Pricing of Volatility Risk in REITs

Transcription

1 Pricing of Volatility Risk in REITs R. Jared DeLisle Department of Finance and Management Science College of Business Washington State University NE Salmon Creek Avenue Vancouver, Washington Ph Fax S. McKay Price Perella Department of Finance College of Business and Economics Lehigh University 621 Taylor Street Bethlehem, PA Ph Fax C.F. Sirmans Department of Risk Management/Insurance, Real Estate, and Legal Studies College of Business Florida State University 821 Academic Way, P.O. Box Tallahassee, FL Ph Fax This Draft: April 30, 2012 Keywords: Risk; Idiosyncratic volatility; Implied systematic volatility; Market volatility factor; Asset pricing; REITs

2 1. Abstract We examine the pricing of volatility risk in the cross-section of equity Real Estate Investment Trust (REIT) stock returns over the period. We consider both aggregate (systematic) volatility and firmspecific (idiosyncratic) volatility. In contrast to the negative and significant price of systematic volatility risk for Non-REIT equities, we find that systematic volatility is not priced in REIT returns. Idiosyncratic volatility, estimated using the Fama and French (1993) three factor model, is negatively priced in the cross-section and is largely independent of Non-REIT idiosyncratic volatility. Within the total volatility risk profile, idiosyncratic volatility dominates aggregate volatility in REIT pricing. 2. Introduction Risk can be defined as the likelihood that an asset s realized returns will differ from that which is expected. According to asset pricing theory, investors require a reward for bearing the portion of an asset s risk which cannot be diversified away. Thus, when decomposing risk into aggregate (or systematic) and firm-specific (or idiosyncratic) volatility, only the volatility that can be explained by systematic factors should be priced. We examine the pricing of systematic and idiosyncratic volatility risk in equity Real Estate Investment Trusts (REITs). Studying the pricing of systematic and idiosyncratic volatility in REIT returns is important for several reasons. First, it has long been debated whether REIT shares behave like typical stocks or the underlying real estate assets which they own (e.g. Wang, Erickson, and Chan, 1995; Ghosh and Sirmans, 1996; Chan, Erickson, and Wang, 2003). The answer to this debate has direct implications for portfolio formation and the hedging properties of REITs. Literature in this area tends to focus on either the degree of REIT and stock market integration or the stability of REIT betas over time (e.g. Ling and Naranjo, 1999; Glascock, Lu, and So, 2000; Chiang, Lee, and Wisen, 2005; Fei, Ding, and Deng, 2010; Liow and Addae-Dapaah, 2010), with mixed results. To date, the extent of potential connections 1

3 between REIT returns and the general stock market have not been decided through analysis that relies heavily on time series correlations and/or market beta. Moreover, market beta is a fairly limited proxy for systematic risk that doesn t directly include aggregate stochastic volatility as a state variable. 1 Simply put, we do not know if REIT exposure to aggregate market volatility is priced in the cross-section of REIT returns. Second, while aggregate volatility is important in understanding risk and return relations in a portfolio context, distinct REIT characteristics renders an understanding of idiosyncratic risk to be of great importance (Chaudry, Maheshwari, and Webb, 2004). Most prominently, REIT assets are all unique with respect to locational attributes. Ooi, Wang, and Webb (hereafter OWW) (2009) state that the inherently localized and segmented nature of real estate markets has led to wide acceptance of the idea that real estate assets and property-related stocks may be more exposed to idiosyncratic risk than typical equities. Despite its importance, REIT idiosyncratic risk has only recently attracted the attention of real estate researchers, with OWW being among the first papers to explicitly study firm-specific volatility in REIT pricing. Overall, the literature is inconclusive with respect to the sign and significance of REIT idiosyncratic volatility. OWW (2009) find that idiosyncratic risk not only matters in REIT pricing, but that idiosyncratic risk is positively priced and dominates market beta in explaining REIT returns. For firm-specific volatility, OWW use Exponential Generalized Auto-Regressive Conditional Heteroskedasticity (EGARCH) models in their estimation of idiosyncratic risk, which unintentionally introduces a look-ahead bias in the computation of idiosyncratic volatility (Guo, Kassa, and Ferguson, 2011). Sun and Yung (2009) do not use EGARCH in their estimation of REIT idiosyncratic volatility and initially find a positive relation with expected returns, although once they incorporate various controls the positive relation loses its significance. Chiang, Jiang, and Lee (2009) study the time-series relation between REIT returns and 2

