Project Report on Term Structure of Interest Rates: Estimation and Interpretation

Description
An interest rate is the rate at which interest is paid by borrowers for the use of money that they borrow from a lender.

BANK OF FINLAND
DISCUSSION PAPERS
Juha SeppaIa - Petri ViertiO
Market Operations Department
4.9.1996
The Term Structure of Interest Rates:
Estimation and Interpretation
19/96
SUOMEN PANKIN KESKUSTELUALOITTEITA • FINLANDS BANKS DISKUSSIONUNDERLAG
Suo men Pankki
Bank of Finland
P.O.Box 160, SF-00I0l HELSINKI, Finland
1f + 358 0 1831
BANK OF FINLAND DISCUSSION PAPERS 19196
Juha SeppaUi - Petri Viertio*
Market Operations Department
4.9.1996
The Term Structure of Interest Rates:
Estimation and Interpretation
* This work was completed during Petri Viertio's visit to the Research Department. We would
like to thank Juha Tarkka, Antti nmanen, Katri Jolma, Antti Ripatti and Jouni Timonen for
several useful discussions and comments. Gregg Moore kindly checked the language. Special
thanks belong to L.E.O. Svensson for providing his GAUSS source code for us. The current
address of Juha SeppaUi is University of Chicago, lllinois, USA and Petri Viertio CD Financial
Technology Ltd., Annankatu 15 B, FIN-OOI20 Helsinki, Finland.
ISBN 951-686-514-3
0785-3572
Suomen Pankin monistuskeskus
Helsinki 1996
The Term Structure of Interest Rates:
Estimation and Interpreatation
Bank of Finland Discussion Papers 19/96
Juha SeppaIa - Petri Viertio
Market Operations Department
Abstract
This document reports the currently used term structure estimation method at the Bank
of Finland ans discusses interpretation of the results it generates. We start by
introducing two widely used term structure estimation methods: the Cubic Spline
Function method and the Nelson-Siegel approach. We compare their results, payIng
special attention to the smoothness of forward interest rates and distribution of pricing
errors. Next, we introduce the Bank of Finland's method, commenting on its strenghts
and weaknesses. Finally, we discuss interpretation of the term structure of interest rates
with emphasis on the inflation expectations and the role of the time-varying risk
premia.
Key words: term structure of interest rates, cubic splines, Nelson-Siegel, forward
interest rates, relative value, inflation expectations, time-varying risk premia.
Tiivistelma
Taman keskustelualoitteen tarkoituksena on esitelUi Suomen Pankissa kaytossaoleva
korkorakenteen estimointimalli seka arvioida sen tuottamia tuloksia. Tutkimuksen en-
simmaisessa osassa tarkastelemme kahta yleisesti kaytettya korkorakenteen estimointi-
tapaa: ns. Cubic-Spline -mallia ja Nelson-Siegel -lahestymistapaa. Vertailemme mal-
lien tuloksia kiinnittaen erityista huomiota termiinikorkojen siistiin kayttaytymiseen
seka hintavirheiden jakaumaan. Tutkimuksen toisessa osassa esittelemme Suomen
Pankissa kaytetyn mallin, vertaillen mallin vahvuuksia ja heikkouksia. Lopuksi kasitte-
lemme korkorakenteen tulkintaa inflaatio-odotusten ja ajassa muuttuvien riskipree-
mioiden kannalta.
Asiasanat: korkorakenteen estimointi, kuutiosplinit, Nelson-Siegel, termiinikorot,
suhteellinen edullisuus, inflaatio-odotukset, ajassa muuttuva riskipreemio.
3
Contents
1 Introduction
2 Estimation of the Term Structure
2.1 Main concepts. . . . . . .
2.2 Other works . . . .
2.3 The Cubic Spline Function Method
2.3.1 B splines. .
2.3.2 Knot points .
2.4 Nelson-Siegel model.
2.4.1 Extended models
2.4.2 The Bank of Finland Method
2.5 Comparison of methods .
2.5.1 General . . . . .. ..
2.5.2 :\lodeling considerations ..
2.5.3 of comparison
2.5.4 Data description· . . .
2.5.5 Results of comparison ..
2.5.6 Discussion . . . .
2.5.7 Enhanced model specification
2.6 The distribution of pricing errors
2.6.1 Properties of the error
2.6.2 Relative value models ...
3 Interpretation of the Term Structure
3.1 Expectat ions and premia. . ....
3.1.1 Expected consumption growth .
3.1.2 Expectations and risk premia
3.1.3 Con\·exit.y bias .. . .
3.1.4 Summary . . . .. .
3.2 Forward interest rates as rough indicators
3.2.1 Interest rate expectations .
3.2.2 Inflation rate expectations .
3.2.3 Exchange rate expectations
3.3 Significance of premia. . . .
7
9
9
11
13
13
14
16
18
18
19
19.
20
20
22
24
27
29
31
31
32
33
33
33
34
36
37
37
37
39
40
40
5
6
4 Conclusions
References
A Appendix
42
43
48
Chapter 1
Introduction
One of the oldest problems in economic theory is the interpretation of the term
structure of interest rates. It has been long recognized that the term struc-
ture of interest rates conveys information about economic agents' expectations
about future interest rates, inflation rates and exchange rates. Indeed, it is
widely seen that the term structure is the best source of information about the
economic agents' inflation expectations for one to four years ahead.
1
Since it
is usually recognized that the monetary policy can only have effect with "long
and variable lags" as Friedman (1968) put it, the term structure provides an
invaluable source of information for the monetary authorities.
2
Moreover, re-
cent empirical studies
3
indicate that the term structure predicts consumption
growth better than vector autoregressions or leading commercial econometric
models.
The empirical success above is, unfortunately, diminished by the fact that
currently there does not exist a theoretical model which could explain all the
implications of the term structure. This means that currently it is not possible
to obtain exactly all information that is hidden in the term structure. This
report tries to give the reader a perspective about what is known about the
term structure, and how we can use the information it contains.
Before we can turn to the question of the interpretation of the term struc-
ture of interest rates we have to specify what we mean by the term structure.
The term structure is something we cannot observe; we observe only some of
its implications such as yields to maturity on coupon-bearing bonds. Thus,the
question of how to pull back the underlying term structure from the market
data is a non-trivial one. Different estimation methods may give very different
term structures, forward interest rates, and, ultimately, indicate contradictory
lSee, eg, Fama (1975, 1990), Mishkin (1981, 1990a, 1992) for studies about the inflation
expectations and the term structure of interest rates using U .5. data. Mishkin (1991) and
Jorion and Mishkin (1991) use international data. Abken (1993) and Blough (1994) provide
nice surveys of the literature.
2Svensson (1994ab) offers excellent discussions about monetary policy and the role of the
term structure of interest rates as a source of information.
3See, e.g., Harvey (1988), Ch en (1991), and Estrella and Hardouvelis (1991).
7
or incongruent implications.
This document reports the currently used method for estimating the term
structure of interest rates at the Bank of Finland, and discusses how we inter-
pret the results. We start in Chapter 2 by introducing the main concepts in
section 2.1 and two widely used term structure estimation methods: the Cubic
Spline Function method in section 2.3 and the N elson-Siegel approach in sec-
tion 2.4. We compare the results they produce in section 2.5, paying special
attention to the smoothness of forward interest rates and distribution of pric-
ing errors. Next, we introduce the Bank of Finland's method in section 2.4.2
and discuss its strengths and weaknesses.
Finally, we discuss about the interpretation of the term structure of interest
rates in Chapter 3. In section 3.1 we explain what should be found from the
forward interest rates: expectations and different term premia. When there
are no premia, obtaining interest, inflation, and exchange rate expectations is
a trivial task as shown in section 3.2. With non-zero term premia the task
is non-trivial and currently unsettled issue in the literature as is emphasized
in section 3.3. That section is a bit more technical than other sections of the
paper, and may be skipped by the reader not interested in empirical research.
The main findings are restated in Chapter 4.
8
Chapter 2
Estimation' of the Term
Structure
2.1 Main concepts
In the introduction we stated that the term structure is something we cannot
observe. \Ve only observe its implications: yields to maturity on coupon-
bearing bonds. Before we continue this discussion, defining some key terms
may be in order. Yield to maturity on a coupon-bearing bond is its internal
rate of return that will set the present value of the bond equal to its price
N CF
i
P
j
= L (1 + Y y. '
t=l J
(2.1 )
where Pj is the price of bond j with N cash flows, CF, so that cash flows
i takes place after ti periods. Such Yj that solves the. above equation given
above parameters is called bond j's yield to maturity. It should be noted that
Eq.2.1 is an implicit function in Yj, and numerical methods (e.g., Newton-
Raphson procedure) are needed to solve for Yj.
Cash flows usually consist of coupon payments, Ci , i = 1, ... ,N - 1, and
the final repayment, CN + F, where F is the nominal value of the bond. To
normalize different payment structures so that they can be compared, it is
common to speak about spot rates, which corresponds to yields to maturity
on zero-coupon bonds
(2.2)
where Sj is the spot rate for the zero-coupon bond j using annual compounding.
The discount function and spot rates are now related in the following way:
d(t) ___ 1 __
- (1 + s(t))t'
(2.3)
9
or with continuous compounding
(2.4)
Assume you have a zero-coupon bond maturing at time t - 1 in the future,
and that you want to increase your investment horizon from t - 1 to t. There
are two ways to accomplish this. You can either sell your current bond and
buy yourself a new bond maturing at time t, or you can keep your current
bond and buy yourself a contract for a period between t - 1 and t. Since
both investments are determined now and the interest rate risk from both
investments is the same, the no-arbitrage principle dictates that the rate of
return from both investments must be the same
(1 + St)t - (1 + St_lY-
1
(1 + st_d
t
-
1
(1 +sd
ft-l,t
(1 + St_l)t-l(l + ft-l,t) - (1 + St_l)t-l
(1 + St-l )t-l
(1 + St_l)t-l(l + ft-l,t)
(1 + sd - (1 + St_lY-
1
(1 + St_l)t-l
where ft-l,t is the internal rate of return for a contract between periods t - 1
and t. Eq.2.5 gives an expression for a forward rate between periods t - 1
and t given the spot rates for period t - 1 and for period t. Generalizations
to different horizons are obvious. Forward rate is an interest rate determined
now (trade date) for an investment beginning in the future (settlement date)
and ending further in the future (maturity date).
