
?
Since
, ) ( 1 1 ) (
12
t CPR t SMM ? ? =
we have
.
) (
)) ( 1 (
12
1 ) (
12
11
? ? ?
?
? =
?
?
?
t CPR
t CPR
t SMM
(2.14)
As discussed earlier, generally CPR(t) is given in the form of a prepayment function,
and there are four main types of prepayment functions (Fabozzi [2000]):
1. Arctangent Model: (An example from the Office of Thrift Supervision (OTS).)
18
))
) (
089 . 1 ( 9518 . 5 arctan( 1389 . 0 2406 . 0 ) (
10
t r
WAC
t CPR ? ? = . (2.15)
2. CPR(S,A,B,M) Model:
); ( ) ( ) ( ) ( ) ( t BM t MM t AGE t RI t CPR = (2.16)
where RI(t) is refinancing incentive;
AGE(t) is the seasoning multiplier;
MM(t) is the monthly multiplier, which is constant for a certain month;
BM(t) is the burnout multiplier.
3. Prepayment models incorporating macroeconomic factors, i.e., the health of
economics, housing market activity, etc.
4. Prepayment models for individual mortgages.
For the last two types of prepayment models, we do not have any explicitly stated
functional forms, mainly because they are proprietary models in the mortgage industry.
But since our approach is general for any type of prepayment function, we can derive the
derivatives once we are given an explicit form for the prepayment function.
We would like to make one claim here: the CPR(t) model we mentioned and we are
going to use in Chapter 2 and Chapter 3 includes both voluntary prepayment (refinance,
house turnover, and cash out) and involuntary prepayment (default). Because default only
makes about 1% of total prepayment, and generally MBS issuer will guarantee the
principal payment to the investor, in case borrower defaults, it is a reasonable not to
model default separately. However, when analyzing MBS backed by high default loans,
such as subprime mortgages, it is desirable to model voluntary prepayment and default
separately.
19
2.2.2 Gradient Estimator for Discounting Factor
We have derived the gradient estimator of cash flow with respect to parameter ?.
Next, we derive the gradient estimator of the discounting factor d(t).
We know that the discounting factor takes the following form from section 2,
when the option adjusted spread (OAS) is not considered. For simplification, we write
d(t) as for d(t, ?):
} ] ) ( [ exp{ ) (
1
0
t i r t d
t
i
? ? =
?
?
=
. (2.17)
Differentiating with respect to ?:
. )
) (
( ) ( )
) (
( } ] ) ( [ exp{
) (
1
0
1
0
1
0
t
i r
t d t
i r
t i r
t d
t
i
t
i
t
i
?
?
?
? = ?
?
?
? ? ? =
?
?
? ? ?
?
=
?
=
?
=
? ? ?
(2.18)
From the gradient estimators for cash flow and discounting factor, we can easily
get the gradient estimator of PV(t):
) , (
) , (
) , (
) , ( )) , ( (
?
?
?
?
?
?
?
?
t d
t c
t c
t d
d
t PV d
?
?
+
?
?
= . (2.19)
The last step would be to apply a specific prepayment model and interest rate
model to arrive at the actual implemented gradient estimators. To illustrate the procedure,
we carry out this exercise in its entirety for one setting in the following section.
20
2.3 Applying the Gradients
We choose our interest model to be the one-factor Hull-White (Hull and White
[1990]) model, for its simplicity and easy calibration to market term structure. For the
prepayment model, we consider a CPR(S,A,B,M) model.
2.3.1 Interest Model Setup
In this section, we briefly discuss the model and the simulation scheme.
In the one-factor Hull-White interest rate model, the underlying process for the
short-term rate r(t) is given by
), ( )) ( ) ( ( ) ( t dB dt t ar t t dr ? ? + ? = (2.20)
where B(t): a standard Brownian motion;
a: mean reverting speed, constant;
?: standard deviation, constant;
?(t): chosen to fit the initial term structure, which is determined by
), 1 (
2
) , 0 (
) , 0 (
) (
2
2
at
e
a
t af
t
t f
t
?
? + +
?
?
=
?
? (2.21)
f(0,t): the instantaneous forward rate, which is determined by
, ) , 0 (
1
) , 0 (
0
?
=
t
du u f
t
t R (2.22)
Differentiating both sides, with respect to t, we have
) , 0 (
) , 0 (
) , 0 ( t R
t
t R
t t f +
?
?
= , (2.23)
where R(0,t): the continuous compounding interest rate from now to time t, i.e. the term
structure.
21
In order to simplify the simulation process, the model can be re-parameterized
from its original to the following:
; 0 ) 0 ( ), ( ) ( ) ( ) ( = + ? = x t dB dt t x t a t dx ? (2.24)
x(t) is determined by
2
2
) 1 (
2
) , 0 ( ) ( ) ( ) (
at
e
a
t f t x t r t a
?
? + = ? =
?
. (2.25)
The process x(t) is called an Ornstein-Uhlenbeck process, and its solution is given
by
?
?
=
t
au at
u dB e e t x
0
) ( ) ( ? , (2.26)
which is a Gaussian Markov process, and can also be represented as
)
2
1
( ) (
2
a
e
W e t x
at
at
?
=
?
? , (2.27)
where {W(t), t?0} is also a Brownian motion.
In this case, the interest rate r(t) can be represented in the following form:
) ) ( ) ( ( ) (
) (t h
W t g t a F t r + = , (2.28)
where a, g: R
+
? R are continuous functions, and the functions F:R ? R and h: R
+
? R
are strictly increasing and continuous. From above we can see that
.
2
1
) (
; ) (
; ) 1 (
2
) , 0 ( ) (
; ) (
2
2
2
2
a
e
t h
e t g
e
a
t f t a
x x F
at
at
at
?
=
=
? + =
=
?
?
?
?
(2.29)
To simulate r(t) given by above, we will first simulate
) (
) ( ) (
t h
W t g t x = ,
22
which is a Gaussian Markov process, and then compute the short-term interest rate by
)) ( ) ( ( ) ( t x t a F t r + = .
For calculating the price of MBS, the short-term rate is not sufficient; the long-
term rate process is also required, especially the 10-year Treasury rate, which is a
deterministic function of r(t) in the Hull-White model. Generally this is the case for
short-term rate models, but not true for more complicated interest rate models, e.g., the
HJM (Heath, Jarrow and Morton [1992]) model and the LIBOR forward rate model
(Jamshidian[1997]). The long-term rate R(t,T) is calculated from the following, :
). 1 ( ) (
4
) , 0 ( ln
) , (
) , 0 (
) , 0 (
ln ) , ( ln
;
1
) , (
; ) , ( ) , (
2 2
3
2
) (
) ( ) , ( ) )( , (
? ? ?
?
?
? =
?
=
= =
? ?
? ?
? ? ?
at at aT
t T a
t r T t B t T T t R
e e e
a t
t P
T t B
t P
T P
T t A
a
e
T t B
e T t A e T t P
?
(2.30)
P(t,T) is the zero coupon bond price at time t, with face value $1, matured at T.
Thus we can derive the R(t,T) as following:
) (
) ( ) , ( ) , ( ln
) , (
t T
t r T t B T t A
T t R
?
?
? = . (2.31)
The standard (forward) path generation method for generating x(t) is given by
, )) ( ( )) ( ( ) ( ) (
) (
) (
))] ( ( )) ( ( )[ ( )) ( ( ) (
) (
) (
) (
1 1 1
1
1 1
1
1
+ + +
+
+ +
+
+
? + =
? + =
i i i i i
i
i
i i i i i
i
i
i
z t h W t h W t g t x
t g
t g
t h W t h W t g t h W t g
t g
t g
t x
(2.32)
where {z
i
} is a series of independent standard normal random variables. In the special
case where x(t) is from the Hull-White model, we have
1
2
1
2
1
) ( ) (
+
? ?
? ?
+
?
+ =
i
t a
i
t a
i
z
a
e
t x e t x
i
i
? , (2.33)
23
where ?t
i
=t
i+1
- t
i
.
2.3.2 Trigonometric Polynomial Shocks
There are multiple factors in the interest rate model that can change and affect the
cash flows and discounting factor along the simulation path. The major changes could be
the initial term structure R(0,t) and the volatility ?.
The most common assumption for term structure change is a parallel shift on all
maturities. However, this is often not an adequate model for the real world, where a shift
in the term structure can take any shape. For example, short-term rates and long-term
rates may change in opposite directions rather than in parallel. We consider a Fourier
series decomposition of the term structure shift.
Our domain of concern is interest rates from time 0 to 30 years, since most
mortgages are amortized in a 30-year term. So for example, we could assume the shift of
term structure takes the following form:
?
?
=
? = ?
0
),
30
cos( ) , 0 (
n
n
t n
t R
?
(2.34)
where ?
n
is the magnitude for the n
th
Fourier function. Figure 4.1 depicts the first four
trigonometric polynomial series. (n=0,1,2,3), which is all that we will consider in our
model. When n=0, the shift is just like a parallel shift in term structure. When n=1, the
short-term and long-term rates move in opposite directions. When n=2, the short-term
and long-term rates move in the same direction, while the middle-term rate moves in the
opposite direction. Thus we decompose any shift in the term structure into the Fourier
functions by Fourier transform. If we have previously calculated the gradients with
respect to the magnitude of each trigonometric polynomial function, we can apply these
24
gradients and get the corresponding changes in the cash flows and discounting factors,
and hence the change in MBS prices.
The Fourier series have a serious drawback: they treat short-term rates the same
as long-term rates. However, from experience, we know that the short-term rates
generally change more frequency than long-term rates. So we would like to change the
shape of the trigonometric polynomial function, which will concentrate more on the
short-term rates, and keep the long-term rates relatively stable. The modified Fourier
function that we adopt takes the following form:
?
?
=
?
? ? = ?
0
/
)) 1 ( cos( ) , 0 (
0
n
T t
n
e n t R ? , (2.35)
where T
0
is a user-specified parameter of the modified Fourier shifts. The smaller T
0
is,
the more likely short rates and long rates are going to act differently. See Figure 2.2 for
the modified Fourier functions, where T
0
=10. Comparing Figures 2.1 and 2.2, the
modified Fourier series concentrate more on the changes with maturities less than T
0,
which is both desirable and easier for analytical purposes.
For a Fourier cosine series that has the following functional form:
?
?
=
+ =
1
0
)
2
cos(
2
) (
n
n
T
t n
a
a
t f
?
, (2.36)
the coefficients are given by a Fourier cosine transform:
?
= =
2 /
0
2 1 0 , )
2
cos( ) (
4
T
n
,...... , , n dt
T
t n
t f
T
a
?
(2.37)
25
Figure 2.1 ?R(0,t) with Original Fourier series
Figure 2.2 ?R(0,t) with T
0
=10 modified Fourier series
26
For our modified Fourier series, perform the following change of variables:
) 1 (
30
'
0
/ T t
e
t
?
? =
and substitute into the expression of ?R(0,t) to get
?
?
=
? = ?
0
)
60
' 2
cos( ) ' , 0 (
n
n
t n
t R
?
, (2.38)
which is a standard Fourier cosine series, and we can use a Fourier transform to estimate
the coefficients. In computer simulation, t is a vector of real time points, evenly
distributed with sample function value ?R(0,t), and t’ is the mapped time point in a new
time scale, which is not evenly distributed, with the same sample function value ?R(0,t’).
However, in order to utilize the discrete cosine transform function provided in
mathematical libraries, we need to resample ?R(0,t’) at even time intervals. This is
carried out by interpolating the function of ?R(0,t’) on the t time scale. Figure 2.3 shows
a sample of ?R(0,t), ?R(0,t’) re-sampled on t, the coefficients estimated on the re-
sampled ?R(0,t’), and the reconstructed Fourier series of ?R(0,t).
From Figure 2.3, if we look at the upper left and lower right sub-figures, we can
see that the reconstructed term structure matches the original sample very well, which
validates our method for estimating the coefficients of the modified Fourier series.
27
Figure 2.3 Coefficients Estimation for Modified Fourier series
2.3.3 Derivation of Gradients with respect to Modified Fourier Functions
Our major task in this section is to derive the gradient estimator with respect to to
specific parameters in the interest rate model and prepayment model. Specifically, for the
former, we are interested in the parameters of the modified Fourier functions (?
n,
n=0, 1,
2, 3).
First we derive the discounting factor gradient estimator. From (3.12), we know
that in order to derive
? ?
? ) (t d
, we must first derive
? ?
? ) (i r
, i=0,…, t-1. Let us recall that in
section 4.1, we have the following simulation scheme for short term rate r(t):
) ( ) ( ) ( t x t a t r + = .
28
So
? ? ? ?
?
+
?
?
=
?
? ) ( ) ( ) ( t x t a t r
, (2.39)
where
? ?
? ) (t a
and
? ?
? ) (t x
are determined as the following in Hull-White model:
. 0
) (
,
) , 0 ( ) (
=
?
?
?
?
=
?
?
?
? ?
t x
t f t a
(2.40)
We also know the relationship between f(0,t) and R(0,t) from (2.23), so
? ?
? ) , 0 ( t f
can be derived as:
.
) , 0 ( ) , 0 (
) , 0 ( ) , 0 ( ) , 0 (
) , 0 (
) , 0 (
) , 0 (
2
2
? ?
? ? ?
? ? ?
?
?
+
? ?
?
=
?
?
+
? ?
?
+
?
?
?
?
=
?
?
+
?
|
.
|
\
|
?
?
?
=
?
?
t R
t
t R
t
t R
t
t R
t
t
t R t
t R t
t R
t
t f
(2.41)
Considering the changes in R(0,t) which takes the form as in (2.38), we can get
the derivatives of R(0,t) (? taken to be ?
n
):
.
) (
)) 1 ( sin(
) (
)) 1 ( sin(
) , 0 (
)), 1 ( cos(
) , 0 (
0
/
/
0
/
/
2
/
0
0
0
0
0
T
e n
e n
T
e n
e n
t
t R
e n
t R
T t
T t
T t
T t
n
T t
n
?
?
?
?
?
? =
?
?
? ? =
? ? ?
?
? =
? ?
?
?
?
?
?
?
(2.42)
We can get the derivatives of r(i):
)). 1 ( cos(
) (
)) 1 ( sin(
) , 0 ( ) (
0
0
0
/
0
/
/ T t
T t
T t
n n
e n
T
e n
e n t
t f i r
?
?
?
? + ? ? =
? ?
?
=
? ?
?
?
?
? (2.43)
And gradient estimator for discounting factor is also obtained, applying (2.18).
29
Next, we are going to derive the cash flow gradient estimator with respect to ?
n
.
From our derivation in section 3, we know that in order to get
? ?
? ) (t c
, we need to derive
? ?
? ) (t CPR
first. We use the second type of prepayment function, among the four
described in section 3. An example for this type of prepayment model is available from
the sample code at the Numerix homepage http://www.numerix.com.
); ( ) ( ) ( ) ( ) ( t BM t MM t AGE t RI t CPR = (2.44)
where
rate. mortgage prevailing ith the w
correlated highly is which t, period of end at the observed rate, year 10 the is ) (
;
) 0 (
) 1 (
7 . 0 3 . 0 ) (
December; in ending January, from starting
0.98], 1.23, 1.22, 1.18, 1.1, 0.98, 0.92, 0.98, 0.95, 0.74, 0.76, [0.94, ) (
);
30
, 1 min( ) (
))); 1 ( ( 430 571 . 8 arctan( 14 . 0 28 . 0 ) (
10
10
t r
B
t B
t BM
t MM
t
t AGE
t r WAC t RI
?
+ =
=
=
? ? + ? + =
From the formulas, only RI(t) and BM(t) depend on ?, when ? is not time t. Thus
we have the following formula for
? ?
? ) (t CPR
:
;
) (
) ( ) ( ) ( ) ( ) ( ) (
) ( ) (
? ? ? ?
?
+
?
?
=
?
? t BM
t MM t AGE t RI t BM t MM t AGE
t RI t CPR
(2.45)
where
.
) 0 (
1 ) 1 (
7 . 0
) (
;
) 1 (
))) 1 ( ( 430 571 . 8 ( 1
) 430 (
14 . 0
) (
10
2
10
B
t B t BM
t r
t r WAC
t RI
? ?
? ?
?
? ?
=
?
?
?
? ?
? ? + ? +
?
=
?
?
(2.46)
30
? ?
? ) (t B
is available, when
? ?
? ) (t c
is calculated out, so the problem is reduced to
calculating
? ?
? ) (
10
t r
. In the one-factor Hull-White framework, as we have discussed in
section 2.3.1, the long-term rate is a deterministic function of r(t), so substituting T=t+10
for (2.30), we have
). 1 ( ) (
4
) , 0 (
1
) , 0 ( ) 10 )( 10 , 0 (
) 1 ( ) (
4
) , 0 ( ln
) 10 , ( ) , 0 ( ln ) 10 , 0 ( ln
) 1 ( ) (
4
) , 0 ( ln
) 10 , (
) , 0 (
) 10 , 0 (
ln ) 10 , ( ln
;
1 1
) 10 , (
; ) 10 , ( ) 10 , (
2 2 ) 10 (
3
2 10
2 2 ) 10 (
3
2
2 2 ) 10 (
3
2
10 ) 10 (
) ( ) 10 , ( ) 10 )( 10 , (
? ? ?
?
+ + + + ? =
? ? ?
?
?
+ ? ? + =
? ? ?
?
?
+ ?
+
= +
?
=
?
= +
+ = = +
? + ?
?
? + ?
? + ?
? ? + ?
+ ? ? + + ?
at at t a
a
at at t a
at at t a
a t t a
t r t t B t t t t R
e e e
a
t R
a
e
t t R t t R
e e e
a t
t P
t t B t P t P
e e e
a t
t P
t t B
t P
t P
t t A
a
e
a
e
t t B
e t t A e t t P
?
?
?
(2.47)
Since
,
10
) (
1
) 1 ( ) (
4
) , 0 (
1
) , 0 ( ) 10 )( 10 , 0 (
10
) ( ) 10 , ( ) 10 , ( ln
) 10 , ( ) (
10
2 2 ) 10 (
3
2 10
10
t r
a
e
e e e
a
t R
a
e
t t R t t R
t r t t B t t A
t t R t r
a
at at t a
a ?
? + ?
?
?
? ? ? ?
?
+ + + + ?
? =
+ ? +
? = + =
?
(2.48)
? ?
? ) (
10
t r
takes the following form, when ? is independent of ? and t:
10
) ( 1 ) , 0 (
)
1
(
) 10 , 0 (
) 10 (
) (
10 10
10 ? ? ?
?
?
? ?
?
?
? ?
+ +
?
+ ?
+ ?
? =
?
?
? ?
t r
a
e t R
a
e
t
t R
t
t r
a a
. (2.49)
Thus we have derived
? ?
? ) (
10
t r
as a function of
? ?
? ) (t R
and
? ?
? ) (t r
derived earlier.
31
2.3.4 Derivation of Gradients with respect to Volatility: Vega
The derivation is straightforward as in section 2.3.3; all we need to do is to
substitute ? with ?, instead of ?
n
. In order to get
? ?
? ) (t d
, we must first derive
? ?
? ) (i r
.
Following the same logic in (2.40), we can get the vega of r(t):
).
2
1
( ) 1 (
) (
so ),
2
1
(
) (
, ) 1 (
) (
2
2
2
2
2
2
a
e
W e e
a
t r
a
e
W e
t x
e
a
t a
at
at at
at
at
at
?
+ ? =
?
?
?
=
?
?
? =
?
?
? ?
?
?
?
?
?
?
?
(2.50)
And vega of d(t) would be:
. )
) (
( ) (
) (
1
0
t
i r
t d
t d
t
i
?
?
?
? =
?
?
?
?
=
? ?
(2.51)
Now we derive
? ?
? ) (t c
, which would require us to derive
? ?
? ) (t CPR
first, which has
the same form as in (2.45), while
? ?
? ) (
10
t r
has the form of:
.
10
) ( 1
) 1 ( ) (
2
10
) ( 1
) 1 ( ) (
2
) , 0 (
)
1
(
) 10 , 0 (
) 10 (
) (
10
2 2 ) 10 (
3
10
2 2 ) 10 (
3
10
10
?
?
?
?
? ?
?
?
? ?
+ ? ?
=
?
? ?
? ? ? ?
?
? ?
+ +
?
+ ?
+ ?
? =
?
?
?
? + ?
?
? + ?
?
t r
a
e
e e e
a
t r
a
e
e e e
a
t R
a
e
t
t R
t
t r
a
at at t a
a
at at t a
a
(2.52)
2.3.5 Derivation of Second Order Gradients: Gamma
Another gradient that interests risk mangers is convexity, or the gamma of MBS,
which is the second order derivative of price against term structure shifts. Now we derive
an estimator for the gamma.
32
In order to calculate the partial second order derivatives (Hessian matrix), we take
? to be the vector, ?=[?
1
?
2
?
3
?
4
?]’. Differentiating (2.1)
1
, we get
. ) (
) (
'
) ( ) (
'
) ( ) (
) (
) ( ) ( ] [
, ) (
) (
) (
) ( ) ( ] [
0
2
2
2
2
0
2
2
2
2
2
2
0 0
(
¸
(
¸
?
?
+
?
?
×
?
?
+
?
?
×
?
?
+
?
?
=
(
¸
(
¸
?
?
=
?
?
=
?
?
(
¸
(
¸
?
?
+
?
?
=
(
¸
(
¸
?
?
=
?
?
=
?
?
? ?
? ?
= =
= =
M
t
M
t
M
t
M
t
t d
t c t d t c t c t d
t c
t d
E
t PV
E
V E P
t d
t c
t c
t d
E
t PV
E
V E P
? ? ? ? ? ? ? ? ?
? ? ? ? ?
(2.53)
where ]' [
4 3 2 1
? ? ?
?
? ?
?
? ?
?
? ?
?
? ?
?
=
?
? P P P P P P
, and
2
2
? ?
? P
is a 5-by-5 matrix, whose (i, j)
th
element is determined by
j i
P
? ? ? ?
?
2
, where ?
I
and ?
I
are the i
th
and j
th
elements of ?,
respectively. The same notation will be used for gradients of other variables, i.e. c(t), d(t),
r(t), etc.
Since we have calculated
? ?
? ) (t c
and
? ?
? ) (t d
in previous sections, now the problem
is reduced to estimate
2
2
) (
? ?
? t c
and
2
2
) (
? ?
? t d
. So we first derive the gamma for the
discounting factor d(t). Differentiating (2.18), we get
t
i r t d
t
i r
t d
t d
t
i
t
i
? ×
?
?
? ×
?
?
+ ? ×
?
?
? × =
?
?
? ?
?
=
?
=
)'
) (
(
) (
)
) (
( ) (
) (
1
0
1
0
2
2
2
2
? ? ? ?
(2.54)
Once we have
2
2
) (
? ?
? i r
, the gamma of d(t) is easily calculated. Now we derive the
gamma for cash flow c(t). From (2.9), we can derive the following gamma equation:
1
Again, we need the first order derivative to be pathwise continous to make the interchange of expectation
and differentiation permissible.
33
{ }
]. ) ( )[ 1 (
) (
] ) ( ][ '
) 1 ( ) (
'
) ( ) 1 (
[
) ( )] ( 1 )[ (
) 1 ( ) (
2
2
2
2
2
2
g t A t B
t SMM
g t A
t B t SMM t SMM t B
t SMM g t SMM t A
t B t c
+ ? ?
?
?
+
+ ?
?
? ?
×
?
?
+
?
?
×
?
? ?
+
+ ?
?
? ?
=
?
?
?
? ? ? ?
? ?
(2.55)
And from (3.5), we can get the gamma of B(t):
2
2
2
2
2
2
) ( ) 1 ( ) (
? ? ? ?
?
?
?
? ?
=
?
? t c
g
t B t B
. (2.56)
Now we calculate gamma of SMM(t):
. ,..., 1 ,
) (
2
2
M t
t SMM
=
?
?
?
As we know from (3.9), we have
.
) (
)) ( 1 (
12
1
) '
) ( ) (
( )) ( 1 (
144
11 ) (
2
2
12
11
12
23
2
2
? ? ? ? ?
?
? +
?
?
×
?
?
? =
?
?
? ?
t CPR
t CPR
t CPR t CPR
t CPR
t SMM
(2.57)
? ?
? ) (t CPR
and
2
2
) (
? ?
? t CPR
will be prepayment model specific.
For discounting factors, if we choose the Hull-White one factor model, we have
the following:
. 5 , 1 ,
) ( ) (
; ]'
) (
) (
) (
) (
) (
[
) (
2
2
2
4 3 2 1
? ?
(
(
¸
(
¸
? ?
?
=
?
?
?
?
? ?
?
? ?
?
? ?
?
? ?
?
=
?
?
j i
i r i r
i r i r i r i r i r i r
j i
? ? ?
? ?
(2.58)
From and (2.43) and (2.49), we can derive the following:
34
.
) 1 ( ) (
; 0
) (
; 0
) (
2
2
2
2
2
2
a
e i r
i r
i r
t ai
i
j i
? ?
?
=
?
?
=
? ? ?
?
=
? ? ? ?
?
?
?
(2.59)
And the gamma of d(t) would be
) ( / '
) ( ) (
)'
) (
(
) (
)'
) (
(
) (
)
) (
( ) (
) (
1
0
1
0
1
0
2
2
2
2
t d
t d t d
t
i r t d
t
i r t d
t
i r
t d
t d
t
i
t
i
t
i
? ?
? ?
? ? ? ?
?
?
×
?
?
=
? ×
?
?
? ×
?
?
=
? ×
?
?
? ×
?
?
+ ? ×
?
?
? × =
?
?
?
? ?
?
=
?
=
?
=
(2.60)
For cash flows, based on the equations (2.45) and (2.48) in the CPR(S, A, B, M)
model, we have:
,
) 0 (
1
*
) 1 (
* 7 . 0
) (
;
) 1 (
))) 1 ( ( 430 571 . 8 ( 1
5 / 301
) '
) 1 ( ) 1 (
)(
))) 1 ( ( 430 571 . 8 ( 1
5 / 301
(
) (
];
) (
) ( '
) ( ) (
'
) ( ) (
) (
) (
)[ ( ) (
) (
2
2
2
2
2
2
10
2
10
10 10
2
10 10
2
2
2
2
2
2
2
2
B
t B t BM
t r
t r WAC
t r t r
t r WAC r
t RI
t BM
t RI
t RI t BM
t BM t RI
t BM
t RI
t MM t AGE
t CPR
? ?
?
? ? ?
? ? ?
? ? ? ?
?
? ?
=
?
?
?
? ?
? ? + ? +
+
?
? ?
×
?
? ?
? ? + ? + ?
?
=
?
?
?
?
+
?
?
×
?
?
+
?
?
×
?
?
+ ×
?
?
=
?
?
(2.61)
where we know from (2.58) that
35
.
10
) ( 1
) (
0;
) 10 , 0 (
; 0
) , 0 (
;
10
) ( 1 ) , 0 (
)
1
(
) 10 , 0 (
) 10 (
) (
2 10
2
10
2
2
2 10 2 10 2
2
10
j i
a
j i
j i
j i
j i
a
j i
a
j i
j i
t r
a
e
t r
t R
t R
t r
a
e t R
a
e
t
t R
t
t r
? ?
? ?
? ?
? ?
? ? ? ? ? ?
? ?
? ?
? ?
=
? ?
?
=
? ?
+ ?
=
? ?
?
? ?
? ?
?
? ?
? ?
+ +
? ?
+ ?
+ ?
? =
? ?
?
?
? ?
(2.62)
Finally, the gamma of price P given by equation (2.53) can be obtained from
equations (2.60), (2.61), and (2.62).
2.3.6 Derivation of ARM PA estimators
In this section, we derive PA estimators for ARMs. We know FRMs only have
two sources of uncertainty:
• Short-term rate r(t), which affects the discounting factor d(t), and
• Long-term rate r
10
(t), which determines the prepayment rate CPR(t), and hence
determines the cash flow C(t).
ARMs introduce one more source of uncertainty, the coupon rate WAC(t), which
affects both the amortization schedule and the prepayment rate CPR(t), and then affects
the cash flow C(t). Coupon rate is determined by many factors:
• The index rate. WAC resets to the index rate plus the margin periodically.
• Margin. The spread between the WAC and the index rate.
• Adjustment period. For fixed period (FP) ARMs, the first adjustment period is
different from subsequent adjustment period.
36
• Period Cap/Floor. The maximum amount the WAC could increase/decrease from
previous period.
• Lifetime Cap/Floor. The maximum/minimum coupon rate over the lifetime of the
mortgage.
In order to derive the PA gradient estimator of C(t) for ARM, we first need to
derive the PA gradient estimator for Index(t) and WAC(t).
The most commonly used index rate is the 1-year Treasury rate. In the Hull-White
model, it is an explicit function of short-term rate r(t) and the term structure R(0, t). As
we have derived the function form of r
10
(t), we can derive the r
lag
(t) for any lag: (in this
case, lag=1)
.
) (
1
) 1 ( ) (
4
1
) , 0 (
1
) , 0 ( ) )( , 0 (
) ( ) , ( ) , ( ln
) , ( ) (
*
2 2 ) (
3
*
lag
t r
a
e
e e e
a
t R
a
e
t t R lag t lag t R
lag
t r lag t t B lag t t A
lag t t R t r
a lag
at at lag t a
a lag
lag
?
? + ?
?
?
? ? ? ?
?
? ? + + ?
? =
+ ? +
? = + =
(2.63)
Thus we have the PA gradient estimator of
? ?
? ) (t Index
in following form:
lag
t r
a
e t R
a
e
t
lag t R
lag t
t r
t Index
a lag a lag
lag
? ? ?
? ?
?
? ?
?
?
? ?
+ ?
?
+ ?
+ ?
? =
?
?
=
?
?
? ?
) ( 1 ) , 0 (
)
1
(
) , 0 (
) (
) (
) (
* *
.
(2.64)
The hard part is to get the
? ?
? ) (t WAC
from
? ?
? ) (t Index
, because of the complicated rules to
determine WAC(t), based on all the factors mentioned above. Given WAC(t-1), Index(t),
37
Margin, Period_Cap
2
, Period_Floor, Life_Cap, Life_Floor, WAC(t) is determined as
follows:
¦
¹
¦
´
¦
? +
+ ?
< + < +
=
+ =
=
? =
; Margin if
; Margin if
; Margin if Margin
1 max
1 min
Otherwise,
moment; adjustment an not is if ), 1 ( ) (
Cap Effective_ Index(t) Cap, Effective_
Index(t) Floor Effective_ Floor, Effective_
Cap Effective_ Index(t) Floor Effective_ , Index(t)
WAC(t)
); Period_Cap ) WAC(t- (Life_Cap, Cap Effective_
loor); )-Period_F r, WAC(t- (Life_Floo Floor Effective_
t t WAC t WAC
(2.65)
Figure 2.4 shows the relationship of WAC with Index.
Figure 2.4 WAC as a function of Index
2
Life_Cap/Life_Floor are absolute numbers, while Period_Cap/Period_Floor are relative.
38
Then we can derive the
? ?
? ) (t WAC
as following:
¹
´
¦
=
¦
¦
¦
¹
¦
¦
¦
´
¦
? + <
?
? ?
+ ? >
?
? ?
< + <
?
?
=
?
?
+ =
=
?
? ?
=
?
?
false. is condition when 0,
true; is condition when , 1
} { where
; Margin if } _ _ { *
) 1 (
; Margin if } _ _ { *
) 1 (
; Margin if
1 max
1 min
Otherwise,
moment; adjustment an not is t if ,
) 1 ( ) (
condition I
Cap Effective_ Index(t) , Cap Life Cap Effective I
t WAC
Index(t) Floor Effective_ , Floor Life Floor Effective I
t WAC
Cap Effective_ Index(t) Floor Effective_ ,
Index(t)
WAC(t)
); Period_Cap ) WAC(t- (Life_Cap, Cap Effective_
loor); )-Period_F r, WAC(t- (Life_Floo Floor Effective_
t WAC t WAC
?
?
?
?
? ?
(2.66)
Note that the gradient is 0, when it is bounded by lifetime cap or floor, because a
perturbation would not change the WAC(t).
Next, we need to derive
? ?
? ) (t CPR
for ARM, assuming ARM borrowers have the
same prepayment behavoir as FRM borrowers (which is not necessarily true, but it does
not affect our analysis), so we are facing the same prepayment function as FRM30 as in
(2.45).
? ?
? ) (t CPR
will be affected because of the uncertainty of WAC(t).
39
.
) 0 (
1 ) 1 (
7 . 0
) (
];
) 1 ( ) (
[
))) 1 ( ) ( ( 430 571 . 8 ( 1
430
14 . 0
) (
;
) (
) ( ) ( ) ( ) ( ) ( ) (
) ( ) (
10
2
10
B
t B t BM
t r t WAC
t r t WAC
t RI
t BM
t MM t AGE t RI t BM t MM t AGE
t RI t CPR
? ?
? ? ?
? ? ?
?
? ?
=
?
?
?
? ?
?
?
?
? ? + ? +
=
?
?
?
?
+
?
?
=
?
?
(2.67)
.
) (
)) ( 1 (
12
1 ) (
; ) ( 1 1 ) (
12
11
12
? ? ?
?
? =
?
?
? ? =
?
t CPR
t CPR
t SMM
t CPR t SMM
(2.68)
Also C(t) will be affected by the introduced uncertainty in WAC(t):
{ } ) ( ) ( )] ( 1 )[ ( ) 1 ( ) ( t SMM t g t SMM t A t B t c + ? ? = , (2.69)
where
);
12
) (
1 ( ) (
;
) 12 / ) ( 1 ( 1
12 / ) (
) (
t WAC
t g
t WAC
t WAC
t A
t WAM
+ =
+ ?
=
+ ?
(2.70)
and
{ }
, ) (
) (
)] ( 1 [
) (
) 1 (
)] ( ) ( )[ 1 (
) (
) ( ) ( )] ( 1 )[ (
) 1 ( ) (
)
`
¹
¹
´
¦
?
?
+ ?
?
?
? +
+ ? ?
?
?
+ + ?
?
? ?
=
?
?
t SMM
t g
t SMM
t A
t B
t g t A t B
t SMM
t SMM t g t SMM t A
t B t c
? ?
? ? ?
(2.71)
where
;
) (
12
1 ) (
;
) (
]
12 / ) ( 1
) 12 / ) ( 1 ( ) ) 12 / ) ( 1 ( 1 (
) (
144
1
) 12 / ) ( 1 ( 1
1
12
1
[
) (
2
? ?
?
?
?
?
=
?
?
?
+
+ ?
+ + ?
+
+ ?
=
?
?
+ ? + ?
+ ?
t WAC t g
t WAC
t WAC
t WAM
t WAC t WAC
t WAC
t WAC
t A
t WAM t WAM
t WAM
(2.72)
40
And the PA gradient estimator for balance B(t) is as the following:
.
) (
12
) 1 ( ) (
)
12
) (
1 (
) 1 ( ) (
); ( )
12
) (
1 )( 1 ( ) (
? ? ? ? ?
? ?
+
?
?
? +
?
? ?
=
?
?
? + ? =
t WAC t B t c t WAC t B t B
t c
t WAC
t B t B
(2.73)
The PA estimator for the discounting factor is unchanged, so we can get the modified
Fourier duration and volatility duration.
41
2.4 Numerical Example
2.4.1 Specification of Numerical Example
We need to specify two sets of data to price the mortgage: the mortgage data and
the interest rate data, which includes the initial term structure and parameters for the
interest rate model.
We price different mortgages to examine the different impacts that a term
structure shift or change in volatility may have on different mortgage products.
The following data are fixed for all products:
Unpaid Balance/UPB =$4,000,000;
WAM =360 months.
Table 2.1 shows the difference between all the products. All the ARM products
have the same subsequent adjustment period of 12 months, period cap/floor of 0.02,
lifetime cap of initial WAC plus 0.06, and no lifetime floor.
Product WAC Index Adjust First
FRM 0.07425 N/A N/A
1 Year ARM 0.06425 Treasury 1 Year 12 month
3/1 FP
1
ARM 0.07425 Treasury 1 Year 36 month
5/1 FPARM 0.07425 Treasury 1 Year 60 month
7/1 FPARM 0.07425 Treasury 1 Year 84 month
10/1 FPARM 0.07425 Treasury 1 Year 120 month
1 Year ARM
2
0.07425 Treasury 10 Year 12 month
Table 2.1 Product Specification for Mortgage Pricing
1
FP ARM refers to Fixed Period ARM, which keep the coupon rate constant for a certain period, and then
adjust periodically, generally once a year. So All the FP ARM products are the same, except different
Adjust First date, which is the first coupon reset date.
2
This ARM is not a mortgage product in the market at present, and is constructed for illustration purpose
only. The following sections will discuss why we introduce this product, and what nice properties it has.
42
We use the same parameters for all the different products in order to have
comparable results. Thus we set all the products to have the same coupon rate, except the
first 1 year ARM with index of Treasury 1 year rate, which has a 100 basis points (bps)
teaser rate. All the ARM products have the same characteristics, except for the Adjust
First date, which is the feature that distinguishes these products.
Our initial term structure is the following:
f(0,t)=ln(150+12t)/100, t=0,1,…,360.
This will produce an upward-sloping curve increasing gradually from 5% to 8.7%
along 30 year maturity, and R(0,t) is acquired by calculating the following:
); 0 , 0 ( ) 0 ( ) 0 , 0 ( ,
) , 0 (
) , 0 (
0
f r R
t
du u f
t R
t
= = =
?
(2.75)
which increases from 5%, to 7.78% gradually.
Our assumptions for interest rate model parameters are the following:
a=0.1; ?=0.1; ?
n
=0.00025, n=0,1,2,3 (used in the FD gradient and gamma
estimator calculation); ??=0.00025, (used in the FD vega estimator calculation).
2.4.2 Comparison of PA and FD gradient estimators
In order to test whether our PA gradient estimators are accurate, and are within
the error tolerance range, we calculate the finite difference (FD) gradient estimators at the
same time during our pricing process. This section will demonstrate the accuracy of our
PA estimators of delta, vega, and gamma for FRM, as well as the delta and gamma for
ARM.
