Description
The current study uses a present value model that allows for a time-varying expected discount rate in conjunction with a VAR process to decompose real-estate risk.
Journal of Real Estate Finance and Economics, 8: 5-20 (1994)
0 1994 Khmer Academic Publishers
An Analysis of Real-Estate Risk Using
the Present Value Model
CROCKER H. LIU
Stern School of Business, New York University, New York, New York 10003
JIANPING ME1
Stem School of Business, New York University, New York, New York 10003
Abstract
The current study uses a present value model that allows for a time-varying expected discount rate in conjunction
with a VAR process to decompose real-estate risk. The study finds that the variance of unexpected returns accounts
for most of the total risk with cash-flow risk accounting for twice as much of the unexplained real-estate risk
although discount rate risk is also an important factor, This dominance of cash-flow risk is found to result in a
weaker mean reversion process for real estate relative to stocks. Another finding is that real estate investors tend
to become apprehensive about the future when news on future cash flow is good, and thus they demand higher
expected future returns.
Key words: Real-Estate Risk, VAR model, Risk Decomposition, REIT
The analysis of real-estate risk has typically focused on either the understatement of risk
arising from the smoothed nature of appraisal-based returns or the systematic components
of risk from an arbitrage pricing (APT) perspective.’ In contrast, little research exists on
partitioning the volatility of total return using a present value framework. The purpose
of the current study, therefore, is to examine real-estate risk by first decomposing total
risk into expected and unexpected movements in asset prices. The unexpected variation
in total risk is further decomposed into two risk components-changes in rational expecta-
tions of future real dividend growth and future expected returns, which we will hereafter
refer to as cash-flow risk and discount-rate risk respectively. To decompose the variance,
the present value model of (Campbell 198/ 1991), which allows the expected discount rate
to vary through time, is used in conjunction with a vector autoregressive (VAR) process.
Focusing on risk using a present value model framework offers several distinct advan-
tages not available with an APT structure. Most important, the present value model is geared
to how real estate is typically valued since the traditional analysis of real estate involves
analyzing the cash flows of a property using a constant discount rate to obtain a value.
By modifying the present value to allow the expected discount rate to vary over time, we
can directly assess how the value of the property fluctuates over time by examining varia-
tions in the cash flows, variations in the expected discount rate, or a combination of the
two. This intertemporal linkage of cash flow to value is more tenuous in applying the APT
framework to real estate because it is difficult to assess how these cash flows will change
over time and in turn how value will vary on an intertemporal basis given that the different
6 CROCKER H. LIU AND JIANPING MEI
macroeconomic factors corresponding to systematic risks are not directly related to the
cash flows of a property. Consequently, the modified present value approach used here
represents a new perspective on linking the traditional fundamental analysis in real estate
to the asset pricing literature.
The study builds on the earlier work of Liu and Mei (1991) who examine the predictabil-
ity of expected returns on equity REITs and their co-movements with other assets. The
current study focuses on movements in unexpected real returns on equity REITs compared
to that for other assets using a VAR approach to decompose the variance of unexpected
asset returns. The importance of each component of unexplained risk does not only depend
on the ability to forecast asset returns but also depends on its intertemporal characteristics
associated with the forecasted component of returns. Hence, this study complements the
study of Liu and Mei (1991).
Another relevant study is the study by Geltner (1990), which uses a discounted cash flow
model of property value. Although Geltner (1990) explores real-estate risk in terms of cash-
flow risk, his model does not allow the discount rate to vary nor are returns decomposed
into expected and unexpected components. This suggests that all movements in real estate
prices must only be due to news about expected future cash flows if one rules out “rational
bubbles.” Thus, only one interpretation of real estate market movements is permitted, and
therefore his study represents a special case of the current study. Geltner (1990) provides
no empirical verification of his theoretical model.
When we first decompose the variance of total real returns for each asset into the variance
associated with contemporaneous expected returns and the variance of unexpected returns,
we find that fluctuations in expected contemporaneous returns account for only a small
portion of the variation in total returns. When we further decompose the variance of unex-
pected returns for each asset into cash-flow risk, discount-rate risk, and the covariance
between these two risks, we find that the relative influence of each of these components
differs for each asset class. For equity REITs, cash-flow risk accounts for the major portion
of unexplained risk. Discount-risk accounts for most of the variation in small stock returns
while cash-flow risk, discount-rate risk, and covariance risk are of equal importance in
accounting for unexplained movements in value-weighted stocks. Consequently, even though
Liu and Mei (1991) and Gyourko and Keim (1991) find that equity REITs behave like small
cap stocks with respect to movements in expected returns, this study suggests that the relative
influence of each of the three risk components differs for EREITs and small stocks with
respect to the variance of their unexpected asset returns. Although cash-flow risk is the
dominant risk characteristic for equity REITs, we find that discount-rate risk and covariance
risk are also important in unexpected real-estate risk components. The study also finds
that mean reversion is much weaker for equity REITs than it is for either value-weighted
or small stocks. This implies that a relatively longer holding period is required for real
estate if a contrarian investment strategy is followed.
The remainder of the paper is organized as follows. The next section discusses the frame-
work used to view the relationship between unexpected returns and movements in expected
returns. The section also describes the VAR approach used to decompose the variance of
stock returns. Section 2 describes our data set. Our empirical results, including the extent
to which the unexpected variation of equity REIT returns is associated with different types
of risks, are reported in section 3. Section 4 concludes the study.
AN ANALYSIS OF REAL-ESTATE RISK 7
1. The Basic Framework and Estimation Process
1.1. The Relationship Between Expected Returns and Unexpected Returns
The log-linear dividend-ratio model of Campbell (1991) and Campbell and Shiller (1988a)
is used to characterize the relationship between the unexpected real asset return in the next
period (t + 1) and changes in rational expectations of future dividend growth and future
asset returns. Thus, the model allows both expected future cash flows and expected returns
(discount rates) to influence asset prices. More formally, the fundamental equation used
in this paper is
h
t+1 - Etht+l = (Et+1 - Et) 2 PjAdt+l+j - (Et+1 - Et) 2 Pjht+l+j (1)
j =O j =O
where Et is the expectation formed at the end of period t, h,,, represents the log of the
real return on an asset held from the end of period t to the end of period t + 1, d,,, is
the log of the real dividend paid during period t + 1, A denotes a one-period backward
difference, and (Et+] - Et) represents a revision in expectations given that new informa-
tion arrived at time t + 1. The parameter p is a constant and is constrained to be smaller
than 1. A more detailed derivation of the model is given in the appendix. The main point
of equation 1 is that if the unexpected return on an asset is negative given that expectations
are internally consistent, then it follows that either the expected future growth in cash flows
(dividends) must decrease, the expected future returns (discount rate) on an asset must
increase, or both. The intuition for this result is that if we consider an asset with fixed
dividends, say real estate leased on a long-term basis with stationary fixed rents whose
price falls, the cap rate on the property will increase, which in turn will increase the prop-
erty’s return unless a further capital loss occurs. However, capital losses on real estate can-
not continue forever, so at some point the property must experience higher returns.
For our study, we will use a more compact version of equation 1 written as follows:
Uh,t+l = vd,t+l - vh,t+l (2)
where “h,t+l
is the unexpected component of the stock return h,,] , r]&+l represents news
about cash flows, and vh,t+l represents news about future returns (discount rate).
1.2. The Estimation Procedure
We model six economic state variables including the asset returns on three asset portfolios
(value-weighted stocks, small stocks and equity REITs), the dividend yield, the yield on
one-month Treasury bills, and the cap rate on real estate, according to a K-order VAR proc-
ess given that Campbell (1988), Campbell and Ammer (1990), Campbell and Mei (1991),
Campbell and Shiller (1988), and others have found that the VAR process provides a useful
framework to summarize data. The VAR approach assumes that the variables in the process
8 CROCKER H. LIU AND JIANPING MEI
are stationary time-series and can thus be modelled using an autoregressive (AR) model?
The fact that lagged state variables are present in an autoregressive process is not necessarily
contradictory to market efficiency if the risk premiums paid on assets vary over time due
to changing economic conditions. If this is the case, then the lagged variables in the VAR
process serves as a proxy for the economic state variables that drive the risk premium,
and this also provides us with some clue as to what relevant variables to include in the
VAR model of asset pricing dynamics. As in previous studies, the dividend yield on an
equally weighted portfolio of all stocks on CRSP, which is measured as the total dividends
paid over the previous year relative to the current stock price, is included in the VAR proc-
ess because it captures information on any changes that may occur in future expected returns
[c.f. Campbell and Shiller (1988a, b), Campbell and Mei (1991), Fama and French (1988b),
Liu and Mei (1991), and Mei and Saunders (1991)].3 We also include the T-bill variable,
which describes the short-term interest rate, and the cap rate, which captures information
on expected future cash flows and required returns in the underlying real estate market?
We include the cap rate as a forecasting variable given the finding in Liu and Mei (1991)
that movements in the cap rate do not necessarily contain the same information as fluctua-
tions in the dividend yield on the stock market.5
To be more specific about the VAR process, the approach involves defining a vector zt+,
that has k elements, the first of which are the real asset returns h,+i in consideration (e.g.,
REITs). Additional elements in this vector are other variables that are known to the market
at the end of period t + 1. Although we initially model asset returns under the assumption
that the vector zi+, follows a first-order VAR process shown in equation 3 below, we later
relax this assumption:
Zt+1 = ht + Wt+1 (3)
Higher-order VAR models that we employ are stacked into this VAR(l) model in the same
manner as discussed in Campbell and Shiller (1988a)P In equation 3, the matrix A is known
as the companion matrix of the VAR.
In addition to the vector zt+i , we also define a k-element vector e, , whose elements are
all equal to zero except the first element, which is equal to 1. The vector e, is used to
separate out real-asset returns h,+i from the vector zt+, (e.g., h,,, = e,‘z,+J and to extract
the unexpected component of real asset returns uh,t+i = h,,, - E,h,+, = e;w,+, from the
residual vector (wtfl) of the VAR process. The VAR(l) approach produces intertemporal
predictions of future expected returns:
Eth+l+j
= eiAj+‘z,.