If we let the difference between the maturity time and the settlement time
approach zero, we obtain the instantaneous forward rate. The instantaneous
forward rate may be seen as the marginal increase in the rate of return from
a marginal increase in the investment horizon. Hence, the spot rate and the
(instantaneous) forward rate are related in the same manner as marginal and
average cost of production are related such that quantity producedcorresponds
to time to maturity. The spot rate at maturity m is the average of the forward
rates
s(m) = - f(t) dt,
1 l
m
m 0
(2.6)
where f( t) denotes the instantaneous forward rate at maturity t.
The meaning of yield curve depends on the author. Here yield curve means
a collection of bond-specific yields versus bond-specific maturities. This is
something that can be unambiguously calculated from the market prices. By
the term structure of interest rates we mean spot curve, or the internal rate of
return for any zero-coupon bond maturing at any time in the future.
In practice, we do not have bonds in general or zero-coupon bonds es-
pecially for each date in the future. Hence, we need first to estimate the
zero-coupon bonds from bonds on the market, then the spot rates from these
"synthetid' zero-coupon bonds for each date in the future, and finally forward
10
rates from spot rates. The first and the last step are unambiguous. The prob-
lem is the middle step: How do we fit a finite number of zero-coupon bonds
into acontinuous and well-behaving spot curve? It is very easy to obtain per-
fect fit, but that will create widely fluctuating forward curve with possible
negative or infinite forward rates.
Finally, it should be mentioned that when we discuss the term structure
we mean spot curve estimated from liquid government nominal bonds. Illiquid
bonds or bonds issued by private banks have risk premia of their own, which
would complicate the discussion in section 3.
The basic formula that is used to estimate the discount function or the
spot rates from bond prices is as follows. The (weighted) sum of bond pricing
errors Ej is minimized
N
E = min L Wj * E/ =
j=l
N 2
mm L Wj * [Fj - Pj] =
mm
j=l
N
LWj*
j=l
[t (exp( -I; ..;(/) * C f;j) _ pj] 2
(2.7)
(2.8)
where N is the number of bonds, Fj is the estimated price and C !ij is the
i:th cash flow of the bond j.
Figure 2.1 presents an example of the term structure. It shows the situation
for Germany on 2 January 1995. The boxes represent observed data from the
market, the straight line represents the estimated term structure, the dotted
line the (instantaneous) forward rate implied by the spot curve, and finally
dots show "theoretical" yields to maturity implied by the spot curve. That
is, once we know the theoretical spot curve and the cash flow structure of the
bonds at market, we can pull back the theoretical price and yield to maturity
for each bond using Eq. 2.1. Figure presents also 95% confidence interval for
the estimates.
2.2 Other works
The most straightforward method to find the term structure is to use a simple
bootstrapping or recursive method. The rates are defined re cursively from the
shortest instrument onwards. Early attempts to fit the term structure were
made by Carleton and Cooper (1976), and Cohen, Kramer and Waugh (1966)
using regression techniques. These discrete point estimation methods resulted
11
_ Fie:ure 2.1: The Term Structure of Interest Rates.
LS/MB/JS Mon JUl 31 16:47':tlz 1995
Yields to Maturity, Spot and Forward Rates. Germany. 2 Jan 95
95% confidence interval
"0
Q)
"0
8
§ 7
o
0..
E
o
C,.)
;>., 6
5
T
-4----t-------------l
-trii]- - . 0
D
o Observed yield to maturity
- - -. Estimated spot rate
-- Estimated forward rate
Estimated yield to maturity
1997 1999 2001 2003 2005 2007 2009 2011 2013 2015 2017
Maturity/settlement
in a good fit, although the rates were not smooth and did not satisfy no-
arbitrage arguments. Further, these methods are only applicable with carefully
selected situations (eg selected maturity dates).
McCulloch was the first to use spline techniques to smooth the rates be-
tween coupon dates. Initially he used a quadratic spline technique (1971) and
later he introduced the cubic spline method (1975) (see following section).
Vasicek and Fong (1982) used exponential splines in their work, and argued
that because the curvature of the discount function is of exponential nature,
it cannot be approximated properly with polynomial functions.
Attempts to find methods for estimating the term structure directly, as
opposite to discount function methods, were made by Fama and Bliss (1987)
and by Nelson and Siegel (1987).
A quite different approach has been proposed by Delbaen and Lorimier
(1992). First they estimate the initial term structure of forward rates (day
1), then they try to minimize the overnight fluctuation of the forward rates
subject on the condition that the pricing errors remain reasonable.
All previous models lack a particular theoretical background beyond the
basic bond mathematics and some empirical findings based on the behaviour of
the rates. The theorethical models that are applicable to the estimation of the
term structure form a different category. Among the most important models
are the one proposed by Cox, Ingersoll and Ross (1985) and the Longstaff-
Schwartz model (1992). The model proposed by Dillen (1994) has also gained
attention recently.
12
In the real world many models are based on a variety of risk factors such
as duration and convexity. Additionally straightforward polynomial approxi-
mation methods are also quite popular.
2.3 The Cubic Spline Function Method
2.3.1 B splines
The various spline methods utilize splines as the building blocks of the dis-
count function. We present here a piecewise polynomial approximation model
based on B splines. Splines provide an extremely high degree of flexibility in
terms of the shapes of the curves. B splines have reasonably good convergence
properties and provide a high degree of derivative continuity. A more detailed
description of the B spline method can be found in Steeley (1991) and Shea
(1985). Spline methods are probably the most widely used methods both in
practice and in theoretical academic studies.
A cubic spline is a piecewise cubic polynomial joined at so-called "knot
points. At each point the polynomials that meet are restricted so that the level
and first two derivatives meet. Because of derivative continuity each additional
knot point in the spline adds only one independent parameter. By increasing
the number of knots, splines provide an increasingly flexible functional form
(Fisher, Nychka, Zervos, 1995).
The functional form of a k-order (for cubic spline k = 3) B spline is as
follows:
p+k+l [P+k+l 1 1
B;(t) = ~ IT ( ) * (t - t l ) ~
L th - tl
I=p h=p,h-:j:.1
-oo<i<oo (2.9)
The subscript p is the spline index number and it denotes that the spline
is non-zero only between the interval [tp, ip+k+ll. The vector [il' i2 , i3 ..• ip+k+ll
is called the knot placement and defines the piecewise use of splines in the
maturity axis. Function (t - t z ) ~ is a truncated power function that has the
property
(t - t l ) ~ = max((i - iz)\ 0). (2.10)
With a sufficient number of piecewise splines any continuously differen-
tiable function can be approximated in an interval within an arbitrary error.
So we can write the discount function
P
d(i) = L wp * Bp(i) (2.11)
p=l
where P is the number of the B splines and Wp are the coefficients (weights)
of the splines. If we insert Eq.2.11 into Eq.2.7, we get
N p=P
Pj = L L CFi * Wp * Bp(t) + Cj,
(2.12)
i=l p=l
13
where the unknowns are the Wp and the pricing errors Ej of the bonds.
Figure 2.2 presents an example of B-splines as approximating functions of
the discount function.
Figure 2.2: B-splines with knot placement vector (1,2,3,5,8,15,23) (years)
0.'8.-----------------------,
0.1e
0.'
o.oa
0.06
0.04
0.02
-0.02.1.-________________ -----1
B,
-92
-93
-B"
--',-,-,,8$
-ea
-B7
-ea
--BO
··B10
The B spline base can, in fact, be a too flexible specification for the estima-
tion of the discount function. The usual problem with spline approximations
is that the derived forward rates tend to be unstable and widely fluctuat-
ing. As a result advanced market participants have turned their attention to
other functional specifications. However it is possible to avoid the "overfitting
problem hrough the use of "smoothing splines. (Fisher, Nychka, Zervos, 1995)
2.3.2 Knot points
Although the spline methods are quite straightforward to apply, one has to
make some critical decisions on the quantity and location of the knot points.
The placement of the knot points defines the explanatory power of the spline
approximation. With respect to knot selection in Poirier (1976, s.151) it is
said that
14
"... the choice of knot positions corresponds very closely to the
selection of functional type in an ordinary curve fitting problem.
... the knots in spline should not be seen as ordinary free parame-
ters, but their specification should rather be seen as analogous to
the choice of functional type. Hence, the knots should be chosen
:as to correspond to the overall behaviour of the data· (number of
observations, positions of maxima and minima, etc. ) ...
Based on this, the following rules-of-thumb can be given when using cubic
splines (Poirier, 1976):
1. Use as few knots as possible to avoid overfitting.
2. Use only one extremum point and one inflexion point per interval.
3. Center extremum points in the intervals.
4. Place inflexion points close to the knots.
Quantity of knots Technically as the quantity of knot points increases the
explanatory power of the discount function also increases. As a result, the
quality of fit increases. Simultaneously, however, the stability of the curves
decreases. If the number of the knots is too small the pricing errors tend to
be highly autocorrelated.
One way to determine an appropriate number of knot points is to use the
well-known rule-of-thumb that the degrees of freedom of the approximation
should equal the square root of the number of observations.
Another method to determine the number of knot points would be to use
economic argumentation, stating that the term structure consists of distinct
"quasi-independent sectors that follow separate patterns of behaviour. Such
natural sectors for example would be the short end, intermediate and long end
of the curve. (Litzenberger and RoHo, 1984). Based on extensive testing of
US Treasury data, Langetiegand Smoot (1988) recommended the use of 3 or
4 carefully selected knots. (See section 2.5.5).