4 idiosyncratic volatility, without EGARCH, and find a positive relation in the vintage REIT era (pre-1992) and a negative relation during the modern REIT era (post-1992). We examine the pricing of systematic volatility risk, and revisit the pricing of idiosyncratic volatility risk, in equity REIT stocks. We avoid the limitations of market beta by utilizing several measures of aggregate volatility which draw upon options data and/or broad, market-wide returns innovations. We also avoid potential EGARCH bias issues by following the idiosyncratic volatility estimation methods of Ang, Hodrick, Xing, and Zhang (hereafter AHXZ) (2006, 2009). Our analysis examines each component of volatility risk to determine whether it is priced in the cross-section of REIT returns, and, if so, the magnitude and nature of the price. We also compare the two components in a multivariate framework to determine the relative importance of each factor. Ultimately, we are able to determine whether variation in the measurement of aggregate and firm-specific risk can improve our understanding of the fundamental REIT risk/return relation and its implications on optimal portfolio formation. For proxies of expected aggregate market volatility, we follow AHXZ (2006), Bakshi, Kapadia, and Madan (hereafter BKM) (2003), and Da and Schaumburg (hereafter DS) (2011). In order to assess whether aggregate market volatility is a state variable, AHXZ estimate stock sensitivity to changes in the Chicago Board Options Exchange s (CBOE) market volatility index (VIX). VIX is constructed using implied volatilities from call and put options on the S&P 500 index and relies on a limited set of at-the-money and out-of-the-money options. 2 AHXZ (2006) show that sensitivity to VIX is a priced risk-factor in the cross section of Non-REIT stock returns. Specifically, they find that stocks with high sensitivity to changes in VIX underperform stocks with low sensitivity to such changes. This negative price of risk is commonly referred to as the negative volatility risk premium. While AHXZ focus exclusively on VIX, we employ the techniques of BKM (2003) to derive additional implied market volatility measures using the full set of tradable S&P 500 and Russell

5 options data. The latter index allows us to examine whether REITs, which are typically smaller firms than those included in the S&P 500, are more sensitive to aggregate volatility in the returns of small stocks. By following BKM, we are also able to compute the higher moments of the returns distribution and control for the effects of implied market skewness. Without controlling for skewness, potentially fat tails in the returns distribution can be misinterpreted as increased volatility. Incorporating skewness allows us to more accurately estimate the shape of the market s return distribution (e.g. BKM, 2003; Chang, Christoffersen, and Jacobs, 2009; DeLisle, Doran, and Peterson, 2011; Chabi-Yo, 2012). We also incorporate the market volatility factor of DS (2011) as an alternative measure that uses monthly, rather than daily, volatility innovations. DS show that augmenting the CAPM by a measure of innovations in market-wide volatility yields a two-factor model that performs well in explaining the cross-section of returns on securities in several asset classes. They construct the volatility factor by extracting the first principal component from the broad cross-section of individual monthly stock volatility innovations. Consistent with what AHXZ (2006) find using implied market volatility, DS (2011) find a negative and highly significant aggregate volatility risk premium using the principal factor in Fama and MacBeth (1973) pricing regressions. We compute idiosyncratic volatility for each REIT using the Fama and French (1993) three factor model in a manner widely used in the extant literature. AHXZ (2006, 2009) provide evidence, using Non- REIT US and international stocks, that idiosyncratic volatility calculated in this way is negatively priced in the cross-section of stock returns. In contrast, OWW (2009) find a positive relation between idiosyncratic volatility and the cross-section of REIT returns by incorporating EGARCH adjustments in their estimation of idiosyncratic volatility. This result is consistent with economic theories which suggest that idiosyncratic volatility and expected returns should be positively related if investors demand a premium for the inability to fully diversify risk (e.g. Merton, 1987; Malkiel and Xu, 2002). However, Fink, Fink, and He (2010) and Guo, Kassa, and Ferguson (2011) show, analytically and empirically, that this 4

6 fairly new estimation strategy inadvertently introduces a look-ahead bias into recursive volatility forecasts by including the contemporaneous stock return in the estimation of month t EGARCH idiosyncratic volatility. Where, in reality, an investor would not have the benefit of knowing month t returns when estimating month t idiosyncratic volatility. Both papers find that without controlling for this EGARCH flaw, idiosyncratic volatility can show up significantly positively related to returns, as in Fu (2009). They further show that EGARCH idiosyncratic volatility can strongly depend on contemporaneous returns in relatively small sample periods, to the point where it affects statistical inference. 3 Our analysis is conducted in the cross-section of REIT returns at both the portfolio- and firmlevel over the period. Our results show that aggregate volatility risk is not priced in REIT returns. REITs do not appear to be sensitive to innovations in implied market volatility using VIX or the full set of tradable options on the S&P 500 and Russell 2000 indices, even when controlling for implied market skewness. Moreover, REITs do not appear to be sensitive to the market volatility factor of DS (2011). These results are strikingly different than the negative and significant relation AHXZ (2006) and DS (2011) find between Non-REIT stocks and aggregate volatility risk. For example, AHXZ find the Non- REIT price of aggregate volatility risk to be approximately -1% per annum, while we find the corresponding REIT price to not be significantly different from zero. In contrast to the cross-sectional REIT results of OWW (2009), but consistent with the crosssectional Non-REIT results of AHXZ (2006, 2009), we find that idiosyncratic volatility risk is negatively priced in the cross-section of REIT returns. 4 AHXZ (2006) find a difference of -1.06% per month between the average returns to stocks sorted into high and low idiosyncratic volatility quintile portfolios for Non- REIT US equities. 5 We find remarkably similar prices of -1.14% and -0.98% per month for equity REITs in portfolio-level and firm-level tests, respectively. Furthermore, we find that idiosyncratic volatility risk pricing remains negative and significant after controlling for various measures of aggregate volatility 5