From equation 4, it follows that news about future returns or discount rates (the present
value of the revisions in forecasted returns) can be defined as
qh,t+l = (Et+1 -
Et) 2 pjhi+i+j = ei 2 pjAjwt+l = eipA(I - PA)-‘w,+I
j = l j = l
= X%+1, (5)
AN ANALYSIS OF REAL-ESTATE RISK 9
where h’ = eipA(1 - PA)-’ is a nonlinear function of the VAR coefficients. In addition
to this, given that the first element of w,+~ is uh,t+i = eiW,+l. equations 5 and 2 imply
that we can calculate cash-flow risk if we obtain consistent estimates of h and the residual
of the VAR process, w~+~ as follows:
vd,t+I = (ei + h’)wt+l. (6)
Equation 6 is important. It implies that we do not need to observe cash flows to calculate
nld. They can be calculated from estimates of the VAR process. We will use the expressions
in equations 5 and 6 to decompose the variance of unexpected asset returns (uh,*+i) into
the cash-flow risk (nd,t+i), discount-rate risk (qh,t+i), and a covariance term.
In addition to decomposing the variance of unexpected returns, we utilize a variance ratio
test to ascertain whether real estate returns display a similar mean-reverting behavior to
returns for value-weighted and small cap stocks? The variance ratio statistic V(K), which
is defined as the ratio of the variance of K-period returns to the variance of one-period
returns, divided by K, can be calculated directly from the autocorrelations of one-period
returns by using the fact that
K - 1
V(K) = 1 + 2 C
J=l
(7)
The variance ratio equals 1 for white noise returns (i.e., there is no serial correlation in
the return series so Corr(h,, h,_j) = 0); it exceeds 1 when returns are mostly positively
autocorrelated, and it is below 1 when negative autocorrelations dominate.
The Generalized Method of Moments of Hansen (1982) is used to jointly estimate the
VAR coefficients and the elements of the variance-covariance matrix of VAR innovations.
This estimation procedure allows for conditional heteroskedasticity and possible serial cor-
relation in the error terms of the VAR process.” To calculate the standard errors associated
with the estimation error for any statistic, we first let y and V represent the entire set of
parameters and the variance-covariance matrix respectively. Next, we write any statistic,
such as the covariance between news about future expected returns and discount rates, as
a nonlinear function f
of the parameter vector y. The standard error for the statistic
is then estimated as J[f,‘Vf,] P
2. Description of the Data
Stock prices and dividends are taken from the Center for Research on Security Prices (CRSP)
monthly stock tape. We study a value-weighted stock index comprised of all New York
Stock Exchange (NYSE) and American Stock Exchange (AMEX) stocks. This value-
weighted stock index is biased towards stocks with large market capitalizations. To adjust
for this bias, we also include a small cap stock index in our study. Both the value-weighted
stock index and the small cap stock index are obtained from the Ibbotson and Associates
Stocks, Bonds, Bills, and Inflation series in CRSI? We also construct an equally weighted
10 CROCKER H. LIU AND JIANPING MEI
equity REIT return series using all equity REITs in CRSP from January 1971 to December
1989 to avoid the problem of survivorship bias. Another advantage to using an EREIT port-
folio (which consists of between 25 to 35 equity REITs on average over most of the study
period) is that we avoid the smoothing issue associated with using appraisal-based returns
even though we recognize that the volatility of real-estate returns might be overstated due
to the closed-end nature of REITs. A REIT is deemed to be an equity REIT it it is listed
as such in at least two of the following three sources: (1) REZT Sourcebook published by
the National Association of Real Estate Investment Trusts, Inc., (2) 7;he Realty Stock Review
published by Audit Investments, and (3) Moody’s Bank and Finance Manual, K&me 2.
The yield on the one-month Treaury bill and the dividend yields on the equally weighted
market portfolio are obtained from Federal Reserve Bulletin and Ibbotson and Associates
(1989). Monthly cap rates on real estate are taken from the American Council of Life Insur-
ance publication Investment Bulletin: Mortgage Commitments on Multifamily and Nonresi-
dential Properties Reported by 20 Life Insurance Companies?o
3. Empirical Results
Table 1 provides summary statistics on real returns for our portfolios of value-weighted
stocks, equity REITs, and small cap stocks as well as for our forecasting variables. Table 1
reveals that although equity REITs resemble large cap stocks (value-weighted stocks) with
Table 1. Summary statistics.
Real return on value-weighted stock portfolio (VWStk)
Real return on Equity REITs portfolio (EREITs)
Real return on small stock portfolio (SmStk)
Dividend yield on equal-weighted portfolio (DivYld)
Capitalization rates on real estate (CapR)
Yield on one-month T-bill (TBill)
Mean SD.
PI
0.386 4.828 0.060
0.832 4.995 0.116
0.736 6.744 0.121
3.722 0.750 0.945
10.440 1.142 0.958
7.374 2.801 0.919
Notes: The sample period for this table is 1971.1-1989.3, with 219 observations. Units on real returns
are percentage per month. Units on the relative T-bill rate and the dividend yield are expressed
as the percentage per annum. p, is the first autocorrelation of the series. The numbers in the pre-
ceding table differ from those in Liu and Mei [1991] since their paper employs excess returns on
a nominal basis whereas real returns are used in the current study.
Correlations of Variables
VWStk EREITs SmStk DivYld CapR Tbill
VWStk 1.000 ,637 ,843 - ,207 ,010 -.I29
EREITs ,637 1.000 ,793 -.174 ,090 -.126
SmStk ,843 ,793 1.000 -.132 ,061 -.104
DivYld - ,207 -.174 -.132 1.000 ,574 ,515
CapR ,010 ,090 .061 .574 1.000 ,698
Tbill -.I29 -.I26 -.I04 ,515 ,698 l.ooo
AN ANALYSIS OF REAL-ESTATE RISK 11
respect to their volatility, EREITs tend to behave more like small cap stocks from the per-
spective of mean returns and correlations. Table 1 also reveals that the returns for equity
REITs and small cap stocks have a higher positive, first-order serial correlation relative
to large cap stocks, which might suggest that the return on these assets are more predict-
able. These findings are consistent with the evidence in Gyourko and Keim (1991) and Liu
and Mei (1991).
The results of regressing the mean adjusted real stock returns on six forecasting variables,
the lag of the return of different stock categories in addition to the returns on Treasury
bills, the dividend yield on the equally weighted market portfolio, and the cap rate are
reported in table 2 for a VAR(l) process.I1 Although we show that this VAR(l) process isn’t
well specified in table 3, we present the results here for simplicity to give the reader some
idea of the estimation of the VAR model since our primary concern is the decomposition
of the unexplained variation in real returns associated with various stock categories rather
than the predictability of returns. The results in table 2 differ slightly from those in Liu
and Mei (1991) since the current study uses real returns in lieu of nominal excess returns
and also uses the lag of the dependent variables as predictors. The use of lagged dependent
variables dampens the impact of some of the other predictor variables such as the dividend
yield, which was significant for both value-weighted and small cap stocks in Liu and Mei
(1991) but is not significant for value-weighted stocks in this study. Like Liu and Mei (1991)
however, the cap rate is a significant forecasting variable with respect to equity REITs and
small cap stocks while the T-bill is a significant predictor of returns for all stock categories.
Table 2. Basic VAR results for real stock returns using generalized method of moments.
One lag, monthly: Period 1971.01 to 1989.03 (T statistics in parentheses).
Dependent
Variable VWStk, EREITs, SmStk, DivYld,
CapR,
TBill, R2 P-Value DW
VWStk,,, 0.080 0.157 -0.108 1.149 0.662 -0.519 ,073 0.01 1.95
(0.56) ( 1. 32) ( - 0. 90) (1.48) ( 1. 65) ( - 3. 21)
EREITs,,, 0.217 0.010 -0.052 1.030 0.966 -0.509 ,088 0.00 1.95
(1.43) ( 0. 07) ( - 0. 49) (0.97) ( 2. 27) ( - 2. 56)
SmStk,,, 0.211 0.049 -0.043 2.538 0.905 -0.752 .109 0.00 1.95
(1.15) ( 0. 29) ( - 0. 27) (2.41) ( 1. 63) ( - 3. 12)
DivYld,,, -0.010 -0.001 0.006 0.906 -0.047 0.035 ,904 0.00 2.00
( - 1 . 3 0 ) ( - 0 . 1 7 ) (0.94) ( 16. 14) ( - 2. 31)
(3.64)
CapR,+I -0.001 -0.004 0.003 0.062 0.868 0.040 ,925 0.00 2.69
( - 0 . 0 7 ) ( - 0 . 6 1 ) (0.40) (1.81) (22.87) (3.42)
TBilI,,, 0.005 0.061 -0.020 -0.074 0.103 0 . 9 0 4 ,851 0.00 2.11
(0.17) ( 2 . 0 9 ) ( - 0 . 7 2 ) ( - 0 . 5 0 ) (1.05) (20.77)
Notes: The T statistics are corrected for heteroskedasticity. The P-Value refers to the signifcance level for a test
of the hypothesis that all regression coefficients are zero. DW is the Durbin-Watson statistic. The numbers in
the preceding table differ from thos in Liu and Mei (1991) since their paper employs excess returns on a nominal
basis whereas real returns are used in the current study.
12 CROCKER H. LIU AND JIANPING ME1
Table 3. F test for joint significance of coefficients of last lag in
each equation.
