Idiosyncratic Risk and REIT Returns
Joseph T.L. OOI*, Jingliang WANG*, and James R. WEBB#
* Department of Real Estate
National University of Singapore
4 Architecture Drive, Singapore 117566
E-mail: rstooitl@nus.edu.sg, Jingliang.wang@nus.edu.sg
# Department of Finance
Cleveland State University
Cleveland, Ohio 44114
Email: j.webb@csuohio.edu
Final revised version dated: September 26, 2007
Journal of Real Estate Finance and Economics (forthcoming)
(Reference # 3287)
Acknowledgement: We would like to thank the anonymous referee and editors, as well
as seminar participants at the National University of Singapore and the 2007 American
Real Estate Society Annual Meeting for helpful comments and suggestions.
Idiosyncratic Risk and REIT Returns
Abstract
The volatility of a stock returns can be decomposed into market and firm-specific volatility, with the
former commonly known as systematic risk and the later as idiosyncratic risk. This study examines the
relevance of idiosyncratic risk in explaining the monthly cross-sectional returns of REIT stocks.
Contrary to the CAPM theory, we find a significant positive relation between idiosyncratic volatility
and their cross-sectional returns. This suggests that firm-specific risk matters in REIT pricing. The
regression results further show that once idiosyncratic risk is controlled for in the asset-pricing model,
the size and book-to-market equity ratio factors ceased to be significant. The explanatory power of
the momentum effect remains robust in the presence of idiosyncratic risk.
Key word:
Idiosyncratic risk, asset pricing, REIT stocks.
Idiosyncratic Risk and REIT Returns
1.
Introduction
The volatility of asset returns can be decomposed into market and firm-specific volatility, with
the former commonly known as systematic risk and the later as idiosyncratic risk. Compared to
the plethora of studies on the relationship between systematic risk and returns, the role of
idiosyncratic volatility in asset pricing has been largely ignored in the literature until recently. This
is not surprising because the capital asset pricing model (CAPM; Sharp, 1964; Lintner, 1965;
Black, 1972) prescribes that only the non-diversifiable systematic risk matters in asset pricing.
Idiosyncratic risk, on the other hand, should not matter because it can be completely diversified
away according to modern portfolio theory. Nevertheless, researchers and investors alike have
started to pay more attention to idiosyncratic risk. It is argued that while idiosyncratic risk can be
eliminated in a well diversified portfolio, most investors care about the firm-specific risk because
they do not hold diversified portfolios, either because of wealth constraints or by choice (Xu and
Malkiel, 2003). Furthermore, the pricing of options and warrants would require knowledge of
total volatility, which includes both market as well as idiosyncratic risks. Meanwhile, Sheleifer and
Vishny (1997) and Ali, Hwang and Trombley (2003) argue that volatility, particularly idiosyncratic
risk, will deter arbitrage activities. A number of studies such as Tinic and West (1986), Malkiel
and Xu (1997, 2006), Goyal and Santa-Clara (2003) and Fu (2005) have observed that portfolios
of common stocks with higher idiosyncratic volatility recorded higher average return. These
studies provide empirical support to Merton’s (1987) contention that in a world of incomplete
information, under-diversified investors are compensated for not holding diversified portfolios.
This paper examines the role of idiosyncratic volatility in the pricing of REIT stocks. Whilst we
do not anticipate the relationship between REIT returns and idiosyncratic volatility to be
significantly different from the broader stock universe, it is widely accepted that real estate assets
and property-related stocks are more exposed to idiosyncratic risk due to the inherently localized
and segmented nature of the real estate space markets. Furthermore, Capozza and Sequin (2003)
1
observe that REITs with greater insider holdings tend to invest in assets with lower systematic
risk. Given that the performance of REITs is intimately linked to underlying illiquid real estate
properties that are prone to booms and busts (Chaudry, Maheshwari and Webb, 2004), a study
focusing on the relationship between idiosyncratic risk and REIT returns is warranted. Whilst
common stock, bond and real estate returns have been employed to explain REIT returns at the
aggregate level, Clayton and MacKinnon (2003) and Anderson et al. (2005) have noted that the
proportion of variance not accounted for by these risk factors has been rising over time. In other
words, they find the influence of idiosyncratic risk on REIT volatility and returns to be growing.
This is consistent with the attempts by REIT and fund managers to outperform the market
benchmark by achieving superior returns (higher alphas) on their investment. For example,
property development activities, which have been identified as one of the future growth engines
of listed property trusts (LPTs) in Australia, will increase considerably their firm-specific risk.
Tan (2004), not surprisingly, observes that the firm-specific risk for LPTs with high exposure to
development activities is much higher than those with minimal development activities. Whilst the
benefits of corporate focus versus diversification are well documented in the REIT literature (see
Capozza and Seguin, 1999), we still do not fully comprehend its implications on stock returns
and risk. Yet in a recent study on listed real estate corporations in the US, British, French, Dutch
and Swedish markets, Boer, Brounen and Veld (2005) observe that although the firm’s systematic
risk is not affected by corporate specialization, there is a strong positive relationship between
corporate focus and firm-specific risk. In other words, firm-specific risk increases with the degree
of corporate focus. A detailed study on the idiosyncratic risk of REITs is, therefore, timely as
REIT managers shift towards a more focused investment strategy.
Prior to examining the relationship between expected returns of REIT stocks and conditional
idiosyncratic volatility at the firm-level, we first track the historic idiosyncratic volatility pattern
of REIT stocks publicly traded in the US between 1990 and 2005 (presented in Figure 1). Several
discernible patterns can also be observed with regards to the behavior of idiosyncratic volatility
of the REIT sector. Firstly, it exhibited a cyclical movement that was repeated twice during the
study period; high from 1990-1993 and then a 5 year drift down from 1993-1999; followed by
another high period 1999-2001 and another 5 year drift down from 2001-2005. Secondly, the
2
sector’s aggregate idiosyncratic volatility exhibited a counter-cyclical pattern in that it moves in
opposite direction from the sector’s performance. In addition, the relationship is asymmetric with
idiosyncratic risk increasing dramatically in bad times, but only reducing marginally in good times.
Overall, the time-varying behavior of idiosyncratic risk has important implications for portfolio
diversification at different stages of the market cycle. Significantly, the sector’s returns variance is
dominated by idiosyncratic risk, which on average constituted 78.3% of the overall volatility
exhibited by REIT stocks between 1990 and 2005.
We then examine whether conditional idiosyncratic volatility of individual REIT stocks is
significantly related to their monthly cross-sectional returns. Our study sample covers 149 REITs,
which were publicly traded in the US between 1990 and 2005. The time-varying idiosyncratic
volatility of individual REIT stocks is measured relative to the standard Fama and French (FF,
1993) three-factor model based on their daily returns over the previous month. Following Fu
(2005), Exponential Generalized Auto-Regressive Conditional Heteroskedasticity (EGARCH)
models are employed to control for the time-varying nature of idiosyncratic risk. We then
estimate month-by-month Fama and MacBeth (FM, 1973) regressions of the cross-section of
REIT returns on the conditional idiosyncratic volatility. By focusing on the cross-sectional
returns of firms operating in the same sector, we can assume away any sector-specific variations
from an econometric perspective. The empirical results indicate that firm-specific idiosyncratic
risk plays a significant role in the pricing of REIT stocks. Contrary to the CAPM theory, but
consistent with extant evidence on the diminishing role of beta, we find that systematic risk does
not significantly explain the expected returns of REIT stocks. The explanatory power of
idiosyncratic risk remains robust when we control for three other well-known asset pricing
anomalies, namely size, value and momentum effects. Interestingly, the explanatory power of size
and value effects dissipated once we control for idiosyncratic risk in the regression models but
the momentum effect is robust to the inclusion of idiosyncratic risk. This is consistent with Fu
(2005) who suggests that the strong size and value effects observed in previous studies could
merely be picking up the effects of omitted idiosyncratic risk in their asset pricing models.
3
The remainder of this study is organized as follows. Section 2 reviews related studies to provide
relevant background for our research design. Section 3 presents the data as well as a descriptive
analysis of the historical trends of idiosyncratic volatility in the REIT market. Section 4 sets up
the econometric models and presents the estimation results on the relationship between
cross-sectional expected returns and the conditional idiosyncratic risk of REITs. Section 5
examines the robustness of the results in the presence of three common market anomalies as
well as different model specification and time period. Section 6 concludes.
2.
Literature Review
The traditional CAPM theory (Sharp, 1964; Lintner, 1965; Black, 1972) prescribes that only
systematic risk matters in asset pricing because it is non-diversifiable. Idiosyncratic risk, on the
other hand, should not be priced because it can be completely diversified away. Nevertheless, risk
diversification through the addition of more stocks in a portfolio involves a trade off between
the benefits of further diversification and higher transaction costs, which rises with the number
of the stocks in the portfolio. In situations where investors do not have complete information of
all the securities in the market, Merton (1987) theorizes that idiosyncratic volatility is relevant to
asset pricing. Since it is costly to learn and follow the performance of individual stocks, he argues
that it is not optimal for an investor to track the information of all the securities in the market.