7 risk. These results hold after controlling for various factors known to affect returns including idiosyncratic skewness, firm size, book-to-market equity, momentum, institutional ownership, and liquidity. Moreover, we find that REIT idiosyncratic volatility is largely independent of Non-REIT idiosyncratic volatility. AHXZ (2009) find that excess returns to idiosyncratic volatility sorted long-short portfolios of international stocks disappear after controlling for exposure to the idiosyncratic volatility of US firms. While the negative price of idiosyncratic volatility risk which we find is similar to AHXZ, our results do not suggest a similarly high level of co-movement between REIT and Non-REIT idiosyncratic volatility. Overall, this study contributes to the literature on REITs and volatility risk in several ways. First, we demonstrate that REITs are not sensitive to innovations in aggregate volatility in the cross-section of expected equity REIT returns using multiple proxies for aggregate volatility risk. This stands in stark contrast to both asset pricing theory and established empirical results for Non-REIT stocks. Second, using empirical methods free from look-ahead bias, we find that equity REIT idiosyncratic volatility is negatively related to expected returns in the cross-section. Third, we show that REITs are not sensitive to aggregate skewness nor idiosyncratic skewness, while in both cases there is opposing theory and Non-REIT evidence. Fourth, we find that Non-REIT idiosyncratic volatility cannot explain REIT idiosyncratic volatility. Finally, we provide additional support for the idea that idiosyncratic volatility risk dominates systematic risk in REIT pricing by showing this to be the case using aggregate volatility measures rather than market beta. The remainder of this study is developed in the following sections. The next section describes the data and variable creation. The third section outlines the analysis and discusses the results. The fourth section concludes. 6

8 3. Data and variable creation Our sample is comprised of the universe of equity REIT firms as identified in Feng, Price, and Sirmans (2011) which includes all equity REITs that are publicly traded on the three major exchanges (NYSE, AMEX, and NASDAQ). We examine the period due to options data availability starting in 1996, with daily prices for options on the S&P 500 and Russell 2000 indices obtained from OptionMetrics IvyDB. VIX index levels are from the CBOE website. 6 Daily and monthly returns data, stock prices, number of shares outstanding, and the number of shares traded are from the Center for Research in Security Prices (CRSP). Excess market returns (MKT), the risk-free rate, and the Fama and French (1993) size (SMB) and book-to-market (HML) factors are from Ken French s website. 7 Book equity is from Standard & Poor s Compustat database and the number of shares owned by institutions is from the Thomson-Reuters Institutional Holdings (13F) Database. In accordance with extant literature, only observations with positive book equity and data available in CRSP for at least one year are kept in the sample. We employ empirical methods in the spirit of AHXZ (2006) with a few modifications to render them more appropriate for our sample and allow for the incorporation of additional, important control variables. While AHXZ rely on the change in VIX to proxy for innovations in systematic volatility risk, we recognize that VIX is somewhat limited in its ability to adequately capture overall expected market volatility. Unlike the VIX computation 8, by following the procedure of BKM (2003) we are able to estimate the implied aggregate volatility of the risk-neutral probability distributions [VOL(SP500) and VOL(R2000)] constructed from all non-zero bid European calls and puts on the S&P 500 and Russell 2000 indices. 9 Exhibit 1 shows daily implied market volatility over the sample period using VIX, VOL(SP500), and VOL(R2000). There are significant differences between these three volatility measures even though they unmistakably follow a similar path. Although not shown, VIX and VOL(SP500) are correlated at 98.9%, while VIX and VOL(R2000) are correlated at 86.8%, and VOL(SP500) and VOL(R2000) are 7