Lag
Equation F Statistic P-Value
3
VWStk,, , 13.18
EREIT,, , 13.96
SmStk, + I
16.61
DivYld,, , 1453.46
CapR,+ I
1526.45
TBill,, , 691.72
, 040
,030
,011
,000
.OOO
,000
VWStk,, , 11.71 ,068
EmIT,+ I
11.36 ,078
SmStk, + I
9.50 ,147
DivYld,, , 10.52 ,104
Car%+ 1
23.43 ,001
TBilI,, , 10.49 ,105
VWStk,,,
EREIT, + I
SmStk,, ,
DivYld,, ,
Cap4+ 1
TBill,, ,
VWStk,, ,
EREIT,, ,
SmStk,, ,
DivYld,, ,
Cap&+ I
TBill,, ,
19.06 ,004
19.08 ,004
18.72 ,005
20.83 ,002
5.62 ,466
3.26 ,775
2.96 ,813
6.32 ,338
1.31 ,970
3.35 ,762
8.94 ,176
6.02 .420
In addition to this, the coefficients associated with the returns on Treasury bills, the divi-
dend yield on the equally weighted market portfolio, and the cap rate for the various assets
appear to be relatively invariant to a change in model specification (e.g., the addition of
lagged dependent variables and a change in the way returns are measured) when compared
to the results of the earlier study by Liu and Mei (1991). This not only suggests that the
results of Liu and Mei (1991) are robust, but it also implies that the dividend yield, the
T-bill, and the cap rate also influence the predictability of real returns. In fact, as much
as 10% of the variation in returns on equity REITs are predictable. From figure 1, we can
see that the conditional risk premium on these assets using a VAR(2) model varies over
time seeming to peak during a recession and to become relatively small during an economic
expansion when these premiums are analyzed with respect to NBER business cycle dates.
In other words, investors in real estate securities demand a higher conditional risk premium
and therefore higher returns during a recession but are comfortable with lower returns (lower
conditional risk premium) when the economy is in an expansion phase.
In table 3 we report the results of our test on the joint significance of including an addi-
tional lag in our VAR model. In other words, we test the incremental effect rather than the
total effect of using a VAR
model where n is the number of lagged dependent variables
AN ANALYSIS OF REAL-ESTATE RISK
13
i
-0
p e a k
Ew
’ 72
U-L
7 4
ij j;:
;: ‘:I j
:: :i.
;j i:
> j.
1.
j: :
:: :
; :
- _x
tz
IIIIIII I,,,,,,,, ,‘
76 78 80
-I, ,,,,,/,,,,,,,,,,,/,,,,,,,,,,
c-peak
t r o u g h
; :!
j .?. .
: :; : : :
:’ i;
:j
:j
: ii
!s return on RElTs
ditional risk premium
8 2 a4 86
Time Period: 197 1 .3- 1989.3
Figure 1. Excess returns on REITs and its conditional-risk premiums.
used in the GMM estimation process. The first panel of table 3 indicates that the first lag
is definitely necessary since the P-Values are all statistically significant. Moreover, the sec-
ond panel reveals that the addition of a second lag of the dependent variable adds valuable
information because the second lag in the value weighted stocks, equity REITs, and cap
rate equations are significant. Panel 3 suggests that a third-order VAR process is also nec-
essary. In contrast to this, the last panel indicates that adding a fourth lag to the VAR proc-
ess does not increase our explanatory power of what accounts for the variation in the vari-
ables at time t.
We therefore use a VAR(2) and alternatively use a VAR(3) process to decompose the
variance of real asset returns into the variance of contemporaneous expected asset returns
and the variance of unexpected asset returns. The results of this decomposition are reported
in table 4. Table 4 reveals that the variance of contemporaneous expected asset returns
accounts for only a minimal portion of the total variation associated with the return on
each asset class regardless of whether a VAR(2) or VAR(3) process is used. More specifically,
the volatility of contemporaneous expected returns on equity REITs account for 9.2 %-13.3 %
of the total real estate risk, while only 5.8 % -9.2 % of the total risk for value-weighted stocks
is due to the variation in expected value-weighted returns. The amount of total risk arising
from the variation in contemporaneous expected returns on small stocks is similar to that
14 CROCKER H. LIU AND JIANPING MEI
Table 4. Variance decomposition of total real asset returns (variance of expected asset returns and variance of
unexpected asset returns).
I. VAR(2) Process
Asset Category R2
Total Variance
02
Expected Unexpected
zz
2
OE
+
2
06
Value-Weighted Stocks ,058 23.310 1.345 21.965
Equity REITs ,092 24.950 2.285 22.665
Small Cap Stocks ,090 45.482 4.075 41.406
II. VAR(3) Process
Asset Category R2
Total Variance
a2
Expected Unexpected
=
2
UE
+
2
UC
Value-Weighted Stocks .092 23.310 2.154 21.156
Equity REITs .I33 24.950 3.318 21.632
Small Cap Stocks .121 45.482 5.503 39.978
Notes: e2 is defined as the variance of total real returns for each asset. This variance can be decomposed into
two components: uz, which is defined as the variance of expected asset returns, and of 2, which represents the
unexplained variance of the residual term associated with each equation in the VAR system or alternatively the
variance of the unexpected asset returns. R2 represents the variation in total asset returns, which is accounted
for by the VAR process where n is the number of lags.
for equity REITs. In summary, most of the variance of returns for each asset class is asso-
ciated with the variance of unexpected asset returns.
Given that the variance of unexpected returns accounts for most of the variance in total
returns, we further decomposed the unexplained variance of the residual term (unexpected
asset returns) associated with each equation in the VAR system denoted uz into three com-
ponents-cash-flow risk [Var(qJ], discount-rate risk [Var(n,)], and the covariance between
future cash flows and discount rates [Cov(~~, qr,)]--and report the results of this decompo-
sition in table 5. For easier interpretation, the three terms Var(qd), Var(n,,), and -~COV(TJ~,
v,,) are given as ratios to the variance of the unexpected real asset returns so that they sum
to one. From table 5, one can observe that a much larger portion of the variance in unex-
pected returns is explained by cash flow risk for real estate compared to either value-weighted
stocks or small cap stocks from a relative standpoint. More specifically, cash-flow risk
accounts for 79.8% (91.1%) of the total variance of unexpected returns in a VAR(2) (VAR(3))
model for equity REITs. i* This result is not unexpected since dividends are a significant
component of REIT returns because REITs are required to pay out 95 % of their earnings.
In contrast, the contribution of cash-flow risk in accounting for the variance of unexpected
returns in small stocks (30-33 X) and value-weighted stocks (33-38%) is smaller. Discount-
rate risk affects the unexplained variance of small cap stocks returns to a greater extent
(79-95 %) than for equity REITs (38-47%) and value-weighted stocks (33-36%). In sum-
mary, the three risk components are roughly equal in accounting for unexplained variance
of value-weighted stock returns, while cash-flow risk accounts for the major portion of
the unexplained variation in equity REIT returns. Discount-rate risk accounts for most of
the unexplained variation in small stock returns. However, the fact that cash-flow risk is
the dominant risk attribute for the variation in unexpected equity REIT returns should not
be interpreted to mean that changes in the expected future discount rate are unimportant,
AN ANALYSIS OF REAL-ESTATE RISK 15
Table 5. Decomposition of the variance of unexpecred real asset returns (standard errors are in parentheses).
Var Specification
2
or
-2covbd, ?h) Corr(Td, ?h)
2 Lugs, Monthly
VWStk 21.97 0.381 0.333 0.286 -0.401
(0.21) (0.20) (0.19) (0.38)
EREIT 22.61 0.798 0.467 -0.265 0.217
(0.40) (0.40) (0.66) (0.41)
SmStk 41.41 0.297 0.947 -0.244 0.230
(0.13) (0.52) (0.61) (0.47)
3 Lags, Monthly
VWStk 21.15 0.327 0.359 0.315 -0. 460
(0.18) (0.17) (0.19) (0.40)
EREIT 21.62 0.911 0.376 -0. 287 0.245
(0.57) (0.37) (0.83) (0.54)
SmStk 39.98 0.329 0.794 -0.123 0.120
(0.16) (0.39) (0.48) (0.42)
Notes: 0,’ represents the unexplained variance of the residual term associated with each equation in the VAR sys-
tem. nd and nh represent news about future cash flows and news about future expected returns respectively. They
are calculated from the VAR system using equations 5 and 6. The three terms Var(n,), Var(qh), and -2Cov(n,, nh)
are given as ratios to the variance of the unexpected asset return so from 2 they sum to one.
since discount rate risk comprises between 38 % and 47 % of the total unexpected real estate
risk.13 The point is that both cash-flow risk and discount-rate risk are important components
of real-estate risk even though the former accounts for twice as much variation in the unex-
pected return as the latter.
From an absolute perspective, the difference in cash-flow risk between small stocks and
EREITs decreases because the total variation in unexpected returns on small stocks is twice
as large as that of EREITs. However, the cash-flow risk associated with EREITs is still
larger than that for small stocks in accounting for 18.09 (.798 * 22.67) of the variance of
unexpected EREIT returns in contrast to 12.30 (41.41 * .297) of the variance of unexpected
returns on small cap stocks in a VAR(2) model. This implies that cash-flow risk is approx-
imately one and a half times (18.09 + 12.30) more important for equity REITs than for
small cap stocks in accounting for the applicable variance of unexpected asset returns. Table
5 also reveals that cash-flow risk is positively correlated with discount-rate risk for equity
REITs and small stocks. This means that whenever there is good news about future cash
flow in real estate and small stocks, the investor will tend to become apprehensive about
the future and thus demand higher expected future returns. This phenomenon tends to
dampen the shock of cash-flow news and discount-rate news to the market because the two
work in opposite directions. News about future cash flow is good for the current asset price,
while news about future expected return is bad for the current asset price.
Figure 2 provides a plot of the variance ratio calculations for EREITs, the value-weighted
market index, and a small stock index. The variance ratios are calculated using six month
16 CROCKER H. LIU AND JIANPING MEI
.6 _ ..................... .. . .....................