43
Comparison of Modified Fourier Gradient Estimators for FRM
Figure 2.5 shows the FD estimator, PA estimator, their difference, and standard
deviation of their difference for
n
t d
? ?
? ) (
. The four curves in each chart are specified as
following, which will be the convention for the rest of the paper:
Blue: Modified Fourier Order 1;
Green: Modified Fourier Order 2;
Red: Modified Fourier Order 3;
Cyan: Modified Fourier Order 4.
We can see that although these two estimators are pretty close, there exists a
pattern in the difference of these two estimators. This will be explained later in the error
analysis section.
Figure 2.6 shows the PA and FD gradient estimators for cash flow c(t): they are
pretty close, and the difference behaves as random noise. Based on
?
?
?
? ) , (t c
and
?
?
?
? ) , (t d
,
we can calculate
?
?
d
t dPV ) , (
, and figure 5.3 shows us the
n
d
t dPV
?
) (
. Figure 5.4 shows the
95% confidence interval for difference between PA and FD estimators of
n
d
t dPV
?
) (
, and
we can see that 0 is generally contained in the 95% confidence interval.
44
Figure 2.5 Gradient Estimator Comparison for ?d(t)/ ??
n
Figure 2.6 Gradient Estimator Comparison for ?c(t)/ ??
n
45
Figure 2.7 Gradient Estimator Comparison for dPV(t)/ d?
n
Figure 2.8 95% Confidence Interval for dPV(t)/d?
n
46
Comparison of Vega Estimators for FRM
In this section, we also compare the FD and PA estimators for the gradient w.r.t.
interest rate volatility: Vega. Figure 2.9 shows the FD estimator, PA estimator, their
difference, and standard deviation of their difference for
? ?
? ) (t d
. Also there exists a
pattern in the difference of these two estimators. This will also be explained later in the
error analysis section. Figure 2.10 shows the gradient estimators for cash flow c(t): they
are pretty close, and the difference behaves as random noise. Figure 2.11 shows us the
? d
t dPV ) (
, and figure 2.12 shows the 95% confidence interval for
? d
t dPV ) (
, and we can see
that 0 is always contained in the 95% confidence interval.
Figure 2.9 Gradient Estimator Comparison for ?d(t)/??
47
Figure 2.10 Gradient Estimator Comparison for ?c(t)/??
Figure 2.11 Gradient Estimator Comparison for dPV(t)/d?
48
Figure 2.12 95% Confidence Interval for dPV(t)/d?
Comparison of Gamma Estimators for FRM
For gamma estimation, ?=[?
1
?
2
?
3
?
4
]’. So
2
2
) (
? ?
? t d
,
2
2
) (
? ?
? t c
, or
2
2
) (
? ?
? t PV
is a
4x4 matrix. If we want to estimate this matrix by the FD method, we would need 144
points to estimate 48 first order derivatives and to estimate 16 second order derivatives.
The following figures show the FD, PA estimators, the difference and STD of
difference for diagonal gamma elements.
49
Figure 2.13 gamma estimators for
2
2
) (
i
t d
? ?
?
, i=1, 2, 3, 4
Figure 2.14 gamma estimators for
2
2
) (
i
t CPR
? ?
?
, i=1, 2, 3, 4
50
Figure 2.15 gamma estimators for
2
2
) (
i
t CF
? ?
?
, i=1, 2, 3, 4
Figure 2.16 gamma estimators for
2
2
) (
i
t PV
? ?
?
, i=1, 2, 3, 4
51
Comparison of ARM gradient estimators
For ARM products, we basically have the same set of PA gradient estimators to
compare with FD gradient estimators, with one additional set of estimators for
i
t WAC
? ?
? ) (
(figure 2.17). To illustrate the accuracy of our simulation in a brief way, we
only show the FD and PA gradient estimator comparison for one ARM product, 1-Year
ARM with index of 1-Year Treasury rate, adjusted annually.
Figure 2.17 Gradient Estimator Comparison for
i
t WAC
? ?
? ) (
, i=1, 2, 3, 4
Figures 2.18 and 2.19 show the FD/PA gradient estimator comparison for
i
t PV
? ?
? ) (
and
? ?
? ) (t PV
for this ARM product, respectively.
52
Figure 2.18 Gradient Estimator Comparison for
i
t PV
? ?
? ) (
, i=1, 2, 3, 4
Figure 2.19 Gradient Estimator Comparison for
? ?
? ) (t PV
53
2.4.3 Result Analysis
Efficiency Analysis
In financial practice, people are more interested in duration, which is the
percentage change for a security, once there is a minor shift in one parameter, which
mathematically is expressed as
NPV d
dNPV
duration
1 ) (
?
?
= . (2.76)
Actually, there should be a minus sign before the expression, since the original
duration of fixed income securities measures the percentage price drop resulting from an
increase in the interest rate. Yet for our analytical purpose, we do not need the duration
always to be positive, since from the following numbers, we see that durations can also
be negative. Table 2.2 shows the FD and PA durations for FRM30, their 95% confidence
interval, and the error range of the mean.
Fourier Order 0 1 2 3 Vega
PA estimator -6.4816±0.1017 3.1012±0.1860 -0.5705±0.1817 0.6269±0.1189 -6.7567±0.6712
FD estimator -6.4814±0.1017 3.1001±0.1860 -0.5695±0.1816 0.6259±0.1188 -6.7565±0.6712
Absolute Error -0.0002 0.0011 -0.001 0.001 -0.0002
Relative Error 0.0031% 0.0355% 0.1753% 0.1595% 0.0030%
Table 2.2 Comparison of PA/FD Duration
We can see that the error size is very small, and the 95% confidence intervals are
almost the same. Thus from the accuracy point of view, we can use PA estimator to
replace FD estimator without causing too much problem. And the improvement in
computation efficiency is enormous. The FD duration estimator works in the following
way:
54
) (
1
2
) ( ) ( 1
*
) , (
? ?
? ? ? ?
?
?
NPV
NPV NPV
NPV d
t dNPV ? ? ? ? +
= . (2.77)
Thus for each parameter, we need two additional simulations. In our case, we
need 2x5+1=11 simulations to estimate the FD duration. However, by PA estimator, we
only need one simulation. Ignoring the costs of middle steps, and middle variables, we
can reduce the computational time by 10/11, or 90.9%. When we consider the second
order derivative, gamma, the computational efficiency improves even more.
The following table shows the comparison of convexity estimators for FRM.
Convexity is calculated as following:
NPV d
NPV d
convexity
1 ) (
2
2
?
?
= . (2.78)
As we have mentioned earlier, we only estimated part of the FD gamma
estimators, via using the PA delta estimators. Because to fully estimate one set of 25
gamma estimators, we would need to simulate 225 times to get all of them. And each
element is a 360 by 300 (time length by simulation path) matrix.
So from the above analysis, we can see that by the conventional FD method, to
estimate one full set of duration and convexity estimators with 5 free variables, would
require 11 plus 225 simulations. Since we achieve almost the same accuracy by a single
simulation in PA analysis, the simulation cost is reduced roughly by more than 99.5%.
However, we also need to contemplate the introduced costs of intermediate variables as a
tradeoff of the PA method.
55
Convexity = Gamma/Mortgage Value
Mortgage Value = 4.22E+08
FD estimator 0 1 2 3 Vol
0 -246.5944896 N/A N/A N/A N/A
1 N/A -1871.407927 N/A N/A N/A
2 N/A N/A -1854.492905 N/A N/A
3 N/A N/A N/A -2000.544882 N/A
Vol N/A N/A N/A N/A -4751.605032
PA estimator
0 -246.6418706 951.9319609 -646.2296558 161.8535453 919.0969179
1 951.9319609 -1871.360546 1251.356282 -233.9200682 -1435.431523
2 -646.2296558 1251.356282 -1854.208619 1106.51252 1223.425174
3 161.8535453 -233.9200682 1106.51252 -2000.92393 -715.5006989
Vol 919.1916799 -1435.407832 1223.377793 -715.4533179 -4755.158608
Fourier Order 0 1 2 3 Vol
PA estimator -246.5944896 -1871.407927 -1854.492905 -2000.544882 -4751.605032
FD estimator -246.6418706 -1871.360546 -1854.208619 -2000.92393 -4755.158608
Absolute Error 0.047381014 -0.047381014 -0.284286087 0.379048115 3.553576082
Relative Error 0.0192% 0.0025% 0.0153% 0.0189% 0.0748%
Table 2.3 Comparison of Convexity Estimators
We did all the simulations on a Pentium III 800 MHz processor, with 512 MB
memory, in Matlab Release 12.0 under Windows 2000. Here is the simulation
comparison.
Method FD PA
Memory Required 17 MB 54 MB
Simulation Time for 300 paths 115.5 765.8
Number of Duration Measures 5 5
Simulation required for estimating
Duration
11 1
Number of Convexity Measures 25 25
Simulation required for estimating
Convexity
225 1
Total Simulation 236 1
Total Expected Simulation Time 27257.7 765.8
Efficiency Improvement 97.2%
Table 2.4 Comparison of Computing Costs
56
Accuracy Analysis
In order to validate the predictive power of our PA estimator, we setup a test case
to compare the predicted percentage change in the MBS price with the real percentage
change.
The test case is set up as following:
. 5 5
; 3 , 2 , 1 , 0 , 5 5
; )) 1 ( cos( 0 0
3
0
/
0
? = ?
= ? = ?
? ? + =
?
=
?
e
n e
e n ,t) R( ,t) R( Perturbed_
n
n
T t
n
?
?
(2.79)
. 004 - -5.0474e =
?
=
?
NPV
NPV NPV Perturbed_
NPV
NPV
While the predicted change in NPV is calculated as following:
004 - -5.0414e
'
2
1
'
) ( '
2
1
)' (
2
2
=
? × × ? + ? × =
? ×
?
?
× ?
+
? ×
?
?
?
?
? ? ?
?
?
? ?
?
convexity duration
NPV
NPV
NPV
NPV
NPV
NPV
(2.80)
where ??=[?
1
?
2
?
3
?
4
??].
We can see that the relative error by using both duration and convexity measures
is only 0.0056%, while using duration measures only would produce a relative error of
0.1403%. So this test validates the predictive power of our PA gradient estimators. In the
next section, we are going to show that PA estimator not only is more efficient than FD
estimator, but also is a more accurate estimator.
57
Error Analysis
Figure 2.5 and 2.9 show that there exists a pattern in the difference of gradient
estimator of discounting factor d(t). Actually this has two reasons: the calculation of
forward rates f(0,t) and the finite difference estimator of d(t). This could be verified by
figure 2.9, which shows the difference of FD and PA
n
t f
? ?
? ) , 0 (
estimators.
We know that in the Hull-White model, f(0,t) is determined by (2.23). However,
generally we do not have an explicit function form for R(0,t). Instead, we only have
discrete points for term structure, so R(0,t) is estimated by interpolation. And f(0,t) is
further estimated by calculating the difference between adjacent points on R(0,t) as
t
t R
?
? ) , 0 (
, which is not so accurate. The detailed calculation is given below.
Figure 2.20 Difference of FD/PA ?f(0, t)/ ??
n
estimators
58
?
?
?
?
?
?
?
?
?
=
?
? ? ?
< <
?
? ? ? ? +
=
?
? ?
=
?
?
term maximum the is T , ,
) , 0 ( ) , 0 (
0 ,
2
) , 0 ( ) , 0 (
0 ,
) 0 , 0 ( ) , 0 (
) , 0 (
T t
t
t T R T R
T t
t
t t R t t R
t
t
R t R
t
t R
(2.81)
So using FD method to calculate the f(0,t) will result inaccuracy in FD estimator
of
n
t r
? ?
? ) (
, and this will result inaccuracy in d(t). Also we know that d(t) takes the
following form:
} ] ) ( [ exp{ ) (
1
0
t i r t d
t
i
? ? =
?
?
=
, and
t
i r
t d t
i r
t i r
t d
t
i
t
i
t
i
?
?
?
? = ?
?
?
? ? ? =
?
?
? ? ?
?
=
?
=
?
=
)
) (
( ) ( )
) (
( } ] ) ( [ exp{
) (
1
0
1
0
1
0
? ? ?
. (2.82)
Figure 2.21 Fuction of xe
-x
59
However, when we use FD method to estimate the first order derivative of e
-x
, the
FD estimator is always greater in the absolute value, because e
-x
is a convex function. So
FD estimator of
? ?
? ) (t d
is always biased, the bias decreases as the FD step width reduces.
The bias increases linearly, while d(t) decreases exponentially. As a result, the bias takes
the form of xe
-x
. Compare the difference of FD/PA
? ?
? ) (t d
gradient estimators and the
figure of xe
-x
as in Figure 2.14, which resembles the error pattern very closely.
For the PA method,
n
t f
? ?
? ) , 0 (
is estimated by the following formula,
0
/
/ /
) (
)) 1 ( sin( )) 1 ( cos(
) , 0 (
0
0 0
T
e n
e n e n
t f
T t
T t T t
n
?
? ?
? + ? =
? ?
? ?
? ? (2.83)
which does not involve the FD estimation of
t
t R
?
? ) , 0 (
. And
? ?
? ) (t d
is directly estimated
using its analytical form of first order derivative. So the PA estimator is more accurate
than the FD estimator.
60
2.5 Interpretation of the Results
In this section, we briefly present the durations for various mortgage products,
which show different trends for modified Fourier duration of different order. And we try
to interpret how the modified Fourier shocks of different order would affect the
discounting factors and the cash flows, and then the present value (PV) of the mortgage.
Then we analyze the relationship of mortgage prepayment option and mortgage duration.
Based on these analysis, we propose a potential new ARM product, which could reduce
the duration over any of the existing mortgages, while having a less volatile index than
most existing mortgages. This product would benefit both the investors who want to
reduce the interest risk, and the mortgage borrowers who want to have a fairly stable
coupon rate.
2.5.1 Overview of the Results
The following table shows the durations for various ARM and FRM products we
specified and priced in section 2.4:
Fourier Order 0 1 2 3 Vega
ARM TSY 1 -1.7761 -4.313 6.5674 -2.3822 -3.4618
FP 3/1 ARM -2.8441 -2.9642 7.1814 -3.4784 -4.1601
FP 5/1 ARM -3.8514 -1.1609 5.9506 -5.3355 -5.2456
FP 7/1 ARM -4.3054 -0.3064 4.9272 -5.4472 -5.6651
FP 10/1 ARM -5.4256 1.6401 1.7592 -2.6933 -6.6163
FRM30 -6.4816 3.1012 -0.5705 0.6269 -6.7567
Table 2.5 Durations of Different Products
The relation can be better illustrated with the figure 6.1. The zero
th
order modified
Fourier duration (with respect to ?
0
) is the same as Option Adjusted Duration (OAD),
which measures the price percentage change to a parallel interest term structure shift.
Other modified Fourier durations are the same measure, with respect to other interest
61
term structure changes. Vega measures the price percentage change to an interest
volatility change. As we can see, for OAD and Vega, the most important hedge measures,
FRM30 has the highest numbers, and 1-Year ARM has the lowest. For everything
between pure FRM and pure ARM, there exists a monotonic relationship with the
product’s approximation to an FRM30. For example, the Fixed Period 10/1 ARM is more
like an FRM30 than a Fixed Period 7/1 ARM, so it has higher OAD, and higher Vega.
This means that ARM products have a lower interest risk than FRM products,
since an ARM borrower takes more interest risk than an FRM borrower. This result is
consistent with Kau et al.[1990,1992,1993] and Chiang [1997].
However, an interesting phenomenon is that the first order modified Fourier duration
(with respect to ?
1
) actually decreases, and changes sign as volatility of the coupon rate
decreases. This indicates that an opposite move of the long-term and short-term rates
would not only affect ARMs with a different magnitude, but also has a reverse effect
from FRMs. Here is the explanation for this. The first order modified Fourier duration
models the following changes in term structure:
• Short-term rate increases;
• Intermediate term rate (e.g. 10 year rate) doesn’t change, or moves only a little bit;
• Long-term rate decreases.
In this scenario, people with a short-term ARM, e.g. 1-year ARM are burnt the
hardest, so they are going to refinance anyway, even if the prevailing mortgage rate does
not change a lot. This will create huge prepayment, and reduce the NPV of the ARM
mortgage. People with FRM, on the other hand, have no incentive to refinance, since the
refinance mortgage rate (highly correlated with 10-Year Treasury rate) does not change a
62
lot. This will make the future cash flow more stable and valuable, since they are
discounted at a lower long-term interest rate, and increase the NPV of the FRM
mortgage.
Figure 2.22 Duration vs. Products
The above analysis is based mainly on intuition, and does not show how will this
term structure shock affect the discounting factors, cash flows, and NPV of MBS. In the
following section, we will see what effect each one of the modified Fourier functions has
on these components of MBS for various mortgage products.
2.5.2 Modified Fourier Shock Impact
The following 8 charts will show different modified Fourier shocks on term
structure R(0,t), and their impact on d(t), CF(t), and PV(t).
Duration vs. Products
-8
-6
-4
-2
0
2
4
6
8
A
R
M
T
S
Y
1
F
P
3
/
1
A
R
M
F
P
5
/
1
A
R
M
F
P
7
/
1
A
R
M
F
P
1
0
/
1
A
R
M
F
R
M
3
0
Products
D
u
r
a
t
i
o
n
Harm Duration 0
Harm Duration 1
Harm Duration 2
Harm Duration 3
Vega
63
Figure 2.23 The Impact of Modified Fourier Function Order 0 on FRM30
Explanation: A parallel shift in the upward slope term structure will have a negative
impact on the discounting factor. Also people are less likely to prepay in the near future,
which reduces the cash flow in the short term, and increase the cash flow in the long term
a little bit. However, the overall effect of such a shift on present value is negative, and
thus reduces the NPV of this MBS.
64
Figure 2.24 The Impact of Modified Fourier Order 1 on FRM30
Explanation: A shift of this shape in the upward slope term structure will have a mixed
impact on the discounting factor: decrease it in the short term, but increase it in the long
term. Also people are more likely to prepay in the near future, which increases the cash
flow in the short term, and reduces the cash flow in the long term a little bit. However,
the overall effect of such a shift on present value is positive, and thus increases the NPV
of this MBS.
65
Figure 2.25 The Impact of Modified Fourier Order 2 on FRM30
Explanation: A shift of this shape in the upward slope term structure will have a mixed
impact on the discounting factor: increase it in the middle term, but decrease it in the long
term. There is little incentive for people to prepay in the near future, and they will also
cling to their current coupon rate in the middle term, because at that time the refinance
rate will increase. However, the overall effect of such a shift on present value is cancelled
out, and has little impact on the NPV of this MBS.
66
Figure 2.26 The Impact of Modified Fourier Order 3 on FRM30
Explanation: same as Modified Fourier Order 2
67
Figure 2.27 The Impact of Modified Fourier Order 0 on ARM TSY 1
Explanation: A parallel shift in the upward slope term structure will have a negative
impact on the discounting factor. Also people with ARM are less likely to prepay in the
near future, because they have a lower ARM rate than the refinance. Yet they will start
prepay in the middle term, because short term rate at that time will increase, due to the
upward slop term structure. This behavior will reduce the cash flow in the short term, and
increase the cash flow in the middle term. However, the overall effect of such a shift on
present value is negative, and thus reduces the NPV of this MBS. Yet the impact will be
much smaller than that for FRM.
68
Figure 2.28 The Impact of Modified Fourier Order 1 on ARM TSY 1
Explanation: A shift of this shape in the upward slope term structure will have a mixed
impact on the discounting factor: decrease it in the short term, but increase it in the long
term. Also people are more likely to prepay in the near future, which increase the cash
flow in the short term, and reduce the cash flow in the long term a little bit. The overall
effect of such a shift on present value is negative, and thus decreases the NPV of this
MBS.
69
Figure 2.29 The Impact of Modified Fourier Order 2 on ARM TSY 1
Explanation: A shift of this shape in the upward slope term structure will have a mixed
impact on the discounting factor: increase it in the middle term, but decrease it in the long
term. People will cling to their low ARM rate for the first few years, but then start to
prepay in the middle term, since short term rate will increase at that time. The overall
effect of such a shift on present value is positive, due to the increase cash flow and
discounting factor in the middle term.
70
Figure 2.30 The Impact of Modified Fourier Order 3 on ARM TSY 1
Explanation: much like Modified Fourier Order 3, yet because the reverse effect of
discounting factor, the overall effect will be negative.
6.3 Potential New ARM Product
Duration is used to measure the interest risk of a fixed income security. The
higher the duration is, the more interest risk that security bears. From the investor’s
perspective, she will benefit if interest rates fall, and suffer if interest rates climb, if the
security is non-callable (no prepayment option). From the mortgage borrower’s point of
view, he will exercise his prepayment option if interest rates drop, and thus reduce the
benefit for the investor. He will be able to lock in the low mortgage rate (for FRM), in
case interest rates climb, and thus hurt the investor more. However, for the ARM
borrower, he benefits from the rate drop, so he does not prepay like the FRM; thus the
71
MBS investor will also benefit. And he also pays the high coupon rate when interest rates
increase, and the ARM MBS investor will not suffer like the FRM MBS investors. From
this perspective, the ARM should have a lower duration compared to FRM.
ARM borrower’s coupon rate fluctuates with the current interest rate, which is
correlated with the prevailing mortgage rate. Because of this, she will have less incentive
to prepay when interest rate drops. So the prepayment option value for a FRM borrower
will be larger than that of an ARM borrower. This is compatible with the market, where
FRM mortgages are sold with the highest rate (borrower pays for the valuable
prepayment option), and ARM, that adjust most frequently are offered with the lowest
rate.
In option theory, we know that option value generally increases as the volatility of
underlying asset increases. However, from the above analysis, we also know that the
option value for a FRM is generally greater than for an ARM, while an ARM bears a
more volatile coupon rate than a FRM. This looks like a contradiction to the option-
volatility relationship. In fact, it’s not, because the underlying asset of a prepayment
option is not its coupon rate, but the difference between the coupon rate and the
prevailing mortgage rate. In most cases, the more volatile the coupon rate is, the less the
difference will be, and the less valuable the option will be. However, the borrower does
not like the volatility, which put her at risk when interest rate jumps. The investor, on the
other hand, does not like the prepayment, which reduce her investment value. It seems
that no product can both reduce the coupon rate volatility and the prepayment option at
the same time. Is this true? We will see that we can achieve both goals in a potential new
ARM product.
72
We have mentioned that the underlying asset for the prepayment function is the
spread between the coupon rate and the prevailing mortgage rate. So an ARM bearing a
volatile index does not necessarily indicate a less volatile spread. From historical data, we
know that 10-Year Treasury rate is highly correlated with conventional (FRM30)
mortgage rate. Figure 6.10 shows the two rates for the period between 1971 and 2001.
The correlation calculated is 97.9%. Figure 6.10 also shows the 10-Year Treasury Rate
and the 1-Year Treasury rate, which is the most commonly used index in ARM. As we
can see, the 1-Year Treasury rate is relatively more volatile than the 10-Year Treasury
Rate. The calculated standard deviation is 2.7890 for the 1-Year Treasury rate, and
2.6309 for the 10-Year Treasury rate. However, the standard deviation of spread of
FRM30 vs. TSY10 is 0.58, compared with the standard deviation of spread of FRM30 vs.
TSY1 at 1.16. Obviously 10-Year Treasury rate has a lower volatility and also a lower
volatile spread. The spread between conventional mortgage rate (FRM30) and 10-Year
Treasury rate and the spread between conventional mortgage rate (FRM30) and 1-Year
Treasury rate are also shown, which indicates that ARM with index of 1-Year Treasury
rate has a more volatile spread.
73
Figure 2.31 10-Year T Rate, 1-Year T Rate, and mortgage rate
Thus if we construct an ARM with index of 10-Year Treasury rate, and reset it
more frequently, we could expect a lower duration. So we construct such an ARM with
the adjustment period of 12 months. This ARM does not exist at present; it is for
illustration purposes only. We then got the modified Fourier duration measures as
following:
Fourier Order 0 1 2 3 Vega
ARM TSY 10 -1.2741 -4.0635 3.1819 -0.4894 -1.8855
Figure 2.32 shows the new ARM product’s duration against duration of other
mortgage products we calculated earlier. We compare this set of durations with table 6.1,
74
and we can see that this product has the smallest durations for modified Fourier function
order 0 and 3, as well as for vega. The durations for modified Fourier function order 1
and 2 are not very high. And we know that generally when there is a shock on the term
structure, the biggest magnitude would be that of the first-order modified Fourier
function, and volatility is also a big impact. So this product would actually have the least
percentage change during a common term structure shift, which satisfies the needs of
investors.
Duration vs. Products
-10
-8
-6
-4
-2
0
2
4
6
8
A
R
M
T
S
Y
1
0
A
R
M
T
S
Y
1
F
P
3
/
1
A
R
M
F
P
5
/
1
A
R
M
F
P
7
/
1
A
R
M
F
P
1
0
/
1
A
R
M
F
R
M
3
0
Products
D
u
r
a
t
i
o
n
Harm Duration 0
Harm Duration 1
Harm Duration 2
Harm Duration 3
Vega
Figure 2.32 New ARM TSY 10 Durations
So we could predict that if there exist such a mortgage, it would have the least
refinancing incentive, which would be a better product to suit investors’ needs, and it will
also have a less volatile index, which suits borrowers’ needs.
75
2.6 Conclusion
This paper applies perturbation analysis (PA) method to estimate MBS
sensitivities. The sensitivity estimators include most interest risk measures like duration
(equivalent to delta), convexity (equivalent to gamma), and vega. MBS products covered
includes fixed rate mortgages (FRMs) and adjustable rate mortgages (ARMs).
We first derive a general framework to derive the PA estimators of MBS, without
restriction to MBS type, interest rate model, or prepayment model. Then we apply the PA
estimator to both FRM and ARM products, in the setup of a one-factor Hull-White model
and a commonly used prepayment model. We compare the PA estimators with finite
difference (FD) estimators, and find that PA method can achieve at least the same
accuracy as FD method, with a much lower computational cost. In the case we presented,
the computational time is reduced by 95.7%, while the memory requirement increases
only by a factor of 3, which can be handled by current computer technology with ease.
Then we analyze the results of PA estimated sensitivity measures for various MBS
products. We justify why and how different term structure shock would affect FRM and
ARM differently. Based these analysis, we propose a potential new ARM product which
could benefit both the MBS investor and the mortgage borrower.
Future research includes applying this method to other MBS-like securities, since
the PA method proposed in section 3 is a very general framework. These include other
asset-backed securities, e.g. securities backed by student loans, car loans, credit card
receivables. It is pretty straightforward to expand this framework to those securities, since
all that is required is to apply a specific interest rate model and prepayment model.
76
Another area for further research is to incorporate more complicated prepayment
and/or default models into the MBS pricing scheme. For MBS investors, the major
concerns are price sensitivities to interest changes, which we have covered in detail.
However, the MBS guarantor/insurer and issuer might have other concerns, e.g., how will
the interest rate change affect the default behavior of the mortgage borrowers? Our
framework would be able to serve this purpose as well. By applying the default model
that same way as we apply a prepayment model, the default cash flow will take the place
of payment cash flow, so the default cost sensitivities could be easily estimated.
77
Chapter 3
Hedging MBS in HJM Framework
3.1 Motivation
As we have pointed out in our first essay, short term rate and long term rate do not
always move in the same direction, it is sometimes misleading to use the conventional
interest rate risk measures like duration and convexity to hedge fixed income instruments,
especially MBS.
One recent event can illustrate this point very well. In July 2003, Federal Reserve
lowered the short term interest rate by another 25 bps, yet just in one month, the long
term 10 year rate jumped upward for more than 100 bps. Part of the reason is that the rate
deduction is lower than market expectation, and market responded with a selling wave in
the bond section. So using a duration measure, which assumes the yield curve moves in
parallel, will produce significant hedging error.
It is natural to hedge against the factors of which any yield curve shift can be
decomposed. We use a series of exponentially decaying modified Fourier series to
approximate any interest rate change in our first essay. However, this is purely for the
generality of modeling convenience, and there is no empirical evidence that such a series
provides a good match of the actual yield curve shift.
78
A lot of literature studying the dynamics of interest rates found that there are three
major factors affecting the yield curve: level, slope, and curvature. A common method to
estimate these factors is Principal Components Analysis (PCA). See details in Litterman
and Scheikman [1991], Litterman, Scheikman, and Weiss [1991), Knez, Litterman, and
Scheikman [1994], Nunes and Webber [1997]. Despite the abundance of research on
identifying the various factors affecting bond prices, there has been little research on
hedging these factors effectively. Golub and Tilman [1999] compared different risk
measures, like PCA, VaR, and key rate duration for yield curve risk, but did not give
hedging performance for these different measures.
In mortgage industry, practitioners generally use effective duration, and empirical
duration in hedging. Goodman and Ho [1999] examined the performance of three
different hedge ratios: effective duration, empirical duration, and option-implied duration,
which is acquired from forward option for a given pass-through MBS. They found that
the average hedging error for a monthly hedge could reach 120 bps in an 18-month
period. And for a daily hedging, it is 25 bps in the same time period. They concluded that
option-implied duration performs the best. However, it does not always outperform the
other hedging measure, and the difference is small. Hayre and Chang [1999] compared
effective duration and empirical duration, and found that effective duration calculated
from OAS model are generally longer than empirical duration, and they challenged a few
assumptions for effective duration calculation in OAS mode. To cite a few, the parallel
yield curve shifts, absence of convexity, etc. These are also issues we addressed in our
79
paper, but in a more systematic way. They proposed a combined duration, which is a
effective duration adjusted for correlations between changes in the yield and prices of
MBS in recent market data, i.e., a combination of effective duration and empirical
duration.
There has not been a unifying framework in hedging MBS with factors affecting
the yield curve shifts, and we would like to pursue in this direction, since we are pretty
confident about its effectiveness in reducing the hedging error, and/or reducing hedging
frequency. In order to incorporate these factors into MBS hedging strategy, we need to
choose an interest rate model, which can handle these factors readily. HJM model is a
good choice, because it is basically driven by volatility structure, and the volatility factors
can take any shape, which easily accommodate the PCA factors we identified from
historical data.
In the rest of this essay, we discuss how to get volatility factors from historical
interest rate data via the PCA method. In section 3, we are give the detailed
implementation of HJM model with these estimated volatility factors. Then we derive the
PA estimators for hedging MBS, which is very similar to Chapter 2, and we will not go
into details to derive PA estimator for each state variable. In section 5, we give the
detailed hedging algorithm with these hedging measures, and we discuss the performance
of our hedging method in section 6. Section 7 concludes the essay, and gives potential
future research directions.
80
3.2 Estimation of Volatility Factors via PCA
The Principal Components Analysis method is generally used to find the
explanatory factors that maximize successive contributions to the variance, effectively
explaining variations as a diagonal matrix. This method has been used in yield curve
analysis for more than 10 years, see Litterman and Scheinkman [1991], Steeley [1990],
Carverhill and Strickland [1992]. Here we give a brief description of PCA method
applied in yield curve analysis:
1. Suppose we have observation of interest rates ) (
j t
i
r ? at time t
i
, i=1, 2, …, n+1, for
different maturity dates ?
j
.
2. Calculate the difference ) ( ) (
1
, j t j t j i
i i
r r d ? ? ? =
+
, where the d
i,j
are regarded as
observations of a random variable, d
j
, that measures the successive variations in the
term structure.
3. Find the covariance matrix ) ,..., cov(
1 k
d d = ? . Write
) , cov( where }, {
, , j i j i j i
d d = ? ? = ? .
4. Find an orthogonal matrix P such that P’=P
-1
and
k k
? ... ? ) , ..., ? diag(? P P ? ? = ?
1 1
where , ' .
5. The column vectors of P are the principal components.
6. Using P, each observation of d
j
can be decomposed into a linear combination of the
principal components. By setting
j i i
d p e ' = , where p
i
is the i
th
column of P, we can
find e
i
, which is the corresponding coefficient for principal component i, i=1, …, k. A
81
small change in e
i
will cause the term structure to alter by a multiple of p
i
along the
time horizon.
We use the weekly data of nominal zero coupon yield from January 1997 to
October 2001 as the term structure data. All data were retrieved from Professor
McLulloch’s web site at the Department of Economics, Ohio State University, at
<http://econ.ohio-state.edu/jhm/ts/ts.html>. For each observation
date, interest rates are provided for maturities in monthly increments from the
instantaneous rate to the 40-year rate, providing a total of 481 interest rates as principal
components. Table 3.1 lists the eigen-values and % variance explained by the first ten
factors, and Figure 3.1 graphs the shapes of the first four factors.
Factor Eigenvalue Explained(%) Cumulative(%)
1 16.38 75.824 75.824
2 4.41 20.432 96.257
3 0.72 3.335 99.592
4 0.087 0.40 99.995
5 0.00088 0.0041 99.999
6 8.67E-05 0.00040 99.9996
7 1.59E-05 7.4E-05 99.99966
8 4.20E-06 1.9E-05 99.99968
9 4.03E-06 1.9E-05 99.99970
10 3.67E-06 1.7E-05 99.99972
Table 3.1. Statistics for Principal Components
82
Figure 3.1 The first four principal components
The statistics indicate that the first three factors explain about 99.6% of the yield
curve changes, and the first four factors explain about 99.995% of the total variance of
yield curve. These results are similar to findings by Litterman and Scheikman [1991], and
Nunes and Webber [1997]. Figures 3.2 and 3.3 plot the matching results with three and
four factors, respectively, for a monthly yield curve shift, as well as for an annual shift.
The figures indicate that four factors provide a substantially improved match, both for the
short term and the long term, over three factors, so in our model we will use four factors.
Thus, hedging against these factors will lead to a considerably more stable portfolio,
thereby reducing hedging transactions and its associated costs.
83
Figure 3.2 Match monthly yield curve shift
Figure 3.3 Match annual yield curve shift
84
3.3 Simulation in HJM Framework
This section gives the detailed implementation of HJM model, using the volatility
factors identified in PCA analysis.
We know that, in a multifactor HJM framework, the dynamics of instantaneous
forward rate looks like:
?
=
? + ? =
N
k
k t k t
t dZ T t dt T t m T t df
1
) ( ) , , ( ) , , ( ) , ( ? , (3.1)
where under no arbitrage assumption, the drift term is determined by volatility structure.
?
?
=
? ? = ?
N
k
T
t
t k t k t
d t T t dt T t m
1
) , , ( ) , , ( ) , , ( ? ? ? ? . (3.2)
Assume our volatility functions take the following form:
) , ( ) , , ( T t PC T t
k k t k
? ? = ? , (3.3)
where PC
i
(t, T) is the principal components we get in last section;
?
i
is a parameter to be calibrated to market price of interest rate derivatives.
Detailed Implementation:
1. Input data include f(0,T), the instantaneous forward curve, and ?
k
(t, T), which
has a specified functional form fitting into our PCA factors.
2. Start loop for maturity, if we need 10 year rate for 30 years, we need maturity at
40 year;
3. Start of time step loop;
4. Start of ? loop, to calculate ?
k
(t, ?) from t to T;
85
5. Calculate ?
k
*
(t, T)=
?
T
t
k
d t ? ? ? ) , ( , using numerical integration technique;
6. Calculate m(t, T)= ?
k
(t, T)* ?
k
*
(t, T);
7. Advance f(t, T) one more step, in our simulation, one month increment:
; ) , ( ) , ( ) , (
, ?
+ ? = ? +
k
k t k
z T t t T t m T t t f ?
where z
t,k
is a series of independent standard normal random variables.
8. End of time step loop;
9. Short rate r(t)=f(t,t); Long rate r
10
(t)=
10
) , (
10
?
+ t
t
d t f ? ?
;
10. End of maturity step loop.
86
3.4 Deriving PA estimators in HJM Framework
Following the logic in Chapter 2, we only need to derive the PA estimator for
short rate r(t) and 10-year rate r
10
(t), since our prepayment model and valuation model are
totally dependent on these two factors.
If we assume that in a short period of time, the principal components for yield
curve volatility are going to be constant, then any interest rate yield curve shift can be
decomposed of these principal components, which is to say:
?
? = ?
k
k k
t PC t R ) ( ) , 0 ( , (3.4)
which is analogous to (2.35). Following the same logics as in (2.51), we can have the
following:
). (
) ( ) , 0 (
,
) , 0 ( ) , 0 (
) , 0 ( ) , 0 ( ) , 0 (
) , 0 (
) , 0 (
) , 0 (
2
2
t PC
t
t PC
t
t f
t R
t
t R
t
t R
t
t R
t
t
t R t
t R t
t R
t
t f
k
k
k
+
?
?
=
? ?
?
?
?
+
? ?
?
=
?
?
+
? ?
?
+
?
?
?
?
=
?
?
+
?
|
.
|
\
|
?
?
?
=
?
?
? ?
? ? ?
? ? ?
(3.5)
We know that in HJM framework:
) , ( ) ( t t f t r = , (3.6)
So
k k
t t f t r
? ?
?
=
? ?
? ) , ( ) (
. (3.7)
87
We also know:
?
=
? + ? =
N
i
i t i t
t dZ T t dt T t m T t df
1
) ( ) , , ( ) , , ( ) , ( ? . (3.8)
When T=t+dt
dt) df(t,d dt) f(t,t dt) dt,t f(t + + + = + + . (3.9)
So
k k k
dt) df(t,d dt) f(t,t dt) dt,t f(t
? ?
+ ?
+
? ?
+ ?
=
? ?
+ + ? ] [
, (3.10)
?
=
? ?
+ ?
+
? ?
+ ?
=
? ?
+ ?
N
i
i
k
i
k k
t dZ
dt t t dt t t m dt) df(t,d
1
) (
) , ( ) , ( ] [ ?
. (3.11)
If we rewrite the drift term as:
) , ( * ) , ( ) , (
*
1
T t T t T t m
k
N
k
k
? ?
?
=
= , (3.12)
where
? ? ? ? d t T t
T
t
k k
?