9 correlated at 87.7%. Each measure renders a slightly different depiction of option implied aggregate volatility. The BKM (2003) procedure also enables us to estimate implied market skewness (SKEW). This is important when working with implied market volatility because the Black and Scholes (1973) model tends to misprice deep in-the-money and deep out-of-the-money options (Black, 1975; Merton, 1976). Furthermore, option-implied volatility and skewness are related (Corrado and Su, 1996, 1997; BKM, 2003). Without controlling for expected skewness, the asymmetry, or fat tails of a returns distribution, may be incorrectly interpreted as additional volatility. This is a potentially critical distinction as studies have shown that investors desire positive skewness in their portfolio, but dislike volatility (Kraus and Litzenberger, 1976; Barberis and Huang, 2008). Additionally, Chabi-Yo (2012) demonstrates the mechanism by which expected aggregate skewness is priced. Chang, Christoffersen, and Jacobs (2009) empirically confirm the predictions about skewness pricing from Chabi-Yo's model. Thus, incorporating the third moment (skewness) into the analysis allows us to better isolate the cross-sectional effects of the second moment (variance). While AHXZ (2006) use simple first differences to capture the changes in VIX, we estimate the actual time series innovations using an autoregressive moving average (ARMA) model with two lags for each component. 10 The ARMA(2,2) innovations are denoted ΔVIX, ΔVOL(SP500), and ΔVOL(R2000) for the three measures of implied market volatility. Similarly, ARMA(2,2) innovations for the BKM (2003) estimated skewness control variables are represented as ΔSKEW(SP500) and ΔSKEW(R2000). DS (2011) note that there is a certain degree of arbitrariness in stock index choice when deriving volatility proxies which can be affected by time-varying portfolio weights and correlations. Theory merely suggests that the index used in a CAPM type of asset pricing framework should be broad-based and, preferably, value-weighted. In order to circumvent this lack of guidance and any potential arbitrariness, DS construct a non-parametric volatility proxy by analyzing the cross-section of realized 8

10 monthly volatility innovations for all US equities. Specifically, they extract the first principal component (F1), each month, of the one-period-ahead ARMA(1,1) innovations in the (log) realized volatilities of each stock in the CRSP value-weighted index. They show the F1 factor to be a state variable which consistently prices volatility risk in stocks, options, and bonds. We incorporate F1 into our analysis as an additional measure of systematic volatility. 11 With estimates for ΔVIX, ΔVOL(SP500), ΔSKEW(SP500), ΔVOL(R2000), and ΔSKEW(R2000) we then obtain factor loadings by regressing daily excess returns (RET) on MKT and each market volatility measure, over the most recent month, τ-1, as follows: (1) where RET i,t and MKT t are as defined above for each firm i on day t. For Volatility t, we substitute ΔVIX t, ΔVOL(SP500) t, and ΔVOL(R2000) t in order to obtain factor loadings for each implied volatility measure. Skewness t is only included in the case of ΔVOL(SP500) t and ΔVOL(R2000) t, where the corresponding skewness measures, ΔSKEW(SP500) and ΔSKEW(R2000), are substituted in. For F1, we obtain factor loadings using monthly excess returns. The loading β ΔVIX represents firm sensitivity to innovations in implied market volatility computed using the AHXZ (2006) method. β ΔVOL(SP500) and β ΔVOL(R2000) signify firm sensitivity to innovations in implied aggregate volatility following the BKM (2003) method with options on the S&P 500 and Russell 2000 indices, respectively. β F1 is the loading on the DS (2011) market volatility factor. Positive (negative) loadings indicate that firms with high positive (negative) sensitivities to ΔVIX, ΔVOL(SP500), ΔVOL(R2000), or F1 have positive (negative) returns when the expected market volatility increases. Following AHXZ (2006, 2009), idiosyncratic volatility (IVOL) is computed relative to the Fama and French (1993) three-factor model estimated over the most recent month, τ-1: 9

11 (2) where RET i,t, MKT t, SMB t, and HML t are as discussed previously and IVOL is equal to the standard deviation of the residuals: ( ) (3) where N is the number of days in the regression, and are the residuals from the regression in equation (2). This technique avoids introducing a look-ahead bias in the calculation of idiosyncratic volatility (Fink, Fink, and He, 2010; Guo, Kassa, and Ferguson, 2011). We compute idiosyncratic skewness (ISKEW) as a control variable because Boyer, Mitton, and Vorkink (2010) find that idiosyncratic skewness helps explain cross-sectional pricing variation in idiosyncratic volatility. ISKEW is calculated as the third central moment of the residuals from the same regression as in Boyer, Mitton, and Vorkink: ( ) (4) We also incorporate several additional control variables in our analysis. A number of firm characteristics are shown to be priced in the cross section of returns, and, thus, we wish to control for these variables. For example, literature shows that size (Banz, 1981; Fama and French, 1992) and the ratio of book-to-market equity (Stattman, 1980; Fama and French, 1992) explain much of the variation in the cross-section of stock returns. Jegadeesh and Titman (1993) demonstrate that stock price momentum has the power to predict future stock returns. Lee and Swaminathan (2000) find liquidity 10