-1
.d
.3j , I-
Q 1 2 3 HoRzon 5 6 7 8
1 OVWStk q EReit ASmStk 1
Figure 2. Implied variance ratios for a VAR(2) process.
intervals and go from six to ninety months. Figure 2 reveals that the general pattern for
both value-weighted and small stocks are similar while the variance ratios for equity REITs
are always larger than either of these stocks. For all three assets, the variance ratios peak
at six months and then decline steadily. The ratio for REITs is greater than one for the
six-month and twelve-month horizon, implying that the autocorrelations for holding period
returns less than six months and one year are predominantly positive. For holding period
returns of greater than one year, the variance ratio becomes smaller than one, implying
that negative autocorrelations dominate for holding period returns that are longer than a
year. This suggests that mean reversion also exists in EREIT returns. This mean reversion
is consistent with the work by Liu and Mei (1991) that returns on EREITs are predictable.
However, mean reversion is much weaker for equity REITs relative to value-weighted and
small cap stocks, which have much smaller variance ratios and thus have a stronger negative
autocorrelation in their returns. The reason that mean reversion is less important for EREITs
is that EREIT returns are mainly driven by news about future cash flows and are much
less influenced by news on future expected returns. The latter type of news is more likely
to be affected by market overreaction or the changing perception of risk by investors.
4. Summary and Conclusions
Concerns over the nature of real-estate risk have tempered institutional investors participation
in real estate. While several studies have addressed the smoothing aspects of risk measure-
ment, this study provides another perspective on the nature of real-estate risk by decomposing
AN ANALYSIS OF REAL-ESTATE RISK 17
real-estate risk using the present-value model of Campbell (1987, 1991), which allows the
discount rate to vary through time in conjunction with a vector autoregressive process.
Total real-estate risk is first decomposed into the variance of contemporaneous expected
asset returns and the variance of unexpected asset returns. From this partitioning, the variance
of unexpected returns is found to account for most of the variance in total returns regardless
of the asset class. Given this result, the unexplained variance of asset returns is further
decomposed into three components: cash-flow risk, discount-rate risk, and the covariation
between cash-flow risk and discount-rate risk. The most interesting result from this decom-
position is that cash-flow risk accounts for twice as much of the variance in unexpected
real-estate returns as discount-rate risk although the latter is also an important risk compo-
nent. In contrast, discount-rate risk accounts for most of the unexplained variation in small
stock returns, while the three risk components are of equal importance in accounting for
the unexplained variance of value-weighted stock returns. Another important contribution
of the study is the finding that mean reversion is much weaker for equity REITs relative
to value-weighted and small stocks, which suggests that a relatively longer holding period
is required for real estate if a contrarian investment strategy is followed.
Acknowledgments
We wish to thank John Campbell for letting us use his variance decomposition algorithm
and Doug Herold and Wayne Ferson for providing data on real-estate cap rates and business
condition factors, respectively.
Appendix: The Dividend-Ratio Model
Campbell and Shiller (1988a) use a first-order Taylor series approximation of the log of
the holding period return equation, h,+i = log((P,+, + D,,,) + Pr) to obtain the following
equation:
h
t+1
- k + 6, - &+I + A&+1, (Al)
where h,+i is the asset return in period t + 1, d, is the log of the real dividend paid during
period t, 6, is the log dividend-price ratio d, - pt, pt is the log real stock price at the end
of period t, p is the average ratio of the stock price to the sum of the stock price and the
dividend, and the constant k is a nonlinear function of p. l4 The log dividend-price ratio
model 6, is the next derived from equation Al by treating equation Al as a difference equa-
tion relating 6, to 6,+, , Ad,,, and h,+i, solving this equation forward, and imposing the
terminal condition that 6t+i does not explode as i increases, e.g., limi,,p’d,+i = 0. The
resulting equation is
4 ~2 d(ht+l+j
- Adt+j+j) - k
j=o
1 - P
(A2)
18 CROCKER H. LIU AND JIANPING MEI
This log dividend-price equation 6, represents the present value of all future returns h,+j
and dividend growth rates Adt + j, discounted at the constant rate p with a constant
k/(1 - p) subtracted from this result. Equation A2 implies that if the dividend yield is
currently large, high future returns will occur unless dividend growth is low in the future.
Although all of the variables in (A2) are measured expost, (A2) also holds ex ante. Conse-
quently, equation 1 in the paper obtains if we use the ex ante version of equation A2 to
substitute 6, and I&+, out of (Al). The reason for this is that 6, is unchanged on the left-
hand side of equation A2 if we take expectations of equation A2, conditional on informa-
tion available at the end of time period t, because 6, is in the information set, and the right-
hand side becomes an expected discounted value.15
Notes
1. Studies that have looked at the understatement of real-estate risk arising from the smoothed nature of appraisal
based returns include Ross and Zisler (1987) and Geltner (1989). On the other hand, Titman and Warga (1986)
and Chan, Hendershott, and Sanders (1990) have examined the risk-adjusted performance of equity REITs
using an arbitrage pricing framework.
2. To test whether the variables in the process are stationary, we performed three versions of the Dickey-Fuller
test on each variable focusing especially on the cap rate variable given one reviewer’s concern that the cap
rate data might be strongly persistent, particularly over monthly intervals. The three versions of the Dickey-
Fuller test include a standard test, the standard test with a constant, and a standard test with a constant and
a time trend [see (Harvey, 1990)]. The Dickey-Fuller test showed that the cap rate variable was stationary
over the time in our study.
3. To see this, consider the ex ante version of equation A2 in the appendix.
4. The cap rate is defined as the ratio of net stabilized earnings to the transaction price (or market value) of
a property. Net stabilized income is calculated under the assumption that the percent occupancy in the building
is equal to or greater than the occupancy rate for comparable existing buildings in the market in which a
property trades. In other words, the vacancy rate has stabilized relative to the vacancy rate that exists during
the leasing up period of a new building. Net stabilized income is computed on a monthly or quarterly basis
and then annualized. The transaction price is the gross price paid for the property. In some cases, the appraisal
value is used in lieu of the transaction price by those who report cap rates. The cap rate used in this paper
is the average of the cap rates for individual properties for which this information is made available to the
ACLI. Alternatively, the cap rate can be thought of as the earnings-price ratio on direct real estate investment.
5. In contrast to Liu and Mei (1991), we do not use a January dummy and the spread between the yields on
long-term AAA corporate bonds and the l-month Treasury bill as forecasting variables. The January dummy
is omitted given the nature of the VAR process while the spread variable is not used because it was not statistically
significant in either Liu and Mei (1991) or in this study for value-weighted stocks, small stocks, and equity
REITs. In addition to this, the parameter estimates appear to be invariant to the inclusion or exclusion of
the spread variable. The parameter estimates also appear to be robust to the inclusion or exclusion of the
dividend yield on an equally weighted portfolio consisting of all equity REITs on the CRSP tapes. The results
of the study, which include the spread variable and the equally weighted dividend yield on equity REITs
are available from the authors on request.
6. It is fairly easy to extend the results from a first-order VAR to a K-order VAR because a K-order VAR can
easily be rewritten in a first-order matrix form as (3). For example, assuming that q+t = AZ, + Bq-t + w,+t ,
by redefining z; = {z,, z,_,}’ and w; = {w,, 0)’ and A’ = [ T i ], we have z;+t = A’z; + w;+i. Thus,
all of the results which we derive for z~+, also applies to z;+t.
7. Campbell (1990), Cochrane (1988), Lo and MacKinlay (1988), and Poterba and Summers (1988), have all
used the variance ratio test to document the mean reverting behavior of stock returns. Kandel and Stambaugh
(1988) also report a number of calculations of this type. However, some controversy exists on the presence
of mean-reversion. The controversy centers around whether the evidence on mean-reversion arises from a
AN ANALYSIS OF REAL-ESTATE RISK 19
slowly reverting component of stock prices or from the weak power associated with the variance ratio test
since this statistic is based on asymptotic theory but is applied to finite samples. For example, Richardson
and Stock (1989) argue that large-sample approximations to sampling distributions perform poorly in practice
because there is not much independent information in a long time series of multi-year returns due to the
small number of nonoverlapping observations.
8. GMM approach used in the paper is designed to alleviate some of the measurement error problems in the
data. Parameter estimates obtained from using GMM will be consistent, as long as the measurement errors
are uncorrelated with lagged instrumental variables.
9. For example, see Chow (1982), Campbell (1991), Campbell and Mei (1991) and Campbell and Shiller (1988a).
10. Unfortunately, the ACLI does not break down monthly cap rates by type of real estate. This is only done
with respect to quarterly cap rates.
11. In contrast to Liu and Mei (1991), we didn’t include bonds because of its distinctive cash flow pattern, which
makes it necessary to use a different variation of the present value model from that used in this paper. However,
the study does include the impact of changes in interest rates on stocks and real estate. We also drop the
term variable used in Liu and Mei (1991) because it was not statistically significant either in Liu and Mei
or in this study. Since our parameter estimates were robust to the inclusion or omission of the term variable,
we omitted it for reasons of parsimony. The January dummy is also excluded given the nature of the VAR
process (e.g., the January effect is implicitly embedded in the model).
12. Changes in future expected cash flows for equity REITs could arise in part from changing expectations in
future rental rates, anticipated vacancies. and tentative absorption rates.
13. The reason that the sum of cash-flow risk and discount-rate risk exceeds 100% of total unexpected real-estate
risk is that the covariance risk accounts for -27% to -28% of the total risk. We should also note that the
standard errors in table 5 are overstated due to insufficient time series data. which in turn leads to understated
T statistics. This is the reason for our claim that discount-rate risk is also an important component of total
unexpected real-estate risk.
14. The equations used in this study differ slightly from those used in Campbell and Shiller (1988a, b) due to a
difference in timing conventions. More specifically, we assume that the stock price at time t and the conditional
expectation of future variables are measured at the end of period t rather than at the beginning of period t.
15. See Campbell and Shiller (1988a) for an evaluation of the quality of the linear approximation in equations
Al and A2.