Consequently, investors (both individuals and institutional) only know a small subset of the
securities in the market and construct their portfolios from these known securities; resulting in
them holding under-diversified portfolios.1 Furthermore, institutional investors, fund managers
and arbitrageurs may also choose not to hold well-diversified portfolios due to contractual
reasons or deliberately structure their portfolios to accept considerably high idiosyncratic risk in
an attempt to gain extraordinary returns. Using a variation of the CAPM model, Malkiel and Xu
(2006) demonstrate that if one group of investors fails to hold the market portfolio for
1
In addition to incomplete information, there are a number of other factors that could also attribute to why
investors hold undiversified portfolios. They include market segmentation and institutional restrictions including
limitations on short sales, taxes, transaction costs, liquidity, imperfect divisibility of securities (Merton, 1987; p. 488)
4
exogenous reasons, the remaining investors will also be unable to hold the market portfolio. In
their model, idiosyncratic risk is priced to compensate rational investors for their inability to hold
the market portfolio.
Empirically, a key study supporting the CAPM theory is Fama and MacBeth (FM, 1973) who
observed that idiosyncratic risk does not play any significant role in explaining the cross-sectional
returns of common stocks. However, more recent studies have yielded contrasting results. Using
the same methodology as FM but over a different time period, Malkiel and Xu (2006) observe a
weakly positive relation between idiosyncratic risk and the cross-section of expected stock
returns. Fu (2005), who uses the more sophisticated generalized autoregressive conditional
heteroskedasticity (GARCH) model to estimate idiosyncratic volatility, finds a stronger positive
relationship. Goyal and Santa-Clara (2003) also find a significant positive relation between
average stock variance, which they demonstrate to be largely idiosyncratic, and the stock market
returns. The positive relation is consistent with Merton (1987) and Malkiel and Xu (2006)
argument that idiosyncratic risk could be priced in an incomplete world where investors hold
under-diversified portfolios either by choice or by constraints.
A puzzling result was, however, observed by Ang et al. (2006). Dividing stocks into five equal size
portfolios according to their idiosyncratic risk in the previous month, they compared the
risk-adjusted returns between the highest risk and lowest risk portfolios. Finding the difference to
be significant negative, they also conclude that idiosyncratic risk is priced. However, the negative
relation is puzzling because it suggests that stocks with lower idiosyncratic volatilities earned
higher average returns! Bali and Cakici (2007) attribute the contrasting results in previous studies
to differences in their methodology, particularly data frequency used to compute idiosyncratic
risk, weighting scheme used to compute average portfolio returns, breakpoints utilized to sort
stocks into quintile portfolios, and screenings for size, price and liquidity. Another drawback of
the portfolio sorting methodology adopted by Ang et al. (2006) and other prior studies on pricing
anomaly is its limited ability to examine the interactive effects of other factors on stock returns.2
2
For example, to allow for variation in beta that is unrelated to firm size, FF (1992) subdivide each size decile into
ten portfolios on the basis of pre-ranking betas for individual stocks. This results in 100 size-beta portfolios.
5
Idiosyncratic risk, by definition, is firm-specific and hence, is not captured by market risk factors.
A common measure for idiosyncratic risk is the standard deviation of the residual ε i in the
regression of either a CAPM model or the following Fama and French (FF, 1993) three-factor
model: Ri − R f = α i + bi ( RM − R f ) + si SMB + hi HML + ε i , where Ri − R f is the return
on the individual stock in excess of the risk-free rate,
RM − R f is the excess return on the
market portfolio, SMB is the difference between the return on a portfolio of small stocks and the
return of a portfolio of large stocks, and HML is the difference between the return on a
portfolio of high book-to-market ratio (B/M) stocks and the return on a portfolio of low B/M
stocks.
One difficulty encountered in empirical tests on asset pricing models is that whilst the models are
framed in expectations (ex-ante), the data employed are usually ex-post. In order to address this,
lagged firm attributes are often employed in cross-sectional studies as a proxy for the expected
value in the subsequent period. For example, FF (1992) used the market equity and B/M of the
previous year to explain the cross-section of the monthly returns of the current year. Although
lagged values of firm characteristics could be used to predict their future values, the same
approach may not be appropriate to measure conditional volatility and returns of stocks due to
their time-varying characteristic (Campbell et al., 2001 and Fu, 2005). Consequently, increasingly
sophisticated statistical models, such as parametric ARCH or stochastic-volatility models, have
been suggested.3 Whilst prior studies such as FM (1973) and Ang et al. (2006) have employed the
lagged values of market risk and idiosyncratic risk as the best estimates of their expected value,
Fu (2005) argues that such approximation is only valid if the stock’s conditional returns and
volatility follows a random walk process.
3
The ARCH model, first proposed by Engle in 1982, relates the variance of the current error term to be a
function of the variances of the previous time period's error terms. However, a limitation of the ARCH model is that
a fairly high lag order (p) is required to obtain a good fit model. Taking advantage of the fact that an autoregressive
moving-average model is a more parsimonious specification, Bollerslev (1986) introduced the generalized ARCH
(GARCH) models of order (p,q) where current volatility is dependent on the volatilities for the previous q days and the
squared returns for the previous q days. Since then, alternative specifications have been considered. The exponential
GARCH (EGARCH) model has been found to be the best specification to model the monthly returns of US stock
(Pagan and Schwert, 1990) and to capture the asymmetric effect of conditional volatilities (Engle and Ng, 1993).
6
To our knowledge, this is the first study that examines the relationship between conditional
idiosyncratic volatility and expected returns of REIT stocks. In a study on the effects of risk on
urban land prices, Capozza and Schwann (1990) suggest that most of the effect of total risk may
be ascribed to unsystematic risk because it is a larger proportion of total risk than systematic risk.
Their empirical results also indicate that unsystematic risk can be a very important determinant
of housing prices. In their decomposition of the variability of REIT returns, Clayton and
MacKinnon (2003) and Anderson et al. (2005) observed a dramatic increase in the proportion of
volatility not accounted for by the three common factors (namely stock, bond and direct real
estate). This suggests that the influence of idiosyncratic risk component in REIT returns is
growing over time, which Clayton and MacKinnon (2003) attribute to the “institutionalization”
of stock ownership and technology changes. It is worthwhile to note that the “idiosyncratic risk”
examined in the two studies is actually sector-specific since the aggregate return (NAREIT Index)
was used in their estimations. In contrast, our current study focuses on idiosyncratic risk at the
firm-level.4
3.
Data & Descriptive Analysis
Our study sample comprises publicly traded REITs between 1990 and 2005. After omitting
REITs that have not traded for more than five years and those with negative equity book equity
value, we are left with a study sample of 149 REITs. The number of REITs in our sample is not
static over the study period; growing from 42 to reach a peak of 149, before finally settling at 146
in the end of our study period (as of December 2005). Table 1 presents the median value of
three financial attributes, namely size, B/M ratio and financial leverage of REITs in our sample at
the start and end of the study period.
[ Table 1 ]
4
In another study, Chaudhry, Maheshwari and Webb (2004) observe that different firm characteristics impact
idiosyncratic risk depending on the time period examined.
7
The data shows that between 1990 and 2005, the median market capitalization of the 42 REITs
in our initial sample grew by 7.57 times, from US$ 59.34 million to US$ 508.37 million, whilst the
median B/M declined from 1.096 to 0.586. This implies that the median REIT has not only
grown larger, but it has also transformed from a value stock to become more of a growth stock.
Over the same period, the debt-equity ratio of the median REIT has increased from 0.946 to
1.875. A comparison of the financial attributes of the initial 42 REITs with that of the final
sample of 146 REITs suggest the new REITs that were listed subsequent to 1990 are generally
larger in terms of market capitalization. They also tend to employ more debt in their capital
structure as compared to the older REITs.
To track the historical pattern, we first measure their idiosyncratic volatility relative to the FF
three-factor model using their daily returns over the past month. In every month of the study
period (January 1990 to December 2005), daily excess returns of individual REITs are regressed
on the daily FF three factors, namely the market excess return ( Rm − rf ), the SMB and the HML.
Daily and monthly returns data of publicly traded REITs are extracted from the Center for
Research in Security Prices (CRSP), whilst data for the three risk factors were downloaded from
Kenneth R. French’s website. Specifically, regressions are conducted every month for each REIT
with its idiosyncratic risk for the particular month represented by the standard deviation of the
regression residual. In order to track the historical movements in the idiosyncratic volatility of
the overall REIT market, we take the average idiosyncratic risk across the individual REITs for
each month using equally-weighted (EW) and value-weighted (VW) measures. The two volatility
series are presented in Figure 1.