The most structured way would perhaps be to use an information criteria
similar than is described in section 2.5.3 and then maximize the adjusted
coefficients of determination subject to the different models.
Location of knots. The choice of knot locations has to be done in conjuction
with choosing the number of knots. McCulloch (1971) proposed a location
scheme whereby there are an equal number of observations (bonds) in each
sector. The resulting curves are then homogeneous in degrees of freedom.
If the number of knots is selected based on economic arguments, then
the natural next question is how the boundaries of the sectors are defined.
Litzenberger and Rolfo (1984) used maturities of 1, 5 and 10 years as a set
of natural knot locations. Langetieg and Smooth (1988) fine-tuned the short
end of the curve by adding a knot at 0.5 years. To determine the optimal
static location scheme properly, one would need to study the historical shapes
of the term structure of interest rates. Unfortunately such studies should then
15
be reviewed from time to time as the behaviour of the market participants
changes and markets develope structurally and technically.
Tests made by Langetieg and Smooth (1988) indicate that the use of a
static knot point location scheme resulted better estimates than the McCulloch
variable location scheme.
The third possibility would be a variant of the McCulloch method, using
optimization to determine the locations. An applicable algorithm can be found
in deBoor (1978 pp. 218 - 222) and Powell (pp. 298 - 311). However if
the location is determined by an optimization algorithm, the possibility to
"intuitively interpret the parameters disappears. Also, the set of parameters
increases and they are far less stable.
Model deficiencies Most criticism against the spline models has focused
either on the fluctuation of estimated forward rates, the asymptotic behaviour
of the rates or the lack of theory giving it a reputation as a "practitioners
approach. In response we can state that the fluctuation of the rates can be
controlled via good choice of knots and locations. However, the asymptotic
behaviour is inevitable: the discount function is a sum of polynomials, so it is
not finite when the term to maturity increases.
2.4 Nelson-Siegel model
The idea of cubic splines was to obtain as good fit as possible to the market
data. It is as atheoretical as possible: the prices reflect all information that is
relevant to the term structure. On the other hand, Nelson and Siegel (1987)
attemted to minimize the number of parameters to be estimated by assuming
that the instantaneous forward rate follows a simple, yet flexible, deterministic
process: second-order linear differential equation with real and equal roots.
1
Hence, the instantaneous forward rate at maturity m is expressed as
J ( m) = $0 + $1 exp ( - 7) + $2 7 exp ( - 7), (2.13)
where T is time constant associated with the equation, and $0, $1, and $2
are determined by initial conditions. In practice, these four parameters are
estimated daily. Calling equation 2.6 that the spot rate is the average of the
forward rates
1i
m
s ( m) = - J ( t) dt.
m 0
By integrating, we obtain
1 - exp( - ~ ) ( m)
s ( m) = $0 + ($1 + $2) '; - $2 exp - --:; .
(2.14)
10riginally Nelson and Siegel assumed that the roots are unequal, but they noticed
quickly that fitting an unequal roots model to the data lead to an overparameterized model.
16
The main advantage of the Nelson-Siegel method are its "intuitive" asymptotic
properties
lim f(m) lim s(m)
m-oo m-oo
lim f(m)
m-O
lim s(m)
m-O
That is, both spot and forward rate approach a constant for both long and
short maturities. Figure 2.3 presents the component functions of the Nelson -
Siegel specification.
Figure 2.3: Components of 6 parameter Nelson-Siegel model
Level
I
t
- - - - Steepness
- - - - - Hun-lp 1
Hurnp2
For"""ord
o k
,,;;-- 10 15 20 2S 30
I
' ".' ,-
l " ..... '
I
' I I" , , ,
-2 +Y ,f

r
_" ...L
Maturity
Another ad\'anta!!(' of their approach is that there seems to be a close
correspondence Lf't\\t'"t'r; the components of the Nelson-Siegel model and the
findings of Litterma.n and Scheinkman (1991), whofound using factor analytic
approach that three factors explain most of the observed variation in bond
returns. These were "level, "steepness and "curvatuT'e
2
.
Nelson and Siegel estimated their model by fixing T, after which ordinary
least square estimation can be used to get the ps. The procedure was re-
peated for a wide number of TS, and the one with best fit was chosen. Another
2The "level factor explains 89.5% of the total variation in US Government Bond sector
returns, "steepness explains 8,5% and "curvature explains 2.0%. For other international
government bond markets the figures are quite similar.
17
approach- is to estimate all parameters simultaneously using a nonlinear es-
timation method (maximum likelihood, nonlinear least squares, generalized
method of moments). Section 3 explains in detail how these methods are
related.
2.4.1 Extended models
The Nelson-Siegel specification could be criticized that it is not flexible enough
to describe all the detailed shapes of the curves. Thus a number of extended
specifications has been proposed. Svensson (1994 a) increased flexibility by
introducing an additional hump term to the original specification. He used
the following extension:
f(m) = /30 + /31 exp (-m) + /32 m exp (- m) + /33 m exp (_ m), (2.15)
71 71 71 72 72 .
Bliss (1991) used the following five parameter version
f(m) = /30 + /31 exp (- m) + /32 m exp (- m).
71 72 72
(2.16)
In the Bliss extension "steepness and "hump terms are completly independent
compared to the original specification. It is a simplyfied version of the six-
parameter form where the third component is dropped off.
2.4.2 The Bank of Finland Method
The old Bank of Finland model used a B-spline approach to construct the
discount function. Due to poor asymptotic behaviour and difficulties in inter-
pretation of model parameters which will be discussed in the following section
2.5.5 we decided to switch to a modified Nelson - Siegel model. As the func-
tional form we selected the extended form proposed by Svensson (1994), where
the spot rates are
s(m)
1 - exp( - [1 - exp( - (m)]
/30 + /31 * + /32 * - exp --
m m T
1.

(m)]
+ /33 * m - exp --
'T2 72 .
(2.17)
The optimization is done using a variable metric method, namely an algo-
rithm called Broyden - Fletcher - Goldfarb - Shanno(BFGS). The model
is written in standard C language for a SUN UNIX environment.
We also tested the implementation made by Svensson(1993), where the
Nelson-Siegel method was implemented in its basic four-parameter configura-
tion as well as in "enhanced six-parameter configuration. The implementation
was done on GAUSS using maximum likelihood methodology. 3
3We would like to thank L.E.O Svensson for providing his GAUSS code for us.
18
The original GAUSS code was not able to handle zero-coupon bonds outside
the money markets (STRIPS) with maturing over one year: We fixed this
shortcoming. Also the computer code was not able to handle instruments that
are in their ex-dividend period. This feature made it impossible for us to test
the model with UK data properly.
We found the implementation quite slow for our purposes: computing time
increased exponentially with the number of instruments. Further the enhanced
six-parameter form often failed to converge. Finally, we found that the estima-
tion methodology (and probably convergence) was highly sensitive to pricing
errors in the input data.
2.5 Comparison of methods
As stated at the start, there is no absolutely correct way to estimate the term
structure. The purpose of the estimation dominates the selection of estimation
methodology. One can only say that there may be correctly estimated term
structures for specific purposes.
2.5.1 General
The properties that are required from a good estimation methodology are
1. Good fit; ie accuracy and precision
2. Parsimony, intuitivity, logicality, opaqueness
3. Robustness and stability.
Bliss(1991) presents an excellent study on testing different estimation method-
ologies from the bond pricing point of view. He first analyzed the methods
based on the coefficients of determination (R2) and hitratios. Then he split
his datasets into two blocks and performed extensive out-of-the-sample tests
to analyze the sensitivity of the estimates. In his tests, the Fama - Bliss ad-
justed bootstrapping method and extended Nelson - Siegel-method ranked high-
est. Spline methods tended to overfit bond prices and their rates were unstable
in out-of-the-sample tests.
Buono, Gregory-Allen and Yaari (1992) generated four "typical shapes of
the term structure of forward rates: fiat, increasing, inverted and humped
(with some realistic restrictions). Then a set of bonds was priced based on
these term structures with a small stochastic bond specific error, allowing
them to use Monte-Carlo simulation to test which of the estimation method-
ologies produced the closest forward rate estimates. They found that discrete
point methods produced more exact estimates than the exponential polyno-
mial method. The methods presented in our paper were not included in their
work.
19
2.5.2 Modeling considerations
Basically there are three major "structural decisions to be made before the
estimation can be done.
First, one must have some underlying assumptions about the functional
form of the discount function or the spot rates. These define the smoothing
principle between rates, and may imply some fundamental mathematical re-
strictions on asymptotic behaviour of the rates. There are theoretical reasons
4 to select such a functional form that produces smooth second derivatives and
finite and even bounded asymptotes. Beyond these, selection is fairly arbitrary
and mainly done ad hoc.
Second, the weighting of the observations must be decided. This issue is
fundamental to the homogeneity (or quality) of the curve. Because of the
nonlinear relationship between (observed) prices and (calculated) curves, the
minimization of pricing errors leads to heteroschedastic yield errors, and hence,
unreliable yield estimates in the short end of the curves. More homogeneous
curves in the yield terms could naturally be reached by minimizing yield errors,
but then the price errors are heteroscedastic. The third possibility is to weight
the bonds by reciprocal of the bid-ask spread, which gives us the "benchmark
or "on the run (or most liquid) curves.
Finally, there is the question of the robustness of the estimation. Especially
when estimated curves are used to provide information for pricing tools in a
realtime environment. they should be robust in the sense that individual errors
in the bond input data do not cause radical change in the results. The obvious
choices are to use standard least squares estimation or more robust estimators
based on median estimation error.
2.5.3 Methodology of comparison
Although the selection of an estimation method is usually made on the basis
of good fit, there are Sf>\·eral other items in the background that are strongly
present in the decision situation. When selecting the methodology to estimate
the term structure of interest rates the alternatives should be thought at least
from the following p o i n t ~ of view:
1. Goodness of fi t
2. Quality of cun·es
3. Asymptotic behaviour of rates
4. Stability of parameters
5. Compactness of the model
4Smoothness, continuity and non-negativity ofthe forward rates are the typical economic
characterizations of a "sensible term structure.