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doc_256590710.pdf
The current study uses a present value model that allows for a time-varying expected discount rate in conjunction with a VAR process to decompose real-estate risk.
Journal of Real Estate Finance and Economics, 8: 5-20 (1994)
0 1994 Khmer Academic Publishers
An Analysis of Real-Estate Risk Using
the Present Value Model
CROCKER H. LIU
Stern School of Business, New York University, New York, New York 10003
JIANPING ME1
Stem School of Business, New York University, New York, New York 10003
Abstract
The current study uses a present value model that allows for a time-varying expected discount rate in conjunction
with a VAR process to decompose real-estate risk. The study finds that the variance of unexpected returns accounts
for most of the total risk with cash-flow risk accounting for twice as much of the unexplained real-estate risk
although discount rate risk is also an important factor, This dominance of cash-flow risk is found to result in a
weaker mean reversion process for real estate relative to stocks. Another finding is that real estate investors tend
to become apprehensive about the future when news on future cash flow is good, and thus they demand higher
expected future returns.
Key words: Real-Estate Risk, VAR model, Risk Decomposition, REIT
The analysis of real-estate risk has typically focused on either the understatement of risk
arising from the smoothed nature of appraisal-based returns or the systematic components
of risk from an arbitrage pricing (APT) perspective.’ In contrast, little research exists on
partitioning the volatility of total return using a present value framework. The purpose
of the current study, therefore, is to examine real-estate risk by first decomposing total
risk into expected and unexpected movements in asset prices. The unexpected variation
in total risk is further decomposed into two risk components-changes in rational expecta-
tions of future real dividend growth and future expected returns, which we will hereafter
refer to as cash-flow risk and discount-rate risk respectively. To decompose the variance,
the present value model of (Campbell 198/ 1991), which allows the expected discount rate
to vary through time, is used in conjunction with a vector autoregressive (VAR) process.
Focusing on risk using a present value model framework offers several distinct advan-
tages not available with an APT structure. Most important, the present value model is geared
to how real estate is typically valued since the traditional analysis of real estate involves
analyzing the cash flows of a property using a constant discount rate to obtain a value.
By modifying the present value to allow the expected discount rate to vary over time, we
can directly assess how the value of the property fluctuates over time by examining varia-
tions in the cash flows, variations in the expected discount rate, or a combination of the
two. This intertemporal linkage of cash flow to value is more tenuous in applying the APT
framework to real estate because it is difficult to assess how these cash flows will change
over time and in turn how value will vary on an intertemporal basis given that the different
6 CROCKER H. LIU AND JIANPING MEI
macroeconomic factors corresponding to systematic risks are not directly related to the
cash flows of a property. Consequently, the modified present value approach used here
represents a new perspective on linking the traditional fundamental analysis in real estate
to the asset pricing literature.
The study builds on the earlier work of Liu and Mei (1991) who examine the predictabil-
ity of expected returns on equity REITs and their co-movements with other assets. The
current study focuses on movements in unexpected real returns on equity REITs compared
to that for other assets using a VAR approach to decompose the variance of unexpected
asset returns. The importance of each component of unexplained risk does not only depend
on the ability to forecast asset returns but also depends on its intertemporal characteristics
associated with the forecasted component of returns. Hence, this study complements the
study of Liu and Mei (1991).
Another relevant study is the study by Geltner (1990), which uses a discounted cash flow
model of property value. Although Geltner (1990) explores real-estate risk in terms of cash-
flow risk, his model does not allow the discount rate to vary nor are returns decomposed
into expected and unexpected components. This suggests that all movements in real estate
prices must only be due to news about expected future cash flows if one rules out “rational
bubbles.” Thus, only one interpretation of real estate market movements is permitted, and
therefore his study represents a special case of the current study. Geltner (1990) provides
no empirical verification of his theoretical model.
When we first decompose the variance of total real returns for each asset into the variance
associated with contemporaneous expected returns and the variance of unexpected returns,
we find that fluctuations in expected contemporaneous returns account for only a small
portion of the variation in total returns. When we further decompose the variance of unex-
pected returns for each asset into cash-flow risk, discount-rate risk, and the covariance
between these two risks, we find that the relative influence of each of these components
differs for each asset class. For equity REITs, cash-flow risk accounts for the major portion
of unexplained risk. Discount-risk accounts for most of the variation in small stock returns
while cash-flow risk, discount-rate risk, and covariance risk are of equal importance in
accounting for unexplained movements in value-weighted stocks. Consequently, even though
Liu and Mei (1991) and Gyourko and Keim (1991) find that equity REITs behave like small
cap stocks with respect to movements in expected returns, this study suggests that the relative
influence of each of the three risk components differs for EREITs and small stocks with
respect to the variance of their unexpected asset returns. Although cash-flow risk is the
dominant risk characteristic for equity REITs, we find that discount-rate risk and covariance
risk are also important in unexpected real-estate risk components. The study also finds
that mean reversion is much weaker for equity REITs than it is for either value-weighted
or small stocks. This implies that a relatively longer holding period is required for real
estate if a contrarian investment strategy is followed.
The remainder of the paper is organized as follows. The next section discusses the frame-
work used to view the relationship between unexpected returns and movements in expected
returns. The section also describes the VAR approach used to decompose the variance of
stock returns. Section 2 describes our data set. Our empirical results, including the extent
to which the unexpected variation of equity REIT returns is associated with different types
of risks, are reported in section 3. Section 4 concludes the study.
AN ANALYSIS OF REAL-ESTATE RISK 7
1. The Basic Framework and Estimation Process
1.1. The Relationship Between Expected Returns and Unexpected Returns
The log-linear dividend-ratio model of Campbell (1991) and Campbell and Shiller (1988a)
is used to characterize the relationship between the unexpected real asset return in the next
period (t + 1) and changes in rational expectations of future dividend growth and future
asset returns. Thus, the model allows both expected future cash flows and expected returns
(discount rates) to influence asset prices. More formally, the fundamental equation used
in this paper is
h
t+1 - Etht+l = (Et+1 - Et) 2 PjAdt+l+j - (Et+1 - Et) 2 Pjht+l+j (1)
j =O j =O
where Et is the expectation formed at the end of period t, h,,, represents the log of the
real return on an asset held from the end of period t to the end of period t + 1, d,,, is
the log of the real dividend paid during period t + 1, A denotes a one-period backward
difference, and (Et+] - Et) represents a revision in expectations given that new informa-
tion arrived at time t + 1. The parameter p is a constant and is constrained to be smaller
than 1. A more detailed derivation of the model is given in the appendix. The main point
of equation 1 is that if the unexpected return on an asset is negative given that expectations
are internally consistent, then it follows that either the expected future growth in cash flows
(dividends) must decrease, the expected future returns (discount rate) on an asset must
increase, or both. The intuition for this result is that if we consider an asset with fixed
dividends, say real estate leased on a long-term basis with stationary fixed rents whose
price falls, the cap rate on the property will increase, which in turn will increase the prop-
erty’s return unless a further capital loss occurs. However, capital losses on real estate can-
not continue forever, so at some point the property must experience higher returns.
For our study, we will use a more compact version of equation 1 written as follows:
Uh,t+l = vd,t+l - vh,t+l (2)
where “h,t+l
is the unexpected component of the stock return h,,] , r]&+l represents news
about cash flows, and vh,t+l represents news about future returns (discount rate).
1.2. The Estimation Procedure
We model six economic state variables including the asset returns on three asset portfolios
(value-weighted stocks, small stocks and equity REITs), the dividend yield, the yield on
one-month Treasury bills, and the cap rate on real estate, according to a K-order VAR proc-
ess given that Campbell (1988), Campbell and Ammer (1990), Campbell and Mei (1991),
Campbell and Shiller (1988), and others have found that the VAR process provides a useful
framework to summarize data. The VAR approach assumes that the variables in the process
8 CROCKER H. LIU AND JIANPING MEI
are stationary time-series and can thus be modelled using an autoregressive (AR) model?
The fact that lagged state variables are present in an autoregressive process is not necessarily
contradictory to market efficiency if the risk premiums paid on assets vary over time due
to changing economic conditions. If this is the case, then the lagged variables in the VAR
process serves as a proxy for the economic state variables that drive the risk premium,
and this also provides us with some clue as to what relevant variables to include in the
VAR model of asset pricing dynamics. As in previous studies, the dividend yield on an
equally weighted portfolio of all stocks on CRSP, which is measured as the total dividends
paid over the previous year relative to the current stock price, is included in the VAR proc-
ess because it captures information on any changes that may occur in future expected returns
[c.f. Campbell and Shiller (1988a, b), Campbell and Mei (1991), Fama and French (1988b),
Liu and Mei (1991), and Mei and Saunders (1991)].3 We also include the T-bill variable,
which describes the short-term interest rate, and the cap rate, which captures information
on expected future cash flows and required returns in the underlying real estate market?
We include the cap rate as a forecasting variable given the finding in Liu and Mei (1991)
that movements in the cap rate do not necessarily contain the same information as fluctua-
tions in the dividend yield on the stock market.5
To be more specific about the VAR process, the approach involves defining a vector zt+,
that has k elements, the first of which are the real asset returns h,+i in consideration (e.g.,
REITs). Additional elements in this vector are other variables that are known to the market
at the end of period t + 1. Although we initially model asset returns under the assumption
that the vector zi+, follows a first-order VAR process shown in equation 3 below, we later
relax this assumption:
Zt+1 = ht + Wt+1 (3)
Higher-order VAR models that we employ are stacked into this VAR(l) model in the same
manner as discussed in Campbell and Shiller (1988a)P In equation 3, the matrix A is known
as the companion matrix of the VAR.
In addition to the vector zt+i , we also define a k-element vector e, , whose elements are
all equal to zero except the first element, which is equal to 1. The vector e, is used to
separate out real-asset returns h,+i from the vector zt+, (e.g., h,,, = e,‘z,+J and to extract
the unexpected component of real asset returns uh,t+i = h,,, - E,h,+, = e;w,+, from the
residual vector (wtfl) of the VAR process. The VAR(l) approach produces intertemporal
predictions of future expected returns:
Eth+l+j
= eiAj+‘z,.