[ Figure 1 ]
Whilst idiosyncratic volatility of the average REIT stock fluctuates over time, several patterns are
discernible from Figure 1. Firstly, the idiosyncratic volatility series exhibited a cyclical movement
that was repeated twice during the study period; high from 1990-1993 and then a 5 year drift
down from 1993-1999; followed by another high period 1999-2001 and another 5 year drift down
8
from 2001-2005. Consistent with Campbell et al. (2001), Figure 1 also reveals a countercyclical
pattern. In particular, the idiosyncratic risk of REITs is especially low during the bullish market
between 1995 and 1998 as reflected by the steadily rising NAREIT index over the period. In
contrast, sudden spikes in the average volatility were registered in late 1990-early 1991, September
1998 and April 2004. These points coincided with periods of decline in the broad REIT market.
It is also interesting to note that the countercyclical pattern is asymmetric; idiosyncratic volatility
decreases marginally in good times, but in bad times, it escalates very quickly. Campbell et al.
(2001) suggest that the countercyclical behavior of volatility has important implications for
diversification of risk at different stages of the business cycle. Since market volatility is
substantially higher in recessions, they argue that even a well diversified portfolio is exposed to
more volatility when the economy turns down. They further argue that increase in volatility is
stronger for an undiversified portfolio because industry and firm-level volatility also increase in
economic downturns. Consequently, diversification is more important and requires more
individual stock holdings to achieve when the economy turns down.5
Given the robust growth of the REIT sector in recent years, it is not surprising that the
idiosyncratic volatility of the sector has declined. The idiosyncratic volatility of REITs can be
expected to rise when the market sentiment settles to a realistic level. Amidst the cyclical pattern,
the volatility series does exhibit a slight downward trend over the long run. Particularly, the
idiosyncratic risk of the average REIT fell from 9.3% at the beginning of the study period to
4.7% by the end of the study period, representing a 50% decrease in the idiosyncratic risk of
individual REITs between 1990 and 2005. This declining trend, which is contrary to that
observed for common stocks (see Xu and Malkiel, 2003; Bennett and Sias, 2005; Fink et al., 2005;
Wei and Zhang, 2006), can be attributed to the dramatic increase in the average size of REITs
after 1990. The average market capitalization of publicly traded REITs grew from just below US$
100 million prior to 1991 to above US$ 1.5 billion in 2004 (Ooi, Webb and Zhou, 2007). Active
5
According to Campbell et al. (2001), the trend decrease in idiosyncratic volatility relative to the market volatility
may imply that the correlations among individual stock returns have increased over the sample period. This in turn
implies that the benefits of portfolio diversification have decreased over time.
9
acquisition and merger activities in the REIT market during the 1990s also resulted in REITs that
were separately listed previously (and hence, their idiosyncratic risks separately measured) being
merged into a single entity; thus, resulting in a lower combined idiosyncratic risk (see Campbell et
al., 2001; Campbell, Petrova and Sirmans, 2003).6 Another indication that the idiosyncratic risk
of larger REITs is lower than smaller REITs, can be observed in Figure 1 where the
value-weighted series are consistently below the equal-weighted series. Chaudhry, Maheswari and
Webb (2004) explain that larger REITs are more likely to be geographically diversified and hence,
they would be more insulated from fluctuations in the market prices of the underlying real estate
properties than smaller firms, which are unable to achieve such a level of diversification.7
To double-check whether the trends observed in Figure 1 are simply due to the increased number
of REITs in the sample, we also construct the idiosyncratic volatility series using only the 42
original REITs that have been trading continuously since January 1990. The resulting series,
which is presented in Figure 2, show similar trends as observed earlier in Figure 2, suggesting that
the observed idiosyncratic volatility pattern for REITs is not driven by the addition of more
REITs over the study period.8
[ Figure 2 ]
6
The rise in firm-specific risk of common stocks can be attributed to two interacting factors, namely a dramatic
increase in the number of new listings and a simultaneous decline in the age of the firm at IPO. Fink et al. (2005), in
particular, argue that since the equity of young firms typically represents a claim on cash flows that are further into the
future, it is not surprising that the idiosyncratic risk of the typical public firm has increased. Xu and Malkiel (2003)
further suggest that the rising idiosyncratic volatility is attributed to more institutional ownership and high growth.
7
Besides size, Chaudhry, Maheshwari and Webb (2004) also observe that efficiency, liquidity and earnings
variability are important determinants of idiosyncratic risk of REITs.
8
In addition, to ensure that the observed patterns in the volatility series are not driven by outliers, we recompute
the two series by omitting 5% observations at both ends of the distribution. The time trend for the reconstructed
series is similar to that observed in Figure 2 and hence, is not reported for brevity. The results show that the results are
also not adversely influenced by extreme observations.
10
Following Anderson et al. (2005), we employ a variance decomposition approach to examine the
significance of the idiosyncratic component of return volatility of REITs. Specifically, its relative
contribution to total REIT return volatility is inferred by calculating the proportion of the
2
variance of REIT returns due to the idiosyncratic component, as follows: σ ε2 / σ REIT
. The
results over the study period are reported in Figure 3. Essentially, the proportion of REIT
volatility unexplained by the three risk factors in FF (1993) asset pricing model appears to be very
dominant. Between 1990 and 2005, 78.3% of the monthly return volatility of the individual
REIT stocks is unrelated to the three risk factors. In other words, idiosyncratic volatility
accounted for most of the total volatility exhibited by REIT stocks over the study period. Our
finding is consistent with Goyal and Santa-Clara (2003) who also observe that the average stock
variance is largely idiosyncratic.9
[ Figure 3 ]
4.
Does Idiosyncratic Volatility Matter?
In view of the dominance of idiosyncratic volatility in the overall volatility exhibited by REIT
stocks, we examine in this section whether the idiosyncratic volatility is priced. In contrast to Ang
et al (2006), we investigate the cross-sectional relation between expected stock returns and
expected idiosyncratic volatilities conditioned on past information and firm-specific variables.
Given the limitation of portfolio sorting approach (adopted in Ang et al., 2006) to control for the
effect of other firm attributes as well as inconsistency in the test results depending on the
methodological issues highlighted by Bali and Cakici (2007), we employ the FM (1973) regression
methodology to examine the cross-sectional relationship between conditional idiosyncratic
9
Goyal and Santa-Clara (2003) find that over the period 1926-1999, idiosyncratic volatility is on average 80% of
total volatility of common stocks. In comparison, Anderson et al (2005) observe that 62% of the monthly return
volatility of the NAREIT index is unrelated to any of the capital market factors, namely large cap stock, small cap
growth stock, small cap value stock, bond and real estate, in their asset pricing model.
11
volatility and expected stock returns. Specifically, the following cross-sectional regression is run
for each month of the sample period:
K
ri ,t = γ 0,t + ∑ γ k ,t X k ,i ,t + ε i ,t , i = 1, 2,L , N t ,
k =1
t = 1, 2,L , T
(1)
where ri ,t is the excess return on security i in month t . X k ,i ,t are the explanatory variables
of the cross-sectional expected returns, such as beta, size, book-to-market equity ratio, past
return, and idiosyncratic risk. The disturbance term, ε i ,t , captures the deviation of the realized
return from its expected value. N t denotes the number of securities in the cross-sectional
regression of month t , which varies from month to month. In our case, the number of
securities, Nt, ranges from 42 to 149; and the maximum number of months, t , is 192. The most
important parameter in Equation (1) is γ$ k ,t , which has the following mean and variance:
γ$ k , t =
1
T
T
∑
t =1
γ$ k , t
T
VAR (γ$ k ,t ) =
∑ (γ$
t =1
k ,t
(2)
− γ$ k ,t ) 2
T (T − 1)
(3)
If under-diversified investors are compensated for their inability to hold well-diversified
portfolios, the conditional idiosyncratic risk would be positively related to cross-sectional returns
of the securities. The t-statistic is the average slope ( γ$ k ,t ) divided by its time-series standard
error, which is the square root of the variance of γ$ k ,t divided by T :
t (γ$ k , t ) =
γ$ k ,t
VAR ( γ$ k , t ) T
(4)
We also carry out preliminary test to determine whether market risk and idiosyncratic risk of
REITs follows a random walk process. The results of our preliminary tests, which are presented
in Table 1, indicate that the mean auto-correlation coefficients for market and firm-specific risks
are 0.86 and 0.90, respectively at the first lag. Furthermore, they have a slow decay rate; indicating
12
that the market risk and idiosyncratic risk of individual REITs are non-stationary. As a
confirmation, both the P-value and Ljung-Box Q-statistic reject the random walk hypothesis for
market risk and idiosyncratic risk at the 1 percent level. Consequently, using lagged values to
approximate their expected values could lead to measurement errors in variables and unreliable
inferences for our study sample.