= ) , ( ) , (
*
. (3.13)
Then gradient of m(t, T) can be written as
?
=
¦
)
¦
`
¹
¦
¹
¦
´
¦
? ?
?
+
? ?
?
=
? ?
?
N
k k
i
i i
k
i
k
T t
T t T t
T t T t m
1
*
*
) , (
* ) , ( ) , ( *
) , ( ) , ( ?
? ?
?
. (3.14)
From the form of ) , ( T t
k
?
) , ( ) , , ( T t PC T t
k k t k
? ? = ? . (3.15)
88
There is no direct relation ship between ) , ( T t
i
? and
k
?
1
, so the above gradients are
zero. This gives us
k k k
dt) ,d f( dt) f(t,t dt) dt,t f(t
? ?
+ ?
= =
? ?
+ ?
=
? ?
+ + ? ] 0 [
... . (3.16)
For the same reason, we can derive the 10 year rate gradient as:
10
) , (
) (
10
10
?
+
? ?
?
=
? ?
?
t
t
k
k
d
t f
t r
?
?
. (3.17)
And follow the same logic, we can get the gradients of discounting factors, prepayment
rate, cash flows, present values, etc.
1
Although observed
k
? and
k
? might have a positive correlation, i.e., when the volatility is high, the
observed shift also might have bigger magnitude. But they have total different meaning,
k
? is the
parameter to calibrate to market price, and
k
? is the observed shift in yield curve.
89
3.5 Hedging MBS in HJM Framework
This section gives a detailed implementation of our hedging algorithm.
Security to be hedged: MBS
Hedging Instruments: Portfolio of {MBS, Treasury bonds with different maturities}
Hedging Method: Dynamic hedging using PCA duration. vs. Conventional duration and
convexity hedging
Hedging Parameters: PCA duration
Hedging Error: The net present value of the portfolio, which has initial value of zero
Hedging Efficiency: Reduce hedging Error
Hedging Strategy: Construct a portfolio, consisting of MBS and various T-notes, bonds,
with 0 face value. Duration matched to 0. Rebalance at each time period to match the
hedging parameters; compare the results with duration and convexity hedging.
There are two issues we need to pay special attention to, in order to effectively execute
the hedging strategy.
Issue 1: With the coupon payment and prepayment of MBS, what needs to be done with
this extra cash flow?
Answer: Use this cash flow to rebalance the portfolio, basically to change the weights of
Treasury bonds holdings. If the position is short in MBS, and long in Treasury bonds, we
need to sell the Treasury bonds to honor the MBS payment.
Issue 2: Some Treasury bonds used to hedging the MBS will expire before the MBS
maturity date. This will hurt the capacity of available hedging instruments.
90
Answer: We only hedge the MBS for a short period of time, e.g. 3 years, and then we can
use Treasury bonds with greater or equal to 3 years maturities. Another solution is to
introduce on extra hedging instrument when there is one expiring at that period.
Hedging Framework
1. At time 0, get the MBS price, gradients (PCA duration) by simulation (360x300
simulation needed). Zero coupon Treasury bonds price and gradients should be
directly available from the yield curve, and the PCA factors;
2. Construct the portfolio, by shorting MBS to finance Treasury bonds; match the
duration, and get the corresponding weights;
3. At time 1, use HJM model to update the yield curve, then get the new price and
gradients of MBS as well as those of Treasury bonds;
4. Use MBS payment to rebalance portfolio (MBS payment is deterministic upon the
last period yield curve);
5. Repeat 3, 4 for next month, till the end of hedging period;
6. Check the effectiveness of hedging strategy.
Implementation of Hedging MBS with Treasury Bonds
1. Get mortgage information;
2. Get historical yield curve data;
3. Get Principal Components Factors;
4. Start clock for hedging period: m=0
5. Calculate MBS_Price(m), MBS_Duration(m)
4x1
, Payment(m),
PrincipalPayment(m);
91
6. Choose hedging instrument: Treas_Portfolio=[12 36 60 84 120], each element
represent months to maturity;
7. Calculate Treasury bond price Treas_Price(m)
5x1
, 5 hedging components are
needed because of 5 factors to hedge: Price, and Duration
4x1
.
Treas_Duration(m)
4x5
.
8. Solving for hedging ratio W(m):
1 5
1 5
5 5
) ( _
) ( _
) (
) ( _
)' ( _
x
x
x
m Duration MBS
m price MBS
m W
m Duration Treas
m price Treas
(
¸
(
¸
=
(
¸
(
¸
, if m=0; (3.18)
1 5
1 5
5 5
) ( _
) 1 ( _ ) 1 ( )' ( _
) (
) ( _
)' ( _
x
x
x
m Duration MBS
m payment MBS m W m price Treas
m W
m Duration Treas
m price Treas
(
¸
(
¸
? ? ?
=
(
¸
(
¸
, if m>0. (3.19)
9. Calculate hedging error:
) 1 ( _ ) 1 ( )' ( _ ) ( _ ) ( ? + ? ? = m payment MBS m W m price Treas m price MBS m error
10. Update loan.UPB=loan.UPB-PrincipalPayment(m);
11. Update loan.WAM=loan.WAM-1/12;
12. Update Treas_Portfolio=Treas_Portfolio-1;
13. m=m+1, go back to 5 until hedging period ends.
92
3.6 Hedging Performance Analysis
In this section, we compare the hedging performance of our PCA-based hedging
and traditional duration and convexity based hedging for a FRM30 MBS instrument.
The principal balance of the MBS is $4 million. We are selling short this MBS at
the market price, and use the proceeds to buy treasury bonds. Initial net present value of
the hedging portfolio is zero. Every month, we try to rebalance the portfolio, and we sell
part of our bonds to meet the payment obligation of the MBS. Hedging error is defined as
the net present value of current portfolio at each time point.
We carry on this practice for 22 months, during which our PCA estimation does
not change dramatically. We repeat the hedging practice for 25 simulations, which is
relatively few, because the simulation scheme takes an extremely long time. The PCA-
based hedging takes around 40 CPU hours to finish, while the duration and convexity
based hedging takes 120 CPU hours to complete the task.
Figure 3.4 shows the hedging performance of three PCA factors, while Figure 3.5
shows the hedging performance of duration and convexity hedging. We can see that the
standard deviation of PCA-based hedging ranges from $4000 to $20000, which is 10 bps
to 50 bps for a $4 million portfolio. Consider the standard deviation of duration and
convexity based hedging, which ranges from $60000 to $200000, i.e. 150 bps to 500 bps
of the hedging balance. The hedging improvement is obvious.
93
Mean Hedging Error
-5.00E+04
-4.00E+04
-3.00E+04
-2.00E+04
-1.00E+04
0.00E+00
1.00E+04
2.00E+04
3.00E+04
0 5 10 15 20 25
month
H
e
d
g
i
n
g
E
r
r
o
r
mean_PCA
mean_D&C
Figure 3.4 Mean Hedging Error of PCA vs. D&C
Standard Deviation of Hedging Error
0
50000
100000
150000
200000
250000
300000
0 5 10 15 20 25
month
H
e
d
g
i
n
g
E
r
r
o
r
std_PCA
std_D&C
Figure 3.5 STD of Hedging Error: PCA vs. D&C
94
3.7 Conclusion
In this essay, we proposed a new method to hedge the interest risk of MBS, based
on PCA factors estimated from historical interest rate data. We estimated the PA
estimators for hedging MBS, and implemented the hedging with a dynamically re-
balancing portfolio of MBS and Treasury bonds. We achieved much better hedging
efficiency, compared with traditional hedging, not only in the measure of mean hedging
error, but also in the standard deviation of hedging error. We made the following
contribution:
• A unified hedging framework for hedging yield curve shift and volatility factors;
• Improved hedging efficiency compared with traditional duration and convexity
based hedging. Our monthly hedging get very close results to daily hedging with
traditional hedging method.
We would like to pursue in the following directions for our future research:
• Apply this hedging method to more sophisticated prepayment models, and
analyze the robustness of this hedging algorithm;
• Improve computational efficiency of the algorithm, which is now very time
consuming.
95
Chapter 4
Hedging the Credit Risk of MBS: A Random
Coefficient Approach
4.1 Motivation
In our previous two chapters, we have assumed that the credit risk of the MBS is
totally absorbed by the MBS issuer, and the MBS investor only needs to hedge the
interest rate risk due to voluntary prepayment, including housing turnover and
refinancing. This assumption is reasonable since in the secondary market for conforming
mortgages, the three major MBS issuers, Ginnie Mae, Fannie Mae, Freddie Mac
1
, all
promise that they will guarantee the principal payment when there is a default event
incurred on the mortgage borrower’s side. The MBS issuers have the following methods
to mitigate the credit risk:
• Mortgage Collateral: Basically when a default occurs, the collateral property will
become REO(Real Estate Owned), and the issuer can foreclose the mortgage and sell
the property, and recover whatever is left;
• Primary Mortgage Insurance (PMI): If a borrower initiate a loan with LTV greater
than 80%, she will be required to purchase mortgage insurance. If default occurs, the
mortgage insurance company pays the owner of the mortgage whatever is promised in
the insurance contract, generally 35% for a 95 LTV loan, and 20% for a 85 LTV loan;
1
However, the credit risks of these agencies are different. Ginnie Mae is guaranteed by the full faith and
credit of the United States government. Both Fannie Mae and Freddie Mac have $2.2 billion line of credit
with the Treasury department. Also they receive an implicit guarantee from the government, since most
96
• Credit Enhancement: The MBS issuer can purchase additional insurance from a
mortgage insurance company for a mortgage pool. This deal is also called pool
insurance, or backend credit enhancement. It is not necessarily purchased from the
same company that provides PMI in the mortgage pool. There is generally an auction
among several insurance companies, and the bidder with the most competitive price
will be awarded the contract.
When hedging the credit risk of the MBS with credit enhancement from a third
party, the issuer is now exposed to the credit risk of the counter party. In order to hedge
the credit risks effectively and efficiently, we not only need to model the default behavior
of the mortgage borrower, but also need to understand the credit worthiness of the
counter party. The credit worthiness of a given counter party for a given time horizon is
generally called a haircut
2
. We need to model the haircuts of the counter party to perform
the following tasks:
• Calculate the insurance premium, i.e., the purchase price for the insurance policy,
to be paid. Apparently, a company with lower credit risk should be charging
higher fees, and vice versa, since lower credit risk means better insurance policy.
• Estimate the credit loss, and report it to external investors and regulators.
Currently the Office of Federal Housing Enterprise Oversight (OFHEO), regulator
of Fannie Mae and Freddie Mac, requires both GSEs to report their risk-based
market participants believe that federal government will interfere whenever any of these two giant GSEs
steps in financial distress.
2
This term is used to determine the reduction applied to promised payment, due to credit risk, e.g., a 25%
haircut means that the promised payment needs to be reduced to 75%.
97
capital calculated by pre-specified haircuts for different rated counter parties.
With the implementation of Basel Accord II
3
, internal credit risk models could be
used to calculate the haircuts, and in-house model for calculating the counter
party credit risk is of extreme importance in reporting the credit risks.
We below show that a haircut is actually a credit risk measure similar to credit
spread. And estimation of a haircut is equivalent to estimation of credit spread. Suppose
we need to take the haircut H(t) for a promised future payment of $1, what would be the
price for this promised payment? In risk neutral probability, the price should be:
P=exp(-r(0,t)*t)[1-H(t)]
Where r(0,t) is the spot rate for maturity t.
If the promised payment can be viewed as a zero-coupon defaultable bond with face
value of $1, its price is given by
P=exp{-[r(0,t)+CS(t)]*t}, where CS(t) is the credit spread for maturity t.
Apparently haircut and credit spread have the following one-to-one relationship:
H(t)=1-exp(-CS(t)*t)
Once we estimated the credit spread of the third party’s defaultable bonds, we will
get the haircut we need to impose on the insurance contract automatically. So it is of
critical importance that we have a good estimation for the credit spread changes of the
counter party. There are generally two ways to model the dynamics of the credit spread:
3
Basel Accord II is the new international banking regulation rule proposed by Basel Committee, which will
be implemented before 2006. It gives more flexibility in treating credit risk, and internal credit risk models
can be used in calculating risk-based capital requirement, which is the capital a financial institution needs
to reserve in order to alleviate the credit risk exposure to counter parties.
98
theoretical approach and empirical approach. There has been a lot of published work on
the literature on the theoretical part of credit risk modeling: either using structural models
(Merton [1974], Longstaff and Schwartz [1995], Collin-Dufresne and Goldstein [2001]),
or reduced form (hazard rate) models (Duffie and Singlton [1999], Madan and Unal
[2000]). In empirical work, different models are estimated and fitted with market data,
and the performances of these models are compared in recent papers, e.g. Eom, Helwege,
and Huang [2003]. Recently there has been interest in using regression to determine the
factors affecting credit spread changes, because neither structural nor reduced form
models can handle the large number of factors affecting credit spread changes. With a
flourishing credit derivative market, there is a great need for identifying the factors that
affect credit spread, in order to find possible financial instruments to hedge credit
derivatives written on credit spreads.
The main model used in these researches is the simple linear regression model,
e.g., Duffee [1998], Collin-Dufresne, Goldstein and Martin [2001], Huang and Kong
[2003]. However, these models generally do not offer very compelling results. In this
essay, we identify the theoretical drawbacks of this type of models, and address these
problems with a new approach: the Random Coefficient Regression (RCR) model, which
we can handle the non-constancy phenomena of credit spread sensitivities.
The rest of this essay is organized as follows. We first give a literature review in
the following section; specifically we are going to discuss several important papers. In
section 3, we introduce the random coefficient regression (RCR) model is given and then
99
apply the model to estimate the dynamics of credit spread changes, using variables from
the simplest structural model. Description of the data is given in section 5, and the
regression results are discussed in the next section. We show that our assumption about
non-linearity and non-constancy of credit spread changes are well supported by the
regression results, also the regression results are consistent with theoretical structural
models, such as Merton [1974]. In the last part of this essay, we give conclusions and
possible future research directions.
100
4.2 Literature Review
There has been a lot of recent interest on identifying the key factors affecting
credit spread. One approach is to add macroeconomic variables into the traditional
structural model. However, by adding new state variables, the model not only becomes
more complicated in the form, but also harder to identify empirical evidence to improve
pricing and hedging practice. Another approach is to concentrate on regression models.
Because of the simplicity and convenience in incorporating any new state variables,
regression is gaining popularity in empirical research for credit spread modeling.
Regression models can be divided into two categories: regression on credit spread
changes and regression on credit spread levels.
Of the first category, there are three major papers: Duffee [1998], Collin-
Dufresne, Goldstein and Martin [2001], and Huang and Kong [2003].
Duffee [1998] did the pioneer work on credit spread changes regression. He
analyzed the credit spread data indexed by different industry, rating group, and maturity.
He used only the interest rate level and slope in the regression, and found that there is a
significant negative correlation between short rate change and credit spread change. He
achieved an average adjusted R
2
around 17%.
Collin-Dufresne et al. [2001] performed similar analysis, but on a lot more
variables. They divided corporate bond data by leverage ratio, rating, and maturity, and
performed multiple regressions. Among many regression models in the literature, this
101
model appears to be the most complicated. Their basic model included six basic
explanatory variables: leverage, interest rate level, interest rate slope, VIX, S&P, jump
probability. They achieved around 25% adjusted R
2
. They then performed principal
components analysis on the residual and found that over 75% variations are due to the
first component. Then they introduced new variables. The total number of variables in
final regression is 19, and the adjusted R
2
improved only to 34%. Eventually they
acknowledged that they could not identify the factor that contributes to the 75% residual
variation, within all the proxy they constructed for liquidity, etc. They claim that the
single factor driving the credit spread variation could be attributed to local
demand/supply fluctuation. Interestingly, while they introduce new variables, none of
these new variables are bond specific; most of them are macroeconomic variables.
Huang and Kong [2003] criticized Collin-Dufresne et al. [2001] for not having
chosen the best proxies for state variables. So they performed regression on credit spread
changes, with similar explanatory variables, while testing multiple proxies for each
variable among eight independent variables, and choosing the best one. Also they choose
to work with credit spread index OAS data (which they claim as cleaner credit spread) of
rating and maturity group. They achieved adjusted R
2
of more than 40% for 5 out of 9
groups. However, the number of observations for each index is merely 67. There is no
theoretical support as to why certain proxies for a state variable should perform better
than other proxies. And using index data in a short time period might have alleviated the
problem.
102
Table 4.1 gives an itemized comparison of the three papers.
Duffee [1998] Collin-Dufresne et al.
[2001]
Huang and Kong
[2003]
Category Industry, rating,
maturity
Leverage, rating,
maturity
Rating and Maturity,
total=9
{All sectors, Industrial,
Utility, Financial}
{<15%, 15-25%, 25-
35%, 45-55%, 55%}
Investment Grade:
{AA-AAA, BBB-A}
{Aaa, Aa, A, Baa} {AAA, AA, A, BBB,
BB, B}
{1-10 yr, 10-15 yr, 15+
yr}
{2-7, 7-15, 15-30} {long (>12 yr),
short (<9 yr)}
High Yield:{BB, B, C}
Data Type Mean corporate yield
vs. corresponding.
Treasury yield(self
constructed index)
Corporate yield vs.
corresponding Treasury
yield
Index
OAS? N N Y
Data
Description
No option embedded,
> 4 yr maturity.
Data Range Monthly,
Jan-85 to March-95
Monthly,
July-88 to Dec-97
Monthly,
Jan-97 to July-02
observations At least 25 observations
for each bond
67 observations for each
index
Adj. R-square Around 17% 19% to 25% by leverage
ratio
>40% for 5 out of 9
17% to 34% by rating
group
67% for B
34% after additional
variables
60% for BB
Table 4.1 Comparison of three papers on credit spread regression
Clearly we can see that all these three papers try to improve the explanatory
power by either adding more state variables or cherry-picking different proxies for the
same state variables. The regression model is fundamentally the same, and the
improvement is marginal.
103
Of the second category of credit spread level regression, one major paper is
Campbell and Taksler [2003]. They claim that equity volatility in the regression is almost
as good as the credit rating variable. What they used in the regression is the excess return
(equity return minus market return) volatility for the last 180 trading days, not the
historical volatility, or implied volatility from options market.
This paper falls into the first category by modeling credit spread changes on
individual bonds. We believe there are several benefits focusing on changes:
• Credit spread changes are more relevant to the modeling of credit spread
dynamics, since regression on credit spread levels will have a large intercept
portion, which is not very informational, because we know that there is always
some credit premium associated with the corporate bond yields;
• Regression in credit spread changes is more useful in developing a hedging
framework, since we can estimate the sensitivities of the credit spread changes to
interest rate, leverage of the company. These sensitivities can be used to derive
hedge ratios.
• Individual bond data contain far more information than the indexed data. All the
firm-relevant data could enter the modeling, especially the leverage, which is a
very important factor in any structural model.
However, realizing the drawbacks from simple linear regression, we adopt a more
flexible approach: Random Coefficient Regression model. Although the RCR model is
not new in statistics (Hildreth and Houck [1968]), it has rarely been used in financial
research.
104
4.3 Introduction to Random Coefficient Model
The most frequently used linear model in statistics might be the following:
i i i
X y ? ? + = , (4.1)
where y
i
is the observed response of dependent variables;
X
i
is the vector of explanatory variables;
? is the vector of coefficients of the linear model;
?
i
is the error term, and ) , 0 ( ~
2
?
? ? N
i
.
For time series data, like those we frequently encounter in financial econometrics, it can
be written as:
. ,..., 2 , 1 , T t x y
t k
k
tk t
= + =
?
? ?
where y
t
is the observed random variable, x
kt
are known explanatory variables, ?
k
are
unknown constants to be estimated, and ?
t
are the error terms, independently and
identically distributed with mean zero, and finite variance. If exact tests of significance
are desired, the error terms, ?
t
, are typically assumed to be normal.
In some applications, the constancy of the coefficients, ?
k
, in consecutive
observations may not hold. For example, a particular ?
k
represents the response of credit
spread change for a bond to interest rate, which depends on the demand/supply ratio and
market liquidity premium. If both demand/supply ratio and market liquidity premium are
relatively stable, the assumption of constancy for ?
k
might be a tolerable approximation.
However, if demand/supply ratio and market liquidity premium vary, but are not
105
observed, assuming ?
k
as the mean of a random response rate may be better than
assuming the response rate to be constant.
Consider the following simple extension of model (4.1):
i i i i
X y ? ? + =
where
i i
v + = ? ? ;
0 ] [ =
i
v E , ? = ] ' [
i i
v v E , and ?
i
is uncorrelated with ?
i
.
As before, y
i
is still the observed random dependent variable and X
i
are known
values of independent variables. In this extension, ? is the mean response of the
dependent variable to the independent variables and (? + v
i
) is the actual response rate in
the ith observation. Combining terms, we have the model:
i i
i i i i i
w X
v X X y
+ =
+ + =
?
? ? ) (
(4.2)
where
0 ] [ =
i
w E ,
i i i i i i
X X I w w E ? = ? + = ' ] ' [
2
?
An important difference between random coefficient and simple regression model
is that the simple linear model assumes the sample is relatively homogeneous. Therefore,
if the estimate for ? is zero, then X will be concluded to have no effect on the
dependent variable. However, random coefficients may indicate that the effect the results
106
from cancellation of positive effects on some observations with negative effects on other
observations. As a result, the randomness of coefficient provides better explanation
power even if the mean of the coefficient ( ? ) is neutral. Dielman, Nantell, and Wright
[1980] emphasized that random coefficient models are very useful in analyzing pooled
cross-sectional and time series data.
In more complex cases,
i
? can be parametrically expressed. For example,
i
? can
be a linear function of several independent variables. While this specification involves
additional assumptions, it is essentially the same as the previous simple extension. This
functional form is appealing in some cases, especially when there is theoretical basis for
the relationship between
i
? and those independent variables, and the relationship are of
interests of researchers.
There are several ways to estimate the model. Details will be given in the next
section.
107
4.4 Random Coefficient Model for Credit Spread Changes
Huang and Kong [2003] mention that the low explanatory power of theoretical
determinants, documented in Collin-Dufresne et al. [2001], could be due to two reasons.
The first reason is that the explanatory variables may not be the best proxies to measure
the changes in default risk. The second reason is that the current existing corporate bond
pricing model might miss some important systematic risk factors. We have a different
opinion as to why the simple linear regression model lacks explanatory power.
We think the fundamental cause lies in the underlying assumptions of the
regression model. When the regression (4.1) is estimated, there is an assumption that the
coefficients are fixed. That is, the marginal effect for one unit change in
i
X has the same
effect ( ? ) on
i
y regardless of the characteristics of instance i .
Suppose credit spread (CS
t
)is a complex function of interest rate (r
t
), firm
leverage (lev
t
), firm asset volatility (?
t
) and other state variables (X
t
), which is compatible
with most structural models. Given
CS
t
=CS(lev
t
, r
t
, ?
t
, {X
t
}),
we can derive the first order approximation for the change of credit spread:
t t t t
X
X
CS CS
r
r
CS
lev
lev
CS
CS ?
?
?
+ ?
?
?
+ ?
?
?
+ ?
?
?
? ? ?
?
(4.3)
In a short time period, if all these state variables, lev, r, ?, X, do not change dramatically,
a simple linear regression model could be used to estimate the coefficients. However,
over a relatively long time period, as in Duffee [1998] and Collin-Dufresne et al. [2001],
108
which uses data spanning a 10-year period, this assumption is no longer valid. Because
all these gradients themselves are functions of each state variable, they are going to
change as well. Specifically, we have:
) , , , (
) , , , (
) , , , (
t t t t
t t t t
t t t t
X r lev h
CS
X r lev g
r
CS
X r lev f
lev
CS
?
?
?
?
=
?
?
=
?
?
=
?
?
Merton [1974] in his seminal paper has calculated the credit spread for a zero-
coupon corporate bond in the following form:
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ?
?
?
/ )] ln(
2
1
[ ) , (
/ )] ln(
2
1
[ ) , (
)] , ( [
1
)] , ( [ ln
1
) (
2 2
2
2 2
1
2
1
2
2
d d h
d d h
d h
d
d h r R H
+ ? =
? ? =
?
?
?
?
?
?
? + ? ? = ? =
where H is the credit spread;
d is a debt ratio measure, defined as d=Be
-r?
/V;
B is the face value of the debt;
V is the asset value of the firm;
? is the volatility for the corporate asset process;
? is the maturity of the zero-coupon bond.
Then he calculates the credit spread gradient to most state variables as follows:
109
; 0
) (
) ( '
] , [
2
1
] , [
*
1
* ] , [
1
; 0 ] , [
1
; 0 ] , [
1
1
2
>
?
?
?
= =
?
?
= ? > ?
< ? =
?
?
?
?
h
h
T d g
T
H
T d g
lev
e T d g
d lev
d
H H T d g
d
H
T d g
r
d
H H
r
d lev d
d r
?
?
? ? ?
where g[d, T] is the ratio of instantaneous bond return volatility to instantaneous asset
return volatility, and is defined as:
) ] , [ /( )] , ( [ ) , (
1
d T d P T d h
F
VF
T d g
V
y
? = = =
?
?
P[d, T] is the price ratio of the defaultable bond to risk-free bond, which is
defined as:
)] , ( [
1
)] , ( [ ] , [
1 2
T d h
d
T d h T d P ? + ? =
T=?
2
?.
Clearly we can see that all these gradients are time varying. If we assume g[d, T]
is constant, or estimate it as ratio of excess return on bond to excess return on asset, then
model can be estimated in a simple form.
To summarize, using a simple linear regression to estimate these coefficients over
a long time horizon can lead to poor results. By adopting a random coefficient method,
the model (4.3) can be restated as:
? ? ?
?
? ? ?
?
?
?
?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ?
+ + + + + =
+ + + + + =
+ + + + + =
+ ? + ? + ? + = ?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
T lev r
T lev r
T lev r
lev r CS
T lev r
CS
lev lev
T
lev lev
lev
lev
r
lev CS
lev
r r
T
r r
lev
r
r
r CS
r
CS CS CS
lev
CS
r
CS
(4.4)
110
Rewriting the original model, we have:
CS lev r
T lev r
lev
T
lev lev
lev
lev
r
r
T
r r
lev
r
r
lev r CS
CS
T lev r
lev lev
T
lev lev
lev
lev
r
lev
r r
T
r r
lev
r
r
r CS
lev r
T lev r
lev T lev lev lev lev r
r T r r lev r r
lev r
T lev r
lev T lev r
r T lev r CS
?
?
?
?
+ ? + ? + ?
+ ? + ? + ? + ?
+ ? + ? + ? + ?
+ ? + ? + ? + ?
+ ? + ? + ? + =
+ ? + + + + +
+ ? + + + + +
+ ? + + + + + + = ?
? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
?
? ?
?
? ?
?
?
?
? ? ?
?
? ? ?
?
?
) (
) (
) (
(4.5)
with standard assumptions:
0 ] [ ] [ ] [ ] [ = = = =
?CS
i i
lev
i
r
i
E E E E ? ? ? ?
?
,
0 ] [ ] [ ] [ = = =
? ? ? CS
i i
CS
i
lev
i
CS
i
r
i
E E E ? ? ? ? ? ?
?
, which basically states that the error
terms are uncorrelated.
The coefficients in the original model now are random, and have their own specifications.
The difference between our model and the simple linear regression model exists
not only in the specification of coefficients, but also in the difference in the assumption of
the error terms. The homoscedasticity assumption in a simple linear regression is relaxed
in our model.
Apparently, the OLS estimates for ? in model (4.5) are still consistent under the
assumption stated above because ?
i
X and
sum
? (sum of all error terms) are uncorrelated.
111
However, the estimates are no longer efficient
4
. Both Feasible Generalized Least Square
(FGLS) and the White robust estimator can provide consistent and efficient estimates
(Greene [1997]). We tried both methods in our application. The difference between the
two estimates is small. We only report the White estimates, because FGLS estimates
involve additional weights from the variance-covariance matrix. If the form of the
heteroscedasticity and parameters involved are known, then FGLS will be a better choice;
otherwise, the White estimator, which is robust to unknown heteroscedasticity, is
certainly appealing, because the weights introduced by FGLS may add additional
variation into the slope estimates.
4
Efficiency of estimators: an unbiased estimator
?
1
? is more efficient than another unbiased estimator
?
2
?
if the sampling variance of
?
1
? is less than that of
?
2
? . That is, ] [ ] [
2 1
? ?
< ? ? Var Var .
112
4.5 Data Description
We extract data from three databases: Warga bond database
5
, CRSP
6
, and
COMPUSTAT
7
for different financial data.
Warga database, which is also known as the Lehman Brothers Fixed Income
Database, contains the most comprehensive bond data for academia. We only choose
those bond that satisfy the following standards:
1. Dealer quoted price, instead of matrix price, since it has been pointed out that
matrix price could produce some problems (Sarig and Warga [1989]);
2. At least 30 consecutive observations;
3. Non-callable and non-putable. This would eliminate the optionality-induced
premium in the yield spread;
4. Bond with maturity greater than four years, since it is well known structural
model is less accurate for short maturity bonds.
Based on these standards, we end up with credit spread time series for 728 bonds, with
45627 observations. We have bond price, yield to maturity, maturity date, and duration
data from this database. These data are used later to construct the credit spread.
5
The Warga Fixed Income Securities Database (FISD) for academia is a collection of publicly offered U.S.
Corporate and Agency bond data. Produced by LJS Global Information Services, Inc., this fixed income
database engine is used by Reuters/Telerate and Bridge/EJV. These vendors collectively account for 83%
of trader screens.
6
The CRSP Database provides access to NYSE, AMEX and Nasdaq daily and monthly securities prices, as
well as to other historical data related to over 20,000 companies. The data is produced, and updated
quarterly, by the Center for Research in Security Prices (CRSP), a financial research center at the Graduate
School of Business at The University of Chicago.
7
The Standard & Poor's COMPUSTAT® databases contain financial, statistical, and market data for
different regions of the world. The databases are searched using Standard & Poor's Research Insight®
software, which enables data queries, retrieval, manipulation and analysis. The software includes
predefined sets for searching different types of data and allows the user to generate this data using
predefined reports.
113
We acquired the equity data from CRSP database. The equity data is linked with
bond data via the CRSP permno (permanent number) index. We retrieved the daily equity
data for 322 companies from January 1987 to March 1998. These data are used later to
construct the mark-to-market equity, as well as stock return volatility.
COMPUSTAT database provided us with the balance sheet information. It is also
linked to the CRSP database via the permno index. We retrieved the quarterly balance
sheet data for the same 322 companies from January 1987 to March 1998. Then we
interpolated the total asset value and total liability value for the months between. These
data are used later to construct leverage ratio.
Here we provide a brief description for the data we constructed in the regression.
Treasury curve is constructed by using linear interpolation. The treasury rate source is
the constant maturity Treasury (CMT) rate of H.15 release from the Federal Reserve web
site. We use the 3-year, 5-year, 7-year, 10-year, 30-year treasury rates. 20-year treasury
rate is disregarded because its discontinuity for the observation period. Interest rate level
is defined as 10-year Treasury rate.
Credit Spread is calculated as the difference between bond yield and treasury rate with
the same maturity
8
. As a convention, only quoted price are used, excluding callable and
putable bonds. The bond yield we use is the yield to maturity. Data ranges from July
1988 to March 1998.
114
Firm leverage is calculated by the following formula:
Liability Total Equity of Value Market
Liability Total
leverage
+
=
Total liability in each quarter is acquired from Compustat database; data in between
months are interpolated linearly. Market value of equity is acquired by multiplying stock
price with shares outstanding. Firm leverage is an important factor in structural models to
calculate distance to default. However, different researchers have been using different
numbers to calculate leverage ratio, e.g. Collin-Dufresne et al. [2001] uses the book value
of debt to calculate leverage, and Moody’s KMV is using short-term debt to calculate
default probability.
Volatility: We considered three different measures for volatility:
1. VIX, which is the volatility index as a weighted average of eight implied
volatilities of near-the-money options on the OEX (S&P 100) index. This
volatility measure is identical to the Collin-Dufresne et al. [2001] paper.
2. Simple estimated standard deviation of last 20 daily returns, for the corresponding
company’s common stock.
3. Excess return volatility for last 180 trading days return. It is the standard deviation
of the last 180 trading day’s excess return, which is defined as the return minus
market return (S&P 500 return). This volatility measure is identical to the
Campbell and Taksler [2003] paper.
The effect of these three different volatility measures will be discussed in later sections.
8
Treasury rate with the same maturity is linearly interpolated from adjacent CMT rates.
115
4.6 Results Analysis
We are going to discuss the regression results of our new model in this section.
First, we compare the coefficients of simple linear regression model with RCR model,
and examine the assumption of dependence between credit spread sensitivities and state
variables. Second, we examine the regression results for different rating and maturity
groups. In the last subsection, we are going to examine the assumption of non-constancy
of credit spread sensitivities.
4.6.1 Dependence of Credit Spread Sensitivities to State Variables
In this section, we are going to discuss the regression results of our RCR model,
compared with simple linear model. Table 4.2 shows the coefficient estimation for both
models, and their t-values. Applying White robust estimator, regression is performed on
individual bond and the average statistics
9
are reported. In the simple linear regression
model, we can find that the sensitivity measures to interest rate change, leverage change,
and volatility changes are significant, and the signs and magnitudes of coefficients are
consistent with structural models and regression results in previously mentioned papers.
In the new model, we find that the following newly constructed interactive
variables are significant (with |t| > 2):
r?r, r??, ???, T??.
9
We followed the convention in Collin-Dufresne et al. [2001] to report these statistics. The reported
coefficient values are average of the regression estimates for the coefficient on each variable. The t-
statistics are calculated by dividing each reported coefficient by the standard deviation of the N estimates
and scaled by sqrt(N).
116
Linear
Model Variable beta
Standard
error t
intercept -0.006 0.02 -9.45
?r -0.058 0.21 -7.37
?lev 0.977 4.77 5.53
?? 0.005 0.01 8.48
Adj. r
2
0.077
RCR
Model Variable beta
t
intercept 0.05 0.002 -9.23
?r 18.47 0.68 -1.95
?lev 295 10.9 -1.38
?? 1.32 0.049 -1.42
r?r 0.60 0.022 6.02
lev?r 18.59 0.69 -0.42
??r 0.09 0.003 1.53
T?r 0.84 0.031 1.42
r?lev 12.44 0.46 -0.45
lev?lev 308 11.4 0.91
??lev 1.68 0.062 -0.32
T?lev 19.41 0.72 1.44
r?? 0.07 0.002 -3.02
lev?? 1.54 0.057 -0.35
??? 0.01 0.000 3.58
T?? 0.11 0.004 2.30
N 728
Adj. r
2
0.297
Table 4.2 Comparison of RCR vs. linear model
These significant interactive terms mean that the level of state variables has a
significant impact on the sensitivity of credit spread to these state variables. For example,
a positive coefficient for r?r means that when interest rate increases, the sensitivity of
credit spread to interest rate change should decrease (because the sensitivity of credit
spread to interest rate is negative). In other words, in a higher interest rate environment,
credit spread will be less sensitive to interest rate, given everything else unchanged. For
the same reason, a negative r?? coefficient means that in a higher interest environment,
credit spread will be less sensitive to volatility, when everything else is kept constant.
117
Also a positive ??? coefficient would stand for high volatility sensitivity in high
volatility environment. The next table summarizes the relationship we found between
levels of state variables and credit spread sensitivities.
Sign of beta
r
CS
?
?
(<0)
lev
CS
?
?
(>0)
? ?
?CS
(>0)
Interest + Not Significant
(N/S)
-
Leverage N/S N/S N/S
Volatility N/S N/S +
Maturity N/S N/S +
Table 4.3 Relationship between state variables and credit spread sensitivities
These findings validate our assumption that sensitivity should be dependent on
state variables. Also we would like to compare these coefficients to structural models,
and to validate whether these findings are consistent with theoretical models. We take the
most straightforward structural model for credit spread, the Merton [1974] model, for
which we have given the derivatives of the credit spread with respect to state variables in
section 4.4. Although it is possible to derive the second order derivative of credit spread
to validate the relationship we found are consistent with structural model or not, we
prefer to demonstrate this in a static analysis, as Merton did in the paper, which will be
more intuitive. The following charts show the results of our static analysis.
118
dCS/dr vs. r
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
3.50%
4.00%
0% 5% 10% 15%
C
S
-0.25%
-0.20%
-0.15%
-0.10%
-0.05%
0.00%
d
C
S
/
d
r
CS dCS/dr
Figure 4.1 Credit Spread vs. Risk-free Rate
Figure 4.1 shows the credit spread and credit spread sensitivity to interest rate at
different interest rate level. This zero-coupon bond is evaluated with 30% leverage, 30%
asset volatility and 5-year maturity, which is pretty representative. We can see that while
the Merton model predicts the credit spread will be decreasing while the interest rate
increases, the credit spread sensitivity to interest rate is an increasing function. However,
since the sensitivity measure itself is negative, being an increasing function actually
means reduced sensitivity at higher interest rate level, which is consistent with our
findings.
119
dCS/dsig vs. sig
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
0% 20% 40% 60% 80%
C
S
0.00%
0.10%
0.20%
0.30%
0.40%
0.50%
0.60%
0.70%
0.80%
0.90%
d
C
S
/
d
s
i
g
CS dCS/dsig
Figure 4.2 Credit Spread vs. Volatility
Figure 4.2 depicts the credit spread and its sensitivity to volatility at different
volatility level. The bond is evaluated at 5% risk-free rate, with 30% leverage and 5-year
maturity. From Figure 4.2, we find that credit spread is an increasing function of
volatility, and sensitivity to volatility is an increasing function for the most volatility
spectrum, from 5% to 60%, and after that, is pretty flat with a slight trend of decreasing.
This result is consistent with our findings.
