Do Hedge Funds Exploit Rare Disaster Concerns?
Gao, George P.;Gao, Pengjie;Song, Zhaogang
2018-03-24 00:00:00
Abstract We find hedge funds that have higher return covariation with a disaster concern index, which we develop through out-of-the-money puts on various economic sector indices, earn significantly higher returns in the cross-section. We provide evidence that these funds’ managers are more skilled at exploiting the market’s ex ante rare disaster concerns (SEDs), which may not be associated with disaster risk. In particular, high-SED funds, on average, outperform low-SED funds by 0.96% per month, but have less exposure to disaster risk. They continue to deliver superior future performance when SEDs are estimated using the disaster concern index purged of disaster risk premiums and have leverage-managing and extreme market-timing abilities. Received June 30, 2014; editorial decision August 26, 2017 by Editor Laura Starks. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web Site next to the link to the final published paper online. Prior research on hedge fund performance and disaster risk focuses on the covariance between fund returns and ex post realized disaster shocks. In the time series, a number of hedge fund investment styles, characterized as de facto sellers of put options, incur substantial losses when the market declines (Mitchell and Pulvino 2001; Agarwal and Naik 2004). In the cross-section, individual hedge funds have heterogeneous disaster risk exposure, and funds with greater exposure to disaster risk usually earn higher returns during normal times, but experience losses during stressful times (Agarwal, Bakshi, and Huij 2010; Jiang and Kelly 2012). At face value, the existing evidence suggests that hedge funds, as a whole, are much like conventional assets in an economy with disaster risk: they earn higher returns simply by being more exposed to disaster risk. We hypothesize that certain hedge fund managers possess skills in exploiting ex ante market disaster concerns and thus deliver superior future fund performance while being less exposed to disaster risk. Our empirical evidence lends strong support to this hypothesis. In particular, it has been documented that market participants, including institutional investors, pay a “fear premium” beyond the compensation for disaster risk to insure against disaster shocks (Bollen and Whaley 2004; Driessen and Maenhout 2007; Han 2008; Bollerslev and Todorov 2011). We hypothesize that if hedge fund managers have better skills in exploiting disaster concerns or the fear premium, they could deliver superior future fund performance. In particular, they could do so because they may be better at identifying market fears that do not result in disaster shocks. By supplying disaster insurance to investors with high disaster concerns, some fund managers profit more than others, who do not possess such skills.1 Second, even when some disaster concerns are subsequently realized as disaster shocks, certain fund managers may be better than others at identifying through research whether the fear premium is overpriced, and thus be able to profit from their research skills. Third, “difficulty in inference regarding ... severity of disasters ... can effectively lead to significant disagreements among investors about disaster risk” (Chen, Joslin, and Tran 2012). That is, disaster concerns can have different levels of fear premiums paid by different investors, regardless of whether a disaster shock is ultimately realized or not. Skilled fund managers can profit from investors who pay a high fear premium and avoid those who pay a low fear premium. In sum, fund managers’ skills in exploiting disaster concerns can contribute to higher returns for certain hedge funds, and at the same time do not necessarily make them more exposed to disaster shocks. To test this hypothesis, we develop a measure of a fund’s skill at exploiting rare disaster concerns (SED) using the covariation between fund returns and a disaster concern index that we construct with out-of-the-money put options on various stock sector indices. We argue that this covariance between hedge fund returns and ex ante disaster concerns can identify skillful fund managers.2 Conceptually, when the market’s disaster concern is high, funds with managers more skilled at identifying the value of the fear premium should earn higher contemporaneous returns than others. Consistent with our conjecture that hedge funds exhibit different levels of skills at exploiting disaster concerns, we find substantial heterogeneity of SED across hedge funds as well as significant persistence in SED. Our main tests focus on the relation between the SED measure and future fund performance. In our baseline results, funds in the highest