Idiosyncratic Risk and REIT Returns

Transcription

1 IRES IRES Working Paper Series Idiosyncratic Risk and REIT Returns OOI Thian Leong, Joseph WANG Jingliang Department of Real Estate National University of Singapore James R. WEBB Department of Finance College of Business Cleveland State University

2 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 rstooitl@nus.edu.sg, Jingliang.wang@nus.edu.sg # Department of Finance Cleveland State University Cleveland, Ohio 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.

3 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.

4 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

5 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 and then a 5 year drift down from ; followed by another high period and another 5 year drift down from Secondly, the 2

6 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 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 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

7 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

8 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

9 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 residualεi in the regression of either a CAPM model or the following Fama and French (FF, 1993) three-factor model: R R = α + b ) + i f i i ( RM R f + sismb + hi HML ε i, where i R f R is the return on the individual stock in excess of the risk-free rate, R R is the excess return on the M f 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

10 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 Data & Descriptive Analysis Our study sample comprises publicly traded REITs between 1990 and 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

11 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$ million to US$ million, whilst the median B/M declined from to 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 to 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 and then a 5 year drift down from ; followed by another high period and another 5 year drift down 8

12 from 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 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 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 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

13 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 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

14 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 variance of REIT returns due to the idiosyncratic component, as follows: 2 2 σ ε / 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 , 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

15 volatility and expected stock returns. Specifically, the following cross-sectional regression is run for each month of the sample period: r K = γ + γ X + ε, i= 1, 2, L, Nt, t = 1, 2, L, T (1) it, 0, t kt, kit,, it, k = 1 where r it, is the excess return on security i in month t. X kit,, 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 it,, captures the deviation of the realized return from its expected value. N denotes the number of securities in the cross-sectional t regression of month t, which varies from month to month. In our case, the number of securities, N t, ranges from 42 to 149; and the maximum number of months, t, is 192. The most important parameter in Equation (1) is $ γ kt,, which has the following mean and variance: 1 T $ γ $ k, t = γ k, t (2) T t = 1 VAR( $ ) = T t = 1 γ kt, $ $ 2 ( γ γ ) kt, kt, 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 kt, ) divided by its time-series standard error, which is the square root of the variance of $ γ kt, divided byt : $ $ γ k, t t( γ k, t) = (4) VAR( $ γ ) T k, t 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

16 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: R r = α + β ( R r ) + s SMB + h HML + ε it, t i i mt, t i t i t it, ε N(0, σ ) (5) 2 it, it, p q 2 2 ε ε lnσit, = αt+ i, jlnσit, j+ ik, θ + γ 2 / π j= 1 k= 1 σ it, k σ it, k it, k it, k b c ( ) 1/2 (6) ε γ σ it, k The term ( ) 1/2 it, k 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 2 square root of conditional varianceσ, which is the function of the past p -period of residual it, 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

17 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 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 EIR ( ) 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

18 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

19 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 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

20 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 [ 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

21 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 and , 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 How can the disappearing influence of the size and value factors in the presence of idiosyncratic 18

22 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, 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

23 [ 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 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, and , 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

24 r r it it = c + γ + 1 E( β ) it + γ 2 ln( MEit ) + γ 3 ln( B / M it ) + γ 4 Ret( 2, 13) it + γ 5E( IR) it ε it (7) = c + γ Ret( 2, 13) + γ 5E( IR) + ε 4 it it 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 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