120
dCS/dsig vs. sig
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
0% 10% 20% 30% 40% 50% 60% 70% 80%
dCS/dsig w/30% lev dCS/dsig w/60% lev
Figure 4.3 Sensitivity to Volatility at different Leverage
In order to test the robustness of the relationship between credit spread sensitivity
to volatility and volatility itself, we choose two different setting for maturity and
leverage. Figure 4.3 shows the credit sensitivity to volatility for a zero coupon bond with
maturity of 15 year, and leverage of 30%, and 60%, at 5% interest rate level. We can see
increased maturity make the curve more flat, compared with Figure 4.2. Also increasing
leverage makes the yield more flat as well. Since the vast majority of our bonds have
maturity less than 15 years, and leverage below 60%, we think the estimated positive
coefficient of volatility on sensitivity to volatility is a valid prediction for the majority of
these bonds.
121
dCS/dsig vs. r
0
0.05
0.1
0.15
0.2
0% 2% 4% 6% 8% 10% 12% 14% 16%
dCS/dsig
Figure 4.4 Sensitivity to Volatility vs. Interest Rate
Figure 4.4 shows the credit sensitivity to volatility for a zero coupon bond with
maturity of 5 year, and leverage of 30%, volatility of 30%, at different interest rate levels.
The Merton [1974] model predicts it to be a decreasing function in interest rate, which
means that the higher the interest rate is, the lower the sensitivity to volatility will be. The
chart is consistent with our coefficient estimator of r??, which is negative.
122
dCS/dsig vs. maturity
0
0.05
0.1
0.15
0.2
0 2 4 6 8 10 12 14 16
dCS/dsig
Figure 4.5 Sensitivity to Volatility vs. Maturity
Figure 4.5 shows the credit sensitivity to volatility for a zero coupon bond with
leverage of 30%, volatility of 30%, at 5% rate level, with different maturities. It shows
how maturity change would affect the credit spread sensitivity to volatility. The
sensitivity will increase rapidly with respect to maturity till 8 years, and then decrease
slightly after maturity passed 8 years. In our estimation, the coefficient is positive, but the
significance is not very strong. We predict it is a mixed result of the rapid increasing and
slow decreasing. Also Merton’s model is based on zero coupon bond, so if applied to
coupon bond, the maturity might be better replaced with duration measure, which is
significantly shorter than the maturity. That would explain why we have the coefficient
estimator to be positive, which is more inclined to the shorter end of the maturity.
123
4.6.2 Results by Rating and Maturity
In this section, we show results for different rating and maturity groups. We have
five rating groups: AAA-AA, A, BBB, BB, B and others (below rating B or not rated),
and three maturity groups: LONG (maturity >12 years), MEDIUM (12 years >maturity>8
years), and SHORT (maturity < 4years). The total number of combinations is 15.
AA_LONG AA_MEDIUM AA_SHORT
beta std_error t beta std_error t beta std_error t
intercept -0.01 0.001 -5.13 -0.003 0.003 -1.33 -0.01 0.003 -3.34
?r -0.53 1.01 -0.52 0.72 0.49 1.48 0.43 0.46 0.94
?lev 8.41 13.12 0.64 8.20 5.13 1.60 2.88 11.59 0.25
?? -0.39 0.19 -2.00 -0.12 0.06 -2.01 -0.24 0.16 -1.51
r?r 0.10 0.02 5.85 0.05 0.04 1.20 -0.03 0.05 -0.66
lev?r 0.31 0.59 0.53 -0.01 0.86 -0.01 0.07 0.71 0.09
??r 0.01 0.00 1.61 -0.02 0.01 -1.85 0.01 0.02 0.75
T?r 0.00 0.02 -0.01 -0.07 0.03 -2.25 -0.07 0.06 -1.28
r?lev -0.32 0.38 -0.84 -0.04 0.48 -0.08 0.93 0.67 1.38
lev?lev 4.42 16.88 0.26 -3.94 14.31 -0.28 -39.72 30.33 -1.31
??lev -0.06 0.09 -0.68 -0.22 0.16 -1.37 0.17 0.26 0.64
T?lev -0.15 0.51 -0.30 -0.39 0.65 -0.60 -0.07 1.17 -0.06
r?? -0.01 0.002 -2.49 -0.004 0.004 -0.95 0.002 0.01 0.25
lev?? 0.07 0.06 1.08 -0.03 0.17 -0.20 0.34 0.24 1.42
??? 0.00 0.00 0.98 0.002 0.001 1.97 -0.001 0.002 -0.67
T?? 0.01 0.01 2.03 0.01 0.003 4.61 0.02 0.01 1.34
N 42 10 53
Adj. r
2
0.266 0.270 0.233
Table 4.4 RCR coefficients for AA-AAA group
For the rating group of AA-AAA, we found that the model performs better (with
higher R
2
) in long maturity group than short maturity group. This is consistent with
previous regression model (Duffee [1998]). Also the average explanatory power for this
group is below average, which is also consistent with previous regression results. AAA
bonds are counted in this group because of the limited numbers in each AAA maturity
group.
124
A_LONG A_MEDIUM A_SHORT
beta std_error t beta std_error t beta std_error t
intercept -0.01 0.002 -4.24 -0.01 0.003 -3.13 -0.01 0.003 -3.73
?r -1.66 0.68 -2.43 -1.67 0.77 -2.16 -0.33 0.30 -1.10
?lev 16.91 10.05 1.68 -2.19 7.53 -0.29 4.91 7.65 0.64
?? 0.00 0.10 0.00 0.08 0.09 0.85 -0.01 0.05 -0.32
r?r 0.11 0.03 3.91 0.12 0.04 3.38 0.11 0.04 2.96
lev?r 0.01 0.51 0.02 1.14 0.92 1.24 -0.02 0.44 -0.05
??r 0.01 0.00 1.94 0.02 0.01 2.78 -0.01 0.01 -1.44
T?r 0.02 0.02 1.05 0.01 0.05 0.15 -0.04 0.03 -1.47
r?lev -0.21 0.37 -0.57 -0.33 0.76 -0.44 -0.86 0.57 -1.51
lev?lev -1.52 9.49 -0.16 -12.68 14.33 -0.88 -21.83 11.84 -1.84
??lev -0.01 0.07 -0.19 -0.10 0.19 -0.54 0.13 0.09 1.45
T?lev -0.60 0.45 -1.35 1.26 0.75 1.68 1.56 0.62 2.51
r?? -0.01 0.00 -3.50 -0.02 0.01 -3.19 -0.005 0.003 -1.69
lev?? -0.05 0.06 -0.79 -0.02 0.12 -0.20 -0.03 0.06 -0.45
??? 0.001 0.0005 2.39 -0.0001 0.001 -0.09 0.002 0.001 2.13
T?? 0.002 0.003 0.59 0.01 0.01 1.12 0.01 0.01 1.28
N 124 35 145
Adj. r
2
0.332 0.298 0.187
Table 4.5 RCR coefficients for A group
For the rating group of A, we also found that the performance of our RCR model
deteriorates as the maturity decreases. The average explanatory power for long and
medium maturity is above and near average (R
2
of 28%), which is also consistent with
previous literature (Duffee [1998]).
125
BBB_LONG BBB_MEDIUM BBB_SHORT
beta std_error t beta std_error t beta std_error t
intercept -0.0175 0.00341 -5.1259 0.0040 0.00686 0.5792 -0.0150 0.00317 -4.7205
?r -1.4606 1.30092 -1.1228 0.4031 3.15080 0.1279 0.4132 0.67383 0.6108
?lev 6.3313 14.42570 0.4389 -25.1286 19.48469 -1.2897 -3.3511 15.10461 -0.2210
?? 0.1922 0.16763 1.1464 0.1464 0.37627 0.3890 0.0037 0.07100 0.0517
r?r 0.1302 0.03744 3.4778 0.0416 0.09990 0.4168 0.1056 0.09241 1.1383
lev?r -0.4979 0.71171 -0.6996 -1.3209 3.10094 -0.4260 -1.2636 1.04115 -1.2089
??r 0.0309 0.00940 3.2876 -0.0354 0.02867 -1.2358 -0.0130 0.00842 -1.5402
T?r 0.0260 0.05094 0.5094 0.1095 0.15163 0.7223 0.0049 0.13122 0.0373
r?lev 0.3805 0.71955 0.5288 1.4138 1.85106 0.7638 -1.8690 1.80103 -1.0337
lev?lev 7.5851 13.19033 0.5751 34.6082 17.93499 1.9296 -0.8355 20.67328 -0.0403
??lev 0.1919 0.14222 1.3493 -0.2079 0.29622 -0.7018 -0.2272 0.33345 -0.6786
T?lev -0.4527 0.70592 -0.6413 0.0419 2.22403 0.0189 3.9359 2.19255 1.7880
r?? -0.0187 0.00617 -3.0315 0.0065 0.01899 0.3438 0.0014 0.00847 0.1649
lev?? 0.0901 0.08254 1.0912 -0.1715 0.30587 -0.5607 -0.0434 0.14752 -0.2927
??? -0.0004 0.00069 -0.5319 0.0033 0.00455 0.7291 0.0031 0.00164 1.8640
T?? -0.0044 0.00747 -0.5856 -0.0151 0.03758 -0.4024 -0.0032 0.00888 -0.3582
N 72 24 126
Adj. r
2
0.3159 0.1873 0.2032
Table 4.6 RCR coefficients for BBB group
For rating group BBB, we found that the performance of our RCR model
deteriorates as the maturity decreases. The average explanatory power for long and
maturity is above average, which is also consistent with previous literature (Duffee
[1998]). The regression results for the BBB medium group is far below average, and
almost none of these variables are statistically significant, which we suspect is due to
limited data problem (only 24 bonds available.)
126
BB_LONG BB_MEDIUM BB_SHORT
beta std_error t beta std_error t beta std_error t
intercept -0.06 0.01 -11.45 N/A N/A N/A -0.07 0.01 -5.68
?r -5.18 1.57 -3.29 N/A N/A N/A -4.61 2.35 -1.96
?lev 21.89 13.93 1.57 N/A N/A N/A -87.91 36.79 -2.39
?? -0.28 0.08 -3.75 N/A N/A N/A 0.48 0.35 1.40
r?r 0.37 0.04 8.32 N/A N/A N/A 0.34 0.16 2.17
lev?r -1.29 2.21 -0.59 N/A N/A N/A 1.03 2.03 0.51
??r 0.03 0.01 2.96 N/A N/A N/A 0.01 0.04 0.16
T?r 0.18 0.15 1.19 N/A N/A N/A 0.15 0.16 0.90
r?lev 0.90 0.70 1.29 N/A N/A N/A -1.99 2.28 -0.87
lev?lev 27.62 13.75 2.01 N/A N/A N/A 66.40 38.54 1.72
??lev -0.78 0.28 -2.80 N/A N/A N/A -0.10 0.83 -0.12
T?lev -1.72 0.66 -2.62 N/A N/A N/A 7.07 3.14 2.25
r?? -0.01 0.00 -3.69 N/A N/A N/A -0.01 0.02 -0.48
lev?? -0.48 0.14 -3.43 N/A N/A N/A -0.66 0.44 -1.51
??? 0.002 0.001 1.35 N/A N/A N/A 0.01 0.002 3.79
T?? 0.03 0.01 3.83 N/A N/A N/A -0.01 0.03 -0.44
N 24 1 27
Adj. r
2
0.123 N/A 0.104
Table 4.7 RCR coefficients for BB group
For the rating group BB, we also see that model became worse when the maturity
decreases. And for BB medium group, there is only one bond, so we cannot draw any
reasonable conclusion about variable significance. Again the explanatory power is both
low for long and short maturity, which could be contributed to limited bond numbers in
both categories.
127
B_LONG B_MEDIUM B_SHORT
beta std_error t beta std_error t beta std_error t
intercept N/A N/A N/A N/A N/A N/A -0.056 0.060 -0.935
?r N/A N/A N/A N/A N/A N/A -29.005 15.404 -1.883
?lev N/A N/A N/A N/A N/A N/A 30.750 309.773 0.099
?? N/A N/A N/A N/A N/A N/A 0.350 2.174 0.161
r?r N/A N/A N/A N/A N/A N/A 1.422 0.609 2.336
lev?r N/A N/A N/A N/A N/A N/A 12.422 10.146 1.224
??r N/A N/A N/A N/A N/A N/A 0.075 0.183 0.409
T?r N/A N/A N/A N/A N/A N/A 1.372 1.538 0.892
r?lev N/A N/A N/A N/A N/A N/A 46.626 20.449 2.280
lev?lev N/A N/A N/A N/A N/A N/A -419.950 376.543 -1.115
??lev N/A N/A N/A N/A N/A N/A -3.616 7.040 -0.514
T?lev N/A N/A N/A N/A N/A N/A 22.297 18.849 1.183
r?? N/A N/A N/A N/A N/A N/A -0.291 0.277 -1.047
lev?? N/A N/A N/A N/A N/A N/A 4.618 5.316 0.869
??? N/A N/A N/A N/A N/A N/A -0.069 0.069 -0.990
T?? N/A N/A N/A N/A N/A N/A -0.162 0.137 -1.186
N 1 1 8
Adj. r
2
N/A N/A 0.500
Table 4.8 RCR coefficients for B and other group
The total number of bonds in B and other group are very limited, so we cannot
make judgments about model performance in each maturity group.
Overall, our model performs best for the A and BBB groups, as well as for longer
maturities. These findings are consistent with Duffee [1998], as well as with theoretical
structural models for credit spreads.
128
The following table shows the significance level of previously identified
interactive terms in each rating and maturity group.
Interactive Terms r?r r?? ??? T??
AA_LONG + - Not Significant
(N/S)
+
AA_MEDIUM + - N/S +
AA_SHORT N/S N/S N/S +
A_LONG + - + N/S
A_MEDIUM + - N/S N/S
A_SHORT + - + +
BBB_LONG + - N/S N/S
BBB_MEDIUM N/S N/S N/S N/S
BBB_SHORT + N/S + N/S
BB_LONG + - + +
BB_SHORT + N/S + N/S
B_SHORT + - N/S N/S
Table 4.9 Summary of RCR coefficients
Not surprisingly, we found that for the interactive terms, which were constructed
in the RCR model, the significance levels and signs of the coefficient estimators are very
consistent for each rating and maturity group. If we remove the three groups with one
sample ecah, the r?r term is significant for 10 of 12 groups, which means that the interest
rate level has a positive impact on the credit spread sensitivity on interest rate, no matter
what rating group or maturity category. Also the impact of interest rate level on credit
spread sensitivity on volatility is very consistent.
129
4.6.3 Non-Constancy of Credit Spread Sensitivities
The non-constancy of credit spread sensitivities would naturally be embedded in
their dependence on state variables in the RCR model. However, we would like to see
how they change over time, and compare it to the simple linear regression sensitivity
estimators, and find out why the RCR estimators provide better accuracy.
Let’s take one bond as example, the bond with CUSIP of "001765AE", one of
American Airlines' long term bonds, and depict its random coefficients and constant
coefficients. The following three figures show the comparison of regression coefficients
with respect to interest rate changes, leverage changes, and volatility changes.
sensitivity to r
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
1
9
9
0
0
6
1
9
9
0
1
0
1
9
9
1
0
2
1
9
9
1
0
6
1
9
9
1
1
0
1
9
9
2
0
2
1
9
9
2
0
6
1
9
9
2
1
0
1
9
9
3
0
2
1
9
9
3
0
6
1
9
9
3
1
0
1
9
9
4
0
2
1
9
9
4
0
6
1
9
9
4
1
0
1
9
9
5
0
2
1
9
9
5
0
6
1
9
9
5
1
0
1
9
9
6
0
2
1
9
9
6
0
6
1
9
9
6
1
0
1
9
9
7
0
2
1
9
9
7
0
6
1
9
9
7
1
0
beta_dr_linear beta_dr_RCR
Figure 4.6 Coefficient for ?r in RCR vs. Linear Model
130
sensitivity to lev
0
1
2
3
4
5
6
7
8
9
1
9
9
0
0
6
1
9
9
0
0
9
1
9
9
0
1
2
1
9
9
1
0
3
1
9
9
1
0
6
1
9
9
1
0
9
1
9
9
1
1
2
1
9
9
2
0
3
1
9
9
2
0
6
1
9
9
2
0
9
1
9
9
2
1
2
1
9
9
3
0
3
1
9
9
3
0
6
1
9
9
3
0
9
1
9
9
3
1
2
1
9
9
4
0
3
1
9
9
4
0
6
1
9
9
4
0
9
1
9
9
4
1
2
1
9
9
5
0
3
1
9
9
5
0
6
1
9
9
5
0
9
1
9
9
5
1
2
1
9
9
6
0
3
1
9
9
6
0
6
1
9
9
6
0
9
1
9
9
6
1
2
1
9
9
7
0
3
1
9
9
7
0
6
1
9
9
7
0
9
1
9
9
7
1
1
beta_dlev_linear beta_dlev_RCR
Figure 4.7 Coefficient for ?vol in RCR vs. Linear Model
sensitivity to sig
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
1
9
9
0
0
6
1
9
9
0
1
0
1
9
9
1
0
2
1
9
9
1
0
6
1
9
9
1
1
0
1
9
9
2
0
2
1
9
9
2
0
6
1
9
9
2
1
0
1
9
9
3
0
2
1
9
9
3
0
6
1
9
9
3
1
0
1
9
9
4
0
2
1
9
9
4
0
6
1
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9
4
1
0
1
9
9
5
0
2
1
9
9
5
0
6
1
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5
1
0
1
9
9
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0
2
1
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6
0
6
1
9
9
6
1
0
1
9
9
7
0
2
1
9
9
7
0
6
1
9
9
7
1
0
beta_dvix_linear beta_dvix_RCR
Figure 4.8 Coefficient for ?lev in RCR vs. Linear Model
There are three major findings from the graphs:
1. Sensitivity to ?r does change over time. In Merton’s model, increased interest rate
would increase the risk neutral drift term, thus decrease the default probability,
and shrink the credit spread. In reality, Fed generally lowers interest rate to
stimulate economy when there is a recession, which is the case during 1990-1992.
Generally higher credit spreads are observed during a recession. That is the main
reason for negative correlation between interest rate and credit spread. However,
131
what about during times of economic recovery or boom? It would be interesting
to compare Figure 4.6 which depicts the sensitivity of credit spread to interest
rate, to Figure 4.9, the history of 3-month Treasury rate, a close reflection of
Fed’s policy on funding rate. When the economy is recovering, lowering interest
rate would have less effect on credit spread. That is exactly the case we found
during 1992-1993, when the Fed continued lowering the short interest rate, and
the sensitivity of credit spread to the interest rate is close to zero. Also when
economy is booming, the Fed is likely to raise the interest rate, and that seems to
have little effect on the credit spread. That is likely the case for 1994-1995.
2. Sensitivity to ?? also changes over time. In structural model, increase in volatility
would increase the default probability, and thus widen the credit spread. However,
comparing Figure 4.8 with 4.10: the history of VIX volatility index, we found that
while volatility is high both in the early 90’s and the late 90’s, their impact on
credit spread sensitivity are quiet different. One explanation for this could be that
during a recession, volatility is a bad thing, because it is likely that the volatility is
a result of dropping equity, and investors will be really concerned with a volatility
spike. However, when the economy is booming, it is likely that high volatility is
introduced by rising stock prices, and investors are less likely to require high
credit spread for this “good volatility”.
132
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Figure 4.10 VIX index from 1990 to 1997
Of the three volatility measures we used, our results show that VIX is better than
both history volatility and excess volatility, which is unanticipated. Originally we thought
that since VIX is a broad market volatility index, replacing it with company specific
volatility should improve our results. The reason for this phenomenon might be that
credit spread response is more sensitive to market perception of risk than to historical
133
volatility. Also we tried the excess return volatility, which Campbell and Taksler (2003)
claims to have significant explanatory power in regression of credit spread levels. The
results are disappointing, and the adjusted R
2
is comparable to historical volatility, but
not as good as VIX index. The reason might be that the credit spread itself already has a
build-in premium associated with the standard deviation of excess return, but the change
of credit spread is not sensitive to its change, so the regression on credit spread levels and
changes will have different explanation.
134
4.7 Conclusions and future work
By using the RCR approach, we not only model the dynamics of credit spread
sensitivities in a more consistent way with current structural model, but also achieve
more explanatory power than simple linear model. Our contributions are the following:
1. The first paper to use RCR model on credit spread data;
2. The first paper to explicitly model the credit spread sensitivities with dependence
against state variables, and empirically validate the dependence relationship
predicted by Merton’s model;
3. Higher explanatory power is achieved without adding new independent state
variables. In this case, we increased the adjusted R
2
from 8% to 30%.
Obviously, there are still some unanswered questions remaining in our work. We
would like to pursue future research in the following directions:
1. We can see from our results analysis from section 4.5, the theoretical sensitivity
changes are not always linear with respect to state variables (Figure 4.5), and
when there is a strong no linear relationship, our predictions of coefficient are
generally weaker. So can we change the functional form in the regression model
for sensitivity parameters and achieve better explanatory power? And which
structural model should we adopt in selecting the functional form? It will be
interesting to compare the regression results for different functional forms of
credit spread sensitivity from different structural models.
2. What would be a better asset volatility proxy than the VIX index? We think that
the option implied volatility for each company’s stock option might be a better
135
indicator of the market perception of risk. However, how to convert the equity
volatility to asset volatility? One way to look into this might be to look at the
combined bond return volatility of the specific company, which means that we
need to group the bonds of the same company, instead of doing individual
regression on each bond issued.
3. It would be interesting to analyze the residuals of the regression error, and find
whether there exists any pattern to discover hidden significant drivers for credit
spread.
136
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doc_228399584.pdf
Credit risk refers to the risk that a borrower will default on any type of debt by failing to make payments which it is obligated to do
ABSTRACT
Title of Dissertation: THREE ESSAYS ON MORTGAGE BACKED
SECURITIES: HEDGING INTEREST RATE AND
CREDIT RISKS
Jian Chen, Doctor of Philosophy, 2003
Dissertation directed by: Professor Michael C. Fu
The Robert H. Smith School of Business
This dissertation includes three essays on hedging the interest rate and credit risks
of Mortgage-Backed Securities (MBS).
Essay one addresses the problem of how to efficiently estimate interest rate
sensitivity parameters of MBS. To do this in Monte Carlo simulation, we derive
perturbation analysis (PA) gradient estimators in a general setting. Then we apply the
Hull-White interest rate model and a common prepayment model to derive the
corresponding specific PA estimators, assuming the shock of interest rate term structure
takes the form of a trigonometric polynomial series. Numerical experiments comparing
finite difference (FD) estimators with our PA estimators indicate that the PA estimators
can provide better accuracy than FD estimators, while using much lower computational
cost. Using the estimators, we analyze the impact of term structure shifts on various
mortgage products. Based these analysis, we propose a new product to mitigate interest
rate risk.
Essay two addresses the problem of how to measure interest rate yield curve shift
more realistically, and how to use these risk measures to hedge the interest rate risk of
MBS. We use a Principal Components Analysis (PCA) approach to analyze historical
interest rate data, and acquire the volatility factors we need in Heath-Jarrow-Morton
interest rate model simulation. Then we propose a hedging algorithm to hedge MBS,
based on PA gradient estimators derived upon these PCA factors. Our results show that
the new hedging method can achieve much better hedging efficiency than traditional
duration and convexity hedging.
Essay three addresses the application a new regression method on credit spread
data. Previous research has shown that variables in traditional structural model have
limited explanatory power in credit spread regression. We argue that this is partially due
to the non-constancy of the credit spread gradients to state variables. We use a Random
Coefficient Regression (RCR) model to accommodate this problem. The explanatory
power increases dramatically with the new RCR model, without adding new independent
variables. This is the first work to address the dependence between credit spread
sensitivities and state variables of structural in a systematic way. Also our estimates are
consistent with prediction from Merton’s structural model.
THREE ESSAYS ON MORTGAGE BACKED SECURITIES:
HEDGING INTEREST RATE AND CREDIT RISKS
By
Jian Chen
Dissertation submitted to the Faculty of the Graduate School
of the University of Maryland in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
November 2003
Advisory Committee:
Professor Michael Fu, Chair
Professor Dilip Madan
Professor Haluk Unal
Professor Nengjiu Ju
Professor Eric Slud
©Copyright by
Jian Chen
2003
ii
Dedicated to
Huixian Jackie Xu, my lovely wife
Li Xueqing and Chen Zhixuan, my parents
Chuan Chen, my son
iii
ACKNOWLEDGMENTS
I would like to sincerely thank my advisor, Dr. Fu, who admitted me six years ago, and
has arduously helped me since then, especially during the lengthy process of my
dissertation writing. I also would like to thank Dr. Unal, for providing me most valuable
advice for my last essay. I also would like to thank Dr. Madan, Dr. Ju, and Dr. Slud for
being my committee members and providing a lot of insightful comments on the
dissertation proposal.
A significant portion of this dissertation is completed during my tenure at Fannie Mae. I
benefited a lot from conversations with my colleagues there. I would like to thank Dr.
Yigao Liang for his inputs on Essay I. I would like to thank Dr. Alex Philipov and Dr.
Arash Sotoodehnia for discussion on HJM model. I want to thank Ms. CJ Zhao on credit
spread discussion. I also want to thank Dr. Jay Guo, who helped me a lot for the last
essay. Also I would like to thank Dr. Levant Guntay of Indiana University, who kindly
provided some data for the last essay, and has given me many research ideas during our
discussions. Needless to say, all errors are mine own.
All opinions expressed in this dissertation are not Fannie Mae’s but mine own.
iv
TABLE OF CONTENTS
List of Figures vii
List of Tables x
List of Abbreviations xi
1. Introduction 1
1.1 Efficient Sensitivity Analysis of Mortgage Backed Securities 3
1.2 Hedging MBS in HJM Framework 9
1.3 Hedging Credit Risk of MBS: A Random Coefficient Approach 10
2. Efficient Sensitivity Analysis of Mortgage Backed Securities 12
2.1 Problem Setting 9
2.2 Derivation of General PA Estimators 14
2.2.1 Gradient Estimator for Cash Flow 15
2.2.2 Gradient Estimator for Discounting Factor 19
2.3 Applying the Gradients 20
2.3.1 Interest Model Setup 20
2.3.2 Trigonometric Polynomial Shocks 23
2.3.3 Derivation of Gradients with respect to Modified Fourier Series 27
2.3.4 Derivation of Gradients with respect to Volatility: Vega 31
2.3.5 Derivation of Second Order Gradients: Gamma 31
2.3.6 Derivation of ARM estimators 35
2.4 Numerical Example 41
2.4.1 Specification of Numerical Example 41
v
2.4.2 Comparison of PA and FD estimators 42
2.4.3 Result Analysis 53
2.5 Interpretation of the Results 60
2.5.1 Overview of the Results 60
2.5.2 Modified Fourier Shock Impact 62
2.5.3 Potential New Product 70
2.6 Conclusion 75
3. Hedging MBS in HJM Framework 77
3.1 Motivation 77
3.2 Estimation of Volatility Factors via PCA 80
3.3 Simulation in HJM Framework 84
3.4 Deriving PA Estimators in HJM Framework 86
3.5 Hedging MBS in HJM Framework 89
3.6 Hedging Performance Analysis 92
3.7 Conclusion 94
4. Hedging Credit Risk of MBS: A Random Coefficient Approach 95
4.1 Motivation 95
4.2 Literature Review 100
4.3 Introduction to Random Coefficient Model 104
4.4 Random Coefficient Model for Credit Spread Changes 107
4.5 Data Description 112
4.6 Results Analysis 115
4.6.1 Dependence of Credit Spread Sensitivities to State Variables 115
vi
4.6.2 Results by Rating and Maturity 123
4.6.3 Non-Constancy of Credit Spread Sensitivities 130
4.7 Conclusion and Future Work 135
Bibliography 136
vii
LIST OF FIGURES
2.1 ?R(0,t) with Original Fourier series 25
2.2 ?R(0,t) with T
0
=10 modified Fourier series 25
2.3 Coefficients Estimation for Modified Fourier series 27
2.4 WAC as a function of Index 37
2.5 Gradient Estimator Comparison for ?d(t)/ ??
n
44
2.6 Gradient Estimator Comparison for ?c(t)/ ??
n
44
2.7 Gradient Estimator Comparison for ?PV(t)/ ??
n
45
2.8 95% Confidence Interval for dPV(t)/d?
n
45
2.9 Gradient Estimator Comparison for ?d(t)/?? 46
2.10 Gradient Estimator Comparison for ?c(t)/?? 47
2.11 Gradient Estimator Comparison for ?PV(t)/?? 47
2.12 95% Confidence Interval for dPV(t)/d? 48
2.13 gamma estimators for
2
2
) (
i
t d
? ?
?
, i=1, 2, 3, 4 49
2.14 gamma estimators for
2
2
) (
i
t CPR
? ?
?
, i=1, 2, 3, 4 49
2.15 gamma estimators for
2
2
) (
i
t CF
? ?
?
, i=1, 2, 3, 4 50
2.16 gamma estimators for
2
2
) (
i
t PV
? ?
?
, i=1, 2, 3, 4 50
2.17 Gradient Estimator Comparison for
i
t WAC
? ?
? ) (
, i=1, 2, 3, 4 51
viii
2.18 Gradient Estimator Comparison for
i
t PV
? ?
? ) (
, i=1, 2, 3, 4 52
2.19 Gradient Estimator Comparison for
? ?
? ) (t PV
52
2.20 Difference of FD/PA ?f(0, t)/ ??
n
estimators 57
2.21 Fuction of xe
-x
58
2.22 Duration vs. Products 62
2.23 The Impact of Modified Fourier Order 0 on FRM30 63
2.24 The Impact of Modified Fourier Order 1 on FRM30 64
2.25 The Impact of Modified Fourier Order 2 on FRM30 65
2.26 The Impact of Modified Fourier Order 3 on FRM30 66
2.27 The Impact of Modified Fourier Order 0 on ARM TSY 1 67
2.28 The Impact of Modified Fourier Order 1 on ARM TSY 1 68
2.29 The Impact of Modified Fourier Order 2 on ARM TSY 1 69
2.30 The Impact of Modified Fourier Order 3 on ARM TSY 1 70
2.31 10-Year T Rate, 1-Year T Rate, and mortgage rate 73
2.32 New ARM TSY 10 Durations 74
3.1 The first four principal components 82
3.2 Match monthly yield curve shift 83
3.3 Match annual yield curve shift 83
3.4 Mean Hedging Error of PCA vs. D&C 93
3.5 STD of Hedging Error: PCA vs. D&C 93
4.1 Credit Spread vs. Risk-free Rate 118
4.2 Credit Spread vs. Volatility 119
ix
4.3 Sensitivity to Volatility at different Leverage 120
4.4 Sensitivity to Volatility vs. Interest Rate 121
4.5 Sensitivity to Volatility vs. Maturity 122
4.6 Coefficient for ?r in RCR vs. Linear Model 130
4.7 Coefficient for ?vol in RCR vs. Linear Model 131
4.8 Coefficient for ?lev in RCR vs. Linear Model 131
4.9 Three-Month Treasury Rate from 1990 to 1997 133
4.10 VIX index from 1990 to 1997 133
x
LIST OF TABLES
2.1 Product Specification for Mortgage Pricing 41
2.2 Comparison of PA/FD Duration 53
2.3 Comparison of Convexity Estimators 55
2.4 Comparison of Computing Costs 55
2.5 Durations of Different Products 60
3.1 Statistics for Principal Components 81
4.1 Comparison of three papers on credit spread regression 102
4.2 Comparison of RCR vs. linear model 116
4.3 Relationship between state variables and credit spread sensitivities 117
4.4 RCR coefficients for AA-AAA group 123
4.5 RCR coefficients for A group 124
4.6 RCR coefficients for BBB group 125
4.7 RCR coefficients for BB group 126
4.8 RCR coefficients for B and other group 127
4.9 Summary of RCR coefficients 128
xi
LIST OF ABBREVIATIONS
A(t) Amortization Factor at time t
AGE(t) Aging Multiplier, a parameter to capture the aging effect in prepayment
rate
ARM Adjustable Rate Mortgage
B(t) Balance at time t
BM(t) Burnout Multiplier, a parameter to capture the burnout effect in
prepayment rate
C(t) Cash flow at time t
CMO Collateralized Mortgage Obligation, a special type of MBS
CPR Conditional Prepayment Rate, annualized prepayment rate
CS(t) Credit Spread at maturity t
D(t) Discounting Factor at time t
f(0,t) Instantaneous forward rate starting from t observed at now
FNMA Federal National Mortgage Association, also known as Fannie Mae
FD Finite Difference
FHLMC Federal Home Loan Mortgage Corporation, also known as Freddie Mac
FRM Fixed Rate Mortgage
GNMA Government National Mortgage Association, also known as Ginnie Mae
GLS Generalized Least Square
GSE Government-Sponsored Enterprise, mainly refers to Fannie Mae,
Freddie Mac.
xii
H(t) Haircut at maturity t
HJM Heath-Jarrow-Morton interest rate model, defined in Heath et al. [1992]
IP(t) Interest Payment at time t
lev
t
Leverage at time t
LTV Loan to Value ratio, an 80 LTV loan means the loan amounts for 80% of
the property value
MBS Mortgage-Backed Securities
MM(t) Monthly Multiplier, a parameter to capture the seasonal effect in
prepayment rate
MP(t) Mortgage Monthly Payment at time t
OFHEO Office of Federal Housing Enterprise Oversight, a government agency
under Department of Housing and Urban Development, regulator of
Fannie Mae and Freddie Mac.
OLS Ordinary Least Square
PA Perturbation Analysis
PCA Principal Components Analysis
PDE Partial Differential Equation
PMI Primary Mortgage Insurance
PP(t) Principal Prepayment at time t
PV(t) Present Value of Cash flow at time t
R(0,t) Spot rate for maturity t observed at now
r(t) Short rate at time t
r
10
(t) 10-year rate at time t
xiii
RCR Random Coefficient Regression
REO Real Estate Owned by the GSEs, in case borrower defaults
RI(t) Refinance Incentive, a parameter to capture the refinance incentive
effect in prepayment rate
SMM Simple Monthly Mortality, monthly prepayment rate
SP(t) Scheduled Principal Payment at time t
TPP(t) Total Principal Payment at time t
Vega The security price sensitivity to volatility
WAC Weighted Average Coupon rate for MBS
WAM Weighted Average Maturity for MBS
w.r.t. with respect to
1
Chapter 1
Introduction
Mortgage-backed securities (MBS) have become increasingly important fixed
income instruments, both because of their volume and the role they play in fund
investment and portfolio management. However, there has not been a very
comprehensive set of risk indicators to measure and manage the risks involved with
MBS. Hedging the interest rate and credit risk of MBS remains a complicated problem in
the fixed income industry. This dissertation develops a set of risk measures for interest
rate risk and credit risk, and then attempts to hedge the risks effectively using such risk
measures. Specifically, the dissertation consists of three essays addressing the following
problems: efficiently estimating these new measures of interest rate risk of MBS, hedging
MBS with these new measures, and hedging the credit risk of MBS with advanced
models for credit spread regression.
The first essay is mainly positioned to answer the following research questions:
• How to measure the interest rate risk in a more comprehensive approach, rather than
simply using the traditional duration
1
and convexity
2
?
1
Duration is the first order derivative of the price of a fixed income security to interest rate, expressed as a
percentage change, see Fabozzi [2001] for more details.
2
Convexity is the second order derivative of the price of a fixed income security to interest rate, expressed
as a percentage change, see Fabozzi [2001] for more details.
2
• How to efficiently estimate the risk measures if more factors are introduced into the
measurement problem?
In answering these two questions, based on the results over a broad spectrum of mortgage
products, we propose a new mortgage product, which could be attractive to MBS
investors and mortgage borrowers.
The second essay tries to answer the following questions:
• What would be a realistic method to measure the term structure shift?
• How can we hedge MBS effectively with these measures, in a general interest rate
model framework?
We use Principal Components Analysis (PCA) method to extract the empirical volatility
factors of term structure, which provides some justification for the form of possible term
structure shifts proposed in the first essay. Then we use the Heath-Jarrow-Morton model
to incorporate the factors in developing new risk measures and show that the hedging
effectiveness is far better than traditional duration/convexity hedging.
The third essay is related to credit risks MBS issuers incur when they purchase
mortgage pool insurance from a third party, and attempts to answer the following
questions:
• How to estimate the sensitivity of credit spread in a regression framework more
effectively than a simple linear regression model?
• What implication does the model have on traditional structural models for credit
spread?
3
We use a Random Coefficient Regression (RCR) model to build our regression model for
credit spread changes. This model has explicit sensitivity measures dependent on state
variables. We acquire much better explanatory power with this new model, without
adding new independent state variables. Also our model supports the dependence of
sensitivity of credit spread on state variables predicted by Merton’s structural model
(Merton [1974]).
1.1 Efficient Sensitivity Analysis of Mortgage Backed Securities
A mortgage-backed security (MBS) is a security collateralized by residential or
commercial mortgage loans. An MBS is generally securitized, guaranteed and issued by
three major MBS originating agencies: Ginnie Mae, Fannie Mae, and Freddie Mac. The
cash flow of an MBS is generally the collected payment from the mortgage borrower,
after the deduction of servicing and guaranty fees. However, the cash flows of an MBS
are not as stable as that of a government or corporate coupon bond. Because the mortgage
borrower has the prepayment option, mainly exercised when moving or refinancing, an
MBS investor is actually writing a call option. Furthermore, the mortgage borrower also
has the default option, which is likely to be exercised when the property value drops
below the mortgage balance, and continuing mortgage payments would not be
economically reasonable. In this case the guarantor is writing the borrower a put option,
and the guarantor absorbs the cost. However, the borrower does not always exercise the
options whenever it is financially optimal to do so, because there are always non-
monetary factors associated with the home, like shelter, sense of stability, etc. And it is
also very hard for the borrower to tell whether it is financially optimal to exercise these
4
options because of lack of complete and unbiased information, e.g., they may not be able
to obtain an accurate home price, unless they are selling it. And there are also some other
fixed/variable costs associated with these options, such as the commission paid to the real
estate agent, the cost to initialize another loan, and the negative credit rating impact when
the borrower defaults on a mortgage. All these factors contribute to the complexity of
MBS cash flows. In practice, the cash flows are generally projected by complicated
prepayment models, which are based on statistical estimation on large historical data sets.