20
Goodness of fit The ability to explain bond prices properly is the most
important feature of the estimation method. It is also the easiest to quan-
tify. One can use standard econometric measures such as the coefficient of
determination (R2), rooted mean square error (RM S E) or mean absolute de-
viation (M AD). RM S E and M AD should be adjusted. Because the bid-ask
. spread produces a range of feasible "zero-error prices, the possibility to fit
prices properly increases and an additional measure of goodness-of-fit can be
used. H itrate is defined as the fraction of bonds with zero error compared to
the total amount of bonds in the sample.
Quality of curves This could be characterized by the amount of fluctuation
or smoothness of the (forward) curves. According to Adams and van Deventer
(1994), the smoothness of the forward curve can be measured by a functional
z = iT [J"( S )]2 ds (2.18)
One should note that this measure penalizes the changes of the slope of
the forward rate in an equal manner in every part of the forward rate. This
might be a little unacceptable based on the assumption that we have more
information or stronger expectations on the behaviour of the curves in a short
time horizon (eg 1-24 months) than longer horizon (eg 15-30 years). Instead
the non-fluctuating criterion should be modified in a way that it is less affected
by strong movements at the short end but is very strict at the long end of the
curve. If we slightly modify equation 2.18 we get a functional
z = iT [s * J"(S)]2 ds (2.19)
The result is that the amount of fluctuation is weighted by maturity and we
achieve the desired flexibility at the short end.
Asymptotic behaviour of rates The mathematical characteristics of the
functional form tend to have a strong impact on estimated rates, especially in
situations where the data sample is limited or heavily concentrated in certain
parts of the maturity axis. This is almost always the case when the rates
beyond 10 years are estimated for markets other than the US.
Stability of parameters The model parameters should be "stable and
somehow predictable in their behaviour. This is normally the case when the
model is properly specified. In the case of an overparametrized model the
fluctuations of the parameters can be large.
21
CompaCtness of the model The compactness of the model can be mea-
sured by Information criteria. Consider equations (2.1) and (2.2). One can
easily see that if the spot rates s(t) in equation (2.2) and yields Yj are exact
transformations of each others the error term Cj vanishes. This situation can
be achieved when either the markets are perfectly efficient or when the estima-
tion model is too flexible and thus overspecified. To avoid such overfitting, the
quality of fit eg the accuracy of the model is measured by adjusted coefficient
of determination R. The first equation is used when standard mean squared
error methodology is used, the second equation gives the adjustment when a
more robust mean absolute price error methodology is used.
n +Kj 2
1 - n _ K. * (1 - Rj )
J
(2.20)
R. n + K
j
(1 R)
J - I-
n
_
K
* - j
J
(2.21 )
where n is the number of observations (bonds), Kj is the number of parameters
in model j and Rj is the coefficient of determination of model j.
2.5.4 Data description
Our test data consists of the prices of USD, DEM, FRF, GBP and JPY gov-
ernment securities. From each of these markets we have included only plain
vanilla domestic issues, ie those with no special characteristics such as em-
bedded options, exchangeable issues or bonds with abnormal coupon periods.
We have separated the strips data (STR) from the US bond data (USD) as
an individual data set, which means that we have actually fitted two different
curves to our US data: US strips and US coupons and bills. The French strips
are not separated from the coupons.
Our bimonthly data covers the period from 25 November 1993 to 17 Jan-
uary 1995 with some dates missing during summer 1994 due to technical prob-
lems. The total amount of sample dates is 23. We have collected the data from
Bloomberg information system using "the most reliable price contributor. We
have not applied any price filtering procedure to the data, so the data may
contain pricing errors and could be characterized as "dirty data. Table 2.1
describes the key characteristics of our data sets
5
•
The DEM data sets are heavily concentrated below the IQ-year sector of
the curve. This is due to the issuing policy of Bunds and Bobls (German
government securities). 10-year Bunds and 5-year Bobls are issued irregularly,
but typically from three to five new issues are introduced during a calendar
year. We included the instruments issued by Treuhandanstalt into our data
sets.
5The Bonds, Maturity, Coupon and Spread columns give the average figures over our
sample period. MaxMat gives the maximum maturity for the whole sample.
22
Currency Bonds Maturity MaxMat Coupon Spread
DEM 114.4 3.9 22.8 7.0 0.083
FRF 98.6 10.1 29.4 3.5 0.161
GBP 44.2 6.5 23.8 10.3 0.087
JPY 107.4 6.8 20.3 5.2 0.001
STR 116.1 14.5 29.7 0 0.268
USD 208.2 5.5 29.7 6.7 0.059
Table 2.1: Characteristics of the data sets
The FRF data consists of four "subsectors: BTFs (treasury bills 1-4 in-
struments per month), BTANs (2-5 year coupons; 2-4 instruments per year)
and OATs (old 10-30 year coupons, 1-2 per year below the IQ-year sector and
1 issue every 4th year beyond the lO-year) and OATstrips (two instruments
per year up to 30 years). The bid-ask spreads of the OATstrips are typically
wider than others.
The GBP data set is the smallest, in part because the market has quite
many excluded issues with special characteristics. The distribution of the
instruments is similar to DEM data.
The JPY data set is exceptional among our test sets because we were
unable to get two-side prices in electronic format for most issues. This can be
seen from the narrowness of the bid-ask spreads.
The STR data consists of evenly distributed quarterly instruments up to
30 years. The spreads are widest which is later reflected in the high hitratios.
The USD data set is the largest and the best in terms of quality.
Shapes of the term structures At the beginning of the sample period
the DEM and FRF curves were humped, with USD, STR and GBP curves
increasing in "normal way. During the period the steepness of the curves in
European markets increased significantly, while the USD curves flattened. At
the end of the period GBP, USD and STR rates were all quite flat.
The DEM and STR curves are usually humped. In the DEM curve it is
due to the futures market. The bond issues that are deliverable as BOBL- and
BUND-futures always tend to be expensive against the neighbouring issues in
the cash market. In the STR curve the hump is due to the high convexity
of the long STRIPs. High convexity means greater possibilities of additional
return and that is penalized by lower yield.
Generally speaking, the shapes of the USD and STR curves are very similar.
This is due because of the constant coupon stipping and reconstitution activity
that prevents these two markets from deviating very far from each other.
The figures in Appendix A presents the development of rates during the
sample period in different markets.
23
2.5.5 Results of comparison
The models that we compared in our tests were B-spline models with variety
of different knot specifications and a N elson-Siegel approach using the basic
four parameter form.
Goodness of fit Goodness of fit is here measured by four measures: the
mean absolute deviation (MAD), rooted mean square error and hitratio. The
fourth measure is non-zero pricing error (NZRMSE), that is calculated as
RMSE of non-zero errors. The underlying idea is that hitratio captures all
perfect fits so NZRMSE should be more sensitive to remaining, true errors.
(This is naturally affected by functional specification and degrees of freedom).
For the spline approach, this means the number and the location of knot points,
in the case of the Nelson-Siegel specification, whether the extended (five-six
parameter) version or the basic model is used.
Table 2.2 presents the RMSE, NZRMSE, Hitratios and MAD for different
methodologies sorted by currency. The general differences between the markets
are quite easily seen. The USD and STR markets are the most efficient, while
in the GBP and JPY markets the errors are high due to taxation and par
effects. Typically, the B-spline method is superior here because of its higher
flexibility. This can be clearly seen from the hitratios. One should, however,
note the small differences, especially in the NZRMSEs reflect the good ability
of the NS4 method to price all issues relatively well.
Quality of the curves Table 2.3 presents a summary of the smoothness
statistics of the estimates. The NS4 methodology gives good results in all
markets. B-spline results are reasonable as long as the model is properly
parametrized.
Asymptotic behaviour of forward rates Table 2.4 summarizes the asymp-
totic behaviour of forward rates in different specifications. One can easily see
that the spline method should never be used for extrapolation due to its poor
asymptotic properties. Note that, although the rates in the Nelson-Siegel ap-
proach are always limited, there are no limitations to guarantee non-negative
forward rates.
Stability of parameters Figures 2.4 and 2.5 present the time path of the
model parameters. Interpretation of the parameters of the NS4 model is fairly
straightforward, especially when the parameters are compared to the evolution
of rates. Note the changes in the level of the yield curve, the increase in
the steepness of the curve in early 1994 and the corresponding changes in
parameters /30 and (31.
24
Currep.cy Label Bsp4 NS4 SP
DEM RMSE 4.4097 4.5632
NZRMSE 5.8246 5.2864
HITRATIO 0.2383 0.137
MAD 17.6346 23.2465
FRF RMSE 3.192 3.4197
NZRMSE 5.0513 4.1233
HITRATIO 0.3893 0.1952
MAD 13.7407 20.19
GBP RMSE 8.786 8.2285
NZRMSE 10.281 9.5544
HITRATIO 0.139 0.1324
MAD 29.8072 32.48
JPY RMSE 7.6059 7.4933
NZRMSE 7.6133 7.4981
HITRATIO 0.0012 0.0004
MAD 46.9511 52.31
STR RMSE 0.8751 2.8899
NZRMSE 3.4449 4.0173
HITRATIO 0.6864 0.2874
MAD 2.1235 16.15
USD RMSE 1.5798 2.2352
NZRMSE 2.1604 2.4887
HITRATIO 0.2621 0.096
MAD 7.4257 16.24
Table 2.2: Goodness of Fit
Currency Bsp4 NS4 SP
DEM 0.0485 0.0418
FRF 0.1006 0.0098
GBP 0.2258 0.0511
JPY 0.0422 0.0327
STR 0.1443 0.0085
USD 0.0957 0.0211
Table 2.3: Quality of the rates
Model fwd(oo) determination
B'fvspline ±oo WN
NS4 limited
130
Table 2.4: Asymptotic behaviour of the models
25
Figure 2.4: DEM Parameters using NS4 model
EVOLUTION OF THE RATES
10.0 -,-----------------------
I
7.5 --'
5.0 -i
2.5....!