SED decile on average outperform funds in the lowest SED decile by $$0.96\%$$ per month (Newey-West t-statistic of $$2.8$$).3 Moreover, high-SED funds exhibit significant performance persistence. The return spread of the high-minus-low SED deciles ranges from $$0.84\%$$ per month (t-statistic of $$2.6$$) for a three-month holding horizon to $$0.44\%$$ per month (t-statistic of $$1.9$$) for a 12-month holding horizon. We also show that the outperformance of high-SED funds is pervasive across almost all hedge fund investment styles. These results run against the interpretation that these hedge funds earn higher returns on average simply by being more exposed to disaster risk. If this were true, that is, if the covariation between fund returns and the disaster concern index (i.e., the SED measure) measures disaster risk exposure, then the high-SED funds should earn lower returns on average (rather than the higher returns we document) because they are good hedges against disaster risk under this interpretation. We provide several pieces of supporting evidence for our hypothesis that SED captures the skill of some hedge fund managers at exploiting disaster concerns. First, if SED captures fund managers’ skill rather than the disaster risk exposure of hedge funds, high-SED funds should be less exposed to disaster risk. We test this possibility by computing the loadings of SED fund deciles on a large set of macroeconomic variables, market risk factors, volatility risk factors, liquidity factors, and option-based risk measures (Ang, Chen, and Xing 2006; Ang et al. 2006; Pastor and Stambaugh 2003; Acharya and Pedersen 2005; Sadka 2006; Hu, Pan, and Wang 2013; Brunnermeier and Pedersen 2009; Mitchell and Pulvino 2012; Bali, Brown, and Caglayan 2011; Bali, Brown, and Caglayan 2012). We find strong evidence that high-SED funds are actually less risky than low-SED funds, consistent with the interpretation that high-SED fund managers are more skilled at exploiting disaster concerns. If our results are driven by missing risk factors, then these factors would have to be nearly uncorrelated with the large set of known risk factors we employ. Second, recognizing that our $$\mathbb{RIX}$$ measure is the price of a disaster insurance contract that contains compensations for both objective disaster shocks (rational disaster risk premiums) and subjective concerns (or fears) about disaster risk, we purge the disaster risk premium from $$\mathbb{RIX}$$ based on the stochastic disaster risk model of Seo and Wachter (2014), and reestimate funds’ SED. These SED estimates capture the managers’ skill in exploiting the pure concerns about disaster risk more directly. We continue to observe that high-SED funds strongly outperform low-SED funds with these revised SED estimates, further supporting our hypothesis that high-SED funds earn higher returns because of their managers’ superior skills at exploiting rare disaster concerns. Third, if SED captures skills, then we expect high-SED fund managers to be better at managing leverage and at timing extreme market conditions. To test this corollary hypothesis, we calculate the leverage implied by $$\mathbb{RIX}$$ and estimate each fund’s ability to manage leverage. The evidence supports this hypothesis, in that high-SED funds’ managers appear to have superior leverage-managing ability: they reduce their fund’s exposure to market-wide leverage when the market leverage condition worsens. This evidence is consistent with the procyclicality of hedge fund leverage documented by Ang, Gorovyy, and van Inwegen (2011) and Jiang (2014), whereas our innovative evidence reveals the intriguing connection between how hedge fund managers exploit disaster concerns and how they manage fund leverage. Moreover, we estimate each fund’s extreme-market-timing ability and find that high-SED funds, on average, have strong bear-market-timing ability. Both results are consistent with the interpretation of SED as measuring fund managers’ skills, though we note that such evidence is only suggestive, because of the lack of fund-level data on portfolio holdings, investment positions, and balance sheets. We also provide strong empirical evidence against alternative interpretations. First, it could be that the higher average returns the high-SED funds earn over the full sample are just a result of better performance during normal times and (hypothetically) worse performance during stressful times that are short in our sample period of 1996–2010. In other words, the high-SED funds may simply be lucky during our sample period. To test this alternative, we perform a conditional portfolio analysis of SED-sorted funds in normal versus stressful market times, and find that high-SED funds outperform low-SED funds even more in stressful