Because of the complicated behaviors of the MBS cash flow, due to the complex
relationships with the underlying interest rate term structures, and path dependencies in
prepayment behaviors, Monte Carlo simulation is generally the only applicable method to
price MBS.
Associated with the uncertainty of cash flows are different kinds of risks.
Treasury bonds only bear interest rate risk, whereas non-callable corporate bonds carry
interest rate and credit risk. MBS are further complicated by prepayment risk (resulting
from both voluntary prepayment and default). Thus risk management is especially critical
for portfolios with large holdings in MBS. Duration and convexity are the main risk
measurements for fixed income portfolio mangers. Many practitioners use either the
Macaulay duration, or modified duration (Kopprasch [1987]) to capture the MBS price
sensitivity with respect to interest rate changes, but these duration measures assume a
constant yield and known deterministic prepayment pattern, which is rarely the case in
practice. So these two approaches to calculate duration can lead to serious errors when
used in hedging. Golub [2001] proposed four different approaches to estimate the
5
duration: Percent of Price (POP), Option-Adjusted Duration (OAD), Implied Duration,
and Coupon Curve Duration (CCD). The first two approaches apply parallel shifts in the
yield curve, which is not a very realistic assumption. The latter two approaches require
large numbers of previous or current accurate MBS prices that are comparable to the
MBS whose duration is to be measured. This might not be practical for on-the-fly pricing
and sensitivity analysis. Another drawback of these approaches is that they handle only
duration and convexity, but not sensitivity to interest rate volatility. OAD method can
estimate the vega (the price sensitivity to volatility) using a finite difference approach,
which requires 3 simulations to estimate one gradient: the base, up and down cases. And
non-parallel yield curve shifts require more parameters to characterize the shift. Thus, in
the setting we consider in Chapter 2, to estimate the duration with respect to yield curve
shift of 4 summed harmonic functions would require 9 (2n+1, n=4) simulations. To
estimate vega requires 2 additional simulations. So estimating the duration and vega
roughly increases the computational cost by a factor of 10. Calculating convexity would
require 75 duration estimators to calculate 25 convexity estimators, increasing the
simulation factor to 225. In other words, if one were to use 10,000 replications to
estimate the MBS price, over 2.25 million simulations would be required to estimate the
various sensitivities. Our work aims to decrease this computational burden dramatically.
Most literature on MBS has concentrated on prepayment model estimations,
although some of the recent work has focused on computational efficiency, e.g.,
dimensionality reduction via Brownian bridge ( Caflisch et al. [1997] ), and quasi-Monte
Carlo (Åkesson and Lehoczky [2000]). However, there is no work that we are aware of
6
that addresses efficient sensitivity analysis of MBS pricing and hedging. Related work in
equities includes Fu and Hu [1995], Broadie and Glasserman [1996], Fu et al. [2000],
[2001], and Wu and Fu [2001]. Perhaps the most relevant paper to our work is
Glasserman [1999], which applied perturbation analysis (PA) method for caplet price
sensitivity analysis. Yet most of these models involve only a single exercise decision with
a one-time payoff, whereas an MBS is a pool of homogenous mortgages rather than an
individual mortgage loan. So the cash flows exist until the maturity of the collateral, and
they are highly path dependent, which makes sensitivity analysis of MBS more
complicated.
The other relevant body of research literature analyzes the duration of different
mortgage products. We know that adjustable rate mortgage (ARM) products will have a
different response from fixed rate mortgage (FRM) products, due to ARMs’ coupon-reset
plan and different prepayment function. In a series of papers, Kau et al.[1990,1992,1993]
priced the ARMs and performed some sensitivity analysis. Chiang [1997] applied a
simple simulation scheme to estimate the modified duration of ARMs. Stanton [1999]
calculated the duration of different indexed ARMs via a scheme like Kau’s. However,
most of these papers are based on solving models based on partial differential equation
(PDE), using simplified assumptions that often miss essential features that can be
captured by Monte Carlo simulation. The three major drawbacks of these models that
make them impractical in the mortgage industry are the following:
• They assume borrowers exercise the prepayment option only when it is financially
optimal to do so. This ignores the fact that people routinely prepay even in financially
7
adverse environments, e.g., house sales. Also seasoning and burnout effects are not
considered.
• By solving the PDE, one can only obtain a set of present values of the MBS along the
interest rate axis. By applying the finite difference method, duration of the MBS
could be acquired. However, it provides no information about the discounting factor
and cash flows along the time horizon. So you will have no information about how
the interest shift affects different components of the present value.
• The PDE method generally uses one-factor interest rate model, which applies the
same interest rate both for discounting and for the prepayment model, which ignores
the difference between short-term and long-term interest rates.
In the first essay, we apply perturbation analysis (PA) to estimate the sensitivities
of MBS. Our work makes the following contribution to MBS literature:
• We decompose any interest rate term structure change into four Fourier-like series,
which is based on Fourier cosine series, and can better measure the yield curve shift;
• We derive PA estimators for these Fourier-like factors, as well as interest rate
volatility, which can largely save computation effort. In our example, we calculate 5
duration estimators and 16 convexity estimators in our simulation, which would
require 155 simulations using a conventional simulation scheme.
• Based on our comprehensive analysis of the sensitivity measures we calculated for a
full spectrum of mortgage product, we propose a new mortgage product, which can
potentially benefit both the MBS investor and mortgage borrower.
8
This essay is organized in the following manner. Section 1 describes the problem
setting. We then derive the framework for PA in a general setting in section 2, without
restrictions to any specific interest rate model or prepayment model. Then we consider
the well-known Hull-White interest rate model (Hull and White [1993]) and a common
prepayment model to derive the corresponding PA sensitivities for FRM and ARM
products in section 3, assuming the shock of interest rate term structure takes the form of
a series of trigonometric polynomial functions. Section 4 presents numerical examples, in
which we compare the performance of FD and PA estimators, indicating that the PA
estimator is at least as good as the FD estimator, while the computation cost is reduced
dramatically. Section 5 gives the insights from our simulation results. Section 6 gives
conclusions.
9
1.2 Hedging MBS in HJM Framework
There is a large body of literature on hedging with different interest risk measures,
like first-order hedging with duration (Ilmanen [1992]), second order hedging with
convexity (Kahn and Lochoff [1990], Lacey and Nawalkha [1993]), principal
components hedging (Golub and Tilman [1997]), key rates hedging (Ho [1992]),
level/slope/curvature hedging (Willner [1996]), etc. Yet there has not been a unifying
effort in combining hedging the term structure together with hedging volatility factors.
This essay tries to extract the empirical volatility factors from historical term
structure data, via principal components analysis (PCA), and apply these factors in a
HJM framework for pricing MBS, while deriving the risk measures for hedging MBS. It
makes the following contribution in the MBS literature:
• The first paper to hedge MBS with PCA factors empirically extracted from historical
interest rate data;
• Hedging efficiency is proved to increase significantly, compared with traditional
duration/convexity hedging.
This essay is organized in the following way. Section 1 gives the motivation for
this research question. Section 2 describes the interest rate data set and PCA method we
used to extract the volatility factors. Section 3 applies these factors in interest rate
simulation within a HJM framework. Section 4 derives the PA estimators in the HJM
framework. Section 5 gives the hedging algorithm for MBS, and Section 6 gives the
performance analysis of this hedging method. Section 7 concludes the essay.
10
1.3 Hedging Credit Risk of MBS: A Random Coefficient
Approach
In order to hedge the credit risk of MBS, the MBS issuer sometimes needs to
purchase pool insurance from a third party, beyond the protection of mortgage collateral,
and primary mortgage insurance. In this case, it is important to model the credit risk of
the third party. Recently there has been increased interest in some research papers to use
regression method to determine what factors affect credit spread. Most of the papers,
which use simple linear regression, found that variables in structural models lack
explanatory power in such regression. We argue that the problem partially results from
non-constant credit spread sensitivities to state variables.
We try to overcome the problem by proposing a Random Coefficient Regression
(RCR) model. We collected data from multiple database, and constructed our data set.
Our regression results show that our assumption of non-constancy of credit spread
sensitivities is correct. As a result of improved regression, we improved adjusted R
2
to
28%, compared with 8% adjusted R
2
for a simple linear regression approach, using the
same set of independent variables. Another important result of our RCR model is that it
validated the relationship between credit spread sensitivities and state variables, which
has been predicted by Merton’s model.
This essay makes the following research contributions to the finance literature:
• The first paper to use the RCR method on credit spread data;
11
• The first paper to explicitly build a dependence relationship between credit spread
sensitivity and state variables;
• The first paper to empirically validate the dependence relationship between credit
spread sensitivity and state variables predicted by a structural model, such as
Merton’s model.
This essay is organized in the following way. Section 1 gives the motivation for
this research question. Section 2 describes several previous papers on this topic. Section 3
gives a brief introduction to the Random Coefficient Regression model. Section 4 applies
this model to changes of credit spread. Section 5 gives the data description used in the
regression, and Section 6 gives the results analysis of this regression method. Section 7
concludes the essay.
12
Chapter 2
Efficient Sensitivity Analysis of MBS
2.1 Problem Setting
Generally the price of any security can be written as the net present value (NPV)
of its discounted cash flows. Specifying the price of an MBS (here we consider only the
pass-through MBS
1
) is as follows:
(
¸
(
¸
=
(
¸
(
¸
= =
? ?
= =
M
t
M
t
t c t d E t PV E V E P
0 0
) ( ) ( ) ( ] [ , (2.1)
where P is the price of the MBS,
V is the value of the MBS, which is a random variable, dependent on the
realization of the economic scenario,
PV(t) is the present value for cash flow at time t,
d(t) is the discounting factor at time t,
c(t) is the cash flow at time t,
M is the maturity of the MBS.
1
A pass-through MBS is an MBS that passes through the principal and interest payments collected from a
mortgage pool, minus the guaranty fee and servicing fee, to the MBS investor directly. This is in contrast to
Collaterized Mortgage Obligations (CMOs), which have multiple tranches and pay the principal payments
according to the seniorities of tranches. In this essay, we assume that mortgages in the MBS pool are
homogenous.
13
Monte Carlo simulation is used to generate cash flows on many paths. By the
strong law of large numbers, we have the following:
| |
?
=
? ?
=
N
i
i N
V
N
V E
1
1
lim , (2.2)
where V
i
is the value calculated out in path i.
The calculation of d(t) is found from the short-term (risk-free) interest rate
process,
? ?
?
=
?
=
? ? = ? ? = ? =
1
0
1
0
} ] ) ( [ exp{ ) ) ( exp( ) , 1 ( ) 2 , 1 ( ) 1 , 0 ( ) (
t
i
t
i
t i r t i r t t d d d t d , (2.3)
where d(i, i+1) is the discounting factor for the end of period i+1 at the end of period i;
r(i) is the short term rate used to generate d(i, i+1), observed at the end of period
i;
?t is the time step in simulation, generally monthly, i.e. ?t= 1 month.
An interest rate model is used to generate the short term-rate r(i); then d(t) is instantly
available when the short-term rate path is generated.
The difficult part is to generate c(t), the path dependent cash flow of MBS for
month t, which is observed at the end of month t. From chapter 19 of Fabozzi [1993], we
have the following formula for c(t):
); ( ) ( ) (
); ( ) ( ) (
); ( ) ( ) ( ) ( ) (
t PP t SP t TPP
t IP t SP t MP
t IP t TPP t PP t MP t c
+ =
+ =
+ = + =
(2.4)
where MP(t): Scheduled Mortgage Payment for month t;
TPP(t): Total Principal Payment for month t;
14
IP(t): Interest Payment for month t;
SP(t): Scheduled Principal Payment for month t;
PP(t): Principal Prepayment for month t.
These quantities are calculated as follows:
; ) ( 1 1 ) (
); ( ) 1 ( ) (
)); ( ) 1 ( )( ( ) (
;
12
) 1 ( ) (
;
) 12 / 1 ( 1
12 /
) 1 ( ) (
12
t CPR t SMM
t TPP t B t B
t SP t B t SMM t PP
WAC
t B t IP
WAC
WAC
t B t MP
t WAM
? ? =
? ? =
? ? =
? =
|
|
.
|
\
|
+ ?
? =
+ ?
(2.5)
B(t): The principal balance of MBS at end of month t;
WAC
2
: Weighted Average Coupon rate for MBS;
WAM
3
: Weighted Average Maturity for MBS;
SMM(t): Single Monthly Mortality for month t, observed at the end of
month t;
CPR(t): Conditional Prepayment Rate for month t, observed at the end of
month t.
In Monte Carlo simulation, along the sample path, CPR(t) is the primary variable
to be simulated. Everything else can be calculated out once CPR(t) is known. Different
prepayment models offer different CPR(t), and it is not our goal to derive a new
2
WAC is the weighted average mortgage rate for a mortgage pool, weighted by the balance of each
mortgage.
3
WAM is the weighted average maturity in month for a mortgage pool, weighted by the balance of each
mortgage.
15
prepayment model or compare existing prepayment models. Instead, our concern is,
given a prepayment model, how can we efficiently estimate the price sensitivities of MBS
against parameters of interest? Generally different prepayment models will lead to
different sensitivity estimates, so it is at the user’s discretion to choose an appropriate
prepayment function, as our method for calculating the “Greeks” is universally
applicable.
2.2 Derivation of General PA Estimators
If P, the price of the MBS, is a continuous function of the parameter of interest,
say ?, and assuming the interchange of expectation and differentiation is permissible
4
, we
have the following PA estimator by differentiating both sides of (2.1):
). , (
) , (
) , (
) , ( )) , ( (
,
) , (
) , (
)] ( [ ) (
1
1
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
t d
t c
t c
t d
d
t PV d
d
t dPV
E
d
t PV d
E
d
V dE
d
dP
M
t
M
t
?
?
+
?
?
=
(
¸
(
¸
=
(
(
(
(
¸
(
¸
= =
?
?
=
=
(2.6)
Now we have reduced the original problem from estimating the gradient of a sum
function to estimating the sum of a bunch of gradients. Actually now we only need to
estimate two gradient estimators,
?
?
?
? ) , (t c
and
?
?
?
? ) , (t d
, at each time step.
4
Strictly speaking, sample pathwise continuity of price function with respect to ? will result in the
interchange being valid.
16
2.2.1 Gradient Estimator for Cash Flow
We first derive
?
?
?
? ) , (t c
. To simplify notation, we write c(t) for c(t, ?).
A simplified expression for c(t) is derived from (2.4) and (2.5) as follows:
{ }, ) ( )] ( 1 )[ ( ) 1 (
) ( )
12
1 )( 1 ( )] ( 1 [
) 12 / 1 ( 1
12 /
) 1 (
) ( )
12
1 )( 1 ( )) ( 1 )( (
) ( )]} ( ) ( [ ) 1 ( { ) (
) ( )] ( ) 1 ( [ ) ( ) ( ) ( ) (
t SMM g t SMM t A t B
t SMM
WAC
t B t SMM
WAC
WAC
t B
t SMM
WAC
t B t SMM t MP
t SMM t IP t MP t B t MP
t SMM t SP t B t MP t PP t MP t c
t WAM
+ ? ? =
+ ? + ?
+ ?
? =
+ ? + ? =
? ? ? + =
? ? + = + =
+ ?
(2.7)
where
).
12
1 (
,
) 12 / 1 ( 1
12 /
) (
WAC
g
WAC
WAC
t A
t WAM
+ =
+ ?
=
+ ?
(2.8)
Then we can derive the gradient for c(t), if WAC and t are independent
5
of ?:
{ } ] ) ( )[ 1 (
) (
) ( )] ( 1 )[ (
) 1 ( ) (
g t A t B
t SMM
t SMM g t SMM t A
t B t c
+ ? ?
?
?
+ + ?
?
? ?
=
?
?
? ? ?
. (2.9)
This leads to recursive equations for calculation of the above gradient estimator
from (2.5) and (3.2):
.
) ( ) 1 ( ) (
); ( ) 1 ( ) ( )
12
1 )( 1 ( ) ( ) 1 ( ) (
); 1 (
12
) ( ) ( ) ( ) (
? ? ? ?
?
?
?
? ?
=
?
?
? ? = ? + ? = ? ? =
? ? = ? =
t c
g
t B t B
t c g t B t c
WAC
t B t TPP t B t B
t B
WAC
t c t IP t c t TPP
(2.10)
5
A fixed Rate Mortgage (FRM) would satisfy this assumption; however an Adjustable Rate Mortgage
(ARM) will not, so we derive the gradient estimator for ARMs later in section 4.
17
Assuming we know that the initial balance is not dependent as ?; we have the
initial conditions:
). ) 1 ( )( 0 (
) 1 ( ) 1 (
; 0
) 0 (
g A B
SMM c
B
+ ?
?
?
=
?
?
=
?
?
? ?
?
(2.11)
This leads to the following
,
) 1 ( ) 0 ( ) 1 (
? ? ? ?
?
?
?
?
=
?
? c
g
B B
(2.12)
{ } ), ) 2 ( )( 1 (
) 2 (
) 2 ( ) 2 ( 1 )( 2 (
) 1 ( ) 2 (
g A B
SMM
SMM g SMM A
B c
+ ?
?
?
+ + ?
?
?
=
?
?
? ? ?
(2.13)
M t
t c
,..., 2 , 1 ,
) (
=
?
?
?
.
Thus the problem of calculating the gradient estimator of cash flow c(t) is reduced
to calculating:
. ,..., 1 ,
) (
M t
t SMM
=
?
?
?
Since
, ) ( 1 1 ) (
12
t CPR t SMM ? ? =
we have
.
) (
)) ( 1 (
12
1 ) (
12
11
? ? ?
?
? =
?
?
?
t CPR
t CPR
t SMM
(2.14)
As discussed earlier, generally CPR(t) is given in the form of a prepayment function,
and there are four main types of prepayment functions (Fabozzi [2000]):
1. Arctangent Model: (An example from the Office of Thrift Supervision (OTS).)
18
))
) (
089 . 1 ( 9518 . 5 arctan( 1389 . 0 2406 . 0 ) (
10
t r
WAC
t CPR ? ? = . (2.15)
2. CPR(S,A,B,M) Model:
); ( ) ( ) ( ) ( ) ( t BM t MM t AGE t RI t CPR = (2.16)
where RI(t) is refinancing incentive;
AGE(t) is the seasoning multiplier;
MM(t) is the monthly multiplier, which is constant for a certain month;
BM(t) is the burnout multiplier.
3. Prepayment models incorporating macroeconomic factors, i.e., the health of
economics, housing market activity, etc.
4. Prepayment models for individual mortgages.
For the last two types of prepayment models, we do not have any explicitly stated
functional forms, mainly because they are proprietary models in the mortgage industry.
But since our approach is general for any type of prepayment function, we can derive the
derivatives once we are given an explicit form for the prepayment function.
We would like to make one claim here: the CPR(t) model we mentioned and we are
going to use in Chapter 2 and Chapter 3 includes both voluntary prepayment (refinance,
house turnover, and cash out) and involuntary prepayment (default). Because default only
makes about 1% of total prepayment, and generally MBS issuer will guarantee the
principal payment to the investor, in case borrower defaults, it is a reasonable not to
model default separately. However, when analyzing MBS backed by high default loans,
such as subprime mortgages, it is desirable to model voluntary prepayment and default
separately.
19
2.2.2 Gradient Estimator for Discounting Factor
We have derived the gradient estimator of cash flow with respect to parameter ?.
Next, we derive the gradient estimator of the discounting factor d(t).
We know that the discounting factor takes the following form from section 2,
when the option adjusted spread (OAS) is not considered. For simplification, we write
d(t) as for d(t, ?):
} ] ) ( [ exp{ ) (
1
0
t i r t d
t
i
? ? =
?
?
=
. (2.17)
Differentiating with respect to ?:
. )
) (
( ) ( )
) (
( } ] ) ( [ exp{
) (
1
0
1
0
1
0
t
i r
t d t
i r
t i r
t d
t
i
t
i
t
i
?
?
?
? = ?
?
?
? ? ? =
?
?
? ? ?
?
=
?
=
?
=
? ? ?
(2.18)
From the gradient estimators for cash flow and discounting factor, we can easily
get the gradient estimator of PV(t):
) , (
) , (
) , (
) , ( )) , ( (
?
?
?
?
?
?
?
?
t d
t c
t c
t d
d
t PV d
?
?
+
?
?
= . (2.19)
The last step would be to apply a specific prepayment model and interest rate
model to arrive at the actual implemented gradient estimators. To illustrate the procedure,
we carry out this exercise in its entirety for one setting in the following section.
20
2.3 Applying the Gradients
We choose our interest model to be the one-factor Hull-White (Hull and White
[1990]) model, for its simplicity and easy calibration to market term structure. For the
prepayment model, we consider a CPR(S,A,B,M) model.
2.3.1 Interest Model Setup
In this section, we briefly discuss the model and the simulation scheme.
In the one-factor Hull-White interest rate model, the underlying process for the
short-term rate r(t) is given by
), ( )) ( ) ( ( ) ( t dB dt t ar t t dr ? ? + ? = (2.20)
where B(t): a standard Brownian motion;
a: mean reverting speed, constant;
?: standard deviation, constant;
?(t): chosen to fit the initial term structure, which is determined by
), 1 (
2
) , 0 (
) , 0 (
) (
2
2
at
e
a
t af
t
t f
t
?
? + +
?
?
=
?
? (2.21)
f(0,t): the instantaneous forward rate, which is determined by
, ) , 0 (
1
) , 0 (
0
?
=
t
du u f
t
t R (2.22)
Differentiating both sides, with respect to t, we have
) , 0 (
) , 0 (
) , 0 ( t R
t
t R
t t f +
?
?
= , (2.23)
where R(0,t): the continuous compounding interest rate from now to time t, i.e. the term
structure.
21
In order to simplify the simulation process, the model can be re-parameterized
from its original to the following:
; 0 ) 0 ( ), ( ) ( ) ( ) ( = + ? = x t dB dt t x t a t dx ? (2.24)
x(t) is determined by
2
2
) 1 (
2
) , 0 ( ) ( ) ( ) (
at
e
a
t f t x t r t a
?
? + = ? =
?
. (2.25)
The process x(t) is called an Ornstein-Uhlenbeck process, and its solution is given
by
?
?
=
t
au at
u dB e e t x
0
) ( ) ( ? , (2.26)
which is a Gaussian Markov process, and can also be represented as
)
2
1
( ) (
2
a
e
W e t x
at
at
?
=
?
? , (2.27)
where {W(t), t?0} is also a Brownian motion.
In this case, the interest rate r(t) can be represented in the following form:
) ) ( ) ( ( ) (
) (t h
W t g t a F t r + = , (2.28)
where a, g: R
+
? R are continuous functions, and the functions F:R ? R and h: R
+
? R
are strictly increasing and continuous. From above we can see that
.
2
1
) (
; ) (
; ) 1 (
2
) , 0 ( ) (
; ) (
2
2
2
2
a
e
t h
e t g
e
a
t f t a
x x F
at
at
at
?
=
=
? + =
=
?
?
?
?
(2.29)
To simulate r(t) given by above, we will first simulate
) (
) ( ) (
t h
W t g t x = ,
22
which is a Gaussian Markov process, and then compute the short-term interest rate by
)) ( ) ( ( ) ( t x t a F t r + = .
For calculating the price of MBS, the short-term rate is not sufficient; the long-
term rate process is also required, especially the 10-year Treasury rate, which is a
deterministic function of r(t) in the Hull-White model. Generally this is the case for
short-term rate models, but not true for more complicated interest rate models, e.g., the
HJM (Heath, Jarrow and Morton [1992]) model and the LIBOR forward rate model
(Jamshidian[1997]). The long-term rate R(t,T) is calculated from the following, :
). 1 ( ) (
4
) , 0 ( ln
) , (
) , 0 (
) , 0 (
ln ) , ( ln
;
1
) , (
; ) , ( ) , (
2 2
3
2
) (
) ( ) , ( ) )( , (
? ? ?
?
?
? =
?
=
= =
? ?
? ?
? ? ?
at at aT
t T a
t r T t B t T T t R
e e e
a t
t P
T t B
t P
T P
T t A
a
e
T t B
e T t A e T t P
?
(2.30)
P(t,T) is the zero coupon bond price at time t, with face value $1, matured at T.
Thus we can derive the R(t,T) as following:
) (
) ( ) , ( ) , ( ln
) , (
t T
t r T t B T t A
T t R
?
?
? = . (2.31)
The standard (forward) path generation method for generating x(t) is given by
, )) ( ( )) ( ( ) ( ) (
) (
) (
))] ( ( )) ( ( )[ ( )) ( ( ) (
) (
) (
) (
1 1 1
1
1 1
1
1
+ + +
+
+ +
+
+
? + =
? + =
i i i i i
i
i
i i i i i
i
i
i
z t h W t h W t g t x
t g
t g
t h W t h W t g t h W t g
t g
t g
t x
(2.32)
where {z
i
} is a series of independent standard normal random variables. In the special
case where x(t) is from the Hull-White model, we have
1
2
1
2
1
) ( ) (
+
? ?
? ?
+
?
+ =
i
t a
i
t a
i
z
a
e
t x e t x
i
i
? , (2.33)
23
where ?t
i
=t
i+1
- t
i
.
2.3.2 Trigonometric Polynomial Shocks
There are multiple factors in the interest rate model that can change and affect the
cash flows and discounting factor along the simulation path. The major changes could be
the initial term structure R(0,t) and the volatility ?.
The most common assumption for term structure change is a parallel shift on all
maturities. However, this is often not an adequate model for the real world, where a shift
in the term structure can take any shape. For example, short-term rates and long-term
rates may change in opposite directions rather than in parallel. We consider a Fourier
series decomposition of the term structure shift.
Our domain of concern is interest rates from time 0 to 30 years, since most
mortgages are amortized in a 30-year term. So for example, we could assume the shift of
term structure takes the following form:
?
?
=
? = ?
0
),
30
cos( ) , 0 (
n
n
t n
t R
?
(2.34)
where ?
n
is the magnitude for the n
th
Fourier function. Figure 4.1 depicts the first four
trigonometric polynomial series. (n=0,1,2,3), which is all that we will consider in our
model. When n=0, the shift is just like a parallel shift in term structure. When n=1, the
short-term and long-term rates move in opposite directions. When n=2, the short-term
and long-term rates move in the same direction, while the middle-term rate moves in the
opposite direction. Thus we decompose any shift in the term structure into the Fourier
functions by Fourier transform. If we have previously calculated the gradients with
respect to the magnitude of each trigonometric polynomial function, we can apply these
24
gradients and get the corresponding changes in the cash flows and discounting factors,
and hence the change in MBS prices.
The Fourier series have a serious drawback: they treat short-term rates the same
as long-term rates. However, from experience, we know that the short-term rates
generally change more frequency than long-term rates. So we would like to change the
shape of the trigonometric polynomial function, which will concentrate more on the
short-term rates, and keep the long-term rates relatively stable. The modified Fourier
function that we adopt takes the following form:
?
?
=
?
? ? = ?
0
/
)) 1 ( cos( ) , 0 (
0
n
T t
n
e n t R ? , (2.35)
where T
0
is a user-specified parameter of the modified Fourier shifts. The smaller T
0
is,
the more likely short rates and long rates are going to act differently. See Figure 2.2 for
the modified Fourier functions, where T
0
=10. Comparing Figures 2.1 and 2.2, the
modified Fourier series concentrate more on the changes with maturities less than T
0,
which is both desirable and easier for analytical purposes.
For a Fourier cosine series that has the following functional form:
?
?
=
+ =
1
0
)
2
cos(
2
) (
n
n
T
t n
a
a
t f
?
, (2.36)
the coefficients are given by a Fourier cosine transform:
?
= =
2 /
0
2 1 0 , )
2
cos( ) (
4
T
n
,...... , , n dt
T
t n
t f
T
a
?
(2.37)
25
Figure 2.1 ?R(0,t) with Original Fourier series
Figure 2.2 ?R(0,t) with T
0
=10 modified Fourier series
26
For our modified Fourier series, perform the following change of variables:
) 1 (
30
'
0
/ T t
e
t
?
? =
and substitute into the expression of ?R(0,t) to get
?
?
=
? = ?
0
)
60
' 2
cos( ) ' , 0 (
n
n
t n
t R
?
, (2.38)
which is a standard Fourier cosine series, and we can use a Fourier transform to estimate
the coefficients. In computer simulation, t is a vector of real time points, evenly
distributed with sample function value ?R(0,t), and t’ is the mapped time point in a new
time scale, which is not evenly distributed, with the same sample function value ?R(0,t’).
However, in order to utilize the discrete cosine transform function provided in
mathematical libraries, we need to resample ?R(0,t’) at even time intervals. This is
carried out by interpolating the function of ?R(0,t’) on the t time scale. Figure 2.3 shows
a sample of ?R(0,t), ?R(0,t’) re-sampled on t, the coefficients estimated on the re-
sampled ?R(0,t’), and the reconstructed Fourier series of ?R(0,t).
From Figure 2.3, if we look at the upper left and lower right sub-figures, we can
see that the reconstructed term structure matches the original sample very well, which
validates our method for estimating the coefficients of the modified Fourier series.
27
Figure 2.3 Coefficients Estimation for Modified Fourier series
2.3.3 Derivation of Gradients with respect to Modified Fourier Functions
Our major task in this section is to derive the gradient estimator with respect to to
specific parameters in the interest rate model and prepayment model. Specifically, for the
former, we are interested in the parameters of the modified Fourier functions (?
n,
n=0, 1,
2, 3).
First we derive the discounting factor gradient estimator. From (3.12), we know
that in order to derive
? ?
? ) (t d
, we must first derive
? ?
? ) (i r
, i=0,…, t-1. Let us recall that in
section 4.1, we have the following simulation scheme for short term rate r(t):
) ( ) ( ) ( t x t a t r + = .
28
So
? ? ? ?
?
+
?
?
=
?
? ) ( ) ( ) ( t x t a t r
, (2.39)
where
? ?
? ) (t a
and
? ?
? ) (t x
are determined as the following in Hull-White model:
. 0
) (
,
) , 0 ( ) (
=
?
?
?
?
=
?
?
?
? ?
t x
t f t a
(2.40)
We also know the relationship between f(0,t) and R(0,t) from (2.23), so
? ?
? ) , 0 ( t f
can be derived as:
.
) , 0 ( ) , 0 (
) , 0 ( ) , 0 ( ) , 0 (
) , 0 (
) , 0 (
) , 0 (
2
2
? ?
? ? ?
? ? ?
?
?
+
? ?
?
=
?
?
+
? ?
?
+
?
?
?
?
=
?
?
+
?
|
.
|
\
|
?
?
?
=
?
?
t R
t
t R
t
t R
t
t R
t
t
t R t
t R t
t R
t
t f
(2.41)
Considering the changes in R(0,t) which takes the form as in (2.38), we can get
the derivatives of R(0,t) (? taken to be ?
n
):
.
) (
)) 1 ( sin(
) (
)) 1 ( sin(
) , 0 (
)), 1 ( cos(
) , 0 (
0
/
/
0
/
/
2
/
0
0
0
0
0
T
e n
e n
T
e n
e n
t
t R
e n
t R
T t
T t
T t
T t
n
T t
n
?
?
?
?
?
? =
?
?
? ? =
? ? ?
?
? =
? ?
?
?
?
?
?
?
(2.42)
We can get the derivatives of r(i):
)). 1 ( cos(
) (
)) 1 ( sin(
) , 0 ( ) (
0
0
0
/
0
/
/ T t
T t
T t
n n
e n
T
e n
e n t
t f i r
?
?
?
? + ? ? =
? ?
?
=
? ?
?
?
?
? (2.43)
And gradient estimator for discounting factor is also obtained, applying (2.18).
29
Next, we are going to derive the cash flow gradient estimator with respect to ?
n
.
From our derivation in section 3, we know that in order to get
? ?
? ) (t c
, we need to derive
? ?
? ) (t CPR
first. We use the second type of prepayment function, among the four
described in section 3. An example for this type of prepayment model is available from
the sample code at the Numerix homepage http://www.numerix.com.
); ( ) ( ) ( ) ( ) ( t BM t MM t AGE t RI t CPR = (2.44)
where
rate. mortgage prevailing ith the w
correlated highly is which t, period of end at the observed rate, year 10 the is ) (
;
) 0 (
) 1 (
7 . 0 3 . 0 ) (
December; in ending January, from starting
0.98], 1.23, 1.22, 1.18, 1.1, 0.98, 0.92, 0.98, 0.95, 0.74, 0.76, [0.94, ) (
);
30
, 1 min( ) (
))); 1 ( ( 430 571 . 8 arctan( 14 . 0 28 . 0 ) (
10
10
t r
B
t B
t BM
t MM
t
t AGE
t r WAC t RI
?
+ =
=
=
? ? + ? + =
From the formulas, only RI(t) and BM(t) depend on ?, when ? is not time t. Thus
we have the following formula for
? ?
? ) (t CPR
:
;
) (
) ( ) ( ) ( ) ( ) ( ) (
) ( ) (
? ? ? ?
?
+
?
?
=
?
? t BM
t MM t AGE t RI t BM t MM t AGE
t RI t CPR
(2.45)
where
.
) 0 (
1 ) 1 (
7 . 0
) (
;
) 1 (
))) 1 ( ( 430 571 . 8 ( 1
) 430 (
14 . 0
) (
10
2
10
B
t B t BM
t r
t r WAC
t RI
? ?
? ?
?
? ?
=
?
?
?
? ?
? ? + ? +
?
=
?
?
(2.46)
30
? ?
? ) (t B
is available, when
? ?
? ) (t c
is calculated out, so the problem is reduced to
calculating
? ?
? ) (
10
t r
. In the one-factor Hull-White framework, as we have discussed in
section 2.3.1, the long-term rate is a deterministic function of r(t), so substituting T=t+10
for (2.30), we have
). 1 ( ) (
4
) , 0 (
1
) , 0 ( ) 10 )( 10 , 0 (
) 1 ( ) (
4
) , 0 ( ln
) 10 , ( ) , 0 ( ln ) 10 , 0 ( ln
) 1 ( ) (
4
) , 0 ( ln
) 10 , (
) , 0 (
) 10 , 0 (
ln ) 10 , ( ln
;
1 1
) 10 , (
; ) 10 , ( ) 10 , (
2 2 ) 10 (
3
2 10
2 2 ) 10 (
3
2
2 2 ) 10 (
3
2
10 ) 10 (
) ( ) 10 , ( ) 10 )( 10 , (
? ? ?
?
+ + + + ? =
? ? ?
?
?
+ ? ? + =
? ? ?
?
?
+ ?
+
= +
?
=
?
= +
+ = = +
? + ?
?
? + ?
? + ?
? ? + ?
+ ? ? + + ?
at at t a
a
at at t a
at at t a
a t t a
t r t t B t t t t R
e e e
a
t R
a
e
t t R t t R
e e e
a t
t P
t t B t P t P
e e e
a t
t P
t t B
t P
t P
t t A
a
e
a
e
t t B
e t t A e t t P
?
?
?
(2.47)
Since
,
10
) (
1
) 1 ( ) (
4
) , 0 (
1
) , 0 ( ) 10 )( 10 , 0 (
10
) ( ) 10 , ( ) 10 , ( ln
) 10 , ( ) (
10
2 2 ) 10 (
3
2 10
10
t r
a
e
e e e
a
t R
a
e
t t R t t R
t r t t B t t A
t t R t r
a
at at t a
a ?
? + ?
?
?
? ? ? ?
?
+ + + + ?
? =
+ ? +
? = + =
?
(2.48)
? ?
? ) (
10
t r
takes the following form, when ? is independent of ? and t:
10
) ( 1 ) , 0 (
)
1
(
) 10 , 0 (
) 10 (
) (
10 10
10 ? ? ?
?
?
? ?
?
?
? ?
+ +
?
+ ?
+ ?
? =
?
?
? ?
t r
a
e t R
a
e
t
t R
t
t r
a a
. (2.49)
Thus we have derived
? ?
? ) (
10
t r
as a function of
? ?
? ) (t R
and
? ?
? ) (t r
derived earlier.
31
2.3.4 Derivation of Gradients with respect to Volatility: Vega
The derivation is straightforward as in section 2.3.3; all we need to do is to
substitute ? with ?, instead of ?
n
. In order to get
? ?
? ) (t d
, we must first derive
? ?
? ) (i r
.
Following the same logic in (2.40), we can get the vega of r(t):
).
2
1
( ) 1 (
) (
so ),
2
1
(
) (
, ) 1 (
) (
2
2
2
2
2
2
a
e
W e e
a
t r
a
e
W e
t x
e
a
t a
at
at at
at
at
at
?
+ ? =
?
?
?
=
?
?
? =
?
?
? ?
?
?
?
?
?
?
?
(2.50)
And vega of d(t) would be:
. )
) (
( ) (
) (
1
0
t
i r
t d
t d
t
i
?
?
?
? =
?
?
?
?
=
? ?
(2.51)
Now we derive
? ?
? ) (t c
, which would require us to derive
? ?
? ) (t CPR
first, which has
the same form as in (2.45), while
? ?
? ) (
10
t r
has the form of:
.
10
) ( 1
) 1 ( ) (
2
10
) ( 1
) 1 ( ) (
2
) , 0 (
)
1
(
) 10 , 0 (
) 10 (
) (
10
2 2 ) 10 (
3
10
2 2 ) 10 (
3
10
10
?
?
?
?
? ?
?
?
? ?
+ ? ?
=
?
? ?
? ? ? ?
?
? ?
+ +
?
+ ?
+ ?
? =
?
?
?
? + ?
?
? + ?
?
t r
a
e
e e e
a
t r
a
e
e e e
a
t R
a
e
t
t R
t
t r
a
at at t a
a
at at t a
a
(2.52)
2.3.5 Derivation of Second Order Gradients: Gamma
Another gradient that interests risk mangers is convexity, or the gamma of MBS,
which is the second order derivative of price against term structure shifts. Now we derive
an estimator for the gamma.