[ Table 2 ]
We, therefore, employ the EGARCH model employed to derive the conditional idiosyncratic
volatility for the individual REITs with the following functions:
Ri ,t − rt = α i + β i ( Rm,t − rt ) + si SMBt + hi HMLt + ε i ,t
p
q
⎧⎪ ⎛ ε
ln σ i2,t = α t + ∑ bi , j ln σ i2,t − j + ∑ ci , k ⎨θ ⎜ i ,t − k
⎜σ
j =1
k =1
⎩⎪ ⎝ i ,t − k
⎡ ε i ,t − k
The term γ ⎢
⎣⎢ σ i ,t − k
⎞
+γ
⎟⎟
⎠
ε i ,t
N (0, σ i2,t )
⎫
⎡ ε i ,t − k
1/ 2 ⎤ ⎪
− (2 / π ) ⎥⎬
⎢
⎢⎣ σ i ,t − k
⎥⎦ ⎭⎪
(5)
(6)
1/ 2 ⎤
− ( 2 / π ) ⎥ is used to capture the asymmetric effect, and when γ < 0 ,
⎦⎥
the return volatility increases after a stock price drop. The monthly excess return process follows
the specification in the FF three factor model as in Equation (5). The idiosyncratic risk is the
square root of conditional variance σ i2,t , which is the function of the past p -period of residual
variance and q -period of shocks as specified by equation (6), where 1 ≤ p, q ≤ 2 . Permutation
of these orders yield four different EGARCH models: EGARCH (1,1), EGARCH (1,2),
EGARCH (2,1) and EGARCH (2,2). We estimate the time-series conditional idiosyncratic
volatility of each individual REIT using all the four E-GARCH models and select the best one
which converges within 500 iterations and yields the lowest Akaike Information Criterion (AIC).
The estimation results are summarized in Table 3 under models 1, 2 and 3. The reported average
slope is the time-series average of the monthly regression slopes for January 1990 to December
2005, and the t-statistic is the average slope divided by its time-series standard error. The number
of stocks in the monthly regressions ranges from 42 to 149. On the influence of beta on the
13
expected returns of REITs, the regression results reported in Table 3 show a relatively flat
relationship with the average slope of expected beta not significantly different from zero. This
indicates that market beta does not help to explain the cross-sectional return of REITs between
1990 and 2005 even when it is the only explanatory variable in the asset pricing model (Model 1).
The insignificant coefficient persists when we include expected idiosyncratic risk as an additional
explanatory variable in the monthly FM regressions (Model 3). The results, although
contradictory to the prediction of the CAPM theory, are consistent with a number of studies
which recorded the diminishing influence of beta on average stock returns (FF, 1992; Goyal and
Santa-Clara, 2003).
[ Table 3 ]
On the other hand, the average slope of conditional idiosyncratic volatility is positive and
statistically significant in Model 2 and Model 3, indicating that REITs with higher expected
idiosyncratic risk do earn higher average returns. In particular, the coefficient estimate is 0.0898
and statistically significant at the 5% level in Model 2. The result continues to hold after we
control for expected market risk in Model 3. Following the inclusion of E ( IR ) in the regression
model, the average R-square which indicates the proportion of variation in the dependent
variable is explained by the independent variables, nearly doubled from 6.65% for Model 1 to
12.88% for Model 3.) In addition, the value of the constant term decreases and becomes not
statistically different from zero in Model 3.
5
Robustness Checks
5.1 Economic Significance
The effect of idiosyncratic risk on expected returns is also economically significant with the
magnitude of the average slope in Model 3 indicating that the REIT’s monthly return will
14
increase by 0.103% with every 1% increase in idiosyncratic volatility. In comparison, Fu (2005)
observed a 0.2% per month rise in common stock returns for a 1% increase in idiosyncratic
volatility. Although the economic effect of idiosyncratic risk on REIT returns appears to be
much smaller than common stock returns, we find that the return from adopting a trading
strategy based on the idiosyncratic risk of REITs is material.
To demonstrate this, we pool the REIT stocks into different equal-sized portfolios (according to
their ranking based on the risk factor) and the returns of the two extreme portfolios are then
examined to determine if they are statistically different.10 To form the portfolios, we rank the
REITs at the beginning of each month in ascending order based on their conditional
idiosyncratic risk for the current month. The REITs in the bottom one-fifth of the sample are
assigned to the low idiosyncratic risk portfolio (denoted as IR
LOW),
while those in the top
one-fifth of the sample are assigned to the high idiosyncratic risk portfolio (denoted as IR HIGH).
The raw and risk-adjusted returns of each of these equal-weighted portfolios are then computed
based on a holding period of 12, 24 and 36 months, respectively. Following the standard practice,
overlapping portfolios are constructed to increase the power of the tests. The idiosyncratic risk
portfolio is a zero-cost, high-minus-low idiosyncratic risk portfolio (IR
HIGH
- IR
LOW).
Table 4
presents the mean excess and risk-adjusted returns for the different portfolios over the three
different holding periods.
[ Table 4 ]
The results in Panel A of Table 4 show that over the over the entire sample period, the
idiosyncratic risk strategy generates an excess return of 0.45%, 0.44% and 0.41% per month for
the 12-, 24- and 36-month holding period. The statistically significant t-statistics suggest that it is
rewarding to follow a trading strategy of constructing portfolios based on the idiosyncratic
volatility of REIT stocks. The risk-adjusted returns (regression intercepts of the FF three-factor
10
Chui, Titman and Wei (2003) and Ooi, Webb and Zhou (2007) adopted a similar approach to examine the
payoffs of REIT portfolios constructed based on the momentum- and value-effect.
15
model), which are presented in Panel B of Table 4, reveal that the payoffs are still significant after
controlling for the three systematic risk factors in the FF model. Further analysis shows that
payoffs from the idiosyncratic strategy are robust to conditions in the broader market.11
5.2 Controlling Size, Value and Momentum Effects
We also test the explanatory power of idiosyncratic risk in the presence of three other
well-known pricing anomalies, namely size, value and momentum effects. The small premium
effect was first highlighted by Banz (1981) who observes that market value of common equity
(ME), not only adds to the explanation of the cross-section of average returns provided by
market risks, but is significantly negatively related to stock returns. Stattman (1980) and
Rosenberg, Reid and Lanstein (1985), who were among the first to document the premium
attached to value stocks, find that average returns of U.S. stocks are positively related to the ratio
of a firm’s book value of common equity to its market equity (B/M).12 Jegadeesh and Titman
(1993) further observe that over an intermediate horizon of three to twelve months, past winners,
on average, continue to outperform past losers. They went on to argue that past returns can be
used to predict future returns. This proposition is now better known as the “momentum effect”
in the literature.
In order to estimate the regressions, we first match the accounting data for all fiscal yearends in
11
Given that Figure 2 and Figure 3 show a countercyclical pattern, we further examine the payoff associated with
adopting the idiosyncratic risk strategy in different market conditions using the portfolio sorting methodology. We
sub-divide the sample period according to whether the market as represented by the NAREIT index is moving
upwards or downwards. The corresponding risk-adjusted returns for the zero-cost idiosyncratic portfolio remain
positive and statistically significant under both rising (0.46%) and declining market conditions ((0.32%). This indicates
that payoffs from the idiosyncratic risk strategy are robust to the overall performance of the market.
12
Although other studies have identified other factors that affect cross-sectional stock returns, such as leverage
(Bhandari, 1988) and earnings-price ratio (Basu, 1983), FF (1992) test the joint role of market equity, book-to-market
equity (BE/ME) ratio, leverage and earnings-price ratio (E/P), and conclude that the combination of market equity
and book-to-market equity ratio seems to absorb the roles of leverage and E/P in average stock returns.
16
calendar year t-1 with the returns for July of year t to June of year t+1. This is to ensure the
accounting variables are known before the returns they are used to explain (FF, 1992). Firm size
is measured by the market value of common equity, which is the product of the monthly closing
price and the number of shares outstanding for June of year t. Book-to-market equity (B/M) is
defined as the fiscal year-end book value of common equity divided by the calendar year-end
market value of common equity. Due to the annual frequency of book equity, this variable is
updated yearly. ME and B/M are transformed to natural logarithm because they are significantly
skewed. In order to capture the momentum effect, we construct a variable called Ret (-2,-13),
which is essentially the cumulative return calculated over the past 12 months beginning with t–2
month, where t presents the current month. Following standard practice, the return of month t-1
is excluded to avoid any spurious association between the prior month return and the current
month return caused by thin trading or the bid-ask spread effect, which may cause returns to
exhibit first order serial correlations. Descriptive statistics of the monthly excess returns,
expected beta, expected idiosyncratic volatility and the three additional variables are presented in
Table 5.13
[ Table 5 ]
The three variables are added one at a time into the month-by-month cross-sectional regressions
in order to examine their joint effect with conditional idiosyncratic volatility and market risk in
explaining the expected return of REIT stocks. The regression results are reported in Table 6.
The positive relation between REIT returns and expected idiosyncratic risk is robust to the
inclusion of new variables, namely beta, size, B/M, and momentum. Conversely, the average
slope for beta consistently remains statistically insignificant for all the regression models.
[ Table 6 ]
13
Following FF (1992) and Fu (2005), the smallest and largest 1% of the observations on ME, B/M and Ret (-2,-13)
are set equal to the next smallest and largest values of the observations (the 0.01 and 0.99 fractiles) to avoid giving
extreme observations heavy weight in the regressions.