12 (as measured by share turnover) is related to future returns. Institutional ownership has also been shown to be correlated with future returns (see Sias, Starks, and Titman, 2006, for an extensive literature review). We define the nomenclature of these variables as follows: SIZE is the natural log of firm market capitalization. BM is the ratio of book-to-market equity. MOM12 represents returns momentum over the most recent twelve month period (non-inclusive of month t = -1) computed as summed excess returns. TURN is calculated as the number of shares traded divided by the number of shares outstanding. Lastly, we define IO as the proportion of shares outstanding held by institutional owners. Exhibit 2 provides sample means and standard deviations for variables considered in the analysis. Note that, while the means of all the aggregate volatility measures are close to zero (due to the mean-reverting nature of market volatility), they exhibit a considerable amount of variation, which is reflected by their standard deviation. Additionally, the sensitivities to the aggregate volatility measures are far less correlated than the measures themselves. Exhibit 3 shows that β ΔVIX and β ΔVOL(SP500) have a correlation coefficient of 0.76, while β ΔVOL(R2000) is only correlated with β ΔVIX and β ΔVOL(SP500) at The highest correlation between β F1 and any of the other measures is Thus, each measure provides a different representation of systematic volatility. As expected, IVOL is not highly correlated with any of the loadings on the systematic factors. 4. Analysis and results At the beginning of each month we separately rank stocks by sensitivity to innovations in implied market volatility, for each of the three measures (β ΔVIX, β ΔVOL(SP500), and β ΔVOL(R2000) ), as well as by idiosyncratic volatility (IVOL). Quintile portfolios are then independently formed each month using the rankings of these characteristics. Next, we create long-short portfolios where we take a long position in equity REITs in the highest quintile (5) of the portfolio formation attribute and a short position in equity REITs in the lowest quintile (1). The long-short portfolios represent a zero-investment strategy based on the attribute of interest. Value-weighted portfolio returns are then examined for the following month. 11

13 This method of portfolio formation is similar to how an investor would use historical information to construct a portfolio, and realize the portfolio returns over the next month. The investor would then rebalance her portfolio based on new information acquired over the past month. To further examine the long-short portfolio returns, we regress the 5-1 portfolio returns on the Fama and French (1993) three factor model and report the alphas. 12 In the idiosyncratic volatility case, following AHXZ (2009), we also augment the model with long-short IVOL returns to the portfolio of all Non-REIT US firms. This allows us to check for co-movement between REIT IVOL and aggregate IVOL. AHXZ (2009) find large and significant co-movements between idiosyncratic volatility portfolio returns in international markets and the US market; where alphas to international IVOL portfolio returns are statistically insignificant when aggregate US IVOL portfolio returns are included in the regression. Exhibit 4 shows value-weighted post-portfolio-formation monthly returns. For the IVOL based portfolios, we find mixed initial results. While the long-short (5-1) monthly returns are negative but not significant, when risk-adjusting the returns by controlling for market returns, size, and book-to-market equity in the Fama and French (1993) three-factor model, the alpha becomes a highly significant -1.14% per month. This negative return is consistent with the idiosyncratic volatility based portfolio returns for Non-REIT equities in AHXZ (2006, 2009), but is contrary to the positive relation between REIT idiosyncratic volatility and returns in OWW (2009). The alpha remains large and significant at -0.89% per month (p-value of 0.059) when controlling for the returns to a portfolio long all Non-REIT US stocks in the highest IVOL quintile and short all Non-REIT US stocks in the lowest IVOL quintile as in AHXZ (2009). 13 Thus, REIT idiosyncratic volatility appears to be largely independent of aggregate idiosyncratic volatility, yet appears to be priced by investors in a similar manner. For the three implied market volatility measures (β ΔVIX, β ΔVOL(SP500), and β ΔVOL(R2000) ), the 5-1 strategy yields monthly returns that are not significantly different from zero. The Fama and French (1993) alphas are insignificant as well. The lack of significant differences stands in stark contrast to the 12