32
In order to calculate the partial second order derivatives (Hessian matrix), we take
? to be the vector, ?=[?
1
?
2
?
3
?
4
?]’. Differentiating (2.1)
1
, we get
. ) (
) (
'
) ( ) (
'
) ( ) (
) (
) ( ) ( ] [
, ) (
) (
) (
) ( ) ( ] [
0
2
2
2
2
0
2
2
2
2
2
2
0 0
(
¸
(
¸
?
?
+
?
?
×
?
?
+
?
?
×
?
?
+
?
?
=
(
¸
(
¸
?
?
=
?
?
=
?
?
(
¸
(
¸
?
?
+
?
?
=
(
¸
(
¸
?
?
=
?
?
=
?
?
? ?
? ?
= =
= =
M
t
M
t
M
t
M
t
t d
t c t d t c t c t d
t c
t d
E
t PV
E
V E P
t d
t c
t c
t d
E
t PV
E
V E P
? ? ? ? ? ? ? ? ?
? ? ? ? ?
(2.53)
where ]' [
4 3 2 1
? ? ?
?
? ?
?
? ?
?
? ?
?
? ?
?
=
?
? P P P P P P
, and
2
2
? ?
? P
is a 5-by-5 matrix, whose (i, j)
th
element is determined by
j i
P
? ? ? ?
?
2
, where ?
I
and ?
I
are the i
th
and j
th
elements of ?,
respectively. The same notation will be used for gradients of other variables, i.e. c(t), d(t),
r(t), etc.
Since we have calculated
? ?
? ) (t c
and
? ?
? ) (t d
in previous sections, now the problem
is reduced to estimate
2
2
) (
? ?
? t c
and
2
2
) (
? ?
? t d
. So we first derive the gamma for the
discounting factor d(t). Differentiating (2.18), we get
t
i r t d
t
i r
t d
t d
t
i
t
i
? ×
?
?
? ×
?
?
+ ? ×
?
?
? × =
?
?
? ?
?
=
?
=
)'
) (
(
) (
)
) (
( ) (
) (
1
0
1
0
2
2
2
2
? ? ? ?
(2.54)
Once we have
2
2
) (
? ?
? i r
, the gamma of d(t) is easily calculated. Now we derive the
gamma for cash flow c(t). From (2.9), we can derive the following gamma equation:
1
Again, we need the first order derivative to be pathwise continous to make the interchange of expectation
and differentiation permissible.
33
{ }
]. ) ( )[ 1 (
) (
] ) ( ][ '
) 1 ( ) (
'
) ( ) 1 (
[
) ( )] ( 1 )[ (
) 1 ( ) (
2
2
2
2
2
2
g t A t B
t SMM
g t A
t B t SMM t SMM t B
t SMM g t SMM t A
t B t c
+ ? ?
?
?
+
+ ?
?
? ?
×
?
?
+
?
?
×
?
? ?
+
+ ?
?
? ?
=
?
?
?
? ? ? ?
? ?
(2.55)
And from (3.5), we can get the gamma of B(t):
2
2
2
2
2
2
) ( ) 1 ( ) (
? ? ? ?
?
?
?
? ?
=
?
? t c
g
t B t B
. (2.56)
Now we calculate gamma of SMM(t):
. ,..., 1 ,
) (
2
2
M t
t SMM
=
?
?
?
As we know from (3.9), we have
.
) (
)) ( 1 (
12
1
) '
) ( ) (
( )) ( 1 (
144
11 ) (
2
2
12
11
12
23
2
2
? ? ? ? ?
?
? +
?
?
×
?
?
? =
?
?
? ?
t CPR
t CPR
t CPR t CPR
t CPR
t SMM
(2.57)
? ?
? ) (t CPR
and
2
2
) (
? ?
? t CPR
will be prepayment model specific.
For discounting factors, if we choose the Hull-White one factor model, we have
the following:
. 5 , 1 ,
) ( ) (
; ]'
) (
) (
) (
) (
) (
[
) (
2
2
2
4 3 2 1
? ?
(
(
¸
(
¸
? ?
?
=
?
?
?
?
? ?
?
? ?
?
? ?
?
? ?
?
=
?
?
j i
i r i r
i r i r i r i r i r i r
j i
? ? ?
? ?
(2.58)
From and (2.43) and (2.49), we can derive the following:
34
.
) 1 ( ) (
; 0
) (
; 0
) (
2
2
2
2
2
2
a
e i r
i r
i r
t ai
i
j i
? ?
?
=
?
?
=
? ? ?
?
=
? ? ? ?
?
?
?
(2.59)
And the gamma of d(t) would be
) ( / '
) ( ) (
)'
) (
(
) (
)'
) (
(
) (
)
) (
( ) (
) (
1
0
1
0
1
0
2
2
2
2
t d
t d t d
t
i r t d
t
i r t d
t
i r
t d
t d
t
i
t
i
t
i
? ?
? ?
? ? ? ?
?
?
×
?
?
=
? ×
?
?
? ×
?
?
=
? ×
?
?
? ×
?
?
+ ? ×
?
?
? × =
?
?
?
? ?
?
=
?
=
?
=
(2.60)
For cash flows, based on the equations (2.45) and (2.48) in the CPR(S, A, B, M)
model, we have:
,
) 0 (
1
*
) 1 (
* 7 . 0
) (
;
) 1 (
))) 1 ( ( 430 571 . 8 ( 1
5 / 301
) '
) 1 ( ) 1 (
)(
))) 1 ( ( 430 571 . 8 ( 1
5 / 301
(
) (
];
) (
) ( '
) ( ) (
'
) ( ) (
) (
) (
)[ ( ) (
) (
2
2
2
2
2
2
10
2
10
10 10
2
10 10
2
2
2
2
2
2
2
2
B
t B t BM
t r
t r WAC
t r t r
t r WAC r
t RI
t BM
t RI
t RI t BM
t BM t RI
t BM
t RI
t MM t AGE
t CPR
? ?
?
? ? ?
? ? ?
? ? ? ?
?
? ?
=
?
?
?
? ?
? ? + ? +
+
?
? ?
×
?
? ?
? ? + ? + ?
?
=
?
?
?
?
+
?
?
×
?
?
+
?
?
×
?
?
+ ×
?
?
=
?
?
(2.61)
where we know from (2.58) that
35
.
10
) ( 1
) (
0;
) 10 , 0 (
; 0
) , 0 (
;
10
) ( 1 ) , 0 (
)
1
(
) 10 , 0 (
) 10 (
) (
2 10
2
10
2
2
2 10 2 10 2
2
10
j i
a
j i
j i
j i
j i
a
j i
a
j i
j i
t r
a
e
t r
t R
t R
t r
a
e t R
a
e
t
t R
t
t r
? ?
? ?
? ?
? ?
? ? ? ? ? ?
? ?
? ?
? ?
=
? ?
?
=
? ?
+ ?
=
? ?
?
? ?
? ?
?
? ?
? ?
+ +
? ?
+ ?
+ ?
? =
? ?
?
?
? ?
(2.62)
Finally, the gamma of price P given by equation (2.53) can be obtained from
equations (2.60), (2.61), and (2.62).
2.3.6 Derivation of ARM PA estimators
In this section, we derive PA estimators for ARMs. We know FRMs only have
two sources of uncertainty:
• Short-term rate r(t), which affects the discounting factor d(t), and
• Long-term rate r
10
(t), which determines the prepayment rate CPR(t), and hence
determines the cash flow C(t).
ARMs introduce one more source of uncertainty, the coupon rate WAC(t), which
affects both the amortization schedule and the prepayment rate CPR(t), and then affects
the cash flow C(t). Coupon rate is determined by many factors:
• The index rate. WAC resets to the index rate plus the margin periodically.
• Margin. The spread between the WAC and the index rate.
• Adjustment period. For fixed period (FP) ARMs, the first adjustment period is
different from subsequent adjustment period.
36
• Period Cap/Floor. The maximum amount the WAC could increase/decrease from
previous period.
• Lifetime Cap/Floor. The maximum/minimum coupon rate over the lifetime of the
mortgage.
In order to derive the PA gradient estimator of C(t) for ARM, we first need to
derive the PA gradient estimator for Index(t) and WAC(t).
The most commonly used index rate is the 1-year Treasury rate. In the Hull-White
model, it is an explicit function of short-term rate r(t) and the term structure R(0, t). As
we have derived the function form of r
10
(t), we can derive the r
lag
(t) for any lag: (in this
case, lag=1)
.
) (
1
) 1 ( ) (
4
1
) , 0 (
1
) , 0 ( ) )( , 0 (
) ( ) , ( ) , ( ln
) , ( ) (
*
2 2 ) (
3
*
lag
t r
a
e
e e e
a
t R
a
e
t t R lag t lag t R
lag
t r lag t t B lag t t A
lag t t R t r
a lag
at at lag t a
a lag
lag
?
? + ?
?
?
? ? ? ?
?
? ? + + ?
? =
+ ? +
? = + =
(2.63)
Thus we have the PA gradient estimator of
? ?
? ) (t Index
in following form:
lag
t r
a
e t R
a
e
t
lag t R
lag t
t r
t Index
a lag a lag
lag
? ? ?
? ?
?
? ?
?
?
? ?
+ ?
?
+ ?
+ ?
? =
?
?
=
?
?
? ?
) ( 1 ) , 0 (
)
1
(
) , 0 (
) (
) (
) (
* *
.
(2.64)
The hard part is to get the
? ?
? ) (t WAC
from
? ?
? ) (t Index
, because of the complicated rules to
determine WAC(t), based on all the factors mentioned above. Given WAC(t-1), Index(t),
37
Margin, Period_Cap
2
, Period_Floor, Life_Cap, Life_Floor, WAC(t) is determined as
follows:
¦
¹
¦
´
¦
? +
+ ?
< + < +
=
+ =
=
? =
; Margin if
; Margin if
; Margin if Margin
1 max
1 min
Otherwise,
moment; adjustment an not is if ), 1 ( ) (
Cap Effective_ Index(t) Cap, Effective_
Index(t) Floor Effective_ Floor, Effective_
Cap Effective_ Index(t) Floor Effective_ , Index(t)
WAC(t)
); Period_Cap ) WAC(t- (Life_Cap, Cap Effective_
loor); )-Period_F r, WAC(t- (Life_Floo Floor Effective_
t t WAC t WAC
(2.65)
Figure 2.4 shows the relationship of WAC with Index.
Figure 2.4 WAC as a function of Index
2
Life_Cap/Life_Floor are absolute numbers, while Period_Cap/Period_Floor are relative.
38
Then we can derive the
? ?
? ) (t WAC
as following:
¹
´
¦
=
¦
¦
¦
¹
¦
¦
¦
´
¦
? + <
?
? ?
+ ? >
?
? ?
< + <
?
?
=
?
?
+ =
=
?
? ?
=
?
?
false. is condition when 0,
true; is condition when , 1
} { where
; Margin if } _ _ { *
) 1 (
; Margin if } _ _ { *
) 1 (
; Margin if
1 max
1 min
Otherwise,
moment; adjustment an not is t if ,
) 1 ( ) (
condition I
Cap Effective_ Index(t) , Cap Life Cap Effective I
t WAC
Index(t) Floor Effective_ , Floor Life Floor Effective I
t WAC
Cap Effective_ Index(t) Floor Effective_ ,
Index(t)
WAC(t)
); Period_Cap ) WAC(t- (Life_Cap, Cap Effective_
loor); )-Period_F r, WAC(t- (Life_Floo Floor Effective_
t WAC t WAC
?
?
?
?
? ?
(2.66)
Note that the gradient is 0, when it is bounded by lifetime cap or floor, because a
perturbation would not change the WAC(t).
Next, we need to derive
? ?
? ) (t CPR
for ARM, assuming ARM borrowers have the
same prepayment behavoir as FRM borrowers (which is not necessarily true, but it does
not affect our analysis), so we are facing the same prepayment function as FRM30 as in
(2.45).
? ?
? ) (t CPR
will be affected because of the uncertainty of WAC(t).
39
.
) 0 (
1 ) 1 (
7 . 0
) (
];
) 1 ( ) (
[
))) 1 ( ) ( ( 430 571 . 8 ( 1
430
14 . 0
) (
;
) (
) ( ) ( ) ( ) ( ) ( ) (
) ( ) (
10
2
10
B
t B t BM
t r t WAC
t r t WAC
t RI
t BM
t MM t AGE t RI t BM t MM t AGE
t RI t CPR
? ?
? ? ?
? ? ?
?
? ?
=
?
?
?
? ?
?
?
?
? ? + ? +
=
?
?
?
?
+
?
?
=
?
?
(2.67)
.
) (
)) ( 1 (
12
1 ) (
; ) ( 1 1 ) (
12
11
12
? ? ?
?
? =
?
?
? ? =
?
t CPR
t CPR
t SMM
t CPR t SMM
(2.68)
Also C(t) will be affected by the introduced uncertainty in WAC(t):
{ } ) ( ) ( )] ( 1 )[ ( ) 1 ( ) ( t SMM t g t SMM t A t B t c + ? ? = , (2.69)
where
);
12
) (
1 ( ) (
;
) 12 / ) ( 1 ( 1
12 / ) (
) (
t WAC
t g
t WAC
t WAC
t A
t WAM
+ =
+ ?
=
+ ?
(2.70)
and
{ }
, ) (
) (
)] ( 1 [
) (
) 1 (
)] ( ) ( )[ 1 (
) (
) ( ) ( )] ( 1 )[ (
) 1 ( ) (
)
`
¹
¹
´
¦
?
?
+ ?
?
?
? +
+ ? ?
?
?
+ + ?
?
? ?
=
?
?
t SMM
t g
t SMM
t A
t B
t g t A t B
t SMM
t SMM t g t SMM t A
t B t c
? ?
? ? ?
(2.71)
where
;
) (
12
1 ) (
;
) (
]
12 / ) ( 1
) 12 / ) ( 1 ( ) ) 12 / ) ( 1 ( 1 (
) (
144
1
) 12 / ) ( 1 ( 1
1
12
1
[
) (
2
? ?
?
?
?
?
=
?
?
?
+
+ ?
+ + ?
+
+ ?
=
?
?
+ ? + ?
+ ?
t WAC t g
t WAC
t WAC
t WAM
t WAC t WAC
t WAC
t WAC
t A
t WAM t WAM
t WAM
(2.72)
40
And the PA gradient estimator for balance B(t) is as the following:
.
) (
12
) 1 ( ) (
)
12
) (
1 (
) 1 ( ) (
); ( )
12
) (
1 )( 1 ( ) (
? ? ? ? ?
? ?
+
?
?
? +
?
? ?
=
?
?
? + ? =
t WAC t B t c t WAC t B t B
t c
t WAC
t B t B
(2.73)
The PA estimator for the discounting factor is unchanged, so we can get the modified
Fourier duration and volatility duration.
41
2.4 Numerical Example
2.4.1 Specification of Numerical Example
We need to specify two sets of data to price the mortgage: the mortgage data and
the interest rate data, which includes the initial term structure and parameters for the
interest rate model.
We price different mortgages to examine the different impacts that a term
structure shift or change in volatility may have on different mortgage products.
The following data are fixed for all products:
Unpaid Balance/UPB =$4,000,000;
WAM =360 months.
Table 2.1 shows the difference between all the products. All the ARM products
have the same subsequent adjustment period of 12 months, period cap/floor of 0.02,
lifetime cap of initial WAC plus 0.06, and no lifetime floor.
Product WAC Index Adjust First
FRM 0.07425 N/A N/A
1 Year ARM 0.06425 Treasury 1 Year 12 month
3/1 FP
1
ARM 0.07425 Treasury 1 Year 36 month
5/1 FPARM 0.07425 Treasury 1 Year 60 month
7/1 FPARM 0.07425 Treasury 1 Year 84 month
10/1 FPARM 0.07425 Treasury 1 Year 120 month
1 Year ARM
2
0.07425 Treasury 10 Year 12 month
Table 2.1 Product Specification for Mortgage Pricing
1
FP ARM refers to Fixed Period ARM, which keep the coupon rate constant for a certain period, and then
adjust periodically, generally once a year. So All the FP ARM products are the same, except different
Adjust First date, which is the first coupon reset date.
2
This ARM is not a mortgage product in the market at present, and is constructed for illustration purpose
only. The following sections will discuss why we introduce this product, and what nice properties it has.
42
We use the same parameters for all the different products in order to have
comparable results. Thus we set all the products to have the same coupon rate, except the
first 1 year ARM with index of Treasury 1 year rate, which has a 100 basis points (bps)
teaser rate. All the ARM products have the same characteristics, except for the Adjust
First date, which is the feature that distinguishes these products.
Our initial term structure is the following:
f(0,t)=ln(150+12t)/100, t=0,1,…,360.
This will produce an upward-sloping curve increasing gradually from 5% to 8.7%
along 30 year maturity, and R(0,t) is acquired by calculating the following:
); 0 , 0 ( ) 0 ( ) 0 , 0 ( ,
) , 0 (
) , 0 (
0
f r R
t
du u f
t R
t
= = =
?
(2.75)
which increases from 5%, to 7.78% gradually.
Our assumptions for interest rate model parameters are the following:
a=0.1; ?=0.1; ?
n
=0.00025, n=0,1,2,3 (used in the FD gradient and gamma
estimator calculation); ??=0.00025, (used in the FD vega estimator calculation).
2.4.2 Comparison of PA and FD gradient estimators
In order to test whether our PA gradient estimators are accurate, and are within
the error tolerance range, we calculate the finite difference (FD) gradient estimators at the
same time during our pricing process. This section will demonstrate the accuracy of our
PA estimators of delta, vega, and gamma for FRM, as well as the delta and gamma for
ARM.
43
Comparison of Modified Fourier Gradient Estimators for FRM
Figure 2.5 shows the FD estimator, PA estimator, their difference, and standard
deviation of their difference for
n
t d
? ?
? ) (
. The four curves in each chart are specified as
following, which will be the convention for the rest of the paper:
Blue: Modified Fourier Order 1;
Green: Modified Fourier Order 2;
Red: Modified Fourier Order 3;
Cyan: Modified Fourier Order 4.
We can see that although these two estimators are pretty close, there exists a
pattern in the difference of these two estimators. This will be explained later in the error
analysis section.
Figure 2.6 shows the PA and FD gradient estimators for cash flow c(t): they are
pretty close, and the difference behaves as random noise. Based on
?
?
?
? ) , (t c
and
?
?
?
? ) , (t d
,
we can calculate
?
?
d
t dPV ) , (
, and figure 5.3 shows us the
n
d
t dPV
?
) (
. Figure 5.4 shows the
95% confidence interval for difference between PA and FD estimators of
n
d
t dPV
?
) (
, and
we can see that 0 is generally contained in the 95% confidence interval.
44
Figure 2.5 Gradient Estimator Comparison for ?d(t)/ ??
n
Figure 2.6 Gradient Estimator Comparison for ?c(t)/ ??
n
45
Figure 2.7 Gradient Estimator Comparison for dPV(t)/ d?
n
Figure 2.8 95% Confidence Interval for dPV(t)/d?
n
46
Comparison of Vega Estimators for FRM
In this section, we also compare the FD and PA estimators for the gradient w.r.t.
interest rate volatility: Vega. Figure 2.9 shows the FD estimator, PA estimator, their
difference, and standard deviation of their difference for
? ?
? ) (t d
. Also there exists a
pattern in the difference of these two estimators. This will also be explained later in the
error analysis section. Figure 2.10 shows the gradient estimators for cash flow c(t): they
are pretty close, and the difference behaves as random noise. Figure 2.11 shows us the
? d
t dPV ) (
, and figure 2.12 shows the 95% confidence interval for
? d
t dPV ) (
, and we can see
that 0 is always contained in the 95% confidence interval.
Figure 2.9 Gradient Estimator Comparison for ?d(t)/??
47
Figure 2.10 Gradient Estimator Comparison for ?c(t)/??
Figure 2.11 Gradient Estimator Comparison for dPV(t)/d?
48
Figure 2.12 95% Confidence Interval for dPV(t)/d?
Comparison of Gamma Estimators for FRM
For gamma estimation, ?=[?
1
?
2
?
3
?
4
]’. So
2
2
) (
? ?
? t d
,
2
2
) (
? ?
? t c
, or
2
2
) (
? ?
? t PV
is a
4x4 matrix. If we want to estimate this matrix by the FD method, we would need 144
points to estimate 48 first order derivatives and to estimate 16 second order derivatives.
The following figures show the FD, PA estimators, the difference and STD of
difference for diagonal gamma elements.
49
Figure 2.13 gamma estimators for
2
2
) (
i
t d
? ?
?
, i=1, 2, 3, 4
Figure 2.14 gamma estimators for
2
2
) (
i
t CPR
? ?
?
, i=1, 2, 3, 4
50
Figure 2.15 gamma estimators for
2
2
) (
i
t CF
? ?
?
, i=1, 2, 3, 4
Figure 2.16 gamma estimators for
2
2
) (
i
t PV
? ?
?
, i=1, 2, 3, 4
51
Comparison of ARM gradient estimators
For ARM products, we basically have the same set of PA gradient estimators to
compare with FD gradient estimators, with one additional set of estimators for
i
t WAC
? ?
? ) (
(figure 2.17). To illustrate the accuracy of our simulation in a brief way, we
only show the FD and PA gradient estimator comparison for one ARM product, 1-Year
ARM with index of 1-Year Treasury rate, adjusted annually.
Figure 2.17 Gradient Estimator Comparison for
i
t WAC
? ?
? ) (
, i=1, 2, 3, 4
Figures 2.18 and 2.19 show the FD/PA gradient estimator comparison for
i
t PV
? ?
? ) (
and
? ?
? ) (t PV
for this ARM product, respectively.
52
Figure 2.18 Gradient Estimator Comparison for
i
t PV
? ?
? ) (
, i=1, 2, 3, 4
Figure 2.19 Gradient Estimator Comparison for
? ?
? ) (t PV
53
2.4.3 Result Analysis
Efficiency Analysis
In financial practice, people are more interested in duration, which is the
percentage change for a security, once there is a minor shift in one parameter, which
mathematically is expressed as
NPV d
dNPV
duration
1 ) (
?
?
= . (2.76)
Actually, there should be a minus sign before the expression, since the original
duration of fixed income securities measures the percentage price drop resulting from an
increase in the interest rate. Yet for our analytical purpose, we do not need the duration
always to be positive, since from the following numbers, we see that durations can also
be negative. Table 2.2 shows the FD and PA durations for FRM30, their 95% confidence
interval, and the error range of the mean.
Fourier Order 0 1 2 3 Vega
PA estimator -6.4816±0.1017 3.1012±0.1860 -0.5705±0.1817 0.6269±0.1189 -6.7567±0.6712
FD estimator -6.4814±0.1017 3.1001±0.1860 -0.5695±0.1816 0.6259±0.1188 -6.7565±0.6712
Absolute Error -0.0002 0.0011 -0.001 0.001 -0.0002
Relative Error 0.0031% 0.0355% 0.1753% 0.1595% 0.0030%
Table 2.2 Comparison of PA/FD Duration
We can see that the error size is very small, and the 95% confidence intervals are
almost the same. Thus from the accuracy point of view, we can use PA estimator to
replace FD estimator without causing too much problem. And the improvement in
computation efficiency is enormous. The FD duration estimator works in the following
way:
54
) (
1
2
) ( ) ( 1
*
) , (
? ?
? ? ? ?
?
?
NPV
NPV NPV
NPV d
t dNPV ? ? ? ? +
= . (2.77)
Thus for each parameter, we need two additional simulations. In our case, we
need 2x5+1=11 simulations to estimate the FD duration. However, by PA estimator, we
only need one simulation. Ignoring the costs of middle steps, and middle variables, we
can reduce the computational time by 10/11, or 90.9%. When we consider the second
order derivative, gamma, the computational efficiency improves even more.
The following table shows the comparison of convexity estimators for FRM.
Convexity is calculated as following:
NPV d
NPV d
convexity
1 ) (
2
2
?
?
= . (2.78)
As we have mentioned earlier, we only estimated part of the FD gamma
estimators, via using the PA delta estimators. Because to fully estimate one set of 25
gamma estimators, we would need to simulate 225 times to get all of them. And each
element is a 360 by 300 (time length by simulation path) matrix.
So from the above analysis, we can see that by the conventional FD method, to
estimate one full set of duration and convexity estimators with 5 free variables, would
require 11 plus 225 simulations. Since we achieve almost the same accuracy by a single
simulation in PA analysis, the simulation cost is reduced roughly by more than 99.5%.
However, we also need to contemplate the introduced costs of intermediate variables as a
tradeoff of the PA method.
55
Convexity = Gamma/Mortgage Value
Mortgage Value = 4.22E+08
FD estimator 0 1 2 3 Vol
0 -246.5944896 N/A N/A N/A N/A
1 N/A -1871.407927 N/A N/A N/A
2 N/A N/A -1854.492905 N/A N/A
3 N/A N/A N/A -2000.544882 N/A
Vol N/A N/A N/A N/A -4751.605032
PA estimator
0 -246.6418706 951.9319609 -646.2296558 161.8535453 919.0969179
1 951.9319609 -1871.360546 1251.356282 -233.9200682 -1435.431523
2 -646.2296558 1251.356282 -1854.208619 1106.51252 1223.425174
3 161.8535453 -233.9200682 1106.51252 -2000.92393 -715.5006989
Vol 919.1916799 -1435.407832 1223.377793 -715.4533179 -4755.158608
Fourier Order 0 1 2 3 Vol
PA estimator -246.5944896 -1871.407927 -1854.492905 -2000.544882 -4751.605032
FD estimator -246.6418706 -1871.360546 -1854.208619 -2000.92393 -4755.158608
Absolute Error 0.047381014 -0.047381014 -0.284286087 0.379048115 3.553576082
Relative Error 0.0192% 0.0025% 0.0153% 0.0189% 0.0748%
Table 2.3 Comparison of Convexity Estimators
We did all the simulations on a Pentium III 800 MHz processor, with 512 MB
memory, in Matlab Release 12.0 under Windows 2000. Here is the simulation
comparison.
Method FD PA
Memory Required 17 MB 54 MB
Simulation Time for 300 paths 115.5 765.8
Number of Duration Measures 5 5
Simulation required for estimating
Duration
11 1
Number of Convexity Measures 25 25
Simulation required for estimating
Convexity
225 1
Total Simulation 236 1
Total Expected Simulation Time 27257.7 765.8
Efficiency Improvement 97.2%
Table 2.4 Comparison of Computing Costs
56
Accuracy Analysis
In order to validate the predictive power of our PA estimator, we setup a test case
to compare the predicted percentage change in the MBS price with the real percentage
change.
The test case is set up as following:
. 5 5
; 3 , 2 , 1 , 0 , 5 5
; )) 1 ( cos( 0 0
3
0
/
0
? = ?
= ? = ?
? ? + =
?
=
?
e
n e
e n ,t) R( ,t) R( Perturbed_
n
n
T t
n
?
?
(2.79)
. 004 - -5.0474e =
?
=
?
NPV
NPV NPV Perturbed_
NPV
NPV
While the predicted change in NPV is calculated as following:
004 - -5.0414e
'
2
1
'
) ( '
2
1
)' (
2
2
=
? × × ? + ? × =
? ×
?
?
× ?
+
? ×
?
?
?
?
? ? ?
?
?
? ?
?
convexity duration
NPV
NPV
NPV
NPV
NPV
NPV
(2.80)
where ??=[?
1
?
2
?
3
?
4
??].
We can see that the relative error by using both duration and convexity measures
is only 0.0056%, while using duration measures only would produce a relative error of
0.1403%. So this test validates the predictive power of our PA gradient estimators. In the
next section, we are going to show that PA estimator not only is more efficient than FD
estimator, but also is a more accurate estimator.
57
Error Analysis
Figure 2.5 and 2.9 show that there exists a pattern in the difference of gradient
estimator of discounting factor d(t). Actually this has two reasons: the calculation of
forward rates f(0,t) and the finite difference estimator of d(t). This could be verified by
figure 2.9, which shows the difference of FD and PA
n
t f
? ?
? ) , 0 (
estimators.
We know that in the Hull-White model, f(0,t) is determined by (2.23). However,
generally we do not have an explicit function form for R(0,t). Instead, we only have
discrete points for term structure, so R(0,t) is estimated by interpolation. And f(0,t) is
further estimated by calculating the difference between adjacent points on R(0,t) as
t
t R
?
? ) , 0 (
, which is not so accurate. The detailed calculation is given below.
Figure 2.20 Difference of FD/PA ?f(0, t)/ ??
n
estimators
58
?
?
?
?
?
?
?
?
?
=
?
? ? ?
< <
?
? ? ? ? +
=
?
? ?
=
?
?
term maximum the is T , ,
) , 0 ( ) , 0 (
0 ,
2
) , 0 ( ) , 0 (
0 ,
) 0 , 0 ( ) , 0 (
) , 0 (
T t
t
t T R T R
T t
t
t t R t t R
t
t
R t R
t
t R
(2.81)
So using FD method to calculate the f(0,t) will result inaccuracy in FD estimator
of
n
t r
? ?
? ) (
, and this will result inaccuracy in d(t). Also we know that d(t) takes the
following form:
} ] ) ( [ exp{ ) (
1
0
t i r t d
t
i
? ? =
?
?
=
, and
t
i r
t d t
i r
t i r
t d
t
i
t
i
t
i
?
?
?
? = ?
?
?
? ? ? =
?
?
? ? ?
?
=
?
=
?
=
)
) (
( ) ( )
) (
( } ] ) ( [ exp{
) (
1
0
1
0
1
0
? ? ?
. (2.82)
Figure 2.21 Fuction of xe
-x
59
However, when we use FD method to estimate the first order derivative of e
-x
, the
FD estimator is always greater in the absolute value, because e
-x
is a convex function. So
FD estimator of
? ?
? ) (t d
is always biased, the bias decreases as the FD step width reduces.
The bias increases linearly, while d(t) decreases exponentially. As a result, the bias takes
the form of xe
-x
. Compare the difference of FD/PA
? ?
? ) (t d
gradient estimators and the
figure of xe
-x
as in Figure 2.14, which resembles the error pattern very closely.
For the PA method,
n
t f
? ?
? ) , 0 (
is estimated by the following formula,
0
/
/ /
) (
)) 1 ( sin( )) 1 ( cos(
) , 0 (
0
0 0
T
e n
e n e n
t f
T t
T t T t
n
?
? ?
? + ? =
? ?
? ?
? ? (2.83)
which does not involve the FD estimation of
t
t R
?
? ) , 0 (
. And
? ?
? ) (t d
is directly estimated
using its analytical form of first order derivative. So the PA estimator is more accurate
than the FD estimator.
60
2.5 Interpretation of the Results
In this section, we briefly present the durations for various mortgage products,
which show different trends for modified Fourier duration of different order. And we try
to interpret how the modified Fourier shocks of different order would affect the
discounting factors and the cash flows, and then the present value (PV) of the mortgage.
Then we analyze the relationship of mortgage prepayment option and mortgage duration.
Based on these analysis, we propose a potential new ARM product, which could reduce
the duration over any of the existing mortgages, while having a less volatile index than
most existing mortgages. This product would benefit both the investors who want to
reduce the interest risk, and the mortgage borrowers who want to have a fairly stable
coupon rate.
2.5.1 Overview of the Results
The following table shows the durations for various ARM and FRM products we
specified and priced in section 2.4:
Fourier Order 0 1 2 3 Vega
ARM TSY 1 -1.7761 -4.313 6.5674 -2.3822 -3.4618
FP 3/1 ARM -2.8441 -2.9642 7.1814 -3.4784 -4.1601
FP 5/1 ARM -3.8514 -1.1609 5.9506 -5.3355 -5.2456
FP 7/1 ARM -4.3054 -0.3064 4.9272 -5.4472 -5.6651
FP 10/1 ARM -5.4256 1.6401 1.7592 -2.6933 -6.6163
FRM30 -6.4816 3.1012 -0.5705 0.6269 -6.7567
Table 2.5 Durations of Different Products
The relation can be better illustrated with the figure 6.1. The zero
th
order modified
Fourier duration (with respect to ?
0
) is the same as Option Adjusted Duration (OAD),
which measures the price percentage change to a parallel interest term structure shift.
Other modified Fourier durations are the same measure, with respect to other interest
61
term structure changes. Vega measures the price percentage change to an interest
volatility change. As we can see, for OAD and Vega, the most important hedge measures,
FRM30 has the highest numbers, and 1-Year ARM has the lowest. For everything
between pure FRM and pure ARM, there exists a monotonic relationship with the
product’s approximation to an FRM30. For example, the Fixed Period 10/1 ARM is more
like an FRM30 than a Fixed Period 7/1 ARM, so it has higher OAD, and higher Vega.
This means that ARM products have a lower interest risk than FRM products,
since an ARM borrower takes more interest risk than an FRM borrower. This result is
consistent with Kau et al.[1990,1992,1993] and Chiang [1997].
However, an interesting phenomenon is that the first order modified Fourier duration
(with respect to ?
1
) actually decreases, and changes sign as volatility of the coupon rate
decreases. This indicates that an opposite move of the long-term and short-term rates
would not only affect ARMs with a different magnitude, but also has a reverse effect
from FRMs. Here is the explanation for this. The first order modified Fourier duration
models the following changes in term structure:
• Short-term rate increases;
• Intermediate term rate (e.g. 10 year rate) doesn’t change, or moves only a little bit;
• Long-term rate decreases.
In this scenario, people with a short-term ARM, e.g. 1-year ARM are burnt the
hardest, so they are going to refinance anyway, even if the prevailing mortgage rate does
not change a lot. This will create huge prepayment, and reduce the NPV of the ARM
mortgage. People with FRM, on the other hand, have no incentive to refinance, since the
refinance mortgage rate (highly correlated with 10-Year Treasury rate) does not change a
62
lot. This will make the future cash flow more stable and valuable, since they are
discounted at a lower long-term interest rate, and increase the NPV of the FRM
mortgage.
Figure 2.22 Duration vs. Products
The above analysis is based mainly on intuition, and does not show how will this
term structure shock affect the discounting factors, cash flows, and NPV of MBS. In the
following section, we will see what effect each one of the modified Fourier functions has
on these components of MBS for various mortgage products.
2.5.2 Modified Fourier Shock Impact
The following 8 charts will show different modified Fourier shocks on term
structure R(0,t), and their impact on d(t), CF(t), and PV(t).
Duration vs. Products
-8
-6
-4
-2
0
2
4
6
8
A
R
M
T
S
Y
1
F
P
3
/
1
A
R
M
F
P
5
/
1
A
R
M
F
P
7
/
1
A
R
M
F
P
1
0
/
1
A
R
M
F
R
M
3
0
Products
D
u
r
a
t
i
o
n
Harm Duration 0
Harm Duration 1
Harm Duration 2
Harm Duration 3
Vega
63
Figure 2.23 The Impact of Modified Fourier Function Order 0 on FRM30
Explanation: A parallel shift in the upward slope term structure will have a negative
impact on the discounting factor. Also people are less likely to prepay in the near future,
which reduces the cash flow in the short term, and increase the cash flow in the long term
a little bit. However, the overall effect of such a shift on present value is negative, and
thus reduces the NPV of this MBS.
64
Figure 2.24 The Impact of Modified Fourier Order 1 on FRM30
Explanation: A shift of this shape in the upward slope term structure will have a mixed
impact on the discounting factor: decrease it in the short term, but increase it in the long
term. Also people are more likely to prepay in the near future, which increases the cash
flow in the short term, and reduces the cash flow in the long term a little bit. However,
the overall effect of such a shift on present value is positive, and thus increases the NPV
of this MBS.
65
Figure 2.25 The Impact of Modified Fourier Order 2 on FRM30
Explanation: A shift of this shape in the upward slope term structure will have a mixed
impact on the discounting factor: increase it in the middle term, but decrease it in the long
term. There is little incentive for people to prepay in the near future, and they will also
cling to their current coupon rate in the middle term, because at that time the refinance
rate will increase. However, the overall effect of such a shift on present value is cancelled
out, and has little impact on the NPV of this MBS.
66
Figure 2.26 The Impact of Modified Fourier Order 3 on FRM30
Explanation: same as Modified Fourier Order 2
67
Figure 2.27 The Impact of Modified Fourier Order 0 on ARM TSY 1
Explanation: A parallel shift in the upward slope term structure will have a negative
impact on the discounting factor. Also people with ARM are less likely to prepay in the
near future, because they have a lower ARM rate than the refinance. Yet they will start
prepay in the middle term, because short term rate at that time will increase, due to the
upward slop term structure. This behavior will reduce the cash flow in the short term, and
increase the cash flow in the middle term. However, the overall effect of such a shift on
present value is negative, and thus reduces the NPV of this MBS. Yet the impact will be
much smaller than that for FRM.
68
Figure 2.28 The Impact of Modified Fourier Order 1 on ARM TSY 1
Explanation: A shift of this shape in the upward slope term structure will have a mixed
impact on the discounting factor: decrease it in the short term, but increase it in the long
term. Also people are more likely to prepay in the near future, which increase the cash
flow in the short term, and reduce the cash flow in the long term a little bit. The overall
effect of such a shift on present value is negative, and thus decreases the NPV of this
MBS.
69
Figure 2.29 The Impact of Modified Fourier Order 2 on ARM TSY 1
Explanation: A shift of this shape in the upward slope term structure will have a mixed
impact on the discounting factor: increase it in the middle term, but decrease it in the long
term. People will cling to their low ARM rate for the first few years, but then start to
prepay in the middle term, since short term rate will increase at that time. The overall
effect of such a shift on present value is positive, due to the increase cash flow and
discounting factor in the middle term.
70
Figure 2.30 The Impact of Modified Fourier Order 3 on ARM TSY 1
Explanation: much like Modified Fourier Order 3, yet because the reverse effect of
discounting factor, the overall effect will be negative.