"'''''''- ......... _----
/I
, ,
/ \ /
'/' \ .. /,'
/, \ \ / I
,1/ ," '_ / ,
, '\ -'.-J
f '/'
,-------1... 'v
,/ I -,....-----,
".,-,'" ,..-- "-,,,,--
__ =--==-==--=: = :=-...-------- _____ ./" ,1
/
0.0 --'---,.,,-_---=-,=-=-_:7_=- ____________ J-, -----------c
-2.5 ...:
-5.0 -c
,/-
,r -_--
/
4 7
... - --,
10
I
/
,
/
'",":.",J
f --
f
13
DATESET
,
[
16
,
-
, ,
,
, ,
,
,
"
BETAI)
SETA'
BET.,
TAUl
SPREAD
--
S<O
[
19 22
Figure 2.6 presents the results using Bsp approach when the number of
knot points are increased. The location of the knot points followed McCulloch
strategy. The hitratios, MAD and fluctuation are plotted against the left axis
and R, AdjustedR and AmemiyaR are plotted against the adjusted righthand
scale (R-99.0). The AdjustedR figures show where the point of the maximum
parametrization has been reached. The overfitting is even more dramatically
reflected in the quality of curves measures.
Table 2.5 presents the maximum parametrization of the spline models in
our sample using different "decision rules (see section 2.3). It can be seen that
the McCoulloch location method is not optimal as the adjusted coefficient
figures always give smaller parametrization. One empirical observation: we
found that the knots should be located farther out than McCulloch proposed.
Information criteria Table 2.6 summarizes our results as far as the infor-
mation criteria is concerned. Both methods give extremely good results in
absolute terms. In relative terms, the differences are surprisingly small.
26
Figure 2.5: DEM Parameters using the Bsp4 model
3S -..... i
: i
20
1
1S I

:t--='- -.-.
I

-s
l---
i
-la ---
2.5.6 Discussion
--B1
--=--- B2
---B3
-84
--ss
---.0.-- B6
--B7
Estimated rates are always a function of the input data, which means that the
input data should be as accurate and as homogeneous as possible. Because
the rates are a complex average of the bond data, interpretation of the rates
is justified only if there are enough observations in that sector of the term
structure.
Because both of our estimation methodologies lack theorethical justifica-
tion, the most basic difference between them is probably that in the B-spline
method the curves are constructed from vertical· pieces (the maturity axis is
spEned 1) whereas in the Nelson-Siegel method curves are constructed from
Currencycy McCulloch AdjRSQ
DEM 14 6-9
FRF 12 6-7
GBP 10 6
JPY 13 11
STR 14 11
USD 17 11
Table 2.5: Maximum number of component functions
27
Currency Label Bsp4 NS4 SP
DEM RSQ 99.99707 99.9969
ADJRSQ 99.99691 99.99681
AmemR2 99.99669 99.99667
R 99.82311 99.76584
ADJR 99.81319 99.75945
AmemR 99.79997 99.74881
FRF RSQ 99.99846 99.9984
ADJRSQ 99.99836 99.99835
AmemR2 99.99822 99.99826
R 99.8623 99.79843
ADJR 99.85322 99.79199
AmemR 99.84112 99.78127
GBP RSQ 99.99605 99.99672
ADJRSQ 99.99541 99.99647
AmemR2 99.99455 99.99606
R 99.70035 99.67389
ADJR 99.65176 99.64944
AmemR 99.58697 99.60867-
JPY RSQ 99.99244 99.99294
ADJRSQ 99.99198 99.99274
AmemR2 99.99138 99.9924
R 99.52615 99.47452
ADJR 99.49772 99.45921
AmemR 99.45981 99.4337
STR RSQ 99.99987 99.99893
ADJRSQ 99.99987 99.9989
AmemR2 99.99986 99.99886
R 99.97873 99.83792
ADJR 99.97756 99.83358
AmemR 99.976 99.82635
USD RSQ 99.99918 99.99874
ADJRSQ 99.99915 99.99872
AmemR2 99.99912 99.99869
R 99.92558 99.83762
ADJR 99.92336 99.83523
AmemR 99.9204 99.83125
Table 2.6: Information criteria
28
Figure 2.6: Effects of parametrization changes in the model
0.6
1
o.s t
I
0 . .4 T
I
I
0.3 t
0.::2
0.1
Porarno1'rtzatlon Bap
0.86
0.78
0.76
--.-------t 0.7.4
ton
o ±::==::::;::::::::==--.--_____ -_------1 0.7
'" "
13 16 le

--MAD
-.0--- Fluc'tuc:::rtlon
---R
---c>- ADJR
Ar"rIerT'lR
horizontal components. This leads the emphasis in the spline methods to be
on flexibility and accuracy, while in the Nelson-Siegel approach the asymptotic
behaviour and the continuity (in its broad sense) of the rates is emphasied.
This difference is also reflected in how the model parameters are interpreted.
One should bear in mind that the ultimate test of the estimation model is
its feasibility as a trading tool. If the model reflects reality, then rates can be
interpreted in a way that is presented in chapter 3.
2.5.7 Enhanced model specification
There are some small shortcomings to our current model. Most of these fea-
tures relate to some market-specific structural feature, which in many cases
due to legislation or taxation. These features could also be characterized as
causes of heterogenities in the input data.
Coupon effect In GBP markets it is commonly known that high coupon
instruments (at least used to) trade at significant discount compared to the low
coupon instruments. The coupon effect explains about 10 % of the remaining
estimation error. This effect has been very stable and is due to the different tax
29
treatmeiIt of coupon income vs. capital gains for certain market participants.
6
The same feature can also be seen in DEM markets to a more limited extent.
Par effect Both the results of Kikugawa and Singleton (1994) and our find-
ings indicate that in the JGB markets there are many market participants
who prefer instruments that trade close to (or just below) par. This is due
to the tax reasons. These par effects typically explain around 30-50 % of the
remaining price errors otherwise unexplained by our estimation model. U n-
fortunately, the functional form of the par effect is most obviously non-linear
and tends to be time variant as well.
Benchmark effect All markets feature certain benchmark bonds, which are
highly liquid and typically, most recently issued. They tend to have lower
transactions costs, so they are used for market trading purposes. Benchmark
bonds are also typically "special in repo markets.
Repo market specials In all markets where well-established repo market
exists (namely USD, STR, JPY, PRP and DEM) some papers (usually bench-
marks or papers belonging to futures deliverable basket) might trade at a
significant premium compared to their theoretical value. At the same time,
however, they tend to trade "speciaF at the repo market, so the total holding
period returns of these papers are similar or even higher than for· issues in
their neighborhood.
Convexity adjustment Highly convex instruments look often expensive
against their theoretical value. High convexity means greater possibilities for
additional return in exchange for lower initial yield. Without any convexity ad-
justment, the rates calculated by models tend to be strongly downward sloping
at the long end of the markets. The value of convexity can be approximated
by
VCx = 0.5 * Cnvx * E(D.y?, (2.22)
where VCx is the price value of convexity, Cnvx is the convexity of the
bond and D.y is the annual yield volatility. 8
6In 1996, the UK tax system was changed so that domestic institutions will also become
neutral as to coupon income and capital gains. The only exception thereafter will be private
individuals holding less than GBP 200 000 in British government bonds (gilts). This group
only accounts 0.25% of gilt turnover.
7If an instrument trades "special ("special collateral) an investor holding the instrument
can lend the instrument and borrow money against it at a very low "special rate and invest
the borrowed money either to purchase a normal instrument ("generic collateral) or to
deposit markets.
8The size of the convexity bias can be quite large. If the annual volatility is 100 bp
and the convexity of 25 year Strip is 6.45, the yield is 8.1% and the rolling yield 6.95%
(downward sloping yield curve), the value of convexity is 2.88% and the expected one-year
return is 6.95% + 2.88% = 9.83% (Ilmanen, 1995)
30
Variable Mean Std Dev Minimum· Maximum
Mean 0.5641 1.3143 -4.7168 2.5136
StdDev 49.1261 26.6884 17.6891 107.5665
Median 0.2689 2.5403 -6.2 9.3
Minimum -303.127 313.0189 -842.51 -34.63
Maximum 167.0104 83.4142 58.67 366.3
Skewness -1.6128 4.2527 -9.28 5.7124
Kurtosis 27.3405 30.8846 0.8895 95.4287
Table 2.7: Characteristics of distribution of DEM pricing errors over observa-
tion days using the NS4 model
2.6 The distribution of pricing errors
2.6.1 Properties of the error
The distribution of the pricing errors is dependent on the quality of input
data as well as the functional form of the approximation, the weighting of the
observations and the norm used in the approximation.
In order to find the market spot rates that describe the bond data well,
we must calculate the pricing error between the observed price and the model
price. Observed price is not a single point, but rather a range of possible prices
bounded by bid and ask prices for each instrument. The range between the
bid and ask prices, or spread, gives traders a buffer against short-term market
fluctuations or uncertainties in the market. Quite often the estimation of the
term structure of interest rates is based on the "one price law. Thus the true
market behaviour and the information presented in the spread is missed. We
define the price error to be
{
Pja - Pj , if PI > Pj
_ b - • b -
tj - Pj - Pj , If Pj < ~ j
0, if pJ < Pj < Pja
(2.23)
Because of the error term definition, the "dirty data issues discussed in the
section 2.5.4 and the shortcomings of the model discussed in the last section
2.5.7, the distribution of the pricing errors does not follow normal distribu-
tion. Table 2.6.1 shows the typical distribution of the pricing errors using
Nelson Siegel four-parameter approach. The high kurtosis can clearly be seen,
but the distribution is not significantly skewed. The distribution is heavily
concentrated towards the cent er but the tails are fat.