14 highly significant portfolio returns differences for Non-REIT equities in AHXZ (2006). This is particularly interesting in the case of β ΔVOL(R2000), which represents the sensitivity to aggregate implied volatility for a large group of small stocks, a segment of the market with which REITs are commonly compared. While some REITs are considered large capitalization stocks, with a few included in the S&P 500 index starting in the fall of 2001, most REITs are relatively small. To disentangle a potential size effect, we break the sample into size and volatility terciles and sequentially sort the sample at the end of each month, first by size and then by volatility. 14 For each volatility measure, the monthly long-short (3-1) value-weighted returns differences are shown for small, medium, and large firms in Exhibit 5. With IVOL, the statistical significance increases as size increases and all long-short returns differences are negative. Similar to the one-dimensional sort returns in Exhibit 4, the IVOL long-short returns in Exhibit 5 are insignificant for small and medium sized firms and only weakly significant for large firms. However, the Fama and French (1993) alphas are significant at close to the 5% level for small firms and are strongly significant at the 1% level for medium and large REITs. When controlling for the returns to a portfolio long all Non-REIT US stocks in the highest IVOL quintile and short all Non-REIT US stocks in the lowest IVOL quintile, the alphas are slightly reduced, but remain economically large, -0.65% and -0.48% per month, and statistically significant at the 5% level for medium and large REITs, respectively. The size sorts confirm the earlier results than any co-movement between REIT idiosyncratic volatility and aggregate idiosyncratic volatility is, at best, only modest. For β ΔVIX, β ΔVOL(SP500), and β ΔVOL(R2000), only one portfolio difference out of the eighteen tested shows up even weakly statistically significant. Across all three aggregate implied volatility measures the returns differences tend to be positive, although decreasing monotonically as size increases. Taken together, the two-way sorts in Exhibit 5 suggest that size is an important characteristic that should be controlled for when pricing volatility risk in REITs. 13

15 While informative, the relatively small number of firms in the equity REIT universe renders portfolios formed on multiple dimensions more susceptible to the influence of outlier observations. Moreover, portfolio-level analysis is limited in the extent to which additional potential influences can be controlled. At the firm-level, we are able to add numerous control variables simultaneously and incorporate the F1 monthly market volatility measure of DS (2011) into the analysis. We use Fama and MacBeth (1973) regressions with Newey and West (1987) standard errors to determine the price of risk. Excess returns are regressed on the variables of interest (IVOL, β ΔVIX, β ΔVOL(SP500), β ΔVOL(R2000), and β F1 ) along with various controls each month. 15 The time series of estimated coefficients on these variables are then used to construct the respective prices of risk and the corresponding test statistics. The results are presented in Exhibit 6. In Panel A, excess returns are regressed on IVOL and firmlevel sensitivity to each aggregate volatility measure individually. The negative and significant (at the 1% level) coefficient on IVOL in regression (1) confirms the portfolio level finding of a negative relation between REIT returns and idiosyncratic volatility. In other words, REITs with low idiosyncratic volatility outperform those with high idiosyncratic volatility. This suggests that the EGARCH look-ahead bias (Fink, Fink, and He, 2010; Guo, Kassa, and Ferguson, 2011) may have affected the positive price manifest in OWW (2009). None of the systematic volatility measures are significantly different from zero in Exhibit 6, Panel A, including firm-level sensitivity to the new F1 volatility factor. While the insignificant coefficients on the sensitivities to aggregate implied volatility measures are consistent with the portfolio-level sorts in Exhibits 4 and 5, the insignificant coefficient on the sensitivity to the F1 factor is curious. DS (2011) find F1 to be priced across multiple asset classes including stocks, bonds, and various derivative securities. 16 In an ICAPM framework (such as Chabi-Yo, 2012), if market volatility is a state variable then it should be priced consistently across asset classes, including REITs. Yet, we do not find evidence of aggregate volatility risk pricing in a REIT setting using numerous measures of firm-level sensitivity to 14

16 systematic volatility (β ΔVIX, β ΔVOL(SP500), β ΔVOL(R2000), and β F1 ). With respect to exposure to aggregate volatility risk, REITs appear to be substantially different than other financial assets, such as industrial equities and bonds. In Exhibit 6, Panel B, we repeat the same regressions and include firm sensitivity to a standard market factor, β MKT, as well as additional firm-level controls (β ΔSKEW(SP500), β ΔSKEW(R2000), ISKEW, SIZE, BM, MOM12, IO, and TURN). 17 Consistent with prior results, the negative and strongly significant coefficient for IVOL shows that idiosyncratic volatility is priced in the same manner as AHXZ (2006) find for Non- REIT equities. Furthermore, like AHXZ (2006, 2009) and OWW (2009), the pricing of idiosyncratic volatility is not explained by exposure to systematic risk. IVOL dominates β ΔVIX, β ΔVOL(SP500), β ΔVOL(R2000), and β F1. However, it is interesting to note that when β F1 is included in the regression, β MKT becomes significant. That is, controlling for market volatility using the DS (2011) F1 measure, individual REIT sensitivity to market returns is negatively priced. This result is consistent with the negative market premium DS find to be associated with Non-REIT equities. To better understand the magnitude of the effect of IVOL on returns in the presence of these controls, we compare the results from Exhibit 6, Panel B, with the value-weighted long-short monthly portfolio returns in Exhibit 4. Using the IVOL coefficient from regression [1], a one unit increase in idiosyncratic volatility has a price of a negative 40 basis points per month during the sample period, holding all else constant. The mean IVOL in Exhibit 4 of the long-short portfolio is ( =) Evaluating the IVOL risk premium coefficient of at the mean of 2.43 results in an equalweighted return of (-0.40 x 2.43 =) , or about -97 basis points per month. When compared to the long-short portfolio Fama and French (1993) alpha of -1.14% per month from Exhibit 4, the equalweighting and additional controls included in the Fama and MacBeth (1973) regressions result in a longshort returns magnitude that is remarkably similar. Likewise, similar results are obtained using the IVOL coefficients from Panel B, regressions [2], [3], and [4], where aggregate implied volatility is controlled for 15