17
Models 4A, 4B and 4C focus on the small size-effect and examine its interactive effect with
conditional idiosyncratic risk. The average slope of -0.13% and -0.12% for ME in Model 4A and
4B, respectively, are significant at the 10% level. This indicates that small REITs earn higher
returns than larger REITs, which is consistent with extant evidence in the finance and real estate
literature (Banz, 1981; McIntosh, Liang, and Tompkins, 1991). However, when conditional
idiosyncratic risk is added to the regression (Model 4C), the average slope on ME losses its
statistical significance. This suggests that the small size-effect dissipates once idiosyncratic risk is
taken into account.
Models 5A, 5B and 5C similarly focus on the premium associated with value stocks and examine
its interactive effect with conditional idiosyncratic risk. The average slope of 0.33% and 0.38%
for B/M in Model 5A and 5B are statistically significant at the 10% and 5% level, respectively.
This result is consistent with Ooi, Webb, and Zhou (2006), who find that value REITs tend to
earn higher excess returns. However, just as we have observed earlier for the small-size effect, the
value effect disappears once idiosyncratic volatility is added to the regression (Model 5C).
The average slope for the Ret (-2, -13) variable in Model 6A and Model 6B is 1.28% and 1.34%,
respectively. Both are statistically significant at the 5% and 1% level, respectively. This indicates
that momentum has a strong influence on REIT returns, which is consistent with the findings of
Chui, Titman and Wei (2003). However, unlike the small-size and value premium, the coefficient
for momentum continues to be significant when we add idiosyncratic conditional volatility and
other risk factors in the regression (Model 6C). When estimated jointly, the coefficients for
momentum and idiosyncratic risk are 0.1370 and 0.0831, respectively. Both are statistically
significant.
The disappearing return premium associated with small firm and value stocks after the addition
of idiosyncratic risk, whilst surprising, is not unique. Chui, Titman and Wei (2003) find that the
small firm and high B/M effects do not exist in REITs. Fu (2005) also reach a similar result for
common stocks traded in NYSE, AMEX and NASDAQ during the period from 1963 to 2002.
How can the disappearing influence of the size and value factors in the presence of idiosyncratic
18
volatility be explained? We think that size and B/M may be capturing the omitted effects of
idiosyncratic risk in the asset pricing models 4A, 4B, 5A and 5B since ME and B/M are both
related to firm size. Table 7 shows that B/M is correlated negatively with ME (-0.49) and both
variables, in turn, are strongly correlated with conditional idiosyncratic volatility, -0.35 for ME
and 0.30 for B/M. The correlation coefficient indicates that REITs with high idiosyncratic risk
tend have smaller market capitalization and valued as growth stocks. Consequently, most of the
relation between size and expected returns can be attributed to the negative correlation between
ME and conditional idiosyncratic risk. Similarly, the relation between M/B and expected returns
is caused by the strong positive correlation between B/M and conditional idiosyncratic risk.
In his critique of size-related anomalies, Berk (1995) shows that firm size will, in general, explain
part of the cross-section of expected returns left unexplained by an incorrectly specified asset
pricing model. His model shows that market value is negatively correlated with all the risk factors
and so long as an omitted risk factor is unrelated to the firm’s operating size, market value will be
negatively correlated with the omitted risk factor.14 Consequently, market value will always
provide additional explanatory power in any test of an asset pricing model that omits relevant
risk factors. Since the size-related variables pick up any unmeasured risks, he suggests that they
can be used in cross-sectional tests to detect model misspecification. In particular, he suggests
that size-related measures provide an indication of how much of the risk premium remains
unexplained by the model being tested; “if a specific asset pricing model claims to explain all relevant risk
factors, then, at a minimum, it must leave any market value related measure with no residual explanatory power.”
The event that the two-sized related variables (ME and BE/ME) become statistically insignificant
in Model 4C and Model 5C after the inclusion of conditional idiosyncratic volatility gives
confidence that the two models are correctly specified in the spirit of Berk (1995).
14
The intuition underlying the theory is best illustrated using the following thought experiment proposed by Berk
(1995): “Consider a one-period economy in which all investors trade off risk and return. Assume that all firms in this economy are exactly
the same size; that is, assume that the expected value of every firm’s end-of-period cashflow is the same. Since the riskiness of each firm’s
cashflow is different…, the market value of each firm must also differ. Given that all firms have the same expected cashflow, riskier firms
will have lower market values and so, by definition, will have higher expected returns. Thus, even though all firms are the same size, if
market value is used as the measure of size, then it will predict return” (p.277).
19
[ Table 7 ]
5.3 Measuring idiosyncratic volatility relative to the single factor model
We have previously measured idiosyncratic volatility relative to the FF three-factor model. To
examine the robustness of our empirical results to an alternative model, we also estimate the
conditional idiosyncratic risk relative to the single factor model. 15 The regression results,
reported in Table 8, show that the findings of the current study are robust to the asset pricing
model employed to derive the conditional idiosyncratic risk of REITs. The significant
relationship between idiosyncratic volatility and cross-sectional returns is also persistent when we
include a binary variable in the regression models to capture for any unique risk factors related to
mortgage REITs.16
[ Table 8 ]
5.4 Sub-period analysis
We also examine the persistence of our empirical results over different time periods. In particular,
we divide our study period into two equal sub-periods covering 120 months each as follows;
January 1990 through December 1999, and January 1996 through December 2005. 17
Month-by-month regressions are carried out based on the following two estimation models:
15
Nevertheless, we would like to point out that the three-factor model is generally more useful than the single-factor
model in explaining the variation in EREIT returns and in providing stable estimates of market betas (Peterson and
Hsieh,1997; Chiang, Lee and Wisen, 2005).
16
For brevity reason, the estimation results are not presented here.
17
Note that the two sub-periods, 1990-1999 and 1996-2005, include overlapping years from 1996 to 1999 to provide
sufficient length of time for the sub-period tests. Due to the substantial month-to-month variability of the parameters of
the risk-return regressions, FM (1973) recommend that a longer time-period of analysis to make the t-statistic value
meaningful (page 624). Consequently, subsequent studies such as Chui, Titman and Wei (2003) and Ang et al (2006) have
carried out sub-period tests using at least ten years data.
20
rit = c + γ 1 E ( β ) it + γ 2 ln(MEit ) + γ 3 ln( B / M it ) + γ 4 Re t (−2,−13) it + γ 5 E ( IR) it + ε it
(7)
rit = c + γ 4 Re t (−2,−13) it + γ 5 E ( IR) it + ε it
(8)
Model (7) incorporates all the risk factors, namely beta, firm size, B/M, past returns and
idiosyncratic volatility, whilst Model (8) is a more parsimonious model for REIT returns
incorporating only the two significant factors, namely past returns and idiosyncratic volatility.
The average slope of the monthly regressions for the full and sub-samples are presented in Table
9.
Consistent with the results obtained for the full sample period, the influence of beta, size and
B/M on the cross sectional REIT returns are muted in the two sub-periods once idiosyncratic
risk is added to the asset pricing model. The sub-period results further support the conclusion
that momentum effect and idiosyncratic volatility are consistently significant factors in explaining
the cross-section of REIT returns. Comparing the explanatory power of past returns over the
two sub-periods, we observe that the momentum effect has weakened in the later sub-period, i.e.
January 1996 through December 2005. Conversely, we observe a stronger relationship with
conditional idiosyncratic risk and expected REIT returns in the second sub-period. Thus, the
results show that our earlier conclusions are robust across different sub-periods.
[ Table 9 ]
6.
Conclusions
This study examines the significance of idiosyncratic risk in explaining the monthly
cross-sectional returns of REIT stocks. The data shows that idiosyncratic risk dominates the total
volatility of REIT returns. More importantly, conditional idiosyncratic volatility is a significant
factor in explaining the cross-sectional returns of REIT stocks. The positive relationship between
expected idiosyncratic risk and the cross-section of average REIT returns continue to persist
after the inclusion of other common explanatory variables, such as size, B/M and momentum
effects. It is also robust to alternative asset pricing models used to derive the conditional
21
idiosyncratic volatility of the individual REITs as well as to categorization of data over different
sub-periods.
Whilst our finding is inconsistent with the prescription of CAPM and modern portfolio theory
that only market risk matters (because idiosyncratic risk can be completely diversified away), it is
consistent with Merton’s (1987) proposition that idiosyncratic risk should be priced because
investors often hold under-diversified portfolios (rather than market portfolios) in the presence
of incomplete information. An important implication of this result is that in addition to
systematic risk, managers should also consider idiosyncratic risk when estimating the required
return or cost of capital on individual stocks or assets. The results also have practical applications
for portfolio formation and performance evaluation. As was shown, a portfolio manager could
have realized exceptional returns with a strategy that tilts towards stocks with high conditional
volatility. This is good news for real estate as an asset class which tends to have high idiosyncratic
risk. Similarly, portfolio returns should be benchmarked against returns of portfolios with
matching idiosyncratic risk.