Based on the definition of the error term, one could assume that the prob-
ability distribution is close to the double exponential distribution in general
shape although the tails should be thinner.
9
Empirically, we found that the
9 A double exponential distribution is basically a normal distribution that is cut, but
because of the trading spread (that affects to our error definition) the middle part of the
distribution is concentrated towards the centre with the tails moved inwards.
31
probability distribution function to be
p(x.) = ). * e-)."'-VX
which hence indicates much thicker tails than one would expect.
(2.24)
In such of situations, a robust estimation methodology is needed. One can
use the sample median as a description of the cent er of the distribution in
favour of the mean. The same issue is reflected in confidence intervals. The
width of the confidence interval range of the median are in our case about 1/4
of the confidence intervals of the mean.
We used the more robust approach in our spline model. When the Lrnorm
is used in approximation the linear programming approach can be used. As
Powell (1980, pp 183-186) has demonstrated the equality of these approaches,
we utilized it with our spline model. Gonin and Money (1989) review differ-
ent guidelines on how to choose the optimal exponent p in general Lp-norm
estimation. Most of these guidelines are based on the kurtosis /32 of the distri-
bution. As an example, the rule proposed by Rarter recommends the use of
L1 if kurtosis /32 > 3.8, L2 if 2.2 < /32 < 3.8 and Loo if /32 < 2.2. Other rules
yield fairly similar results.
Sometimes the estimation errors for the short-term instruments are signifi-
cantly skewed. This is due to the fact that the cash flows of these instruments
are discounted at the same rate as the coupon cash flows of the longer ones.
Note that the individual cash flows of the long papers are not as liquid as the
short instruments. Chambers, Carleton and Waldman (1984) have reported
similar observations.
2.6.2 Relative value models
Relative value arbitrage models used to identify small temporary price dis-
crepancies' are mainly based on the assumption that the pricing errors of each
individual bond are mean re\·erting. It is assumed that all instrument-specific
features (as discussed in section 2.5.7) determine an individual "fair yield
spread ~ j (t) at time t. It is calculated as an average of the bond pricing
errors fj(t) ... fj(t - T j in !Jltld terms from the last T observations. The relative
value of an individuall,ond j at time t+ 1 is given by the Studentized deviation
(2.25)
where CTtj(t) is the standard de\·iation of the previous yield errors. Several trad-
ing rules can be formed based on the relative value measures. The performance
of the rules is, of course. highly dependent on transactions costs, because the
deviations are surprisingly small in the marketplace. One can, however, say
that the ultimate test of a term structure model is its applicability as a trading
model.
32
Chapter 3
Interpretation of the Term
Structure
3.1 Expectations and premia
3.1.1 Expected consumption growth
In the introduction we mentioned that recent empirical studies indicate that
the term structure predicts consumption growth better than vector autoregres-
sions or leading commercial econometric models. This may sound surprising,
but actually it is simple consequence of basic (neoclassical) economic models.
Almost all dynamic macroeconomic models produce a Euler equation which
links the current consumption with the future consumption according to a
relation
l
(3.1 )
where Et denotes the expectation operator with respect to It, the common
information set of the agents at period t, Ct is the consumption at period i,
f3 E (0,1) is a constant discount factor, u(·) is the household's one-period
utility function, and Rt is the It measurable (risk-free real) gross rate of return
on bond holding.
The economic content of Eq. 3.1 is same as in all economic models with
optimizing agents: the decisions are varied until the marginal losses (costs)
are same as marginal gains (returns). That is, Eq. 3.1 states that the current
consumption decision is optimal if the marginal loss in utility today (left-hand
side of equation), ie when one unit of consumption is allocated from today to
tomorrow, is same as the marginal gain in utility tomorrow (right-hand side
of equation).
By rearranging Eq. 3.1, we obtain
1 = f3R E [u
l
( CHI )] (3.2)
t t u
'
(et) .
lThis approach to asset pricing was pioneered by Lucas (1978). For a very readable
introduction, see Sargent (1987, ch. 3).
33
This is justified as ut (Ct) is in the agents' information set at time t. Moreover,
we assume that the agents have a constant relative risk-averse utility function
Cl-er - 1
u(c)= 1 '
-0-
where 0- is the agents' constant coefficient of relative risk-aversion. It is usually
assumed
2
that 0- is a constant between 1 and 10. Using this specification, Eq.
3.2 reduces to
(3.3)
Suppose now that the agents expect that CH1 will go down relative to Ct or
it is expected that (ct!Ct+1Y will go up. Since fJ is constant, the only way that
(3.3) can hold is if Rt goes down. Economic reasoning is simple: if interest
rates are high when consumption is high relative to the future consumption,
everyone wants to save. In aggregate they cannot, because the endowment of
the economy is fixed. Therefore, they will bid the interest rates down until
everyone is happy consuming Ct. Thus, we can make a simple rule-of-thumb:
upward-sloping term structure forecasts economic recoveries, downward-sloping
term structure forecasts economic recessions.
3.1.2 Expectations and risk premia
The oldest and simplest theory about the information content of the term
structure is the (pure) expectations hypothesis. According to the pure expec-
tations theory, forward rates are unbiased predictors of future spot rates. It is
also common to modify the theory so that constant risk-premium is allowed-
this is usually called the expectations hypothesis. Next we will investigate this
assumption using modern macroeconomic theory.
Assume that the representative agent has access to both one-period and
two-period bonds.
3
The Euler equations associated with them are
(3R E [u
t
( C
t
+1)] 1
1t t ut(Ct)
(3
2R E [U
t
(CH2)] = 1,
2t t t() ,
U Ct
2See, e.g., Mehra and Prescott (1985) for references.
3Some of the discussion below follows Sargent (1987, section 3.5) very closely. For a very
readable continuous-time discussion, see Ingersoll (1987). The most important continuous-
time models are by Cox, Ingersoll, and Ross (1985) and Longstaff and Schwartz (1992).
Duffle (1992) presents a general framework for continuous-time models. Den Haan (1995)
compares continuous-time and discrete-time models. For other important theoretical dis-
cussions of the term structure, see Cox, Ingersoll, and Ross (1981), Breeden (1986) and
Campbell (1986).
34
where- Rlt and R2t are It measurable gross rate of return on one-period and
two-period bonds, respectively. These imply
f3E
t
[U'(Ct+
l
)]
u'( Ct)
13
2
Et [U'(Ct+2)].
u'( Ct)
Recalling the definition of the spot rate Eq. 2.2
P- _ Pj
J - (l+sj)tN
By normalizing the nominal value of bond, Pj , for both bonds to one, and
letting Rlt = (1 + SI) and R2t = (1 + S2)2, we obtain the following pricing
relations
- f3E
t
[u'(Ct+1)]
u'( Ct)
13
2
E [U'(Ct+2)]
t u'(Ct) ,
(3.4)
(3.5)
Next, using Eqs.3.4, 3.5, and the law of iterated expectations, Et [Et+l [Xt+2]] =
Et[Xt+2], we obtain
P2t = 13
2
Et [U'(Ct+2)]
u'( Ct)
132 E [u'(Ct+l ) . u'(Ct+2)]
t u'(Ct) U'(Ct+l)
_ E [13 u'( Ct+1) . 13 u'( Ct+2)]
t u'(Ct) u'(Ct+1)
Et .Plt+1].
(3.6)
Eq.3.6 can be further decomposed using the definition of conditional co-
variance, COVt[Xt+b Yt+1] = Et[Xt+1Yt+1] - Et[Xt+1]Et[Yt+l], and eq. 3.4
P2t = Et Et[PIt+1] + COVt 'PIt+1]
P"E,[P"+l] + cov, [" , P"H].
(3.7)
Eq. 3.7 is a generalized version of the expectations theory of the term
structure. The first term is the expectations model. The second term is a
risk, liquidity or term premium. If one-period bonds have a high price when
U'(Ct+l)/U'(Ct) is low (note that COVt[,BU'(Ct+l)/U'(Ct),Plt+l] is negative), and
hence consumption growth is high, then holding a two-period bond is not a
good strategy. If at t + 1 you do happen to get a negative income shock,
35
you would like to sell your bond and consume a little more. But with nega-
tive correlation the price of two-period bond will be especially low. It would
have been better to buy two one-period bonds instead. Hence, when there
is positive covariance between the consumption growth and the price of the
one-period bond, the two-period bond price is driven down (it becomes worth
less than earlier). Using eq. 2.2, this means that the two-period yield is driven
up. Two-period bonds have to promise a higher yield to compensate for this
risk. Summarizing: when there is positive covariance between the consumption
growth and the short-term bond prices, long-term bonds will carry a positive
risk premium. Moreover, as conditional covariance is taken with respect to all
available information at time t and as this information changes over time, so
will the conditional covariance. That is, risk premium by its nature must vary
over time.
Eq.3.7 implies also that the expectations model holds only in special cases.
One special case is when the utility function is linear in consumption. That
is, people are risk neutral with respect to consumption. This means that
U'(CHI)/U'(Ct) = 1 Vt and COVt[,Bu'(CHl)/U'(Ct),Plt+1] = 0. A second case is
when there is no uncertainty: COVt[,Bu'( CHl)/U'( Ct), P1t+1] = O. Hence, as long
as people are risk averse, the world is uncertain and bond prices correlate with
consumption growth, bond prices will carry a risk premium.
3.1.3 Convexity bias
Suppose now, for the sake of an argument, that people are risk-neutral. Eq.
3.7 reduces to
or
(3.8)
Let FIt - 1 denote the forward rate at period t of one-period bond from pe-
riod t + 1 to period t + 2. Using eqs. 2.2 and 2.5, we obtain
or
1
FIt = .
Et [Rl!+J
(3.9)
Note that due to Jensen's inequality (E[x-
1
] > (E[X])-l for x E (0,00))
36
which implies
Hence, the implication of eq. 3.9 is that even when the agents are risk neutral,
The result is called convexity premium or bias. Due to the convex relationship
between the bond price and the bond yield, forward rates are not equal to the
expected spot rates even when we assume risk-neutral investors.