17 using β ΔVIX, β ΔVOL(SP500) and β ΔSKEW(SP500), and β ΔVOL(R2000) and β ΔSKEW(R2000), respectively. When controlling for firm sensitivity to market volatility using β F1 in regression [5], the magnitude of the IVOL risk premium is reduced by a little less than half. However, while the IVOL risk premium is lowered, it remains statistically significant at the 5% level. 5. Conclusions We investigate whether volatility risk, both aggregate (systematic) and firm-specific (idiosyncratic), is priced in the cross-section of expected equity REIT returns. For aggregate volatility risk we use several distinct measures which draw upon either options data or innovations in the broad crosssection of individual firm returns. By incorporating empirical methods in the spirit of AHXZ (2006) we model equity REIT sensitivity to implied systematic volatility risk using the Chicago Board Options Exchange s VIX index. We also utilize the techniques of BKM (2003) to derive two additional implied aggregate volatility measures using the full-spectrum of options available on the S&P 500 and Russell 2000 indices. The BKM (2003) methods allow us to calculate the higher moments of the returns distribution in order to control for, and determine the price of, implied aggregate skewness. We also incorporate the market volatility factor of DS (2011), which is derived using the first principal component of the innovations in realized volatilities of all available firms in the cross-section. For firmspecific volatility risk, we follow AHXZ (2006, 2009) and estimate idiosyncratic volatility without the unintentional look-ahead bias introduced by EGARCH models (Fink, Fink, and He, 2010; Guo, Kassa, and Ferguson, 2011). Employing these methods enables us to avoid the limitations of 1.) relying on market beta to proxy for systematic risk and 2.) biased estimates of idiosyncratic risk. We find that systematic volatility risk is not priced in equity REIT stocks. This result holds across all four measures of aggregate volatility in portfolio- and firm-level tests, and both univariate and multivariate analyses. The lack of aggregate volatility risk pricing is in sharp contrast to that which is observed by AHXZ (2006) and DS (2011) for typical Non-REIT equities. This robust result suggests that 16

18 REITs are distinctively different from Non-REIT equities, with portfolio formation implications. This finding has important portfolio implications, as investors should be able to use REITs to hedge their portfolios against innovations in aggregate market volatility. Using REITs to hedge aggregate volatility should also be particularly attractive to size-style investors, since REITs are not sensitive to aggregate volatility in neither large stocks (S&P 500), nor small stocks (Russell 2000). Similarly, REITs diverge from both theory and Non-REIT empirical results in that aggregate skewness is not priced (Chang, Christoffersen, and Jacobs, 2009; Chabi-Yo, 2012), and its inclusion does not change aggregate volatility pricing. We find that idiosyncratic volatility risk is priced in the cross-section of equity REIT returns and that the price is negative. REITs with low idiosyncratic volatility outperform those with high idiosyncratic volatility. While this is consistent with the negative price for Non-REIT stocks (AHXZ 2006, 2009), we do not observe significant co-movement between REIT idiosyncratic volatility and Non-REIT idiosyncratic volatility. These results add to the recent mixed evidence in the REIT literature where three papers reach three separate conclusions. Chiang, Jiang, and Lee (2009) find a negative time-series relation between idiosyncratic volatility and REIT returns; Sun and Young (2009) do not find robust support for idiosyncratic volatility pricing; and, most prominently, OWW (2009) find a positive price. We attribute the sign difference between our results and OWW to the inadvertent look-ahead bias that is introduced into the estimation of idiosyncratic volatility using EGARCH techniques (Fink, Fink, and He, 2010; Guo, Kassa, and Ferguson, 2011). We find the method used in this study convincing as it uses only information available to investors at the time of portfolio formation. The negative price of equity REIT idiosyncratic volatility risk documented here, for US Non-REIT equities in AHXZ (2006), and for international Non-REIT equities in AHXZ (2009) is puzzling. Since the market should only reward investors for bearing that portion of volatility risk which cannot be diversified away, we are left to conjecture that market frictions and incomplete information create an 17