From equation 4, it follows that news about future returns or discount rates (the present
value of the revisions in forecasted returns) can be defined as
qh,t+l = (Et+1 -
Et) 2 pjhi+i+j = ei 2 pjAjwt+l = eipA(I - PA)-‘w,+I
j = l j = l
= X%+1, (5)
AN ANALYSIS OF REAL-ESTATE RISK 9
where h’ = eipA(1 - PA)-’ is a nonlinear function of the VAR coefficients. In addition
to this, given that the first element of w,+~ is uh,t+i = eiW,+l. equations 5 and 2 imply
that we can calculate cash-flow risk if we obtain consistent estimates of h and the residual
of the VAR process, w~+~ as follows:
vd,t+I = (ei + h’)wt+l. (6)
Equation 6 is important. It implies that we do not need to observe cash flows to calculate
nld. They can be calculated from estimates of the VAR process. We will use the expressions
in equations 5 and 6 to decompose the variance of unexpected asset returns (uh,*+i) into
the cash-flow risk (nd,t+i), discount-rate risk (qh,t+i), and a covariance term.
In addition to decomposing the variance of unexpected returns, we utilize a variance ratio
test to ascertain whether real estate returns display a similar mean-reverting behavior to
returns for value-weighted and small cap stocks? The variance ratio statistic V(K), which
is defined as the ratio of the variance of K-period returns to the variance of one-period
returns, divided by K, can be calculated directly from the autocorrelations of one-period
returns by using the fact that
K - 1
V(K) = 1 + 2 C
J=l
(7)
The variance ratio equals 1 for white noise returns (i.e., there is no serial correlation in
the return series so Corr(h,, h,_j) = 0); it exceeds 1 when returns are mostly positively
autocorrelated, and it is below 1 when negative autocorrelations dominate.
The Generalized Method of Moments of Hansen (1982) is used to jointly estimate the
VAR coefficients and the elements of the variance-covariance matrix of VAR innovations.
This estimation procedure allows for conditional heteroskedasticity and possible serial cor-
relation in the error terms of the VAR process.” To calculate the standard errors associated
with the estimation error for any statistic, we first let y and V represent the entire set of
parameters and the variance-covariance matrix respectively. Next, we write any statistic,
such as the covariance between news about future expected returns and discount rates, as
a nonlinear function f
of the parameter vector y. The standard error for the statisticis then estimated as J[f,
‘Vf,] P2. Description of the Data
Stock prices and dividends are taken from the Center for Research on Security Prices (CRSP)
monthly stock tape. We study a value-weighted stock index comprised of all New York
Stock Exchange (NYSE) and American Stock Exchange (AMEX) stocks. This value-
weighted stock index is biased towards stocks with large market capitalizations. To adjust
for this bias, we also include a small cap stock index in our study. Both the value-weighted
stock index and the small cap stock index are obtained from the Ibbotson and Associates
Stocks, Bonds, Bills, and Inflation series in CRSI? We also construct an equally weighted
10 CROCKER H. LIU AND JIANPING MEI
equity REIT return series using all equity REITs in CRSP from January 1971 to December
1989 to avoid the problem of survivorship bias. Another advantage to using an EREIT port-
folio (which consists of between 25 to 35 equity REITs on average over most of the study
period) is that we avoid the smoothing issue associated with using appraisal-based returns
even though we recognize that the volatility of real-estate returns might be overstated due
to the closed-end nature of REITs. A REIT is deemed to be an equity REIT it it is listed
as such in at least two of the following three sources: (1) REZT Sourcebook published by
the National Association of Real Estate Investment Trusts, Inc., (2) 7;he Realty Stock Review
published by Audit Investments, and (3) Moody’s Bank and Finance Manual, K&me 2.
The yield on the one-month Treaury bill and the dividend yields on the equally weighted
market portfolio are obtained from Federal Reserve Bulletin and Ibbotson and Associates
(1989). Monthly cap rates on real estate are taken from the American Council of Life Insur-
ance publication Investment Bulletin: Mortgage Commitments on Multifamily and Nonresi-
dential Properties Reported by 20 Life Insurance Companies?o
3. Empirical Results
Table 1 provides summary statistics on real returns for our portfolios of value-weighted
stocks, equity REITs, and small cap stocks as well as for our forecasting variables. Table 1
reveals that although equity REITs resemble large cap stocks (value-weighted stocks) with
Table 1. Summary statistics.
Real return on value-weighted stock portfolio (VWStk)
Real return on Equity REITs portfolio (EREITs)
Real return on small stock portfolio (SmStk)
Dividend yield on equal-weighted portfolio (DivYld)
Capitalization rates on real estate (CapR)
Yield on one-month T-bill (TBill)
Mean SD.
PI
0.386 4.828 0.060
0.832 4.995 0.116
0.736 6.744 0.121
3.722 0.750 0.945
10.440 1.142 0.958
7.374 2.801 0.919
Notes: The sample period for this table is 1971.1-1989.3, with 219 observations. Units on real returns
are percentage per month. Units on the relative T-bill rate and the dividend yield are expressed
as the percentage per annum. p, is the first autocorrelation of the series. The numbers in the pre-
ceding table differ from those in Liu and Mei [1991] since their paper employs excess returns on
a nominal basis whereas real returns are used in the current study.
Correlations of Variables
VWStk EREITs SmStk DivYld CapR Tbill
VWStk 1.000 ,637 ,843 - ,207 ,010 -.I29
EREITs ,637 1.000 ,793 -.174 ,090 -.126
SmStk ,843 ,793 1.000 -.132 ,061 -.104
DivYld - ,207 -.174 -.132 1.000 ,574 ,515
CapR ,010 ,090 .061 .574 1.000 ,698
Tbill -.I29 -.I26 -.I04 ,515 ,698 l.ooo
AN ANALYSIS OF REAL-ESTATE RISK 11
respect to their volatility, EREITs tend to behave more like small cap stocks from the per-
spective of mean returns and correlations. Table 1 also reveals that the returns for equity
REITs and small cap stocks have a higher positive, first-order serial correlation relative
to large cap stocks, which might suggest that the return on these assets are more predict-
able. These findings are consistent with the evidence in Gyourko and Keim (1991) and Liu
and Mei (1991).
The results of regressing the mean adjusted real stock returns on six forecasting variables,
the lag of the return of different stock categories in addition to the returns on Treasury
bills, the dividend yield on the equally weighted market portfolio, and the cap rate are
reported in table 2 for a VAR(l) process.I1 Although we show that this VAR(l) process isn’t
well specified in table 3, we present the results here for simplicity to give the reader some
idea of the estimation of the VAR model since our primary concern is the decomposition
of the unexplained variation in real returns associated with various stock categories rather
than the predictability of returns. The results in table 2 differ slightly from those in Liu
and Mei (1991) since the current study uses real returns in lieu of nominal excess returns
and also uses the lag of the dependent variables as predictors. The use of lagged dependent
variables dampens the impact of some of the other predictor variables such as the dividend
yield, which was significant for both value-weighted and small cap stocks in Liu and Mei
(1991) but is not significant for value-weighted stocks in this study. Like Liu and Mei (1991)
however, the cap rate is a significant forecasting variable with respect to equity REITs and
small cap stocks while the T-bill is a significant predictor of returns for all stock categories.
Table 2. Basic VAR results for real stock returns using generalized method of moments.
One lag, monthly: Period 1971.01 to 1989.03 (T statistics in parentheses).
Dependent
Variable VWStk, EREITs, SmStk, DivYld,
CapR,
TBill, R2 P-Value DW
VWStk,,, 0.080 0.157 -0.108 1.149 0.662 -0.519 ,073 0.01 1.95
(0.56) ( 1. 32) ( - 0. 90) (1.48) ( 1. 65) ( - 3. 21)
EREITs,,, 0.217 0.010 -0.052 1.030 0.966 -0.509 ,088 0.00 1.95
(1.43) ( 0. 07) ( - 0. 49) (0.97) ( 2. 27) ( - 2. 56)
SmStk,,, 0.211 0.049 -0.043 2.538 0.905 -0.752 .109 0.00 1.95
(1.15) ( 0. 29) ( - 0. 27) (2.41) ( 1. 63) ( - 3. 12)
DivYld,,, -0.010 -0.001 0.006 0.906 -0.047 0.035 ,904 0.00 2.00
( - 1 . 3 0 ) ( - 0 . 1 7 ) (0.94) ( 16. 14) ( - 2. 31)
(3.64)
CapR,+I -0.001 -0.004 0.003 0.062 0.868 0.040 ,925 0.00 2.69
( - 0 . 0 7 ) ( - 0 . 6 1 ) (0.40) (1.81) (22.87) (3.42)
TBilI,,, 0.005 0.061 -0.020 -0.074 0.103 0 . 9 0 4 ,851 0.00 2.11
(0.17) ( 2 . 0 9 ) ( - 0 . 7 2 ) ( - 0 . 5 0 ) (1.05) (20.77)
Notes: The T statistics are corrected for heteroskedasticity. The P-Value refers to the signifcance level for a test
of the hypothesis that all regression coefficients are zero. DW is the Durbin-Watson statistic. The numbers in
the preceding table differ from thos in Liu and Mei (1991) since their paper employs excess returns on a nominal
basis whereas real returns are used in the current study.
12 CROCKER H. LIU AND JIANPING ME1
Table 3. F test for joint significance of coefficients of last lag in
each equation.