3.1.4 Summary
Summarizing the results of this section, we note that the forward rates are
equal to the sum of expected spot rates, risk premium, and the convexity bias.
Risk premium will tend to make forward rates higher than expected spot rates,
whereas the convexity bias will tend to make forward rates lower than expected
spot rates.
3.2 Forward interest rates as rough indica-
tors
3.2.1 Interest rate expectations
In this section we will assume that neither risk premia nor convexity bias exist.
Forward interest rates are unbiased predictors of future spot rates. We will
show how to read interest, inflation, and exchange rate expectations in this
unrealistic case. However, keeping in mind that the premia do exist, some
useful intuition about the real expectations should be possible to obtain using
(pure) expectations hypothesis as a guide.
Figure 3.1 shows one-month forward interest rates for Germany and Fin-
land as of 10 August 1995. The x-axes show the settlement day as years ahead.
That is, settlement day 1 corresponds to 10 August 1996. If we assume that
the pure expectations hypothesis holds, on 10 August 1996 the one-month in-
terest rate in Finland will (or, to be more precise, is expected to be) roughly
5% in Germany and roughly 7.5% in Finland. Moreover, the Figure shows
the whole time-path of one-month interest rates: settlement 0 is the current
one-month interest rate, 4.5% in Germany and 5.8% in Finland, and from then
on they are expected to rise to 7% in Germany and to 9% in Finland in five
years. Their difference is shown in Figure 3.2. Again, if we assume that the
term premia are zero, we get the expected time-path of difference between
Finnish and German one-month interest rates.
37
Figure 3.1: Forward rates as indicators
One Month Forvvarct Rates
:
:=. -------

o 0.5 2 2.5 3 3.5
$eH_em_ft' CV_Clra to the Futur.>
'-FIM -DEMI
.... 5 5
However, we probably can talk with more confidence about the expecta-
tions when we consider differences. Remember from the previous section that
the forward rates are equal to the sum of expected spot rates, risk premium,
and the convexity bias
where 'Pt,t+1 denote5 the term premium, the sum of risk premium and the
convexity bias, for one· period bond from period t to period t + 1. Now, let
* 's denote foreign (German) variables. The difference between the Finnish
forward rate and German forward rate will be
or
Et [Rlt+l] - = Fu - FI*t - ['Pt,HI - 'P;,t+I] ,
If we assume that YU+l :::::: Y;.t+1 we get
Et[RIH1] - :::::: FIt - Fl*t·
(3.10)
Eq. 3.10 show that. assuming that the term premia are roughly the same
in Finland and Germany, the expected time-path of difference between the
Finnish and German one-month interest rates can be seen from Figure 3.2.
38
Figure 3.2: Forward rate differentials as indicators
One Month Forvvard Rate Difference (FIM-
DEM)
:::t

"'" '.6 T "'____.____
"1.4 i
'.2
,
o 0.5 '1.5:2 2.5 :3 3.5 4.5 5
S_ttl_ment' <V_c. ... 1'0 'th_ FU1'ur.>
3.2.2 Inflation rate expectations
Assume now that we have a market for real or index-linked bonds, ie bonds
whose interest rate depends on the current inflation rate. Using the analysis
of previous section, *'s denoting the index-linked variables, we observe that
the difference between the forward rate of the nominal and the real bond is
approximately the expected difference between their future spot rates. Since
the nominal and index-linked bonds are issued by the same government, it
may seem reasonable to assume that the only source of difference between
their expected spot rates is due to the inflation expectations. In other words,
the Fisher equation would hold and the expected future inflation rate would
be the difference between the expected nominal interest rate and the real rate.
However, as Svensson (1993) points out, just as forward rates have a risk
premia over expected spot rates, nominal bonds have a inflation risk premium
over real bonds.
Moreover, the analysis is complicated by the fact that currently only the
Bank of England issues index-linked bonds. British real rates have usually
fluctuated between 3 and 4 per cent. Hence, Svensson (1994b) assumes that
an expected future Swedish short real rate is around 4%, and the inflation risk
premium is zero. If we make the same assumptions, by subtracting 4% from
the lines in Figure 3.1 we obtain the expected time-path of future inflation
39
rates in Germany and Finland. In any case, as Barro (1995) states
"The best and most objective sign that inflation is about to rise
is a rise of yields on conventional gilts relative to those on indexed
gilts. ( ... )
Such information is available to inform policy because the Bank
of England is the world leader at issuing, studying and perfecting
index-linked securities. I wish the US Federal Reserve was as ad-
vanced."
3.2.3 Exchange rate expectations
Finally, assuming that in addition to zero term premia uncovered interest
parity also holds-ie the risk premium from foreign exchange rate risk is zero-
the differences between forward rates of two countries equals the expected
future depreciation rate of the domestic country relative to the foreign country.
Under above assumptions Figure 3.2 shows the time-path of how much Finnish
markka is expected to depreciate relative to German mark.
3.3 Significance of premia
The empirical reserch on the term structure of interest rates has concentrated
on the (pure) expectations hypothesis. That is, the question has been if for-
ward rates are unbiased predictors of future spot rates. The most popular
way to test the hypothesis has been running a linear regression (error term
omitted)
St+l - St = a + b(ft+1,t - St).
The pure expectations hypothesis implies that a = 0 and b = 1. Rejection of
the first restriction, a = 0, gives the expectations hypothesis: term premium
is nonzero but constant.
Yet, from the earlier discussion we have seen that even in principle this
should not be the case. By and large the literature dismisses both restrictions.
4
Rejection of the second restriction, b = 1, requires, under the alternative, a
risk premiumS that varies through time and is correlated with the forward
premium, jt+1,t - St. Both implications are consistent with the theory pre-
sented above, and most studies-( eg Fama and Bliss (1987) and Fama and
4The literature is huge. Useful surveys are provided by Melino (1988), Shiller (1990), and
Mishkin (1990b). The most important individual studies are probably Shiller (1979), Shiller,
Campbell, and Schoenholtz (1983), Fama (1984, 1990), Fama and Bliss (1987), Froot (1989),
Campbell and Shiller (1991), and Campbell (1995).
5The effect of convexity bias is fairly easy to take into account. For more details about
the effect of convexity, see Cox, Ingersoll, and Ross (1981), Ho (1990), and Gilles (1994).
40
French (1989))-take this to indicate the existence of time-varying risk pre-
mium; Therefore, we should ask if there are models which are capable of
generating similar risk premiums to the ones observed in the real time series.
This question is broaded in Backus, Gregory, and Zin (1989). Using a
complete markets model, tehy conclude that the model can account for nei-
ther the sign nor the magnitude of average risk premiums in forward prices and
holding-period returns. Similar puzzles have been obtained for equity premi-
ums by Mehra and Prescott (1985) and for holding-period yields by Grossman,
Melino, and Shiller (1987).
A recent study by Heaton and Lucas (1992) may provide an answer to these
puzzles. They use a three-period incomplete markets model with trading costs
to address the same question. Their answer is that "uninsurable income shocks
may help explain one of the more persistent term structure puzzles" but "the
question remains whether the prediction of a relatively large forward premium
will obtain in a long horizon model.,,6
6See also Shenn and Starr (1994) for evidence about importance of trading costs.
41
Chapter 4
Conclusions
We have described and compared two different methods to describe the term
structure of interest rates. For our purposes an extended Nelson-Siegel ap-
proach seems to be preferred methodology. If, however, a spline methodology
is used one must be careful with parametrization strategy.
We analyzed the distribution of the pricing errors of a simple model and
found that they do not follow normal distribution. Some explanation of non-
normality was explained by the economic and legislative environment as well
as from the behaviour of the market participants.
After deciding on the Nelson-Siegel estimation methodology is suitable as
a base of further analysis, we turned our focus on the interpretation of the
term structure. A couple of simple models of the predicted evolution of the
rates were introduced. After that we went through some of the components
that affect and distort the basic models. As a conclusion we found that a
more comprehensive model that allows time-varying risk premia and captures
all market pecularities is needed to analyze the behaviour of forward rates.