Another striking result of our empirical tests is that once idiosyncratic risk is controlled for in the
asset-pricing model, the influence of size and B/M on REIT cross-sectional returns become
insignificant. The FM regression results show significant small-size and value premium when ME
and B/M are used alone or together with market beta to explain REIT returns. However, the
observed premium is not robust to the inclusion of idiosyncratic risk in the pricing model. The
explanatory power of a third pricing anomaly, namely the momentum effect, remains robust in
the presence of idiosyncratic risk. Idiosyncratic risk appears to have absorbed the influence of
these two common factors which have become standard in asset pricing models. In their
influential paper, FF (1992) propose that size and B/M proxy for risk factors in returns, related to
relative earning prospects that are priced in expected returns. Our empirical evidence suggests
that the common risk factor proxied by size and B/M may be none other than the omitted
conditional idiosyncratic risk in previous asset pricing models. The correlation analysis indicates
that smaller and value REITs tend to have higher idiosyncratic risk.
22
Reference
Ali, Ashiq, Lee-Seok Hwang, and Mark A. Trombley, 2003, Arbitrage risk and the book-to-market anomaly,
Journal of Financial Economics 69, 355-373.
Anderson, Randy, Jim Clayton, Greg MacKinnon, and Rajneesh Sharma, 2005, REIT returns and pricing:
The small cap value factor, Journal of Property Research 22:4, 267-286.
Ang, Andrew, Robert J. Hodrick, Y. Xing, and X. Zhang, 2006, The cross-section of volatility and expected
returns, Journal of Finance 61, 259 – 299.
Bali, Turan G., and Nusret Cakici, 2006, Idiosyncratic volatility and the cross-section of expected returns,
Journal of Financial and Quantitative Analysis (forthcoming).
Banz, Rolf W., 1981, The relationship between return and market value of common stocks, Journal of
Financial Economics 9, 3 – 18.
Basu, S., 1983, The relationship between earnings yield, market value, and return for NYSE common
stocks: Further evidence”, Journal of Financial Economics 12, 129-156.
Bennett, James A., and Richard W. Sias, 2005, Why does firm-specific risk change over time, Working paper,
University of Southern Maine.
Berk B. Jonathan, 1995, A critique of size-related anomalies, Review of Financial Studies 8, 275 – 286.
Bhandari, L.C., 1988, Debt/Equity ratio and expected common stock returns: empirical evidence, Journal of
Finance 43, 507-528.
Black, Fischer, 1972, Capital market equilibrium with restricted borrowing, Journal of Business 45, 444 –
455.
Boer, Dick, Dirk Brounen, and Hans O. Veld, 2005, Corporate focus and stock performance: international
evidence from listed property markets, Journal of Real Estate Finance and Economics 31, 263 – 281.
Campbell, John Y., Martin Lettau, Burton G. Malkiel, and Y. Xu, 2001, Have individual stocks become
more volatile? An empirical exploration of idiosyncratic risk, Journal of Finance 56, 1 – 43.
Campbell, Robert D., Milena T. Petrova, and C.F. Sirmans, 2003, Value creation in REIT property sell-offs,
Real Estate Economics 34(2), 329-342.
Capozza, Dennis R, and Gregory M. Schwann, 1990, The value of risk in real estate markets, Journal of
Real Estate Finance and Economics 3, 117-140.
Capozza, Dennis R., and Paul J. Seguin, 1999, Focus, transparency and value: the Reit evidence, Real Estate
23
Economics 27, 587 – 619.
Capozza, Dennis R., and Paul J. Seguin, 2003, Inside ownership, risk sharing and Tobin’s q-ratios: Evidence
from REITs, Real Estate Economics 31(4), 367-404.
Chaudhry, Mukesh K., Suneel Maheshwari, and James R. Webb, 2004, Reits and idiosyncratic risk, Journal
of Real Estate Research 26, 207 – 222.
Chiang, Kevin C.H., Ming-Long Lee and Craig H. Wisen, 2005, On the time-series properties of real estate
investment trust betas, Real Estate Economics 33(2), 381-396.
Chui, Andy E.W., Sheridan Titman, and K.C. John Wei, 2003, The cross section of expected Reit returns,
Real Estate Economics 31, 451 – 479.
Clayton, Jim, and Greg Mackinnon, 2003, The relative importance of stock, bond and real estate factors in
explaining Reit returns, Journal of Real Estate Finance and Economics 27, 39 – 60.
Fama, Eugene F., and Kenneth R. French, 1992, The cross-section of expected stock return, Journal of
Finance 47, 427– 465.
Fama, Eugene F., and Kenneth R. French, 1993, Common risk factors in the returns on stocks and bonds,
Journal of Financial Economics 33, 3 – 56.
Fama, Eugene F., and James D. MacBeth, 1973, Risk, return, and equilibrium: empirical tests, Journal of
Political Economy 81, 607 – 636.
Fink, Jason, Kristin E. Fink, Gustavo Grullon, and James P. Weston, 2005, Firm age and fluctuations in
idiosyncratic risk, Working paper, James Madison University.
Fu, F., 2005, Idiosyncratic risk and the cross-section of expected stock returns, Working paper, University
of Rochester.
Goyal, Amit and P. Santa-Clara, 2003, Idiosyncratic risk matters!, Journal of Finance 58, 975-1007.
Jegadeesh, Narasimhan and Sheridan Titman, 1993, Returns to buying winners and selling losers:
implications for stock market efficiency, Journal of Finance 48, 65 – 91.
Linter, John, 1965, The valuation of risk assets and the selection of risky investments in stock portfolios
and capital budgets, Review of Economics and Statistics 47, 13 – 37.
Malkiel, Burton G. and Y. Xu, 1997, Risk and return revisited, Journal of Portfolio Management 24, 9-14.
Malkiel, Burton G. and Y. Xu, 2006, Idiosyncratic risk and security returns, Working Paper, School of
Management, The University of Texas at Dallas.
24
Merton, Robert C., 1987, A simple model of capital market equilibrium with incomplete information,
Journal of Finance 42, 483 – 510.
McIntosh, Willard, Youguo Liang, and Daniel L. Tompkins, 1991, An examination of the small-firm effect
within the REIT industry, Journal of Real Estate Research 6, 9 – 17.
Ooi, Joseph T.L., James R. Webb, and Dingding Zhou, 2007, Extrapolation theory and the pricing of
REIT stocks, Journal of Real Estate Research 29(1), 27-55.
Peterson, J. and C. Hsieh, 1997, Do common risk factors in the returns on stocks and bonds explain returns on
REITs? Real Estate Economics 25: 321–345.
Rosenberg, Barr, Kenneth Reid, and Ronald Lanstein, 1985, Persuasive evidence of market inefficiency,
Journal of Portfolio Management 9, 18 – 28.
Sharp, William F., 1964, Capital asset prices: a theory of market equilibrium under conditions of risk,
Journal of Finance 19, 425 – 442.
Shleifer, A. and Vishny, R., 1997, The limits of arbitrage, Journal of Finance 52, 35-55.
Stattman, Dennis, 1980, Book values and stock returns, The Chicago MBA: A Journal of Selected Papers 4,
25 – 45.
Tan, Yen Keng, 2004, Is development goods for LPTs?, Property Australia (November issue), 50-52.
Tinic, S.M. and R.R. West, 1986, Risk, Return and Equilibrium: A Revisit, Journal of Political Economy 94,
126-147.
Wei, Steven X., and C. Zhang, 2006, Why did individual stocks become more volatile? Journal of Business
79, 259 – 292.
Xu, Y., Burton G. Malkiel, 2003, Investigating the behaviour of idiosyncratic volatility, Journal of Business
76, 613 – 644.
25
Figure 1: Time path of idiosyncratic risk of REITs (Full Sample; 1990-2005)
The figure shows the equal-weighted and value-weighted average observed idiosyncratic risk and the annual
percentage change of the NAREIT Index. The idiosyncratic risk is estimated as follows: Each month
between January 1990 and December 2005, the REIT’s excess daily returns are regressed on the
Fama-French (1992) three factors with the idiosyncratic risk of the individual REIT for the particular
month represented by the standard deviation of the regression residuals. The daily residuals are
transformed to monthly residuals by multiplying them with the square root of 22, which is the average
number of trading days in one month.
1.00
0.14
0.80
0.1
0.60
0.08
0.40
0.06
7
0.20
0.04
0.02
0
NAREIT Index (% Change)
Idiosyncratic Risk
0.12
0.00
-0.20
0
1
2
6
3
5
4
7
8
9
0
2
1
3
4
5
n-9 an-9 an-9 an-9 an-9 an-9 an-9 an-9 an-9 an-9 an-0 an-0 an-0 an-0 an-0 an-0
Ja
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
EW
VW
% Change of NAREIT Index
26
Figure 2: Time path of idiosyncratic risk of 42 REITs (Initial Sample)
The figure shows the equal-weighted (EV) and value-weighted (VW) average observed idiosyncratic risk
for the initial sample of REITs that have been traded on the US market since January 1990. The
idiosyncratic risk is estimated as follows: Each month between January 1990 and December 2005, the
REIT’s excess daily returns are regressed on the Fama-French (1992) three factors with the idiosyncratic
risk of the individual REIT for the particular month represented by the standard deviation of the
regression residuals. The daily residuals are transformed to monthly residuals by multiplying them with the
square root of 22, which is the average number of trading days in one month.