market times, including the severe 2008 financial crisis. Such evidence is inconsistent with the luck interpretation and supports our skill-based explanation of hedge fund performance. Second, as the spikes in the $$\mathbb{RIX}$$ factor often occur when disaster shocks hit the market, it is possible that some of our high-SED funds earn profits by purchasing—rather than selling—disaster insurance before the disaster shock; these funds then realize large positive payoffs when such disastrous outcomes hit the market. Among the credit-style hedge fund sample, we identify a potential set of such funds and find even stronger SED effects on future fund performance after excluding them from our portfolio analysis. Moreover, we explore a general identification condition for the funds purchasing disaster insurance: the time $$t-1$$ returns of these funds, who pay a cost to buy disaster insurance before disastrous events at time $$t$$ should have significant negative loadings on the $$\mathbb{RIX}$$ at time $$t$$ Accordingly, we identify fund managers who purchase disaster insurance by regressing the fund’s monthly excess return at $$t-1$$ on the next-period $$\mathbb{RIX}$$ at $$t$$. We find no significant return difference between low- and high-exposure funds, a finding that contradicts the interpretation that the high-SED funds purchase disaster insurance. These results support our theory that the skills of high-SED fund managers are to identify the existence and magnitude of the fear premium and sell insurance contracts accordingly, rather than to forecast the disaster event and buy disaster insurance beforehand. Third, we investigate whether high-SED funds are those that exploit insurance associated with the intermediate rather than the extreme tails of the market. The answer is unequivocally no. In particular, we capture the intermediate tails of the market using the VIX, given that it is less sensitive to the extreme tail events captured by $$\mathbb{RIX}$$. We find that hedge fund portfolios formed on the covariation between fund excess returns and the VIX (analogous to SED) have no significant return spreads. Moreover, sequential sorts show that the SED well explains cross-sectional hedge fund returns in the presence of potential fund managers’ skill at exploiting intermediate tails, but not vice versa. Collectively, these results suggest that it is fund managers’ skill at exploiting disaster concerns, rather than concerns about intermediate tail events, that explain cross-sectional hedge fund performance. Fourth, and finally, as our paper primarily contributes to the literature about hedge fund managers’ skills and cross-sectional fund performance, we show that the SED measure is distinct from other fund manager skill variables, including the skill of hedging systematic risk (Titman and Tiu 2011), the skill of adopting innovative strategies (Sun, Wang, and Zheng 2012), the skill of timing market liquidity (Cao et al. 2013), and the conditional performance measure based on downside returns (Sun, Wang, and Zheng 2013), using both Fama and MacBeth 1973 regressions and double-sorting portfolios.4 Throughout the paper, we compute risk-adjusted abnormal returns using the Fung and Hsieh (2001) 8-factor model and the 10-factor model recently developed by Namvar et al. (2014) (NPPR, hereafter). The difference in alpha between the high- and low-SED funds remains highly significant: 1.27% and 0.80% per month, with Newey-West t-statistics of 3.8 and 2.8, relative to the Fung-Hsieh and NPPR models, respectively. Our results also survive a battery of robustness checks, including alternative measures of ex ante disaster concerns using 90-day options and S&P 500 index options, different choices of portfolio weight, fund size, fund backfilling bias, delisted fund returns, fund December and non-December returns, different benchmark models, and different hedge fund databases. In addition, we control for a large set of hedge fund characteristics using Frazzini and Pedersen (2014) regressions. 1. The Conceptual Framework and Hypothesis Our main hypothesis is that certain hedge fund managers are more skilled at exploiting disaster concerns, in the sense that they can better reap a high premium in disaster risk insurance. It has been well documented that market participants, including institutional investors, usually pay a high premium over the compensation for disaster risk to insure against disaster risk (Bollen and Whaley 2004; Driessen and Maenhout 2007; Han 2008; Constantinides, Jackwerth, and Perrakis 2009; Bollerslev and Todorov 2011). If certain hedge fund managers are better at exploiting disaster concerns and reaping the overpaid high premium, they could deliver superior future fund performance.5 Our hypothesis is not