42
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Figure A.I: An example of relative value report
Ticker Maturity Cp Bid Ask MD YTM Model Avg Std Min Max Z-value
bps bps bps bps bps
T_4.2500_10/94 3l. 10. 1994 4.25 100.59 100.63 0.88 3.58 5.5 5.4 3.4 1.4 14.6 0.04
T_11.625_11/94 15.11.1994 11.6 107.44 107.47 0.91 3.59 6.2 9.7 7.2 3.6 30.8 -0.49
5_0_11/15/94 15.11.1994 0 96.648 96.73 0.93 3.62 2.9 2.3 2.7 0.0 7.9 0.25
T_6_11/15/94 15.1l.1994 6 102.19 102.22 0.92 3.64 2.1 3.4 4.0 0.0 14.8 -0.32
T _8.2500_11/94 15.1l.1994 8.25 104.27 104.3 0.91 3.64 1.5 3.8 5.0 -0.8 19.2 -0.46
T_10.125_11/94 15.11.1994 10.1 106 106.06 0.91 3.64 1.2 3.8 5.4 0.0 19.8 -0.48
T_4.6250_11/94 30.11.1994 4.63 100.95 100.97 0.96 3.64 4.2 -7.6 52.7 -225.3 9.4 0.22
T_4.6250_12/94 31.12.1994 4.63 100.98 101.02 1.02 3.68 7.2 7.3 2.1 4.8 12.2 -0.04
T_7.6250_12/94 31.12.1994 7.63 104.11 104.14 1.01 3.69 5.9 8.3 3.1 3.8 14.9 -0.78
T_8.6250_01/95 15.1.1995 8.63 105.27 105.3 1.04 3.76 0.2 2.2 2.6 0.0 8.7 -0.79
T_4.2500_01/95 3l.1.1995 4.25 100.56 100.59 1.11 3.75 3.5 3.0 1.7 0.0 6.9 0.29
T_3_02/15/95 15.2.1995 3 100.3 102.3 1.17 2.75 105.6 115.8 3.8 108.0 122.3 -2.69
Tj.8750_02/00 15.2.1995 7.88 104.41 104.53 1.12 4.07 -16.7 -20.5 12.7 -40.5 -1.1 0.29
T_5.5OO0_02/95 15.2.1995 5.5 102 102.03 1.14 3.78 2.7 1.0 1.5 -0.3 5.0 1.13
T_11.250_02/95 15.2.1995 11.3 108.69 108.72 1.11 3.77 2.2 2.5 3.0 -0.4 9.9 -0.11
T_7.7500_02/95 15.2.1995 7.75 104.61 104.64 1.13 3.78 1.8 1.7 2.0 -0.3 6.1 0.07
5_0_02/15/95 15.2.1995 0 95.585 95.686 1.18 3.8 0.6 0.0 0.0 0.0 0.0 55.10
T_10.500_02/95 15.2.1995 10.5 107.78 107.84 1.11 3.8 0.0 l.6 2.7 0.0 8.8 -0.60
T_3.8750_02/95 28.2.1995 3.88 100.08 100.09 1.19 3.81 0.7 0.5 1.1 -0.9 3.6 0.14
T_3.8750_03/95 31.3.1995 3.88 100.02 100.03 l.27 3.86 0.2 0.6 l.0 -0.3 3.7 -0.40
T_8.3750_04/95 15.4.1995 8.38 105.88 105.91 l.28 3.91 0.0 0.4 1 .1 -1.3 3.8 -0.33
T_3.8750_04/95 30.4.1995 3.88 99.953 99.969 1.35 3.91 0.0 -0.4 0.7 -1.9 0.5 0.59
T_12.625_05/95 15.5.1995 12.6 112.23 112.3 l.34 3.87 3.6 1.9 3.5 -2.1 10.7 0.47
T_5.8750_05/95 15.5.1995 5.88 102.72 102.73 1.38 3.93 -0.3 -1.0 1.5 -5.4 0.3 0.43
T_l1.250_05/95 15.5.1995 11.3 110.25 110.31 1.35 3.91 0.0 0.9 2.6 -3.1 6.9 -0.35
T_l0.375_05/95 15.5.1995 10.4 109.02 109.08 1.36 3.92 0.0 0.4 2.1 -1.8 6.7 -0.21
T_8.5000_05/95 15.5.1995 8.5 106.36 106.42 1.37 3.94 0.0 -0.3 1.7 -4.4 3.2 0.16
5_0_05/15/95 15.5.1995 0 94.519 94.64 1.42 3.93 0.0 0.0 0.0 0.0 0.0 ODO
T_4.1250_05/95 31.5.1995 4.13 100.25 100.27 1.43 3.95 0.0 -8.2 3l.5 -138.0 0.0 0.26·
T_4.1250_06/95 30.6.1995 4.13 100.2 100.23 1.49 3.99 0.0
. -0.3
0.5 -1.6 0.0 0.65
T_8.8750_07/95 15.7.1995 8.88 107.45 107.52 1.47 4.06 -1.9 0.6 1.7 -0.7 6.6 -1.47
T_4.2500_07/95 31.7.1995 4.25 100.31 100.36 l.57 4.05 -1.8 -1.5 0.6 -2.6 0.0 -0.56
T_4.6250_08/95 15.8.1995 4.63 100.91 100.97 1.6 4.07 0.0 -0.7 1.0 -3.3 0.0 0.67
T_l0.500_08/95 15.8.1995 10.5 110.52 110.58 1.54 4.03 0.0 0.9 2.2 0.0 8.4 -0.43
T _8.5000_08/95 15.8.1995 8.5 107.23 107.3 1.56 4.05 0.0 0.3 1.2 -0.9 4.7 -0.27
S_0_08/15/95 15.8.1995 0 93.422 93.563 1.67 4.04 0.0 0.0 0.0 0.0 0.1 -0.31
T_3.8750_08/95 31.8.1995 3.88 99.672 99.703 1.65 4.07 0.0 -0.1 0.6 -1.6 1.9 0.12
T_3.8750_09/95 30.9.1995 3.88 99.594 99.625 1.73 4.11 1.0 -0.2 0.9 -2.6 l.9 1.40
T_8.6250_10/95 15.10.1995 8.63 107.98 108.05 1.72 4.13 0.0 0.8 1.6 0.0 5.5 -0.49
T_3.8750_10/95 31.10.1995 3.88 99.484 99.516 1.82 4.16 -0.6 -1.2 1.0 -3.5 0.0 0.61
S_0_11/15/95 15.11.1995 0 92.337 92.496 1.91 4.13 2.8 0.7 1.0 0.0 2.2 2.19
T_9.5000_11/95 15.11.1995 9.5 109.95 110.02 1.79 4.13 0.8 1.3 1.9 0.0 6.4 -0.27
T_8.5000_11/95 15.11.1995 8.5 108.05 108.11 1.8 4.16 0.0 0.5 1.2 -0.1 4.6 -0.39
T_11.500_11/95 15.11.1995 11.5 113.64 113.7 1.77 4.14 0.0 0.7 1.9 -1.Q 6.3 -0.37
T_5.1250_11/95 15.11.1995 5.13 101.78 101.84 1.84 4.16 0.0 0.3 0.8 O.t:l 3.4 -0.33
T_4.2500_11/95 30.11.1995 4.25 100.11 100.14 1.89 4.19 -0.3 0.0 0.0 0.0 0.0 -25.40
T_9.2500_01/96 15.1.1996 9.25 110.03 110.09 1.88 4.24 0.0 -0.6 0.6 -2.0 0.0 1.04
T_7.5000_01/96 31.1.1996 7.5 106.61 106.67 1.95 4.26 0.0 -0.1 0.6 -1.3 2.0 0.11
49
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EVOLUTION OF DEM RATES
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EVOLUTION OF GBP RATES
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EVOLUTION OF JPV RATES
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55
BANK OF FINLAND DISCUSSION PAPERS
ISSN 0785-3572
1/96 Kari T. SipiUi A Data Communication Network for Administrative Purposes within the
EO. 1996.53 p. ISBN 951-686-492-9. (TK)
2196 Veikko Saarinen - Kirsti Tanila - Kimmo Virolainen Payment and Settlement Systems
in Finland 1995. 1996. 60 p. ISBN 951-686-493-7. (RM)
3/96 Harri Kuussaari Systemic Risk in the Finnish Payment System: an Empirical
Investigation. 1996.32 p. ISBN 951-686-494-5. (RM)
4/96 Janne Lauha OTC-johdannaisetja Suomen oikeus (OTe Derivatives and Finnish Law
1996.98 p. ISBN 951-686-495-3. (RATA)
5/96 Jukka Ahonen- llmo Pyyhtia Suomen teollisuuden rakenne ja hairioalttius suhteessa
muihin EU-maihin (The structure of Finnish industry and its sensitivity to shocks in
comparison to other EU countries). 1996.37 p. ISBN 951-686-496-1. (RP)
6/96 Pekka llmakunnas - Jukka Topi Microeconomic and Macroeconomic InDuences on
Entry and Exit of Finns. 1996.33 p. ISBN 951-686-497-X. (TU)
7/96 Jaakko Autio Korot Suomessa 1862-1952 (Interest Rates in Finland). 1996. 55 p.
ISBN 951-686-498-8. (TU)
8/96 Vesa VihriaIa Theoretical Aspects to the Finnish Credit Cycle. 1996.63 p.
ISBN 951-686-500-3. (TU)
9/96 Vesa VihriaIa Bank Capital, Capital Regulation and Lending. 1996. 52 p.
ISBN 951-686-501-1. (TU)
10/96 Vesa VIhriaIa Credit Growth and Moral Hazard. An Empirical Study of the Causes of
Credit Expansion by the Finnish Local Banks in 1986-1990. 1996. 53 p.
ISBN 951-686-502-X. (TU)
11196 Vesa VIhriaIa Credit Cnmch or Collateral Squeeze? An Empirical Analysis of Credit
Supply of the Finnish Local Banks in 1990-1992.1996.50 p. ISBN 951-686-503-8.
(TU)
12196 Peter Redward Structural Reform in New Zealand: A Review. 1996. 35 p.
ISBN 951-686-504-6. (KT)
13/96 Kaare Guttorm Andersen - Karlo Kauko A Cross-Country Study of Market-Based
Housing Finance. 1996. 45p. ISBN 951-686-505-4.·(RM)
14/96 Kristina Rantalainen Valtion luotonotto ja velkapaperimarkkinat ltaliassa (Government
borrowing and the debt instrument markets in Italy). 1996.29 p. ISBN 951-686-506-2.
(RM)
15/96 Tuomas Saarenheimo Monetary Policy for Smoothing Real Fluctuations? - Assessing
Finnish Monetary Autonomy. 1996.36 p. ISBN 951-686-508-9. (KT)
16/96 Mikko Ayub Elikerahoitus ja talouskasvu. Taloutta kohtaavien sokkien vaikutus
kasvuun ja hyvinvointiin osittain rahastoivassa elakejirjesteimassa (Pension Financing
and Economic Growth. The Effect of Exogenous Shocks on Economic Growth and Utility
in a Partly Funded Pension System). 1996.72 p. ISBN 951-686-509-7. (TU)
17/96 Veikko Saarinen Maksujarjestelmat ja -vaIineet Suomessa (Payment Systems and
Payment Instruments in Finland). 1996.34 p. ISBN 951-686-511-9. (RM)
18/96 Antti Ripatti Stability of the Demand for Ml and Harmonized M3 in Finland. 1996.
31 p. ISBN 951-686-513-5. (TU)
19/96 Juha Seppala - Petri Viertio The Structure of Interest Rates: Estimation and
Interpretation. 1996.55 p. ISBN 951-686-514-3. (MO)

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