6.3 Potential New ARM Product
Duration is used to measure the interest risk of a fixed income security. The
higher the duration is, the more interest risk that security bears. From the investor’s
perspective, she will benefit if interest rates fall, and suffer if interest rates climb, if the
security is non-callable (no prepayment option). From the mortgage borrower’s point of
view, he will exercise his prepayment option if interest rates drop, and thus reduce the
benefit for the investor. He will be able to lock in the low mortgage rate (for FRM), in
case interest rates climb, and thus hurt the investor more. However, for the ARM
borrower, he benefits from the rate drop, so he does not prepay like the FRM; thus the
71
MBS investor will also benefit. And he also pays the high coupon rate when interest rates
increase, and the ARM MBS investor will not suffer like the FRM MBS investors. From
this perspective, the ARM should have a lower duration compared to FRM.
ARM borrower’s coupon rate fluctuates with the current interest rate, which is
correlated with the prevailing mortgage rate. Because of this, she will have less incentive
to prepay when interest rate drops. So the prepayment option value for a FRM borrower
will be larger than that of an ARM borrower. This is compatible with the market, where
FRM mortgages are sold with the highest rate (borrower pays for the valuable
prepayment option), and ARM, that adjust most frequently are offered with the lowest
rate.
In option theory, we know that option value generally increases as the volatility of
underlying asset increases. However, from the above analysis, we also know that the
option value for a FRM is generally greater than for an ARM, while an ARM bears a
more volatile coupon rate than a FRM. This looks like a contradiction to the option-
volatility relationship. In fact, it’s not, because the underlying asset of a prepayment
option is not its coupon rate, but the difference between the coupon rate and the
prevailing mortgage rate. In most cases, the more volatile the coupon rate is, the less the
difference will be, and the less valuable the option will be. However, the borrower does
not like the volatility, which put her at risk when interest rate jumps. The investor, on the
other hand, does not like the prepayment, which reduce her investment value. It seems
that no product can both reduce the coupon rate volatility and the prepayment option at
the same time. Is this true? We will see that we can achieve both goals in a potential new
ARM product.
72
We have mentioned that the underlying asset for the prepayment function is the
spread between the coupon rate and the prevailing mortgage rate. So an ARM bearing a
volatile index does not necessarily indicate a less volatile spread. From historical data, we
know that 10-Year Treasury rate is highly correlated with conventional (FRM30)
mortgage rate. Figure 6.10 shows the two rates for the period between 1971 and 2001.
The correlation calculated is 97.9%. Figure 6.10 also shows the 10-Year Treasury Rate
and the 1-Year Treasury rate, which is the most commonly used index in ARM. As we
can see, the 1-Year Treasury rate is relatively more volatile than the 10-Year Treasury
Rate. The calculated standard deviation is 2.7890 for the 1-Year Treasury rate, and
2.6309 for the 10-Year Treasury rate. However, the standard deviation of spread of
FRM30 vs. TSY10 is 0.58, compared with the standard deviation of spread of FRM30 vs.
TSY1 at 1.16. Obviously 10-Year Treasury rate has a lower volatility and also a lower
volatile spread. The spread between conventional mortgage rate (FRM30) and 10-Year
Treasury rate and the spread between conventional mortgage rate (FRM30) and 1-Year
Treasury rate are also shown, which indicates that ARM with index of 1-Year Treasury
rate has a more volatile spread.
73
Figure 2.31 10-Year T Rate, 1-Year T Rate, and mortgage rate
Thus if we construct an ARM with index of 10-Year Treasury rate, and reset it
more frequently, we could expect a lower duration. So we construct such an ARM with
the adjustment period of 12 months. This ARM does not exist at present; it is for
illustration purposes only. We then got the modified Fourier duration measures as
following:
Fourier Order 0 1 2 3 Vega
ARM TSY 10 -1.2741 -4.0635 3.1819 -0.4894 -1.8855
Figure 2.32 shows the new ARM product’s duration against duration of other
mortgage products we calculated earlier. We compare this set of durations with table 6.1,
74
and we can see that this product has the smallest durations for modified Fourier function
order 0 and 3, as well as for vega. The durations for modified Fourier function order 1
and 2 are not very high. And we know that generally when there is a shock on the term
structure, the biggest magnitude would be that of the first-order modified Fourier
function, and volatility is also a big impact. So this product would actually have the least
percentage change during a common term structure shift, which satisfies the needs of
investors.
Duration vs. Products
-10
-8
-6
-4
-2
0
2
4
6
8
A
R
M
T
S
Y
1
0
A
R
M
T
S
Y
1
F
P
3
/
1
A
R
M
F
P
5
/
1
A
R
M
F
P
7
/
1
A
R
M
F
P
1
0
/
1
A
R
M
F
R
M
3
0
Products
D
u
r
a
t
i
o
n
Harm Duration 0
Harm Duration 1
Harm Duration 2
Harm Duration 3
Vega
Figure 2.32 New ARM TSY 10 Durations
So we could predict that if there exist such a mortgage, it would have the least
refinancing incentive, which would be a better product to suit investors’ needs, and it will
also have a less volatile index, which suits borrowers’ needs.
75
2.6 Conclusion
This paper applies perturbation analysis (PA) method to estimate MBS
sensitivities. The sensitivity estimators include most interest risk measures like duration
(equivalent to delta), convexity (equivalent to gamma), and vega. MBS products covered
includes fixed rate mortgages (FRMs) and adjustable rate mortgages (ARMs).
We first derive a general framework to derive the PA estimators of MBS, without
restriction to MBS type, interest rate model, or prepayment model. Then we apply the PA
estimator to both FRM and ARM products, in the setup of a one-factor Hull-White model
and a commonly used prepayment model. We compare the PA estimators with finite
difference (FD) estimators, and find that PA method can achieve at least the same
accuracy as FD method, with a much lower computational cost. In the case we presented,
the computational time is reduced by 95.7%, while the memory requirement increases
only by a factor of 3, which can be handled by current computer technology with ease.
Then we analyze the results of PA estimated sensitivity measures for various MBS
products. We justify why and how different term structure shock would affect FRM and
ARM differently. Based these analysis, we propose a potential new ARM product which
could benefit both the MBS investor and the mortgage borrower.
Future research includes applying this method to other MBS-like securities, since
the PA method proposed in section 3 is a very general framework. These include other
asset-backed securities, e.g. securities backed by student loans, car loans, credit card
receivables. It is pretty straightforward to expand this framework to those securities, since
all that is required is to apply a specific interest rate model and prepayment model.
76
Another area for further research is to incorporate more complicated prepayment
and/or default models into the MBS pricing scheme. For MBS investors, the major
concerns are price sensitivities to interest changes, which we have covered in detail.
However, the MBS guarantor/insurer and issuer might have other concerns, e.g., how will
the interest rate change affect the default behavior of the mortgage borrowers? Our
framework would be able to serve this purpose as well. By applying the default model
that same way as we apply a prepayment model, the default cash flow will take the place
of payment cash flow, so the default cost sensitivities could be easily estimated.
77
Chapter 3
Hedging MBS in HJM Framework
3.1 Motivation
As we have pointed out in our first essay, short term rate and long term rate do not
always move in the same direction, it is sometimes misleading to use the conventional
interest rate risk measures like duration and convexity to hedge fixed income instruments,
especially MBS.
One recent event can illustrate this point very well. In July 2003, Federal Reserve
lowered the short term interest rate by another 25 bps, yet just in one month, the long
term 10 year rate jumped upward for more than 100 bps. Part of the reason is that the rate
deduction is lower than market expectation, and market responded with a selling wave in
the bond section. So using a duration measure, which assumes the yield curve moves in
parallel, will produce significant hedging error.
It is natural to hedge against the factors of which any yield curve shift can be
decomposed. We use a series of exponentially decaying modified Fourier series to
approximate any interest rate change in our first essay. However, this is purely for the
generality of modeling convenience, and there is no empirical evidence that such a series
provides a good match of the actual yield curve shift.
78
A lot of literature studying the dynamics of interest rates found that there are three
major factors affecting the yield curve: level, slope, and curvature. A common method to
estimate these factors is Principal Components Analysis (PCA). See details in Litterman
and Scheikman [1991], Litterman, Scheikman, and Weiss [1991), Knez, Litterman, and
Scheikman [1994], Nunes and Webber [1997]. Despite the abundance of research on
identifying the various factors affecting bond prices, there has been little research on
hedging these factors effectively. Golub and Tilman [1999] compared different risk
measures, like PCA, VaR, and key rate duration for yield curve risk, but did not give
hedging performance for these different measures.
In mortgage industry, practitioners generally use effective duration, and empirical
duration in hedging. Goodman and Ho [1999] examined the performance of three
different hedge ratios: effective duration, empirical duration, and option-implied duration,
which is acquired from forward option for a given pass-through MBS. They found that
the average hedging error for a monthly hedge could reach 120 bps in an 18-month
period. And for a daily hedging, it is 25 bps in the same time period. They concluded that
option-implied duration performs the best. However, it does not always outperform the
other hedging measure, and the difference is small. Hayre and Chang [1999] compared
effective duration and empirical duration, and found that effective duration calculated
from OAS model are generally longer than empirical duration, and they challenged a few
assumptions for effective duration calculation in OAS mode. To cite a few, the parallel
yield curve shifts, absence of convexity, etc. These are also issues we addressed in our
79
paper, but in a more systematic way. They proposed a combined duration, which is a
effective duration adjusted for correlations between changes in the yield and prices of
MBS in recent market data, i.e., a combination of effective duration and empirical
duration.
There has not been a unifying framework in hedging MBS with factors affecting
the yield curve shifts, and we would like to pursue in this direction, since we are pretty
confident about its effectiveness in reducing the hedging error, and/or reducing hedging
frequency. In order to incorporate these factors into MBS hedging strategy, we need to
choose an interest rate model, which can handle these factors readily. HJM model is a
good choice, because it is basically driven by volatility structure, and the volatility factors
can take any shape, which easily accommodate the PCA factors we identified from
historical data.
In the rest of this essay, we discuss how to get volatility factors from historical
interest rate data via the PCA method. In section 3, we are give the detailed
implementation of HJM model with these estimated volatility factors. Then we derive the
PA estimators for hedging MBS, which is very similar to Chapter 2, and we will not go
into details to derive PA estimator for each state variable. In section 5, we give the
detailed hedging algorithm with these hedging measures, and we discuss the performance
of our hedging method in section 6. Section 7 concludes the essay, and gives potential
future research directions.
80
3.2 Estimation of Volatility Factors via PCA
The Principal Components Analysis method is generally used to find the
explanatory factors that maximize successive contributions to the variance, effectively
explaining variations as a diagonal matrix. This method has been used in yield curve
analysis for more than 10 years, see Litterman and Scheinkman [1991], Steeley [1990],
Carverhill and Strickland [1992]. Here we give a brief description of PCA method
applied in yield curve analysis:
1. Suppose we have observation of interest rates ) (
j t
i
r ? at time t
i
, i=1, 2, …, n+1, for
different maturity dates ?
j
.
2. Calculate the difference ) ( ) (
1
, j t j t j i
i i
r r d ? ? ? =
+
, where the d
i,j
are regarded as
observations of a random variable, d
j
, that measures the successive variations in the
term structure.
3. Find the covariance matrix ) ,..., cov(
1 k
d d = ? . Write
) , cov( where }, {
, , j i j i j i
d d = ? ? = ? .
4. Find an orthogonal matrix P such that P’=P
-1
and
k k
? ... ? ) , ..., ? diag(? P P ? ? = ?
1 1
where , ' .
5. The column vectors of P are the principal components.
6. Using P, each observation of d
j
can be decomposed into a linear combination of the
principal components. By setting
j i i
d p e ' = , where p
i
is the i
th
column of P, we can
find e
i
, which is the corresponding coefficient for principal component i, i=1, …, k. A
81
small change in e
i
will cause the term structure to alter by a multiple of p
i
along the
time horizon.
We use the weekly data of nominal zero coupon yield from January 1997 to
October 2001 as the term structure data. All data were retrieved from Professor
McLulloch’s web site at the Department of Economics, Ohio State University, at
<http://econ.ohio-state.edu/jhm/ts/ts.html>. For each observation
date, interest rates are provided for maturities in monthly increments from the
instantaneous rate to the 40-year rate, providing a total of 481 interest rates as principal
components. Table 3.1 lists the eigen-values and % variance explained by the first ten
factors, and Figure 3.1 graphs the shapes of the first four factors.
Factor Eigenvalue Explained(%) Cumulative(%)
1 16.38 75.824 75.824
2 4.41 20.432 96.257
3 0.72 3.335 99.592
4 0.087 0.40 99.995
5 0.00088 0.0041 99.999
6 8.67E-05 0.00040 99.9996
7 1.59E-05 7.4E-05 99.99966
8 4.20E-06 1.9E-05 99.99968
9 4.03E-06 1.9E-05 99.99970
10 3.67E-06 1.7E-05 99.99972
Table 3.1. Statistics for Principal Components
82
Figure 3.1 The first four principal components
The statistics indicate that the first three factors explain about 99.6% of the yield
curve changes, and the first four factors explain about 99.995% of the total variance of
yield curve. These results are similar to findings by Litterman and Scheikman [1991], and
Nunes and Webber [1997]. Figures 3.2 and 3.3 plot the matching results with three and
four factors, respectively, for a monthly yield curve shift, as well as for an annual shift.
The figures indicate that four factors provide a substantially improved match, both for the
short term and the long term, over three factors, so in our model we will use four factors.
Thus, hedging against these factors will lead to a considerably more stable portfolio,
thereby reducing hedging transactions and its associated costs.
83
Figure 3.2 Match monthly yield curve shift
Figure 3.3 Match annual yield curve shift
84
3.3 Simulation in HJM Framework
This section gives the detailed implementation of HJM model, using the volatility
factors identified in PCA analysis.
We know that, in a multifactor HJM framework, the dynamics of instantaneous
forward rate looks like:
?
=
? + ? =
N
k
k t k t
t dZ T t dt T t m T t df
1
) ( ) , , ( ) , , ( ) , ( ? , (3.1)
where under no arbitrage assumption, the drift term is determined by volatility structure.
?
?
=
? ? = ?
N
k
T
t
t k t k t
d t T t dt T t m
1
) , , ( ) , , ( ) , , ( ? ? ? ? . (3.2)
Assume our volatility functions take the following form:
) , ( ) , , ( T t PC T t
k k t k
? ? = ? , (3.3)
where PC
i
(t, T) is the principal components we get in last section;
?
i
is a parameter to be calibrated to market price of interest rate derivatives.
Detailed Implementation:
1. Input data include f(0,T), the instantaneous forward curve, and ?
k
(t, T), which
has a specified functional form fitting into our PCA factors.
2. Start loop for maturity, if we need 10 year rate for 30 years, we need maturity at
40 year;
3. Start of time step loop;
4. Start of ? loop, to calculate ?
k
(t, ?) from t to T;
85
5. Calculate ?
k
*
(t, T)=
?
T
t
k
d t ? ? ? ) , ( , using numerical integration technique;
6. Calculate m(t, T)= ?
k
(t, T)* ?
k
*
(t, T);
7. Advance f(t, T) one more step, in our simulation, one month increment:
; ) , ( ) , ( ) , (
, ?
+ ? = ? +
k
k t k
z T t t T t m T t t f ?
where z
t,k
is a series of independent standard normal random variables.
8. End of time step loop;
9. Short rate r(t)=f(t,t); Long rate r
10
(t)=
10
) , (
10
?
+ t
t
d t f ? ?
;
10. End of maturity step loop.
86
3.4 Deriving PA estimators in HJM Framework
Following the logic in Chapter 2, we only need to derive the PA estimator for
short rate r(t) and 10-year rate r
10
(t), since our prepayment model and valuation model are
totally dependent on these two factors.
If we assume that in a short period of time, the principal components for yield
curve volatility are going to be constant, then any interest rate yield curve shift can be
decomposed of these principal components, which is to say:
?
? = ?
k
k k
t PC t R ) ( ) , 0 ( , (3.4)
which is analogous to (2.35). Following the same logics as in (2.51), we can have the
following:
). (
) ( ) , 0 (
,
) , 0 ( ) , 0 (
) , 0 ( ) , 0 ( ) , 0 (
) , 0 (
) , 0 (
) , 0 (
2
2
t PC
t
t PC
t
t f
t R
t
t R
t
t R
t
t R
t
t
t R t
t R t
t R
t
t f
k
k
k
+
?
?
=
? ?
?
?
?
+
? ?
?
=
?
?
+
? ?
?
+
?
?
?
?
=
?
?
+
?
|
.
|
\
|
?
?
?
=
?
?
? ?
? ? ?
? ? ?
(3.5)
We know that in HJM framework:
) , ( ) ( t t f t r = , (3.6)
So
k k
t t f t r
? ?
?
=
? ?
? ) , ( ) (
. (3.7)
87
We also know:
?
=
? + ? =
N
i
i t i t
t dZ T t dt T t m T t df
1
) ( ) , , ( ) , , ( ) , ( ? . (3.8)
When T=t+dt
dt) df(t,d dt) f(t,t dt) dt,t f(t + + + = + + . (3.9)
So
k k k
dt) df(t,d dt) f(t,t dt) dt,t f(t
? ?
+ ?
+
? ?
+ ?
=
? ?
+ + ? ] [
, (3.10)
?
=
? ?
+ ?
+
? ?
+ ?
=
? ?
+ ?
N
i
i
k
i
k k
t dZ
dt t t dt t t m dt) df(t,d
1
) (
) , ( ) , ( ] [ ?
. (3.11)
If we rewrite the drift term as:
) , ( * ) , ( ) , (
*
1
T t T t T t m
k
N
k
k
? ?
?
=
= , (3.12)
where
? ? ? ? d t T t
T
t
k k
?
= ) , ( ) , (
*
. (3.13)
Then gradient of m(t, T) can be written as
?
=
¦
)
¦
`
¹
¦
¹
¦
´
¦
? ?
?
+
? ?
?
=
? ?
?
N
k k
i
i i
k
i
k
T t
T t T t
T t T t m
1
*
*
) , (
* ) , ( ) , ( *
) , ( ) , ( ?
? ?
?
. (3.14)
From the form of ) , ( T t
k
?
) , ( ) , , ( T t PC T t
k k t k
? ? = ? . (3.15)
88
There is no direct relation ship between ) , ( T t
i
? and
k
?
1
, so the above gradients are
zero. This gives us
k k k
dt) ,d f( dt) f(t,t dt) dt,t f(t
? ?
+ ?
= =
? ?
+ ?
=
? ?
+ + ? ] 0 [
... . (3.16)
For the same reason, we can derive the 10 year rate gradient as:
10
) , (
) (
10
10
?
+
? ?
?
=
? ?
?
t
t
k
k
d
t f
t r
?
?
. (3.17)
And follow the same logic, we can get the gradients of discounting factors, prepayment
rate, cash flows, present values, etc.
1
Although observed
k
? and
k
? might have a positive correlation, i.e., when the volatility is high, the
observed shift also might have bigger magnitude. But they have total different meaning,
k
? is the
parameter to calibrate to market price, and
k
? is the observed shift in yield curve.
89
3.5 Hedging MBS in HJM Framework
This section gives a detailed implementation of our hedging algorithm.
Security to be hedged: MBS
Hedging Instruments: Portfolio of {MBS, Treasury bonds with different maturities}
Hedging Method: Dynamic hedging using PCA duration. vs. Conventional duration and
convexity hedging
Hedging Parameters: PCA duration
Hedging Error: The net present value of the portfolio, which has initial value of zero
Hedging Efficiency: Reduce hedging Error
Hedging Strategy: Construct a portfolio, consisting of MBS and various T-notes, bonds,
with 0 face value. Duration matched to 0. Rebalance at each time period to match the
hedging parameters; compare the results with duration and convexity hedging.
There are two issues we need to pay special attention to, in order to effectively execute
the hedging strategy.
Issue 1: With the coupon payment and prepayment of MBS, what needs to be done with
this extra cash flow?
Answer: Use this cash flow to rebalance the portfolio, basically to change the weights of
Treasury bonds holdings. If the position is short in MBS, and long in Treasury bonds, we
need to sell the Treasury bonds to honor the MBS payment.
Issue 2: Some Treasury bonds used to hedging the MBS will expire before the MBS
maturity date. This will hurt the capacity of available hedging instruments.
90
Answer: We only hedge the MBS for a short period of time, e.g. 3 years, and then we can
use Treasury bonds with greater or equal to 3 years maturities. Another solution is to
introduce on extra hedging instrument when there is one expiring at that period.
Hedging Framework
1. At time 0, get the MBS price, gradients (PCA duration) by simulation (360x300
simulation needed). Zero coupon Treasury bonds price and gradients should be
directly available from the yield curve, and the PCA factors;
2. Construct the portfolio, by shorting MBS to finance Treasury bonds; match the
duration, and get the corresponding weights;
3. At time 1, use HJM model to update the yield curve, then get the new price and
gradients of MBS as well as those of Treasury bonds;
4. Use MBS payment to rebalance portfolio (MBS payment is deterministic upon the
last period yield curve);
5. Repeat 3, 4 for next month, till the end of hedging period;
6. Check the effectiveness of hedging strategy.
Implementation of Hedging MBS with Treasury Bonds
1. Get mortgage information;
2. Get historical yield curve data;
3. Get Principal Components Factors;
4. Start clock for hedging period: m=0
5. Calculate MBS_Price(m), MBS_Duration(m)
4x1
, Payment(m),
PrincipalPayment(m);
91
6. Choose hedging instrument: Treas_Portfolio=[12 36 60 84 120], each element
represent months to maturity;
7. Calculate Treasury bond price Treas_Price(m)
5x1
, 5 hedging components are
needed because of 5 factors to hedge: Price, and Duration
4x1
.
Treas_Duration(m)
4x5
.
8. Solving for hedging ratio W(m):
1 5
1 5
5 5
) ( _
) ( _
) (
) ( _
)' ( _
x
x
x
m Duration MBS
m price MBS
m W
m Duration Treas
m price Treas
(
¸
(
¸
=
(
¸
(
¸
, if m=0; (3.18)
1 5
1 5
5 5
) ( _
) 1 ( _ ) 1 ( )' ( _
) (
) ( _
)' ( _
x
x
x
m Duration MBS
m payment MBS m W m price Treas
m W
m Duration Treas
m price Treas
(
¸
(
¸
? ? ?
=
(
¸
(
¸
, if m>0. (3.19)
9. Calculate hedging error:
) 1 ( _ ) 1 ( )' ( _ ) ( _ ) ( ? + ? ? = m payment MBS m W m price Treas m price MBS m error
10. Update loan.UPB=loan.UPB-PrincipalPayment(m);
11. Update loan.WAM=loan.WAM-1/12;
12. Update Treas_Portfolio=Treas_Portfolio-1;
13. m=m+1, go back to 5 until hedging period ends.
92
3.6 Hedging Performance Analysis
In this section, we compare the hedging performance of our PCA-based hedging
and traditional duration and convexity based hedging for a FRM30 MBS instrument.
The principal balance of the MBS is $4 million. We are selling short this MBS at
the market price, and use the proceeds to buy treasury bonds. Initial net present value of
the hedging portfolio is zero. Every month, we try to rebalance the portfolio, and we sell
part of our bonds to meet the payment obligation of the MBS. Hedging error is defined as
the net present value of current portfolio at each time point.
We carry on this practice for 22 months, during which our PCA estimation does
not change dramatically. We repeat the hedging practice for 25 simulations, which is
relatively few, because the simulation scheme takes an extremely long time. The PCA-
based hedging takes around 40 CPU hours to finish, while the duration and convexity
based hedging takes 120 CPU hours to complete the task.
Figure 3.4 shows the hedging performance of three PCA factors, while Figure 3.5
shows the hedging performance of duration and convexity hedging. We can see that the
standard deviation of PCA-based hedging ranges from $4000 to $20000, which is 10 bps
to 50 bps for a $4 million portfolio. Consider the standard deviation of duration and
convexity based hedging, which ranges from $60000 to $200000, i.e. 150 bps to 500 bps
of the hedging balance. The hedging improvement is obvious.
93
Mean Hedging Error
-5.00E+04
-4.00E+04
-3.00E+04
-2.00E+04
-1.00E+04
0.00E+00
1.00E+04
2.00E+04
3.00E+04
0 5 10 15 20 25
month
H
e
d
g
i
n
g
E
r
r
o
r
mean_PCA
mean_D&C
Figure 3.4 Mean Hedging Error of PCA vs. D&C
Standard Deviation of Hedging Error
0
50000
100000
150000
200000
250000
300000
0 5 10 15 20 25
month
H
e
d
g
i
n
g
E
r
r
o
r
std_PCA
std_D&C
Figure 3.5 STD of Hedging Error: PCA vs. D&C
94
3.7 Conclusion
In this essay, we proposed a new method to hedge the interest risk of MBS, based
on PCA factors estimated from historical interest rate data. We estimated the PA
estimators for hedging MBS, and implemented the hedging with a dynamically re-
balancing portfolio of MBS and Treasury bonds. We achieved much better hedging
efficiency, compared with traditional hedging, not only in the measure of mean hedging
error, but also in the standard deviation of hedging error. We made the following
contribution:
• A unified hedging framework for hedging yield curve shift and volatility factors;
• Improved hedging efficiency compared with traditional duration and convexity
based hedging. Our monthly hedging get very close results to daily hedging with
traditional hedging method.
We would like to pursue in the following directions for our future research:
• Apply this hedging method to more sophisticated prepayment models, and
analyze the robustness of this hedging algorithm;
• Improve computational efficiency of the algorithm, which is now very time
consuming.
95
Chapter 4
Hedging the Credit Risk of MBS: A Random
Coefficient Approach
4.1 Motivation
In our previous two chapters, we have assumed that the credit risk of the MBS is
totally absorbed by the MBS issuer, and the MBS investor only needs to hedge the
interest rate risk due to voluntary prepayment, including housing turnover and
refinancing. This assumption is reasonable since in the secondary market for conforming
mortgages, the three major MBS issuers, Ginnie Mae, Fannie Mae, Freddie Mac
1
, all
promise that they will guarantee the principal payment when there is a default event
incurred on the mortgage borrower’s side. The MBS issuers have the following methods
to mitigate the credit risk:
• Mortgage Collateral: Basically when a default occurs, the collateral property will
become REO(Real Estate Owned), and the issuer can foreclose the mortgage and sell
the property, and recover whatever is left;
• Primary Mortgage Insurance (PMI): If a borrower initiate a loan with LTV greater
than 80%, she will be required to purchase mortgage insurance. If default occurs, the
mortgage insurance company pays the owner of the mortgage whatever is promised in
the insurance contract, generally 35% for a 95 LTV loan, and 20% for a 85 LTV loan;
1
However, the credit risks of these agencies are different. Ginnie Mae is guaranteed by the full faith and
credit of the United States government. Both Fannie Mae and Freddie Mac have $2.2 billion line of credit
with the Treasury department. Also they receive an implicit guarantee from the government, since most
96
• Credit Enhancement: The MBS issuer can purchase additional insurance from a
mortgage insurance company for a mortgage pool. This deal is also called pool
insurance, or backend credit enhancement. It is not necessarily purchased from the
same company that provides PMI in the mortgage pool. There is generally an auction
among several insurance companies, and the bidder with the most competitive price
will be awarded the contract.
When hedging the credit risk of the MBS with credit enhancement from a third
party, the issuer is now exposed to the credit risk of the counter party. In order to hedge
the credit risks effectively and efficiently, we not only need to model the default behavior
of the mortgage borrower, but also need to understand the credit worthiness of the
counter party. The credit worthiness of a given counter party for a given time horizon is
generally called a haircut
2
. We need to model the haircuts of the counter party to perform
the following tasks:
• Calculate the insurance premium, i.e., the purchase price for the insurance policy,
to be paid. Apparently, a company with lower credit risk should be charging
higher fees, and vice versa, since lower credit risk means better insurance policy.
• Estimate the credit loss, and report it to external investors and regulators.
Currently the Office of Federal Housing Enterprise Oversight (OFHEO), regulator
of Fannie Mae and Freddie Mac, requires both GSEs to report their risk-based
market participants believe that federal government will interfere whenever any of these two giant GSEs
steps in financial distress.
2
This term is used to determine the reduction applied to promised payment, due to credit risk, e.g., a 25%
haircut means that the promised payment needs to be reduced to 75%.
97
capital calculated by pre-specified haircuts for different rated counter parties.
With the implementation of Basel Accord II
3
, internal credit risk models could be
used to calculate the haircuts, and in-house model for calculating the counter
party credit risk is of extreme importance in reporting the credit risks.
We below show that a haircut is actually a credit risk measure similar to credit
spread. And estimation of a haircut is equivalent to estimation of credit spread. Suppose
we need to take the haircut H(t) for a promised future payment of $1, what would be the
price for this promised payment? In risk neutral probability, the price should be:
P=exp(-r(0,t)*t)[1-H(t)]
Where r(0,t) is the spot rate for maturity t.
If the promised payment can be viewed as a zero-coupon defaultable bond with face
value of $1, its price is given by
P=exp{-[r(0,t)+CS(t)]*t}, where CS(t) is the credit spread for maturity t.
Apparently haircut and credit spread have the following one-to-one relationship:
H(t)=1-exp(-CS(t)*t)
Once we estimated the credit spread of the third party’s defaultable bonds, we will
get the haircut we need to impose on the insurance contract automatically. So it is of
critical importance that we have a good estimation for the credit spread changes of the
counter party. There are generally two ways to model the dynamics of the credit spread:
3
Basel Accord II is the new international banking regulation rule proposed by Basel Committee, which will
be implemented before 2006. It gives more flexibility in treating credit risk, and internal credit risk models
can be used in calculating risk-based capital requirement, which is the capital a financial institution needs
to reserve in order to alleviate the credit risk exposure to counter parties.
98
theoretical approach and empirical approach. There has been a lot of published work on
the literature on the theoretical part of credit risk modeling: either using structural models
(Merton [1974], Longstaff and Schwartz [1995], Collin-Dufresne and Goldstein [2001]),
or reduced form (hazard rate) models (Duffie and Singlton [1999], Madan and Unal
[2000]). In empirical work, different models are estimated and fitted with market data,
and the performances of these models are compared in recent papers, e.g. Eom, Helwege,
and Huang [2003]. Recently there has been interest in using regression to determine the
factors affecting credit spread changes, because neither structural nor reduced form
models can handle the large number of factors affecting credit spread changes. With a
flourishing credit derivative market, there is a great need for identifying the factors that
affect credit spread, in order to find possible financial instruments to hedge credit
derivatives written on credit spreads.
The main model used in these researches is the simple linear regression model,
e.g., Duffee [1998], Collin-Dufresne, Goldstein and Martin [2001], Huang and Kong
[2003]. However, these models generally do not offer very compelling results. In this
essay, we identify the theoretical drawbacks of this type of models, and address these
problems with a new approach: the Random Coefficient Regression (RCR) model, which
we can handle the non-constancy phenomena of credit spread sensitivities.
The rest of this essay is organized as follows. We first give a literature review in
the following section; specifically we are going to discuss several important papers. In
section 3, we introduce the random coefficient regression (RCR) model is given and then
99
apply the model to estimate the dynamics of credit spread changes, using variables from
the simplest structural model. Description of the data is given in section 5, and the
regression results are discussed in the next section. We show that our assumption about
non-linearity and non-constancy of credit spread changes are well supported by the
regression results, also the regression results are consistent with theoretical structural
models, such as Merton [1974]. In the last part of this essay, we give conclusions and
possible future research directions.
100
4.2 Literature Review
There has been a lot of recent interest on identifying the key factors affecting
credit spread. One approach is to add macroeconomic variables into the traditional
structural model. However, by adding new state variables, the model not only becomes
more complicated in the form, but also harder to identify empirical evidence to improve
pricing and hedging practice. Another approach is to concentrate on regression models.
Because of the simplicity and convenience in incorporating any new state variables,
regression is gaining popularity in empirical research for credit spread modeling.
Regression models can be divided into two categories: regression on credit spread
changes and regression on credit spread levels.
Of the first category, there are three major papers: Duffee [1998], Collin-
Dufresne, Goldstein and Martin [2001], and Huang and Kong [2003].
Duffee [1998] did the pioneer work on credit spread changes regression. He
analyzed the credit spread data indexed by different industry, rating group, and maturity.
He used only the interest rate level and slope in the regression, and found that there is a
significant negative correlation between short rate change and credit spread change. He
achieved an average adjusted R
2
around 17%.
Collin-Dufresne et al. [2001] performed similar analysis, but on a lot more
variables. They divided corporate bond data by leverage ratio, rating, and maturity, and
performed multiple regressions. Among many regression models in the literature, this
101
model appears to be the most complicated. Their basic model included six basic
explanatory variables: leverage, interest rate level, interest rate slope, VIX, S&P, jump
probability. They achieved around 25% adjusted R
2
. They then performed principal
components analysis on the residual and found that over 75% variations are due to the
first component. Then they introduced new variables. The total number of variables in
final regression is 19, and the adjusted R
2
improved only to 34%. Eventually they
acknowledged that they could not identify the factor that contributes to the 75% residual
variation, within all the proxy they constructed for liquidity, etc. They claim that the
single factor driving the credit spread variation could be attributed to local
demand/supply fluctuation. Interestingly, while they introduce new variables, none of
these new variables are bond specific; most of them are macroeconomic variables.
Huang and Kong [2003] criticized Collin-Dufresne et al. [2001] for not having
chosen the best proxies for state variables. So they performed regression on credit spread
changes, with similar explanatory variables, while testing multiple proxies for each
variable among eight independent variables, and choosing the best one. Also they choose
to work with credit spread index OAS data (which they claim as cleaner credit spread) of
rating and maturity group. They achieved adjusted R
2
of more than 40% for 5 out of 9
groups. However, the number of observations for each index is merely 67. There is no
theoretical support as to why certain proxies for a state variable should perform better
than other proxies. And using index data in a short time period might have alleviated the
problem.
102
Table 4.1 gives an itemized comparison of the three papers.
Duffee [1998] Collin-Dufresne et al.
[2001]
Huang and Kong
[2003]
Category Industry, rating,
maturity
Leverage, rating,
maturity
Rating and Maturity,
total=9
{All sectors, Industrial,
Utility, Financial}
{<15%, 15-25%, 25-
35%, 45-55%, 55%}
Investment Grade:
{AA-AAA, BBB-A}
{Aaa, Aa, A, Baa} {AAA, AA, A, BBB,
BB, B}
{1-10 yr, 10-15 yr, 15+
yr}
{2-7, 7-15, 15-30} {long (>12 yr),
short (<9 yr)}
High Yield:{BB, B, C}
Data Type Mean corporate yield
vs. corresponding.
Treasury yield(self
constructed index)
Corporate yield vs.
corresponding Treasury
yield
Index
OAS? N N Y
Data
Description
No option embedded,
> 4 yr maturity.
Data Range Monthly,
Jan-85 to March-95
Monthly,
July-88 to Dec-97
Monthly,
Jan-97 to July-02
observations At least 25 observations
for each bond
67 observations for each
index
Adj. R-square Around 17% 19% to 25% by leverage
ratio
>40% for 5 out of 9
17% to 34% by rating
group
67% for B
34% after additional
variables
60% for BB
Table 4.1 Comparison of three papers on credit spread regression
Clearly we can see that all these three papers try to improve the explanatory
power by either adding more state variables or cherry-picking different proxies for the
same state variables. The regression model is fundamentally the same, and the
improvement is marginal.
103
Of the second category of credit spread level regression, one major paper is
Campbell and Taksler [2003]. They claim that equity volatility in the regression is almost
as good as the credit rating variable. What they used in the regression is the excess return
(equity return minus market return) volatility for the last 180 trading days, not the
historical volatility, or implied volatility from options market.
This paper falls into the first category by modeling credit spread changes on
individual bonds. We believe there are several benefits focusing on changes:
• Credit spread changes are more relevant to the modeling of credit spread
dynamics, since regression on credit spread levels will have a large intercept
portion, which is not very informational, because we know that there is always
some credit premium associated with the corporate bond yields;
• Regression in credit spread changes is more useful in developing a hedging
framework, since we can estimate the sensitivities of the credit spread changes to
interest rate, leverage of the company. These sensitivities can be used to derive
hedge ratios.
• Individual bond data contain far more information than the indexed data. All the
firm-relevant data could enter the modeling, especially the leverage, which is a
very important factor in any structural model.
However, realizing the drawbacks from simple linear regression, we adopt a more
flexible approach: Random Coefficient Regression model. Although the RCR model is
not new in statistics (Hildreth and Houck [1968]), it has rarely been used in financial
research.
104
4.3 Introduction to Random Coefficient Model
The most frequently used linear model in statistics might be the following:
i i i
X y ? ? + = , (4.1)
where y
i
is the observed response of dependent variables;
X
i
is the vector of explanatory variables;
? is the vector of coefficients of the linear model;
?
i
is the error term, and ) , 0 ( ~
2
?
? ? N
i
.
For time series data, like those we frequently encounter in financial econometrics, it can
be written as:
. ,..., 2 , 1 , T t x y
t k
k
tk t
= + =
?
? ?
where y
t
is the observed random variable, x
kt
are known explanatory variables, ?
k
are
unknown constants to be estimated, and ?
t
are the error terms, independently and
identically distributed with mean zero, and finite variance. If exact tests of significance
are desired, the error terms, ?
t
, are typically assumed to be normal.
In some applications, the constancy of the coefficients, ?
k
, in consecutive
observations may not hold. For example, a particular ?
k
represents the response of credit
spread change for a bond to interest rate, which depends on the demand/supply ratio and
market liquidity premium. If both demand/supply ratio and market liquidity premium are
relatively stable, the assumption of constancy for ?
k
might be a tolerable approximation.
However, if demand/supply ratio and market liquidity premium vary, but are not
105
observed, assuming ?
k
as the mean of a random response rate may be better than
assuming the response rate to be constant.
Consider the following simple extension of model (4.1):
i i i i
X y ? ? + =
where
i i
v + = ? ? ;
0 ] [ =
i
v E , ? = ] ' [
i i
v v E , and ?
i
is uncorrelated with ?
i
.