inconsistent with the prior research showing that hedge funds earn returns simply by taking on exposure to disaster risk on the whole (Mitchell and Pulvino 2001; Agarwal and Naik 2004), but differs by emphasizing that one group of skilled hedge fund managers should be able to take on the high premium in disaster insurance with less disaster risk exposure than others. To identify these skilled fund managers and capture their skill at exploiting disaster concerns, we use the covariation between fund returns and ex ante disaster concerns (SED). When the market’s disaster concern level is high, funds with managers more skilled at identifying the value of the fear premium should earn higher contemporaneous returns than the rest by selling disaster insurance. Therefore, our main hypothesis predicts that high-SED hedge funds should earn higher future returns in the cross-section. This prediction runs directly against the interpretation that these hedge funds earn higher returns simply by being more exposed to disaster risk. If this were true, that is, if the covariation between fund returns and ex ante disaster concerns captures disaster risk exposure, then the high-SED funds should earn lower returns because they are good hedges against disaster risk under this interpretation. We further test several important aspects of our main hypothesis. First, if SED captures fund managers’ skill rather than the fund’s disaster risk exposure, high-SED funds should be less exposed to disaster risk. Second, our hypothesis is that high-SED fund managers are skilled at identifying the existence and magnitude of the fear premium and selling insurance contracts accordingly. As a result, the SED effect is expected to be most significant when we use measures of pure concerns or fears about disaster risk. Third, as both the leverage and the extreme market conditions are integrated parts of any disaster episodes, we expect high-SED fund managers to be better at managing leverage and at timing extreme market conditions, as two specific types of skills. Because of the lack of fund-level data on portfolio holdings, investment positions, and balance sheets, it is unrealistic to prove that the covariation between fund returns and disaster concerns entirely captures fund managers’ skill at exploiting disaster concerns. In consequence, it is important to rule out alternative mechanisms that SED may be related to. The first alternative is that the higher returns earned by high-SED funds over the full sample are just a result of better performance during normal times and (hypothetically) worse performance during stressful times that are short in our sample. In other words, the high-SED funds may simply be lucky during our sample period. Second, it is possible that some high-SED funds earn profits by purchasing—rather than selling—disaster insurance before the disaster shock, because disaster concerns usually spike when disaster shocks hit the market. Third, high-SED funds could be those that exploit insurance associated with the intermediate rather than the extreme tails of the market. 2. Data and Estimation 2.1 Measuring rare disaster concerns In this section, we develop a rare disaster concern index ($$\mathbb{RIX}$$) to estimate the ex ante market expectation regarding future disaster events. We build on the model-free implied volatility measures of Carr and Madan (1998), Britten-Jones and Neuberger (2000), Carr and Wu (2009), and Du and Kapadia (2012). In particular, the value of $$\mathbb{RIX}$$ depends on the price difference between two option-based replication portfolios of variance swap contracts. The first portfolio accounts for mild market volatility shocks, and the second accounts for extreme volatility shocks induced by market jumps associated with rare event risk. By construction, the $$\mathbb{RIX}$$ is equal to the insurance price against extreme downside market movements in the future. Over time, the $$\mathbb{RIX}$$ signals variations in ex ante disaster concerns. 2.1.1 Construction of $$\mathbb{RIX}$$ Consider an underlying asset whose time-$$t$$ price is $$S_{t}$$. We assume for simplicity that the asset does not pay dividends. An investor holding this security is concerned about its price fluctuations over a time period $$[t,T]$$. One way to protect him- or herself against price changes is to buy a contract that delivers payments equal to the extent of price variations over $$[t,T]$$, minus a prearranged price. Such a contract is called a “variance swap contract,” as the price variations are essentially about the stochastic variance of the price process.6 The standard variance swap contract in practice pays \begin{equation*} \left( \ln \frac{S_{t+\Delta }}{S_{t}}\right) ^{2}+\left( \ln \frac{ S_{t+2\Delta }}{S_{t+\Delta }}\right) ^{2}+\cdots +\left( \ln \frac{S_{T}}{ S_{T-\Delta }}\right) ^{2} \end{equation*} minus the prearranged price $$\mathbb{VP}$$. In principle, replication portfolios consisting of out-of-the-money (OTM) options written on $$\left. S_{t}\right.