0.16
Idiosyncratic Risk
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
n-9 an-9 an-9 an-9 an-9 an-9 an-9 an-9 an-9 an-9 an-0 an-0 an-0 an-0 an-0 an-0
Ja
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
EW
VW
27
Figure 3: Idiosyncratic risk as a proportion of total volatility
The figure shows the proportion of idiosyncratic volatility over total variance of REIT stocks between
January 1990 and December 2005. The idiosyncratic risk is estimated as follows: In every month, excess
daily returns of each individual REIT are regressed on the Fama-French (1992) three factors and the
monthly idiosyncratic risk of the REIT is the variance of the regression residuals. Total volatility is defined
as the variance of the returns over the same period.
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
4
5
1
3
2
5
8
0
9
2
1
4
7
0
3
6
n-0
n-0
n-0
n-0
n-0
n-9
n-9
n-0
n-9
n-9
n-9
n-9
n-9
n-9
n-9
n-9
Ja
Ja
Ja
Ja
Ja
Ja
Ja
Ja
Ja
Ja
Ja
Ja
Ja
Ja
Ja
Ja
Proportion (EW)
28
Table 1: Financial attributes of REITs in the sample
The median value of three financial attributes, namely size, book-to-market equity ratio and financial
leverage of the sampled REITs are presented at the beginning (January 1990) and end (December 2005) of
the study period. The initial sample comprises 42 REITs, whilst the final sample comprises 146 REITs.
Change refers to the magnitude of the change in the particular attribute between 1990 and 2005.
Characteristics
Size (ME) (US $ million)
Final Sample, 2005
Initial Sample, 1990
(146 REITs)
(42 REITs)
2005
1990
2005
Change
1,061.14
59.34
508.37
7.57 x
Book-to-market equity
0.538
1.096
0.586
-0.47 x
Debt-equity ratio
2.069
0.946
1.875
0.98 x
29
Table 2: Random walk tests of monthly beta and idiosyncratic risk
This table summarizes the random walk test statistics for the market and idiosyncratic risks of individual
REITs. The reported figures are the mean statistics across all the REITs. The beta and idiosyncratic risk
reported in Panel A and Panel B are estimated in the spirit of Fama-Mac Beth (1973) at the individual
REIT level using 60-month rolling window market model regressions. The idiosyncratic risk reported in
Panel C is estimated as in Ang et. al (2006): Every month, excess daily returns of each individual REIT are
regressed on the Fama-French (1992) three factors and the monthly idiosyncratic risk of the REIT is
represented by the standard deviation of the regression residuals of the previous month. AC is the
autocorrelation coefficients; Q-statistic is the Ljung-Box Q-statistic with 12 lags; and P-value is the lowest
significance level at which the random walk hypothesis can be rejected.
Lags
1
2
3
4
5
6
7
8
11
12
Panel A: Random walk test for beta estimated in the spirit of F-M (1973)
AC
Q-statistic
P-value
0.86
0.74
0.65
0.59
0.53
0.48
0.43
0.40
0.30
0.27
85.72
159.75
225.94
285.71
340.10
389.52
434.56
475.72
578.37
606.37
0.00
0.00
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.02
Panel B: Random walk test for idiosyncratic risk of F-M (1973)
AC
Q-statistic
P-value
0.90
0.80
0.72
0.65
0.58
0.52
0.47
0.43
0.33
0.30
89.88
169.78
241.59
306.52
365.40
418.72
467.19
511.31
620.79
650.72
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.01
Panel C: Random walk test for idiosyncratic risk (Ang et al. 2006)
AC
Q-statistic
P-value
0.39
0.30
0.26
0.21
0.18
0.19
0.17
0.14
0.14
0.14
29.33
49.80
66.65
80.16
92.04
103.69
114.27
123.66
151.24
159.39
0.02
0.03
0.02
0.03
0.03
0.04
0.04
0.05
0.05
0.05
30
Table 3: Average slopes (t-statistics) from month-by-month regressions
of REIT returns on conditional beta and idiosyncratic volatility
The average slope is the time-series average of the monthly regression slopes for January 1990 through
December 2005, and the t -statistic is the average slope divided by its time-series standard error. The
dependent variable is the percentage monthly excess return. C refers to the regression intercept. E(BETA)
is one month ahead expected market risk, which is estimated using the bivariate GARCH (1,1) model. E(IR)
is the one month ahead expected idiosyncratic risk estimated using the exponential GARCH model relative
to the Fama-French (1992) three factor model.
MODEL
C
E(BETA)
1
0.0107***
-0.0013
(4.72)
(-0.39)
2
0.0043
0.0898**
(1.40)
(1.98)
3
E(IR)
0.0045
-0.0027
0.1028**
(1.59)
(-0.94)
(2.38)
R2 (%)
Adj. R2 (%)
6.65
5.53
8.04
6.95
12.88
10.78
Note: *, **, and *** denotes significance at the 10% level, 5% and 1% level, respectively.
31
Table 4: Returns from idiosyncratic risk portfolio (January 1990 to December 2005)
Panel A reports the average monthly excess returns for the low idiosyncratic risk (IRLOW), high
idiosyncratic risk (IRHIGH) and no-cost idiosyncratic (IRHIGH - LOW) portfolios over three different holding
periods, namely 12, 24 and 36 months. Portfolios are formed at the beginning of every month based on
the conditional idiosyncratic risk estimated using GARCH-type model. Portfolio LIR comprises stocks
ranked in the bottom one-fifth, portfolio HIR comprises stocks ranked in the top one-fifth, and portfolio
HIR-LIR is the zero-cost high-minus-low idiosyncratic risk portfolio. Panel B reports the portfolios’
risk-adjusted returns, which are essentially intercepts (alphas) of the Fama-French (1992) three-factor
model regressions. The numbers in the parenthesis are robust Newey-West (1987) t-statistics, which
corrects for the serial correlation caused by overlapping portfolios.
Holding Period
12 months
24months
36months
0.82%
0.75%
0.68%
(5.06)
(5.14)
(5.31)
1.27%
1.18%
1.09%
(3.67)
(4.16)
(4.35)
0.45%
0.44%
0.41%
(1.89)
(2.45)
(2.76)
0.71%
0.67%
0.63%
(4.46)
(4.37)
(4.40)
1.13%
1.08%
1.02%
(3.16)
(3.50)
(3.78)
0.42%
0.41%
0.39%
(1.68)
(2.18)
(2.61)
Panel A: Raw excess returns
IRLOW
IRHIGH
IRHIGH - LOW
Panel B: Risk-adjusted returns
IRLOW
IRHIGH
IRHIGH - LOW
32
Table 5: Descriptive statistics
The descriptive statistics for the pooled sample (comprising 20,353 observations) from January 1990
through December 2005 are presented. ER(%) is the monthly percentage excess return, which is the total
return net of the one-month T-bill rate. E(BETA) is the one month ahead expected market risk, which is
estimated using the bivariate GARCH (1,1) model. E(IR) is the one month ahead expected idiosyncratic
risk estimated using the exponential GARCH model relative to the Fama-French (1992) three-factor model.
Ln (ME) is the natural logarithm of market equity (price times number of shares outstanding), which is
computed in June of year t and updated monthly. Ln (B/M) is the natural logarithm of book-to-market
equity, where BE is the stockholder’s book equity, plus balance sheet deferred taxes and investment tax
credit, minus the book value of preferred stock, and is for each REIT’s latest fiscal year end of calendar
year t-1. The B/M ratio is measured using market equity ME in the end of December of year t-1 and is
updated annually. Ret (-2,-13) is the cumulative return calculated over the past the 12 months beginning in
the second to last month.
Variables
Mean
Median
Maximum
Minimum
Std Dev
Skewness
Kurtosis
ER(%)
0.0106
0.0089
1.6913
-0.8472
0.0855
2.0499
38.4080
E(BETA)
0.3589
0.2955
20.9597
-11.4119
0.6155
6.3506
185.4888
E(IR)
0.0682
0.0543
1.8031
0.0082
0.0517
6.6856
110.2237
Ln(ME)
5.6346
5.9442
9.7708
-0.6992
1.7665
-0.5625
2.7967
Ln(B/M)
-0.2704
-0.3118
2.3217
-6.2500
0.6388
-0.8000
12.0462
0.1751
0.1522
6.8571
-0.9223
0.3514
2.9445
31.1433
Ret(-2,-13) (%)
33
Table 6: Average slopes (t-statistics) from month-by-month regressions of REIT returns on beta,
idiosyncratic volatility, size, book-to-market equity and momentum.