19 environment where investors are unable to fully diversify away firm-specific risk (Merton, 1987). However, in such a case, we would expect the price of idiosyncratic volatility risk to be positive. Nonetheless, AHXZ (2006, 2009) rule out several possible explanations for the negative price including the potential for idiosyncratic volatility to proxy for transactions costs, analyst coverage, and price delay. Moreover, other possible economic explanations for the negative price could include the potential for idiosyncratic volatility to proxy for the overpricing of positive skewness, the presence of uninformed traders, or liquidity risk. We control for each of these possibilities by including measures of skewness, institutional ownership, and share turnover, and still find a strongly negative price in the cross-section. Despite the puzzling sign, we find that REIT idiosyncratic volatility risk pricing is also robust to the inclusion of various controls for aggregate volatility risk. In short, aggregate volatility risk is dominated by firm-specific volatility risk in the cross-section of REIT pricing. Thus, in the context of portfolio formation, investors should consider the negative pricing of idiosyncratic volatility when choosing equity REIT stocks to hedge aggregate volatility risk. 18

20 6. Endnotes 1 Market beta is widely found to be an insignificant variable in the presence of other factors, and papers with this finding are too numerous to be referenced completely. However, Fama and French (1992, 1993) provide convincing evidence of the inadequacies of market beta. 2 See Whaley (2000) for a complete description of VIX. 3 The precision of EGARCH estimates is another concern. EGARCH is a data hungry process which often does not converge in small samples and short windows (Cumby, Figlewski, and Hasbrouk, 1993; Fink, Fink, and He, 2010; Guo, Kassa, and Ferguson, 2011). 4 Chiang, Jiang, and Lee (2009) also find a negative relation in time-series tests of REIT returns, but do not determine an actual price. 5 AHXZ (2009) find this difference to be even more pronounced internationally at -1.31% As explained in the CBOE white paper located at if options with two consecutive strike prices both have zero-bids, all options beyond those strike prices are disregarded, even if there are options with non-zero bids. 9 Essentially, risk-neutral probabilities represent the only arbitrage free price for a given redundant security in a complete market. We refer interested readers to Gisiger (2010) for a primer on risk-neutral probabilities. 10 The optimal lags were determined to be (2,2) using Bayesian Information Criteria. 11 We thank Ernst Schaumburg for providing the F1 market volatility factor time-series, which runs through December, Although not shown, we repeat the analysis while controlling for skewness, momentum, and using equalweighted returns and obtain consistent results with each. 13 AHXZ (2009) find highly significant coefficients on aggregate IVOL long-short portfolio returns, whereas the aggregate IVOL long-short portfolio returns coefficient in our regression (not shown) is nowhere near significant at any conventional level (p-value of 0.225). 19

21 14 Given the relatively small number of firms in the REIT industry, we use terciles for the two-dimensional sorts in order to increase the number of observations in each portfolio. 15 Brennan, Chordia, and Subrahmanyam (1998) find firm-level characteristics to have more explanatory power than the sensitivity to the Fama and French (1993) factors themselves, thus we use the actual firm characteristics when possible. 16 We note that DS (2011) examine the pricing of the market wide measure, F1, on portfolios of assets, whereas we examine the pricing of individual firm sensitivity to F1. 17 We control for outliers by Winsorizing all variables at the 1% and 99% level. 20

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25 8. Acknowledgements The authors acknowledge the helpful comments and suggestions of Ko Wang (the editor), three anonymous referees, Chia Chien, Haim Kassa, and seminar participants at the University of Georgia, the Pennsylvania State University, the 2011 AsRES & AREUEA Joint International Conference, and the 2012 ARES Conference. We thank Ernst Schaumburg for providing the F1 market volatility factor time-series data. All remaining errors are our own. 24

26 Exhibit 1 Daily Implied Market Volatility as Measured by VIX and Annualized VOL for the S&P500 (SP500) and Russell 2000 (R2000) 80 VIX VOL(SP500) 70 VOL(R2000) Jan Sep Jun Mar Dec Sep-09 Notes: VIX is the CBOE market volatility index shown on a daily basis over the sample period. Following BKM (2003), VOL(SP500) is calculated as the standard deviation of the risk neutral density using a continuum of European call and put options on the S&P 500 index and represents estimated implied market volatility. VOL(R2000) is calculated in the same manner using a options on the Russell 2000 index. 25