Lag
Equation F Statistic P-Value
3
VWStk,, , 13.18
EREIT,, , 13.96
SmStk, + I
16.61
DivYld,, , 1453.46
CapR,+ I
1526.45
TBill,, , 691.72
, 040
,030
,011
,000
.OOO
,000
VWStk,, , 11.71 ,068
EmIT,+ I
11.36 ,078
SmStk, + I
9.50 ,147
DivYld,, , 10.52 ,104
Car%+ 1
23.43 ,001
TBilI,, , 10.49 ,105
VWStk,,,
EREIT, + I
SmStk,, ,
DivYld,, ,
Cap4+ 1
TBill,, ,
VWStk,, ,
EREIT,, ,
SmStk,, ,
DivYld,, ,
Cap&+ I
TBill,, ,
19.06 ,004
19.08 ,004
18.72 ,005
20.83 ,002
5.62 ,466
3.26 ,775
2.96 ,813
6.32 ,338
1.31 ,970
3.35 ,762
8.94 ,176
6.02 .420
In addition to this, the coefficients associated with the returns on Treasury bills, the divi-
dend yield on the equally weighted market portfolio, and the cap rate for the various assets
appear to be relatively invariant to a change in model specification (e.g., the addition of
lagged dependent variables and a change in the way returns are measured) when compared
to the results of the earlier study by Liu and Mei (1991). This not only suggests that the
results of Liu and Mei (1991) are robust, but it also implies that the dividend yield, the
T-bill, and the cap rate also influence the predictability of real returns. In fact, as much
as 10% of the variation in returns on equity REITs are predictable. From figure 1, we can
see that the conditional risk premium on these assets using a VAR(2) model varies over
time seeming to peak during a recession and to become relatively small during an economic
expansion when these premiums are analyzed with respect to NBER business cycle dates.
In other words, investors in real estate securities demand a higher conditional risk premium
and therefore higher returns during a recession but are comfortable with lower returns (lower
conditional risk premium) when the economy is in an expansion phase.
In table 3 we report the results of our test on the joint significance of including an addi-
tional lag in our VAR model. In other words, we test the incremental effect rather than the
total effect of using a VAR
model where n is the number of lagged dependent variablesAN ANALYSIS OF REAL-ESTATE RISK
13
i
-0
p e a k
Ew
’ 72
U-L
7 4
ij j;:
;: ‘:I j
:: :i.
;j i:
> j.
1.
j: :
:: :
; :
- _x
tz
IIIIIII I,,,,,,,, ,‘
76 78 80
-I, ,,,,,/,,,,,,,,,,,/,,,,,,,,,,
c-peak
t r o u g h
; :!
j .?. .
: :; : : :
:’ i;
:j
:j
: ii
!s return on RElTs
ditional risk premium8 2 a4 86
Time Period: 197 1 .3- 1989.3
Figure 1. Excess returns on REITs and its conditional-risk premiums.
used in the GMM estimation process. The first panel of table 3 indicates that the first lag
is definitely necessary since the P-Values are all statistically significant. Moreover, the sec-
ond panel reveals that the addition of a second lag of the dependent variable adds valuable
information because the second lag in the value weighted stocks, equity REITs, and cap
rate equations are significant. Panel 3 suggests that a third-order VAR process is also nec-
essary. In contrast to this, the last panel indicates that adding a fourth lag to the VAR proc-
ess does not increase our explanatory power of what accounts for the variation in the vari-
ables at time t.
We therefore use a VAR(2) and alternatively use a VAR(3) process to decompose the
variance of real asset returns into the variance of contemporaneous expected asset returns
and the variance of unexpected asset returns. The results of this decomposition are reported
in table 4. Table 4 reveals that the variance of contemporaneous expected asset returns
accounts for only a minimal portion of the total variation associated with the return on
each asset class regardless of whether a VAR(2) or VAR(3) process is used. More specifically,
the volatility of contemporaneous expected returns on equity REITs account for 9.2 %-13.3 %
of the total real estate risk, while only 5.8 % -9.2 % of the total risk for value-weighted stocks
is due to the variation in expected value-weighted returns. The amount of total risk arising
from the variation in contemporaneous expected returns on small stocks is similar to that
14 CROCKER H. LIU AND JIANPING MEI
Table 4. Variance decomposition of total real asset returns (variance of expected asset returns and variance of
unexpected asset returns).
I. VAR(2) Process
Asset Category R2
Total Variance
02
Expected Unexpected
zz
2
OE
+
2
06
Value-Weighted Stocks ,058 23.310 1.345 21.965
Equity REITs ,092 24.950 2.285 22.665
Small Cap Stocks ,090 45.482 4.075 41.406
II. VAR(3) Process
Asset Category R2
Total Variance
a2
Expected Unexpected
=
2
UE
+
2
UC
Value-Weighted Stocks .092 23.310 2.154 21.156
Equity REITs .I33 24.950 3.318 21.632
Small Cap Stocks .121 45.482 5.503 39.978
Notes: e2 is defined as the variance of total real returns for each asset. This variance can be decomposed into
two components: uz, which is defined as the variance of expected asset returns, and of 2, which represents the
unexplained variance of the residual term associated with each equation in the VAR system or alternatively the
variance of the unexpected asset returns. R2 represents the variation in total asset returns, which is accounted
for by the VAR
process where n is the number of lags.for equity REITs. In summary, most of the variance of returns for each asset class is asso-
ciated with the variance of unexpected asset returns.
Given that the variance of unexpected returns accounts for most of the variance in total
returns, we further decomposed the unexplained variance of the residual term (unexpected
asset returns) associated with each equation in the VAR system denoted uz into three com-
ponents-cash-flow risk [Var(qJ], discount-rate risk [Var(n,)], and the covariance between
future cash flows and discount rates [Cov(~~, qr,)]--and report the results of this decompo-
sition in table 5. For easier interpretation, the three terms Var(qd), Var(n,,), and -~COV(TJ~,
v,,) are given as ratios to the variance of the unexpected real asset returns so that they sum
to one. From table 5, one can observe that a much larger portion of the variance in unex-
pected returns is explained by cash flow risk for real estate compared to either value-weighted
stocks or small cap stocks from a relative standpoint. More specifically, cash-flow risk
accounts for 79.8% (91.1%) of the total variance of unexpected returns in a VAR(2) (VAR(3))
model for equity REITs. i* This result is not unexpected since dividends are a significant
component of REIT returns because REITs are required to pay out 95 % of their earnings.
In contrast, the contribution of cash-flow risk in accounting for the variance of unexpected
returns in small stocks (30-33 X) and value-weighted stocks (33-38%) is smaller. Discount-
rate risk affects the unexplained variance of small cap stocks returns to a greater extent
(79-95 %) than for equity REITs (38-47%) and value-weighted stocks (33-36%). In sum-
mary, the three risk components are roughly equal in accounting for unexplained variance
of value-weighted stock returns, while cash-flow risk accounts for the major portion of
the unexplained variation in equity REIT returns. Discount-rate risk accounts for most of
the unexplained variation in small stock returns. However, the fact that cash-flow risk is
the dominant risk attribute for the variation in unexpected equity REIT returns should not
be interpreted to mean that changes in the expected future discount rate are unimportant,
AN ANALYSIS OF REAL-ESTATE RISK 15
Table 5. Decomposition of the variance of unexpecred real asset returns (standard errors are in parentheses).
Var Specification
2
or
-2covbd, ?h) Corr(Td, ?h)
2 Lugs, Monthly
VWStk 21.97 0.381 0.333 0.286 -0.401
(0.21) (0.20) (0.19) (0.38)
EREIT 22.61 0.798 0.467 -0.265 0.217
(0.40) (0.40) (0.66) (0.41)
SmStk 41.41 0.297 0.947 -0.244 0.230
(0.13) (0.52) (0.61) (0.47)
3 Lags, Monthly
VWStk 21.15 0.327 0.359 0.315 -0. 460
(0.18) (0.17) (0.19) (0.40)
EREIT 21.62 0.911 0.376 -0. 287 0.245
(0.57) (0.37) (0.83) (0.54)
SmStk 39.98 0.329 0.794 -0.123 0.120
(0.16) (0.39) (0.48) (0.42)
Notes: 0,’ represents the unexplained variance of the residual term associated with each equation in the VAR sys-
tem. nd and nh represent news about future cash flows and news about future expected returns respectively. They
are calculated from the VAR system using equations 5 and 6. The three terms Var(n,), Var(qh), and -2Cov(n,, nh)
are given as ratios to the variance of the unexpected asset return so from 2 they sum to one.
since discount rate risk comprises between 38 % and 47 % of the total unexpected real estate
risk.13 The point is that both cash-flow risk and discount-rate risk are important components
of real-estate risk even though the former accounts for twice as much variation in the unex-
pected return as the latter.
From an absolute perspective, the difference in cash-flow risk between small stocks and
EREITs decreases because the total variation in unexpected returns on small stocks is twice
as large as that of EREITs. However, the cash-flow risk associated with EREITs is still
larger than that for small stocks in accounting for 18.09 (.798 * 22.67) of the variance of
unexpected EREIT returns in contrast to 12.30 (41.41 * .297) of the variance of unexpected
returns on small cap stocks in a VAR(2) model. This implies that cash-flow risk is approx-
imately one and a half times (18.09 + 12.30) more important for equity REITs than for
small cap stocks in accounting for the applicable variance of unexpected asset returns. Table
5 also reveals that cash-flow risk is positively correlated with discount-rate risk for equity
REITs and small stocks. This means that whenever there is good news about future cash
flow in real estate and small stocks, the investor will tend to become apprehensive about
the future and thus demand higher expected future returns. This phenomenon tends to
dampen the shock of cash-flow news and discount-rate news to the market because the two
work in opposite directions. News about future cash flow is good for the current asset price,
while news about future expected return is bad for the current asset price.
Figure 2 provides a plot of the variance ratio calculations for EREITs, the value-weighted
market index, and a small stock index. The variance ratios are calculated using six month
16 CROCKER H. LIU AND JIANPING MEI
.6 _ ..................... .. . .....................