As before, y
i
is still the observed random dependent variable and X
i
are known
values of independent variables. In this extension, ? is the mean response of the
dependent variable to the independent variables and (? + v
i
) is the actual response rate in
the ith observation. Combining terms, we have the model:
i i
i i i i i
w X
v X X y
+ =
+ + =
?
? ? ) (
(4.2)
where
0 ] [ =
i
w E ,
i i i i i i
X X I w w E ? = ? + = ' ] ' [
2
?
An important difference between random coefficient and simple regression model
is that the simple linear model assumes the sample is relatively homogeneous. Therefore,
if the estimate for ? is zero, then X will be concluded to have no effect on the
dependent variable. However, random coefficients may indicate that the effect the results
106
from cancellation of positive effects on some observations with negative effects on other
observations. As a result, the randomness of coefficient provides better explanation
power even if the mean of the coefficient ( ? ) is neutral. Dielman, Nantell, and Wright
[1980] emphasized that random coefficient models are very useful in analyzing pooled
cross-sectional and time series data.
In more complex cases,
i
? can be parametrically expressed. For example,
i
? can
be a linear function of several independent variables. While this specification involves
additional assumptions, it is essentially the same as the previous simple extension. This
functional form is appealing in some cases, especially when there is theoretical basis for
the relationship between
i
? and those independent variables, and the relationship are of
interests of researchers.
There are several ways to estimate the model. Details will be given in the next
section.
107
4.4 Random Coefficient Model for Credit Spread Changes
Huang and Kong [2003] mention that the low explanatory power of theoretical
determinants, documented in Collin-Dufresne et al. [2001], could be due to two reasons.
The first reason is that the explanatory variables may not be the best proxies to measure
the changes in default risk. The second reason is that the current existing corporate bond
pricing model might miss some important systematic risk factors. We have a different
opinion as to why the simple linear regression model lacks explanatory power.
We think the fundamental cause lies in the underlying assumptions of the
regression model. When the regression (4.1) is estimated, there is an assumption that the
coefficients are fixed. That is, the marginal effect for one unit change in
i
X has the same
effect ( ? ) on
i
y regardless of the characteristics of instance i .
Suppose credit spread (CS
t
)is a complex function of interest rate (r
t
), firm
leverage (lev
t
), firm asset volatility (?
t
) and other state variables (X
t
), which is compatible
with most structural models. Given
CS
t
=CS(lev
t
, r
t
, ?
t
, {X
t
}),
we can derive the first order approximation for the change of credit spread:
t t t t
X
X
CS CS
r
r
CS
lev
lev
CS
CS ?
?
?
+ ?
?
?
+ ?
?
?
+ ?
?
?
? ? ?
?
(4.3)
In a short time period, if all these state variables, lev, r, ?, X, do not change dramatically,
a simple linear regression model could be used to estimate the coefficients. However,
over a relatively long time period, as in Duffee [1998] and Collin-Dufresne et al. [2001],
108
which uses data spanning a 10-year period, this assumption is no longer valid. Because
all these gradients themselves are functions of each state variable, they are going to
change as well. Specifically, we have:
) , , , (
) , , , (
) , , , (
t t t t
t t t t
t t t t
X r lev h
CS
X r lev g
r
CS
X r lev f
lev
CS
?
?
?
?
=
?
?
=
?
?
=
?
?
Merton [1974] in his seminal paper has calculated the credit spread for a zero-
coupon corporate bond in the following form:
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ?
?
?
/ )] ln(
2
1
[ ) , (
/ )] ln(
2
1
[ ) , (
)] , ( [
1
)] , ( [ ln
1
) (
2 2
2
2 2
1
2
1
2
2
d d h
d d h
d h
d
d h r R H
+ ? =
? ? =
?
?
?
?
?
?
? + ? ? = ? =
where H is the credit spread;
d is a debt ratio measure, defined as d=Be
-r?
/V;
B is the face value of the debt;
V is the asset value of the firm;
? is the volatility for the corporate asset process;
? is the maturity of the zero-coupon bond.
Then he calculates the credit spread gradient to most state variables as follows:
109
; 0
) (
) ( '
] , [
2
1
] , [
*
1
* ] , [
1
; 0 ] , [
1
; 0 ] , [
1
1
2
>
?
?
?
= =
?
?
= ? > ?
< ? =
?
?
?
?
h
h
T d g
T
H
T d g
lev
e T d g
d lev
d
H H T d g
d
H
T d g
r
d
H H
r
d lev d
d r
?
?
? ? ?
where g[d, T] is the ratio of instantaneous bond return volatility to instantaneous asset
return volatility, and is defined as:
) ] , [ /( )] , ( [ ) , (
1
d T d P T d h
F
VF
T d g
V
y
? = = =
?
?
P[d, T] is the price ratio of the defaultable bond to risk-free bond, which is
defined as:
)] , ( [
1
)] , ( [ ] , [
1 2
T d h
d
T d h T d P ? + ? =
T=?
2
?.
Clearly we can see that all these gradients are time varying. If we assume g[d, T]
is constant, or estimate it as ratio of excess return on bond to excess return on asset, then
model can be estimated in a simple form.
To summarize, using a simple linear regression to estimate these coefficients over
a long time horizon can lead to poor results. By adopting a random coefficient method,
the model (4.3) can be restated as:
? ? ?
?
? ? ?
?
?
?
?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ?
+ + + + + =
+ + + + + =
+ + + + + =
+ ? + ? + ? + = ?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
T lev r
T lev r
T lev r
lev r CS
T lev r
CS
lev lev
T
lev lev
lev
lev
r
lev CS
lev
r r
T
r r
lev
r
r
r CS
r
CS CS CS
lev
CS
r
CS
(4.4)
110
Rewriting the original model, we have:
CS lev r
T lev r
lev
T
lev lev
lev
lev
r
r
T
r r
lev
r
r
lev r CS
CS
T lev r
lev lev
T
lev lev
lev
lev
r
lev
r r
T
r r
lev
r
r
r CS
lev r
T lev r
lev T lev lev lev lev r
r T r r lev r r
lev r
T lev r
lev T lev r
r T lev r CS
?
?
?
?
+ ? + ? + ?
+ ? + ? + ? + ?
+ ? + ? + ? + ?
+ ? + ? + ? + ?
+ ? + ? + ? + =
+ ? + + + + +
+ ? + + + + +
+ ? + + + + + + = ?
? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
?
? ?
?
? ?
?
?
?
? ? ?
?
? ? ?
?
?
) (
) (
) (
(4.5)
with standard assumptions:
0 ] [ ] [ ] [ ] [ = = = =
?CS
i i
lev
i
r
i
E E E E ? ? ? ?
?
,
0 ] [ ] [ ] [ = = =
? ? ? CS
i i
CS
i
lev
i
CS
i
r
i
E E E ? ? ? ? ? ?
?
, which basically states that the error
terms are uncorrelated.
The coefficients in the original model now are random, and have their own specifications.
The difference between our model and the simple linear regression model exists
not only in the specification of coefficients, but also in the difference in the assumption of
the error terms. The homoscedasticity assumption in a simple linear regression is relaxed
in our model.
Apparently, the OLS estimates for ? in model (4.5) are still consistent under the
assumption stated above because ?
i
X and
sum
? (sum of all error terms) are uncorrelated.
111
However, the estimates are no longer efficient
4
. Both Feasible Generalized Least Square
(FGLS) and the White robust estimator can provide consistent and efficient estimates
(Greene [1997]). We tried both methods in our application. The difference between the
two estimates is small. We only report the White estimates, because FGLS estimates
involve additional weights from the variance-covariance matrix. If the form of the
heteroscedasticity and parameters involved are known, then FGLS will be a better choice;
otherwise, the White estimator, which is robust to unknown heteroscedasticity, is
certainly appealing, because the weights introduced by FGLS may add additional
variation into the slope estimates.
4
Efficiency of estimators: an unbiased estimator
?
1
? is more efficient than another unbiased estimator
?
2
?
if the sampling variance of
?
1
? is less than that of
?
2
? . That is, ] [ ] [
2 1
? ?
< ? ? Var Var .
112
4.5 Data Description
We extract data from three databases: Warga bond database
5
, CRSP
6
, and
COMPUSTAT
7
for different financial data.
Warga database, which is also known as the Lehman Brothers Fixed Income
Database, contains the most comprehensive bond data for academia. We only choose
those bond that satisfy the following standards:
1. Dealer quoted price, instead of matrix price, since it has been pointed out that
matrix price could produce some problems (Sarig and Warga [1989]);
2. At least 30 consecutive observations;
3. Non-callable and non-putable. This would eliminate the optionality-induced
premium in the yield spread;
4. Bond with maturity greater than four years, since it is well known structural
model is less accurate for short maturity bonds.
Based on these standards, we end up with credit spread time series for 728 bonds, with
45627 observations. We have bond price, yield to maturity, maturity date, and duration
data from this database. These data are used later to construct the credit spread.
5
The Warga Fixed Income Securities Database (FISD) for academia is a collection of publicly offered U.S.
Corporate and Agency bond data. Produced by LJS Global Information Services, Inc., this fixed income
database engine is used by Reuters/Telerate and Bridge/EJV. These vendors collectively account for 83%
of trader screens.
6
The CRSP Database provides access to NYSE, AMEX and Nasdaq daily and monthly securities prices, as
well as to other historical data related to over 20,000 companies. The data is produced, and updated
quarterly, by the Center for Research in Security Prices (CRSP), a financial research center at the Graduate
School of Business at The University of Chicago.
7
The Standard & Poor's COMPUSTAT® databases contain financial, statistical, and market data for
different regions of the world. The databases are searched using Standard & Poor's Research Insight®
software, which enables data queries, retrieval, manipulation and analysis. The software includes
predefined sets for searching different types of data and allows the user to generate this data using
predefined reports.
113
We acquired the equity data from CRSP database. The equity data is linked with
bond data via the CRSP permno (permanent number) index. We retrieved the daily equity
data for 322 companies from January 1987 to March 1998. These data are used later to
construct the mark-to-market equity, as well as stock return volatility.
COMPUSTAT database provided us with the balance sheet information. It is also
linked to the CRSP database via the permno index. We retrieved the quarterly balance
sheet data for the same 322 companies from January 1987 to March 1998. Then we
interpolated the total asset value and total liability value for the months between. These
data are used later to construct leverage ratio.
Here we provide a brief description for the data we constructed in the regression.
Treasury curve is constructed by using linear interpolation. The treasury rate source is
the constant maturity Treasury (CMT) rate of H.15 release from the Federal Reserve web
site. We use the 3-year, 5-year, 7-year, 10-year, 30-year treasury rates. 20-year treasury
rate is disregarded because its discontinuity for the observation period. Interest rate level
is defined as 10-year Treasury rate.
Credit Spread is calculated as the difference between bond yield and treasury rate with
the same maturity
8
. As a convention, only quoted price are used, excluding callable and
putable bonds. The bond yield we use is the yield to maturity. Data ranges from July
1988 to March 1998.
114
Firm leverage is calculated by the following formula:
Liability Total Equity of Value Market
Liability Total
leverage
+
=
Total liability in each quarter is acquired from Compustat database; data in between
months are interpolated linearly. Market value of equity is acquired by multiplying stock
price with shares outstanding. Firm leverage is an important factor in structural models to
calculate distance to default. However, different researchers have been using different
numbers to calculate leverage ratio, e.g. Collin-Dufresne et al. [2001] uses the book value
of debt to calculate leverage, and Moody’s KMV is using short-term debt to calculate
default probability.
Volatility: We considered three different measures for volatility:
1. VIX, which is the volatility index as a weighted average of eight implied
volatilities of near-the-money options on the OEX (S&P 100) index. This
volatility measure is identical to the Collin-Dufresne et al. [2001] paper.
2. Simple estimated standard deviation of last 20 daily returns, for the corresponding
company’s common stock.
3. Excess return volatility for last 180 trading days return. It is the standard deviation
of the last 180 trading day’s excess return, which is defined as the return minus
market return (S&P 500 return). This volatility measure is identical to the
Campbell and Taksler [2003] paper.
The effect of these three different volatility measures will be discussed in later sections.
8
Treasury rate with the same maturity is linearly interpolated from adjacent CMT rates.
115
4.6 Results Analysis
We are going to discuss the regression results of our new model in this section.
First, we compare the coefficients of simple linear regression model with RCR model,
and examine the assumption of dependence between credit spread sensitivities and state
variables. Second, we examine the regression results for different rating and maturity
groups. In the last subsection, we are going to examine the assumption of non-constancy
of credit spread sensitivities.
4.6.1 Dependence of Credit Spread Sensitivities to State Variables
In this section, we are going to discuss the regression results of our RCR model,
compared with simple linear model. Table 4.2 shows the coefficient estimation for both
models, and their t-values. Applying White robust estimator, regression is performed on
individual bond and the average statistics
9
are reported. In the simple linear regression
model, we can find that the sensitivity measures to interest rate change, leverage change,
and volatility changes are significant, and the signs and magnitudes of coefficients are
consistent with structural models and regression results in previously mentioned papers.
In the new model, we find that the following newly constructed interactive
variables are significant (with |t| > 2):
r?r, r??, ???, T??.
9
We followed the convention in Collin-Dufresne et al. [2001] to report these statistics. The reported
coefficient values are average of the regression estimates for the coefficient on each variable. The t-
statistics are calculated by dividing each reported coefficient by the standard deviation of the N estimates
and scaled by sqrt(N).
116
Linear
Model Variable beta
Standard
error t
intercept -0.006 0.02 -9.45
?r -0.058 0.21 -7.37
?lev 0.977 4.77 5.53
?? 0.005 0.01 8.48
Adj. r
2
0.077
RCR
Model Variable beta
t
intercept 0.05 0.002 -9.23
?r 18.47 0.68 -1.95
?lev 295 10.9 -1.38
?? 1.32 0.049 -1.42
r?r 0.60 0.022 6.02
lev?r 18.59 0.69 -0.42
??r 0.09 0.003 1.53
T?r 0.84 0.031 1.42
r?lev 12.44 0.46 -0.45
lev?lev 308 11.4 0.91
??lev 1.68 0.062 -0.32
T?lev 19.41 0.72 1.44
r?? 0.07 0.002 -3.02
lev?? 1.54 0.057 -0.35
??? 0.01 0.000 3.58
T?? 0.11 0.004 2.30
N 728
Adj. r
2
0.297
Table 4.2 Comparison of RCR vs. linear model
These significant interactive terms mean that the level of state variables has a
significant impact on the sensitivity of credit spread to these state variables. For example,
a positive coefficient for r?r means that when interest rate increases, the sensitivity of
credit spread to interest rate change should decrease (because the sensitivity of credit
spread to interest rate is negative). In other words, in a higher interest rate environment,
credit spread will be less sensitive to interest rate, given everything else unchanged. For
the same reason, a negative r?? coefficient means that in a higher interest environment,
credit spread will be less sensitive to volatility, when everything else is kept constant.
117
Also a positive ??? coefficient would stand for high volatility sensitivity in high
volatility environment. The next table summarizes the relationship we found between
levels of state variables and credit spread sensitivities.
Sign of beta
r
CS
?
?
(<0)
lev
CS
?
?
(>0)
? ?
?CS
(>0)
Interest + Not Significant
(N/S)
-
Leverage N/S N/S N/S
Volatility N/S N/S +
Maturity N/S N/S +
Table 4.3 Relationship between state variables and credit spread sensitivities
These findings validate our assumption that sensitivity should be dependent on
state variables. Also we would like to compare these coefficients to structural models,
and to validate whether these findings are consistent with theoretical models. We take the
most straightforward structural model for credit spread, the Merton [1974] model, for
which we have given the derivatives of the credit spread with respect to state variables in
section 4.4. Although it is possible to derive the second order derivative of credit spread
to validate the relationship we found are consistent with structural model or not, we
prefer to demonstrate this in a static analysis, as Merton did in the paper, which will be
more intuitive. The following charts show the results of our static analysis.
118
dCS/dr vs. r
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
3.50%
4.00%
0% 5% 10% 15%
C
S
-0.25%
-0.20%
-0.15%
-0.10%
-0.05%
0.00%
d
C
S
/
d
r
CS dCS/dr
Figure 4.1 Credit Spread vs. Risk-free Rate
Figure 4.1 shows the credit spread and credit spread sensitivity to interest rate at
different interest rate level. This zero-coupon bond is evaluated with 30% leverage, 30%
asset volatility and 5-year maturity, which is pretty representative. We can see that while
the Merton model predicts the credit spread will be decreasing while the interest rate
increases, the credit spread sensitivity to interest rate is an increasing function. However,
since the sensitivity measure itself is negative, being an increasing function actually
means reduced sensitivity at higher interest rate level, which is consistent with our
findings.
119
dCS/dsig vs. sig
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
0% 20% 40% 60% 80%
C
S
0.00%
0.10%
0.20%
0.30%
0.40%
0.50%
0.60%
0.70%
0.80%
0.90%
d
C
S
/
d
s
i
g
CS dCS/dsig
Figure 4.2 Credit Spread vs. Volatility
Figure 4.2 depicts the credit spread and its sensitivity to volatility at different
volatility level. The bond is evaluated at 5% risk-free rate, with 30% leverage and 5-year
maturity. From Figure 4.2, we find that credit spread is an increasing function of
volatility, and sensitivity to volatility is an increasing function for the most volatility
spectrum, from 5% to 60%, and after that, is pretty flat with a slight trend of decreasing.
This result is consistent with our findings.
120
dCS/dsig vs. sig
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
0% 10% 20% 30% 40% 50% 60% 70% 80%
dCS/dsig w/30% lev dCS/dsig w/60% lev
Figure 4.3 Sensitivity to Volatility at different Leverage
In order to test the robustness of the relationship between credit spread sensitivity
to volatility and volatility itself, we choose two different setting for maturity and
leverage. Figure 4.3 shows the credit sensitivity to volatility for a zero coupon bond with
maturity of 15 year, and leverage of 30%, and 60%, at 5% interest rate level. We can see
increased maturity make the curve more flat, compared with Figure 4.2. Also increasing
leverage makes the yield more flat as well. Since the vast majority of our bonds have
maturity less than 15 years, and leverage below 60%, we think the estimated positive
coefficient of volatility on sensitivity to volatility is a valid prediction for the majority of
these bonds.
121
dCS/dsig vs. r
0
0.05
0.1
0.15
0.2
0% 2% 4% 6% 8% 10% 12% 14% 16%
dCS/dsig
Figure 4.4 Sensitivity to Volatility vs. Interest Rate
Figure 4.4 shows the credit sensitivity to volatility for a zero coupon bond with
maturity of 5 year, and leverage of 30%, volatility of 30%, at different interest rate levels.
The Merton [1974] model predicts it to be a decreasing function in interest rate, which
means that the higher the interest rate is, the lower the sensitivity to volatility will be. The
chart is consistent with our coefficient estimator of r??, which is negative.
122
dCS/dsig vs. maturity
0
0.05
0.1
0.15
0.2
0 2 4 6 8 10 12 14 16
dCS/dsig
Figure 4.5 Sensitivity to Volatility vs. Maturity
Figure 4.5 shows the credit sensitivity to volatility for a zero coupon bond with
leverage of 30%, volatility of 30%, at 5% rate level, with different maturities. It shows
how maturity change would affect the credit spread sensitivity to volatility. The
sensitivity will increase rapidly with respect to maturity till 8 years, and then decrease
slightly after maturity passed 8 years. In our estimation, the coefficient is positive, but the
significance is not very strong. We predict it is a mixed result of the rapid increasing and
slow decreasing. Also Merton’s model is based on zero coupon bond, so if applied to
coupon bond, the maturity might be better replaced with duration measure, which is
significantly shorter than the maturity. That would explain why we have the coefficient
estimator to be positive, which is more inclined to the shorter end of the maturity.
123
4.6.2 Results by Rating and Maturity
In this section, we show results for different rating and maturity groups. We have
five rating groups: AAA-AA, A, BBB, BB, B and others (below rating B or not rated),
and three maturity groups: LONG (maturity >12 years), MEDIUM (12 years >maturity>8
years), and SHORT (maturity < 4years). The total number of combinations is 15.
AA_LONG AA_MEDIUM AA_SHORT
beta std_error t beta std_error t beta std_error t
intercept -0.01 0.001 -5.13 -0.003 0.003 -1.33 -0.01 0.003 -3.34
?r -0.53 1.01 -0.52 0.72 0.49 1.48 0.43 0.46 0.94
?lev 8.41 13.12 0.64 8.20 5.13 1.60 2.88 11.59 0.25
?? -0.39 0.19 -2.00 -0.12 0.06 -2.01 -0.24 0.16 -1.51
r?r 0.10 0.02 5.85 0.05 0.04 1.20 -0.03 0.05 -0.66
lev?r 0.31 0.59 0.53 -0.01 0.86 -0.01 0.07 0.71 0.09
??r 0.01 0.00 1.61 -0.02 0.01 -1.85 0.01 0.02 0.75
T?r 0.00 0.02 -0.01 -0.07 0.03 -2.25 -0.07 0.06 -1.28
r?lev -0.32 0.38 -0.84 -0.04 0.48 -0.08 0.93 0.67 1.38
lev?lev 4.42 16.88 0.26 -3.94 14.31 -0.28 -39.72 30.33 -1.31
??lev -0.06 0.09 -0.68 -0.22 0.16 -1.37 0.17 0.26 0.64
T?lev -0.15 0.51 -0.30 -0.39 0.65 -0.60 -0.07 1.17 -0.06
r?? -0.01 0.002 -2.49 -0.004 0.004 -0.95 0.002 0.01 0.25
lev?? 0.07 0.06 1.08 -0.03 0.17 -0.20 0.34 0.24 1.42
??? 0.00 0.00 0.98 0.002 0.001 1.97 -0.001 0.002 -0.67
T?? 0.01 0.01 2.03 0.01 0.003 4.61 0.02 0.01 1.34
N 42 10 53
Adj. r
2
0.266 0.270 0.233
Table 4.4 RCR coefficients for AA-AAA group
For the rating group of AA-AAA, we found that the model performs better (with
higher R
2
) in long maturity group than short maturity group. This is consistent with
previous regression model (Duffee [1998]). Also the average explanatory power for this
group is below average, which is also consistent with previous regression results. AAA
bonds are counted in this group because of the limited numbers in each AAA maturity
group.
124
A_LONG A_MEDIUM A_SHORT
beta std_error t beta std_error t beta std_error t
intercept -0.01 0.002 -4.24 -0.01 0.003 -3.13 -0.01 0.003 -3.73
?r -1.66 0.68 -2.43 -1.67 0.77 -2.16 -0.33 0.30 -1.10
?lev 16.91 10.05 1.68 -2.19 7.53 -0.29 4.91 7.65 0.64
?? 0.00 0.10 0.00 0.08 0.09 0.85 -0.01 0.05 -0.32
r?r 0.11 0.03 3.91 0.12 0.04 3.38 0.11 0.04 2.96
lev?r 0.01 0.51 0.02 1.14 0.92 1.24 -0.02 0.44 -0.05
??r 0.01 0.00 1.94 0.02 0.01 2.78 -0.01 0.01 -1.44
T?r 0.02 0.02 1.05 0.01 0.05 0.15 -0.04 0.03 -1.47
r?lev -0.21 0.37 -0.57 -0.33 0.76 -0.44 -0.86 0.57 -1.51
lev?lev -1.52 9.49 -0.16 -12.68 14.33 -0.88 -21.83 11.84 -1.84
??lev -0.01 0.07 -0.19 -0.10 0.19 -0.54 0.13 0.09 1.45
T?lev -0.60 0.45 -1.35 1.26 0.75 1.68 1.56 0.62 2.51
r?? -0.01 0.00 -3.50 -0.02 0.01 -3.19 -0.005 0.003 -1.69
lev?? -0.05 0.06 -0.79 -0.02 0.12 -0.20 -0.03 0.06 -0.45
??? 0.001 0.0005 2.39 -0.0001 0.001 -0.09 0.002 0.001 2.13
T?? 0.002 0.003 0.59 0.01 0.01 1.12 0.01 0.01 1.28
N 124 35 145
Adj. r
2
0.332 0.298 0.187
Table 4.5 RCR coefficients for A group
For the rating group of A, we also found that the performance of our RCR model
deteriorates as the maturity decreases. The average explanatory power for long and
medium maturity is above and near average (R
2
of 28%), which is also consistent with
previous literature (Duffee [1998]).
125
BBB_LONG BBB_MEDIUM BBB_SHORT
beta std_error t beta std_error t beta std_error t
intercept -0.0175 0.00341 -5.1259 0.0040 0.00686 0.5792 -0.0150 0.00317 -4.7205
?r -1.4606 1.30092 -1.1228 0.4031 3.15080 0.1279 0.4132 0.67383 0.6108
?lev 6.3313 14.42570 0.4389 -25.1286 19.48469 -1.2897 -3.3511 15.10461 -0.2210
?? 0.1922 0.16763 1.1464 0.1464 0.37627 0.3890 0.0037 0.07100 0.0517
r?r 0.1302 0.03744 3.4778 0.0416 0.09990 0.4168 0.1056 0.09241 1.1383
lev?r -0.4979 0.71171 -0.6996 -1.3209 3.10094 -0.4260 -1.2636 1.04115 -1.2089
??r 0.0309 0.00940 3.2876 -0.0354 0.02867 -1.2358 -0.0130 0.00842 -1.5402
T?r 0.0260 0.05094 0.5094 0.1095 0.15163 0.7223 0.0049 0.13122 0.0373
r?lev 0.3805 0.71955 0.5288 1.4138 1.85106 0.7638 -1.8690 1.80103 -1.0337
lev?lev 7.5851 13.19033 0.5751 34.6082 17.93499 1.9296 -0.8355 20.67328 -0.0403
??lev 0.1919 0.14222 1.3493 -0.2079 0.29622 -0.7018 -0.2272 0.33345 -0.6786
T?lev -0.4527 0.70592 -0.6413 0.0419 2.22403 0.0189 3.9359 2.19255 1.7880
r?? -0.0187 0.00617 -3.0315 0.0065 0.01899 0.3438 0.0014 0.00847 0.1649
lev?? 0.0901 0.08254 1.0912 -0.1715 0.30587 -0.5607 -0.0434 0.14752 -0.2927
??? -0.0004 0.00069 -0.5319 0.0033 0.00455 0.7291 0.0031 0.00164 1.8640
T?? -0.0044 0.00747 -0.5856 -0.0151 0.03758 -0.4024 -0.0032 0.00888 -0.3582
N 72 24 126
Adj. r
2
0.3159 0.1873 0.2032
Table 4.6 RCR coefficients for BBB group
For rating group BBB, we found that the performance of our RCR model
deteriorates as the maturity decreases. The average explanatory power for long and
maturity is above average, which is also consistent with previous literature (Duffee
[1998]). The regression results for the BBB medium group is far below average, and
almost none of these variables are statistically significant, which we suspect is due to
limited data problem (only 24 bonds available.)
126
BB_LONG BB_MEDIUM BB_SHORT
beta std_error t beta std_error t beta std_error t
intercept -0.06 0.01 -11.45 N/A N/A N/A -0.07 0.01 -5.68
?r -5.18 1.57 -3.29 N/A N/A N/A -4.61 2.35 -1.96
?lev 21.89 13.93 1.57 N/A N/A N/A -87.91 36.79 -2.39
?? -0.28 0.08 -3.75 N/A N/A N/A 0.48 0.35 1.40
r?r 0.37 0.04 8.32 N/A N/A N/A 0.34 0.16 2.17
lev?r -1.29 2.21 -0.59 N/A N/A N/A 1.03 2.03 0.51
??r 0.03 0.01 2.96 N/A N/A N/A 0.01 0.04 0.16
T?r 0.18 0.15 1.19 N/A N/A N/A 0.15 0.16 0.90
r?lev 0.90 0.70 1.29 N/A N/A N/A -1.99 2.28 -0.87
lev?lev 27.62 13.75 2.01 N/A N/A N/A 66.40 38.54 1.72
??lev -0.78 0.28 -2.80 N/A N/A N/A -0.10 0.83 -0.12
T?lev -1.72 0.66 -2.62 N/A N/A N/A 7.07 3.14 2.25
r?? -0.01 0.00 -3.69 N/A N/A N/A -0.01 0.02 -0.48
lev?? -0.48 0.14 -3.43 N/A N/A N/A -0.66 0.44 -1.51
??? 0.002 0.001 1.35 N/A N/A N/A 0.01 0.002 3.79
T?? 0.03 0.01 3.83 N/A N/A N/A -0.01 0.03 -0.44
N 24 1 27
Adj. r
2
0.123 N/A 0.104
Table 4.7 RCR coefficients for BB group
For the rating group BB, we also see that model became worse when the maturity
decreases. And for BB medium group, there is only one bond, so we cannot draw any
reasonable conclusion about variable significance. Again the explanatory power is both
low for long and short maturity, which could be contributed to limited bond numbers in
both categories.
127
B_LONG B_MEDIUM B_SHORT
beta std_error t beta std_error t beta std_error t
intercept N/A N/A N/A N/A N/A N/A -0.056 0.060 -0.935
?r N/A N/A N/A N/A N/A N/A -29.005 15.404 -1.883
?lev N/A N/A N/A N/A N/A N/A 30.750 309.773 0.099
?? N/A N/A N/A N/A N/A N/A 0.350 2.174 0.161
r?r N/A N/A N/A N/A N/A N/A 1.422 0.609 2.336
lev?r N/A N/A N/A N/A N/A N/A 12.422 10.146 1.224
??r N/A N/A N/A N/A N/A N/A 0.075 0.183 0.409
T?r N/A N/A N/A N/A N/A N/A 1.372 1.538 0.892
r?lev N/A N/A N/A N/A N/A N/A 46.626 20.449 2.280
lev?lev N/A N/A N/A N/A N/A N/A -419.950 376.543 -1.115
??lev N/A N/A N/A N/A N/A N/A -3.616 7.040 -0.514
T?lev N/A N/A N/A N/A N/A N/A 22.297 18.849 1.183
r?? N/A N/A N/A N/A N/A N/A -0.291 0.277 -1.047
lev?? N/A N/A N/A N/A N/A N/A 4.618 5.316 0.869
??? N/A N/A N/A N/A N/A N/A -0.069 0.069 -0.990
T?? N/A N/A N/A N/A N/A N/A -0.162 0.137 -1.186
N 1 1 8
Adj. r
2
N/A N/A 0.500
Table 4.8 RCR coefficients for B and other group
The total number of bonds in B and other group are very limited, so we cannot
make judgments about model performance in each maturity group.
Overall, our model performs best for the A and BBB groups, as well as for longer
maturities. These findings are consistent with Duffee [1998], as well as with theoretical
structural models for credit spreads.
128
The following table shows the significance level of previously identified
interactive terms in each rating and maturity group.
Interactive Terms r?r r?? ??? T??
AA_LONG + - Not Significant
(N/S)
+
AA_MEDIUM + - N/S +
AA_SHORT N/S N/S N/S +
A_LONG + - + N/S
A_MEDIUM + - N/S N/S
A_SHORT + - + +
BBB_LONG + - N/S N/S
BBB_MEDIUM N/S N/S N/S N/S
BBB_SHORT + N/S + N/S
BB_LONG + - + +
BB_SHORT + N/S + N/S
B_SHORT + - N/S N/S
Table 4.9 Summary of RCR coefficients
Not surprisingly, we found that for the interactive terms, which were constructed
in the RCR model, the significance levels and signs of the coefficient estimators are very
consistent for each rating and maturity group. If we remove the three groups with one
sample ecah, the r?r term is significant for 10 of 12 groups, which means that the interest
rate level has a positive impact on the credit spread sensitivity on interest rate, no matter
what rating group or maturity category. Also the impact of interest rate level on credit
spread sensitivity on volatility is very consistent.
129
4.6.3 Non-Constancy of Credit Spread Sensitivities
The non-constancy of credit spread sensitivities would naturally be embedded in
their dependence on state variables in the RCR model. However, we would like to see
how they change over time, and compare it to the simple linear regression sensitivity
estimators, and find out why the RCR estimators provide better accuracy.
Let’s take one bond as example, the bond with CUSIP of "001765AE", one of
American Airlines' long term bonds, and depict its random coefficients and constant
coefficients. The following three figures show the comparison of regression coefficients
with respect to interest rate changes, leverage changes, and volatility changes.
sensitivity to r
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
1
9
9
0
0
6
1
9
9
0
1
0
1
9
9
1
0
2
1
9
9
1
0
6
1
9
9
1
1
0
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9
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2
0
2
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9
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2
0
6
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9
2
1
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9
3
0
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9
3
0
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1
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0
2
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4
0
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4
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0
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5
0
2
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0
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1
0
1
9
9
7
0
2
1
9
9
7
0
6
1
9
9
7
1
0
beta_dr_linear beta_dr_RCR
Figure 4.6 Coefficient for ?r in RCR vs. Linear Model
130
sensitivity to lev
0
1
2
3
4
5
6
7
8
9
1
9
9
0
0
6
1
9
9
0
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1
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2
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3
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1
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3
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1
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4
0
3
1
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4
0
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4
0
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1
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4
1
2
1
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5
0
3
1
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5
0
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1
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0
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5
1
2
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0
3
1
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0
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6
0
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1
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6
1
2
1
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9
7
0
3
1
9
9
7
0
6
1
9
9
7
0
9
1
9
9
7
1
1
beta_dlev_linear beta_dlev_RCR
Figure 4.7 Coefficient for ?vol in RCR vs. Linear Model
sensitivity to sig
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
1
9
9
0
0
6
1
9
9
0
1
0
1
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9
1
0
2
1
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1
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9
7
0
2
1
9
9
7
0
6
1
9
9
7
1
0
beta_dvix_linear beta_dvix_RCR
Figure 4.8 Coefficient for ?lev in RCR vs. Linear Model
There are three major findings from the graphs:
1. Sensitivity to ?r does change over time. In Merton’s model, increased interest rate
would increase the risk neutral drift term, thus decrease the default probability,
and shrink the credit spread. In reality, Fed generally lowers interest rate to
stimulate economy when there is a recession, which is the case during 1990-1992.
Generally higher credit spreads are observed during a recession. That is the main
reason for negative correlation between interest rate and credit spread. However,
131
what about during times of economic recovery or boom? It would be interesting
to compare Figure 4.6 which depicts the sensitivity of credit spread to interest
rate, to Figure 4.9, the history of 3-month Treasury rate, a close reflection of
Fed’s policy on funding rate. When the economy is recovering, lowering interest
rate would have less effect on credit spread. That is exactly the case we found
during 1992-1993, when the Fed continued lowering the short interest rate, and
the sensitivity of credit spread to the interest rate is close to zero. Also when
economy is booming, the Fed is likely to raise the interest rate, and that seems to
have little effect on the credit spread. That is likely the case for 1994-1995.
2. Sensitivity to ?? also changes over time. In structural model, increase in volatility
would increase the default probability, and thus widen the credit spread. However,
comparing Figure 4.8 with 4.10: the history of VIX volatility index, we found that
while volatility is high both in the early 90’s and the late 90’s, their impact on
credit spread sensitivity are quiet different. One explanation for this could be that
during a recession, volatility is a bad thing, because it is likely that the volatility is
a result of dropping equity, and investors will be really concerned with a volatility
spike. However, when the economy is booming, it is likely that high volatility is
introduced by rising stock prices, and investors are less likely to require high
credit spread for this “good volatility”.
132
3m Treasury Rate
0
1
2
3
4
5
6
7
8
9
J
a
n
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u
l
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a
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u
l
-
9
6
J
a
n
-
9
7
J
u
l
-
9
7
3m
Figure 4.9 Three-Month Treasury Rate from 1990 to 1997
vix
0
5
10
15
20
25
30
35
40
1
9
9
0
0
5
1
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Figure 4.10 VIX index from 1990 to 1997
Of the three volatility measures we used, our results show that VIX is better than
both history volatility and excess volatility, which is unanticipated. Originally we thought
that since VIX is a broad market volatility index, replacing it with company specific
volatility should improve our results. The reason for this phenomenon might be that
credit spread response is more sensitive to market perception of risk than to historical
133
volatility. Also we tried the excess return volatility, which Campbell and Taksler (2003)
claims to have significant explanatory power in regression of credit spread levels. The
results are disappointing, and the adjusted R
2
is comparable to historical volatility, but
not as good as VIX index. The reason might be that the credit spread itself already has a
build-in premium associated with the standard deviation of excess return, but the change
of credit spread is not sensitive to its change, so the regression on credit spread levels and
changes will have different explanation.
134
4.7 Conclusions and future work
By using the RCR approach, we not only model the dynamics of credit spread
sensitivities in a more consistent way with current structural model, but also achieve
more explanatory power than simple linear model. Our contributions are the following:
1. The first paper to use RCR model on credit spread data;
2. The first paper to explicitly model the credit spread sensitivities with dependence
against state variables, and empirically validate the dependence relationship
predicted by Merton’s model;
3. Higher explanatory power is achieved without adding new independent state
variables. In this case, we increased the adjusted R
2
from 8% to 30%.
Obviously, there are still some unanswered questions remaining in our work. We
would like to pursue future research in the following directions:
1. We can see from our results analysis from section 4.5, the theoretical sensitivity
changes are not always linear with respect to state variables (Figure 4.5), and
when there is a strong no linear relationship, our predictions of coefficient are
generally weaker. So can we change the functional form in the regression model
for sensitivity parameters and achieve better explanatory power? And which
structural model should we adopt in selecting the functional form? It will be
interesting to compare the regression results for different functional forms of
credit spread sensitivity from different structural models.
2. What would be a better asset volatility proxy than the VIX index? We think that
the option implied volatility for each company’s stock option might be a better
135
indicator of the market perception of risk. However, how to convert the equity
volatility to asset volatility? One way to look into this might be to look at the
combined bond return volatility of the specific company, which means that we
need to group the bonds of the same company, instead of doing individual
regression on each bond issued.
3. It would be interesting to analyze the residuals of the regression error, and find
whether there exists any pattern to discover hidden significant drivers for credit
spread.
136
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