The average slope is the time-series average of the monthly regression slopes for January 1990 through
December 2005, and the t -statistic is the average slope divided by its time-series standard error. Firm size,
ln(ME), is measured in June of year t and updated monthly (price times shares outstanding). BE is the
stockholder’s book equity, plus balance sheet deferred taxes and investment tax credit, minus the book
value of preferred stock, and is for each REIT’s latest fiscal year end of calendar year t-1. The BE/ME
ratio is measured using market equity ME in the end of December of year t-1 and is updated monthly. In
the monthly regressions, these values of the explanatory variables for individual REITs are matched with
the excess returns for the months from July of year t to June of year t+1. The gap between the accounting
data and the excess returns ensures that the accounting data are available prior to the corresponding excess
returns. Ret(-2,-13), which proxies the momentum effect, is the cumulative return calculated over the past
the 12 months beginning in the second to last month. This measure was computed excluding the data of
the immediate prior month in order to avoid any spurious association between the prior month data and
the current month data caused by thin trading or bid-ask spread effects. In order to avoid giving extreme
observations a heavy weight in the cross-section regressions, we set the smallest and largest 1% of the
explanatory variables equal to the next smallest or largest values.
MODEL
C
E(BETA)
ln(ME)
ln(BE/ME)
Ret(-2,-13)
E(IR)
R2 (Adj. R2 )
Size-effect
4A
4B
4C
0.0168***
-0.0013*
(3.85)
(-1.70)
4.14 (2.97)
0.0166***
-0.0007
-0.0012*
(4.38)
(-0.19)
(1.65)
9.83 (7.63)
0.0077**
-0.0024
-0.0004
0.0858**
(2.23)
(-0.78)
(-0.56)
(2.01)
14.36 (11.21)
Value-effect
5A
0.0104***
0.0033*
5B
0.0111***
-0.0015
0.0038**
(5.10)
(-0.46)
(2.17)
0.0065*
-0.0026
-0.0001
0.0016
0.0845**
(1.85)
(-0.83)
(-0.18)
(1.14)
(1.98)
(4.02)
5C
2.88 (1.70)
(1.72)
8.70 (6.47)
15.72 (11.53)
Momentum-effect
6A
0.0080***
0.0128**
6B
0.0086***
-0.0015
0.0134***
(3.75)
(-0.44)
(2.94)
0.0069**
-0.0024
-0.0007
0.0005
0.0137***
0.0831**
(1.97)
(-0.80)
(-0.97)
(0.33)
(3.09)
(2.01)
(3.12)
6C
4.40 (3.25)
(2.52)
9.90 (7.71)
19.04 (13.96)
Note: *, **, and *** denotes significance at the 10% level, 5% and 1% level, respectively.
34
Table 7: Cross-sectional pearson correlations
The time-series means of the cross-sectional Pearson correlations between the variables defined in Table 5
are presented. The significance level is decided according to the t-statistics computed by the time-series
means of the cross-sectional Pearson correlations divided by the corresponding time-series standard error.
Variables
Ln(ME)
Ln(BE/ME)
Ret(-2,-13)
E(IR)
E(BETA)
0.11
0.06
-0.07
0.14
-0.49
0.12
-0.35
0.00
0.30
Ln(ME)
Ln(BE/ME)
Ret(-2,-13)
-0.06
35
Table 8: Average slopes (t-statistics) from month-by-month regressions of REIT returns on beta,
idiosyncratic volatility (CAPM-based), size, book-to-market equity and momentum.
The average slope is the time-series average of the monthly regression slopes for January 1990 through
December 2005, and the t -statistic is the average slope divided by its time-series standard error. E(BETA)
is the one month ahead expected market risk, which is estimated using a bivariate GARCH (1,1) model.
E(IR)(CAPM) is one month ahead expected idiosyncratic risk estimated using an exponential GARCH
model relative to CAPM. Firm size, ln(ME), is measured in June of year t and updated monthly (price
times shares outstanding). BE is the stockholder’s book equity, plus balance sheet deferred taxes and
investment tax credit, minus the book value of the preferred stock, and is for each REIT’s latest fiscal year
end of calendar year t-1. The BE/ME ratio is measured using market equity ME in the end of December
of year t-1 and is updated monthly. In the monthly regressions, the values of the explanatory variables for
individual REITs are matched with the excess returns for the months from July of year t to June of year
t+1. The gap between the accounting data and the excess returns ensures that the accounting data are
available prior to the corresponding excess returns. Ret(-2,-13), which proxies the momentum effect, is the
cumulative return calculated over the past the 12 months beginning in the second to last month. This
measure was computed excluding the data of the immediate prior month in order to avoid any spurious
association between the prior month data and the current month data caused by thin trading or bid-ask
spread effects. In order to avoid giving extreme observations a heavy weight in the cross-section
regressions, we set the smallest and largest 1% of the explanatory variables equal to the next smallest or
largest values.
MODEL
C
E(BETA)
1
0.0107***
-0.0013
(4.72)
(-0.39)
2
ln(ME)
ln(BE/ME)
Ret(-2,-13)
E(IR)
6.65 (5.53)
0.0043
0.0832*
(1.45)
(1.82)
3
0.0045*
(1.66)
(-0.82)
4C
0.0066*
-0.0017
-0.0004
0.0870**
(1.95)
(-0.59)
(-0.61)
(2.01)
0.0060*
-0.0020
-0.0003
0.0004
0.0888**
(1.76)
(-0.67)
(-0.44)
(0.33)
(2.02)
0.0060*
-0.0013
-0.0008
-0.0008
0.0141***
0.0888**
(1.73)
(-0.44)
(-1.17)
(-0.66)
(3.28)
(2.08)
5C
6C
R2 (Adj. R2 )
-0.0024
0.1028**
7.92 (6.82)
12.79 (10.68)
(2.38)
15.76 (12.66)
17.03 (12.9)
20.64 (15.63)
Note: *, **, and *** denotes significance at the 10% level, 5% and 1% level, respectively.
36
Table 9: Average slopes (t-statistics) from month-by-month regressions of REIT returns on beta,
idiosyncratic volatility, size, book-to-market equity and momentum (sub-period analysis)
The table presents the time series averages of Fama-MacBeth (1973) slopes for two equal sub-periods
(January 1990 – December 1999 and January 1996 – December 2005) from two regressions: (a) the
cross-section of excess REIT returns on momentum factor and idiosyncratic risk; (b) the cross-section of
excess REIT returns on conditional beta, size, book-to-market equity ratio, momentum factor and
conditional idiosyncratic risk. The numbers in the parenthesis are the t-statistic values of the
corresponding coefficients, which is the average slope divided by its time series standard errors. Firm size
ln(ME) is measure in June of year t and updated monthly (price times shares outstanding). BE is the
stockholder’s book equity, plus balance sheet deferred taxes and investment tax credit, minus the book
value of preferred stock, and is for each REIT’s latest fiscal year end of calendar year t-1. The BE/ME
ratio is measured using market equity ME in the end of December of year t-1 and is updated annually. In
the monthly regressions, these values of the explanatory variables for individual REITs are matched with
the excess returns for the months from July of year t to June of year t+1. The gap between the accounting
data and the excess returns ensures that the accounting data are available prior to the corresponding excess
returns. Ret(-2,-13), which proxies the momentum effect, is the cumulative return calculated over the past
the 12 months beginning in the second to last month. This measure was computed excluding the data of
the immediate prior month in order to avoid any spurious association between the prior month data and
the current month data caused by thin trading or bid-ask spread effects.
Period
Variable
01/90-12/05(192 months)
Mean
St.dev
t-stat
01/90-12/99(120 months)
Mean
01/96-12/05(120 months)
St.dev
t-stat
Mean
St.dev
t-stat
rit = c + γ 4 Re t ( − 2, −13) it + γ 5 E ( IR ) it + ε it
c
γ4
0.0033
0.04
1.05
-0.0020
0.05
-0.49
0.003
0.04
0.78
0.0122
0.06
3.02
0.0167
0.06
3.13
0.0081
0.05
1.65
γ5
0.0789
0.60
1.84
0.1139
0.69
1.81
0.1016
0.51
2.17
rit = c + γ 1E ( β )it + γ 2 ln(MEit ) + γ 3 ln( BE / MEit ) + γ 4 Re t (−2, −13)it + γ 5 E ( IR)it + ε it
c
γ1
γ2
γ3
γ4
γ5
0.0064
0.05
1.77
0.0031
0.06
0.60
0.0057
0.04
1.48
-0.0020
0.04
-0.68
-0.0005
0.04
-0.12
-0.0010
0.04
-0.30
-0.0007
0.01
-0.94
-0.0011
0.01
-1.09
-0.0006
0.01
-0.77
0.0002
0.02
0.18
-0.0007
0.01
-0.53
0.0004
0.02
0.22
0.0131
0.06
3.19
0.0175
0.06
3.15
0.0087
0.05
1.79
0.0891
0.58
2.14
0.1192
0.69
1.90
0.1116
0.47
2.59
Note: critical value of t-stat: 2.58 (1% level); 1.96 (5% level); 1.65 (10% level).
37