-1
.d
.3j , I-
Q 1 2 3 HoRzon 5 6 7 8
1 OVWStk q EReit ASmStk 1
Figure 2. Implied variance ratios for a VAR(2) process.
intervals and go from six to ninety months. Figure 2 reveals that the general pattern for
both value-weighted and small stocks are similar while the variance ratios for equity REITs
are always larger than either of these stocks. For all three assets, the variance ratios peak
at six months and then decline steadily. The ratio for REITs is greater than one for the
six-month and twelve-month horizon, implying that the autocorrelations for holding period
returns less than six months and one year are predominantly positive. For holding period
returns of greater than one year, the variance ratio becomes smaller than one, implying
that negative autocorrelations dominate for holding period returns that are longer than a
year. This suggests that mean reversion also exists in EREIT returns. This mean reversion
is consistent with the work by Liu and Mei (1991) that returns on EREITs are predictable.
However, mean reversion is much weaker for equity REITs relative to value-weighted and
small cap stocks, which have much smaller variance ratios and thus have a stronger negative
autocorrelation in their returns. The reason that mean reversion is less important for EREITs
is that EREIT returns are mainly driven by news about future cash flows and are much
less influenced by news on future expected returns. The latter type of news is more likely
to be affected by market overreaction or the changing perception of risk by investors.
4. Summary and Conclusions
Concerns over the nature of real-estate risk have tempered institutional investors participation
in real estate. While several studies have addressed the smoothing aspects of risk measure-
ment, this study provides another perspective on the nature of real-estate risk by decomposing
AN ANALYSIS OF REAL-ESTATE RISK 17
real-estate risk using the present-value model of Campbell (1987, 1991), which allows the
discount rate to vary through time in conjunction with a vector autoregressive process.
Total real-estate risk is first decomposed into the variance of contemporaneous expected
asset returns and the variance of unexpected asset returns. From this partitioning, the variance
of unexpected returns is found to account for most of the variance in total returns regardless
of the asset class. Given this result, the unexplained variance of asset returns is further
decomposed into three components: cash-flow risk, discount-rate risk, and the covariation
between cash-flow risk and discount-rate risk. The most interesting result from this decom-
position is that cash-flow risk accounts for twice as much of the variance in unexpected
real-estate returns as discount-rate risk although the latter is also an important risk compo-
nent. In contrast, discount-rate risk accounts for most of the unexplained variation in small
stock returns, while the three risk components are of equal importance in accounting for
the unexplained variance of value-weighted stock returns. Another important contribution
of the study is the finding that mean reversion is much weaker for equity REITs relative
to value-weighted and small stocks, which suggests that a relatively longer holding period
is required for real estate if a contrarian investment strategy is followed.
Acknowledgments
We wish to thank John Campbell for letting us use his variance decomposition algorithm
and Doug Herold and Wayne Ferson for providing data on real-estate cap rates and business
condition factors, respectively.
Appendix: The Dividend-Ratio Model
Campbell and Shiller (1988a) use a first-order Taylor series approximation of the log of
the holding period return equation, h,+i = log((P,+, + D,,,) + Pr) to obtain the following
equation:
h
t+1
- k + 6, - &+I + A&+1, (Al)
where h,+i is the asset return in period t + 1, d, is the log of the real dividend paid during
period t, 6, is the log dividend-price ratio d, - pt, pt is the log real stock price at the end
of period t, p is the average ratio of the stock price to the sum of the stock price and the
dividend, and the constant k is a nonlinear function of p. l4 The log dividend-price ratio
model 6, is the next derived from equation Al by treating equation Al as a difference equa-
tion relating 6, to 6,+, , Ad,,, and h,+i, solving this equation forward, and imposing the
terminal condition that 6t+i does not explode as i increases, e.g., limi,,p’d,+i = 0. The
resulting equation is
4 ~2 d(ht+l+j
- Adt+j+j) - k
j=o
1 - P
(A2)
18 CROCKER H. LIU AND JIANPING MEI
This log dividend-price equation 6, represents the present value of all future returns h,+j
and dividend growth rates Adt + j, discounted at the constant rate p with a constant
k/(1 - p) subtracted from this result. Equation A2 implies that if the dividend yield is
currently large, high future returns will occur unless dividend growth is low in the future.
Although all of the variables in (A2) are measured expost, (A2) also holds ex ante. Conse-
quently, equation 1 in the paper obtains if we use the ex ante version of equation A2 to
substitute 6, and I&+, out of (Al). The reason for this is that 6, is unchanged on the left-
hand side of equation A2 if we take expectations of equation A2, conditional on informa-
tion available at the end of time period t, because 6, is in the information set, and the right-
hand side becomes an expected discounted value.15
Notes
1. Studies that have looked at the understatement of real-estate risk arising from the smoothed nature of appraisal
based returns include Ross and Zisler (1987) and Geltner (1989). On the other hand, Titman and Warga (1986)
and Chan, Hendershott, and Sanders (1990) have examined the risk-adjusted performance of equity REITs
using an arbitrage pricing framework.
2. To test whether the variables in the process are stationary, we performed three versions of the Dickey-Fuller
test on each variable focusing especially on the cap rate variable given one reviewer’s concern that the cap
rate data might be strongly persistent, particularly over monthly intervals. The three versions of the Dickey-
Fuller test include a standard test, the standard test with a constant, and a standard test with a constant and
a time trend [see (Harvey, 1990)]. The Dickey-Fuller test showed that the cap rate variable was stationary
over the time in our study.
3. To see this, consider the ex ante version of equation A2 in the appendix.
4. The cap rate is defined as the ratio of net stabilized earnings to the transaction price (or market value) of
a property. Net stabilized income is calculated under the assumption that the percent occupancy in the building
is equal to or greater than the occupancy rate for comparable existing buildings in the market in which a
property trades. In other words, the vacancy rate has stabilized relative to the vacancy rate that exists during
the leasing up period of a new building. Net stabilized income is computed on a monthly or quarterly basis
and then annualized. The transaction price is the gross price paid for the property. In some cases, the appraisal
value is used in lieu of the transaction price by those who report cap rates. The cap rate used in this paper
is the average of the cap rates for individual properties for which this information is made available to the
ACLI. Alternatively, the cap rate can be thought of as the earnings-price ratio on direct real estate investment.
5. In contrast to Liu and Mei (1991), we do not use a January dummy and the spread between the yields on
long-term AAA corporate bonds and the l-month Treasury bill as forecasting variables. The January dummy
is omitted given the nature of the VAR process while the spread variable is not used because it was not statistically
significant in either Liu and Mei (1991) or in this study for value-weighted stocks, small stocks, and equity
REITs. In addition to this, the parameter estimates appear to be invariant to the inclusion or exclusion of
the spread variable. The parameter estimates also appear to be robust to the inclusion or exclusion of the
dividend yield on an equally weighted portfolio consisting of all equity REITs on the CRSP tapes. The results
of the study, which include the spread variable and the equally weighted dividend yield on equity REITs
are available from the authors on request.
6. It is fairly easy to extend the results from a first-order VAR to a K-order VAR because a K-order VAR can
easily be rewritten in a first-order matrix form as (3). For example, assuming that q+t = AZ, + Bq-t + w,+t ,
by redefining z; = {z,, z,_,}’ and w; = {w,, 0)’ and A’ = [ T i ], we have z;+t = A’z; + w;+i. Thus,
all of the results which we derive for z~+, also applies to z;+t.
7. Campbell (1990), Cochrane (1988), Lo and MacKinlay (1988), and Poterba and Summers (1988), have all
used the variance ratio test to document the mean reverting behavior of stock returns. Kandel and Stambaugh
(1988) also report a number of calculations of this type. However, some controversy exists on the presence
of mean-reversion. The controversy centers around whether the evidence on mean-reversion arises from a
AN ANALYSIS OF REAL-ESTATE RISK 19
slowly reverting component of stock prices or from the weak power associated with the variance ratio test
since this statistic is based on asymptotic theory but is applied to finite samples. For example, Richardson
and Stock (1989) argue that large-sample approximations to sampling distributions perform poorly in practice
because there is not much independent information in a long time series of multi-year returns due to the
small number of nonoverlapping observations.
8. GMM approach used in the paper is designed to alleviate some of the measurement error problems in the
data. Parameter estimates obtained from using GMM will be consistent, as long as the measurement errors
are uncorrelated with lagged instrumental variables.
9. For example, see Chow (1982), Campbell (1991), Campbell and Mei (1991) and Campbell and Shiller (1988a).
10. Unfortunately, the ACLI does not break down monthly cap rates by type of real estate. This is only done
with respect to quarterly cap rates.
11. In contrast to Liu and Mei (1991), we didn’t include bonds because of its distinctive cash flow pattern, which
makes it necessary to use a different variation of the present value model from that used in this paper. However,
the study does include the impact of changes in interest rates on stocks and real estate. We also drop the
term variable used in Liu and Mei (1991) because it was not statistically significant either in Liu and Mei
or in this study. Since our parameter estimates were robust to the inclusion or omission of the term variable,
we omitted it for reasons of parsimony. The January dummy is also excluded given the nature of the VAR
process (e.g., the January effect is implicitly embedded in the model).
12. Changes in future expected cash flows for equity REITs could arise in part from changing expectations in
future rental rates, anticipated vacancies. and tentative absorption rates.
13. The reason that the sum of cash-flow risk and discount-rate risk exceeds 100% of total unexpected real-estate
risk is that the covariance risk accounts for -27% to -28% of the total risk. We should also note that the
standard errors in table 5 are overstated due to insufficient time series data. which in turn leads to understated
T statistics. This is the reason for our claim that discount-rate risk is also an important component of total
unexpected real-estate risk.
14. The equations used in this study differ slightly from those used in Campbell and Shiller (1988a, b) due to a
difference in timing conventions. More specifically, we assume that the stock price at time t and the conditional
expectation of future variables are measured at the end of period t rather than at the beginning of period t.
15. See Campbell and Shiller (1988a) for an evaluation of the quality of the linear approximation in equations
Al and A2.
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