$$ can be used to replicate the time-varying payoff associated with the variance swap contract and, hence, to determine the price $$\left. \mathbb{VP}\right.$$. We now introduce two replication portfolios and their implied prices for the variance swap contract. The first, which underlies the CBOE’s construction of VIX and focuses on the limit of the discrete sum of squared log returns, determines $$\mathbb{VP}$$ as \begin{equation} \mathbb{IV}\equiv \frac{2e^{r\tau }}{\tau }\left\{ \int_{K>S_{t}}\frac{1}{ K^{2}}C(S_{t};K,T)dK+\int_{K<S_{t}}\frac{1}{K^{2}}P(S_{t};K,T)dK\right\} , \label{f:IV} \end{equation} (1) where $$\left. r\right.$$ is the constant risk-free rate, $$\left. \tau \equiv T-t\right.$$ is the time-to-maturity, and $$\left. C(S_{t};K,T)\right.$$ and $$\left. P(S_{t};K,T)\right.$$ are the prices of call and put options with strike $$\left. K\right.$$ and maturity date $$\left. T\right.$$, respectively. As observed in Equation (1), this replication portfolio contains positions in OTM calls and puts with a weight inversely proportional to their squared strikes. The second replication portfolio relies on $$\left. Var_{t}^{\mathbb{Q}}\left( \ln S_{T}/S_{t}\right) \right.$$, which avoids the discrete sum approximation, and determines $$\left. \mathbb{VP}\right.$$ as \begin{align} \mathbb{V} &\equiv \frac{2e^{r\tau }}{\tau }\left\{ \int_{K>S_{t}}\frac{1-\ln\left( K/S_{t}\right)}{K^{2}}C(S_{t};K,T)dK \right. \notag \\ &\quad\left. +\int_{K<S_{t}}\frac{1-\ln\left( K/S_{t}\right) }{K^{2}}P(S_{t};K,T)dK\right\} . \label{f:V} \end{align} (2) This replication portfolio differs from the first in Equation (1) by assigning larger (smaller) weights to more deeply OTM put (call) options. As strike price $$\left. K\right.$$ declines (increases), that is, put (call) options become more out of the money, $$\left. 1-\ln \left( K/S_{t}\right)\right.$$ becomes larger (smaller). Since more deeply OTM options protect investors against larger price changes, it is intuitive that the difference between $$\mathbb{IV}$$ and $$\mathbb{V}$$ captures investors’ expectations about the distribution of large price variations. Our measure of disaster concerns is essentially equal to the difference between $$\mathbb{V}$$ and $$\mathbb{IV}$$, which is due to extreme deviations of $$S_{T}$$ from $$S_{t}$$. However, both upside and downside price jumps contribute to this difference. In view of the many recent studies that indicate that investors are more concerned about downside price swings (Liu, Pan, and Wang 2005; Ang, Chen, and Xing 2006; Barro 2006; Gabaix 2012; Wachter 2013), we focus on downside rare events associated with unlikely, but extremely negative, price jumps. In particular, we consider the downside versions of both $$\mathbb{IV}$$ and $$\mathbb{V}$$: \begin{align} \mathbb{IV}^{-} &\equiv \frac{2e^{r\tau }}{\tau }\int_{K<S_{t}}\frac{1}{ K^{2}}P(S_{t};K,T)dK, \notag \\ \mathbb{V}^{-} &\equiv \frac{2e^{r\tau }}{\tau }\int_{K<S_{t}}\frac{1-\ln \left( K/S_{t}\right) }{K^{2}}P(S_{t};K,T)dK, \label{f:downsideV_IV} \end{align} (3) where only OTM put options that protect investors against negative price jumps are used. We then define our rare disaster concern index ($$\mathbb{RIX}$$) as \begin{equation} \mathbb{RIX}\equiv \mathbb{V}^{-}-\mathbb{IV}^{-}=\frac{2e^{r\tau }}{\tau } \int_{K<S_{t}}\frac{\ln \left( S_{t}/K\right) }{K^{2}}P(S_{t};K,T)dK. \label{f:RIX_portfolio} \end{equation} (4) Assume the price process follows the Merton (1976) jump-diffusion model, with $$dS_{t}/S_{t}=\left( r-\lambda \mu _{J}\right) dt+\sigma dW_{t}+dJ_{t}$$, where $$\left. r\right.$$ is the constant risk-free rate, $$\left. \sigma\right.$$ is the volatility, $$\left. W_{t}\right.$$ is a standard Brownian motion, $$\left. J_{t}\right.$$ is a compound Poisson process with jump intensity $$\left. \lambda \right.$$, and the compensator for the Poisson random measure $$\left. \omega \left[ dx,dt\right] \right.$$ is equal to $$\left. \lambda \frac{1}{\sqrt{2\pi }\sigma _{J}}\exp \left( -\left( x-\mu_{J}\right) ^{2}/2\right) \right.$$. We can show that \begin{equation} \mathbb{RIX}\equiv 2\mathbb{E}_{t}^{\mathbb{Q}}\int_{t}^{T}\int_{R_{0}} \left( 1+x+x^{2}/2-e^{x}\right) \omega ^{-}\left[ dx,dt\right] , \label{f:RIX} \end{equation} (5) where $$\left. \omega ^{-}\left[ dx,dt\right] \right.$$ is the Poisson random measure associated with negative price jumps. Therefore, $$\mathbb{RIX}$$ captures all the high-order ($$\geq 3$$) moments of the jump distribution with negative sizes, given that $$e^{x}-(1+x+x^{2}/2)$$ = $$x^{3}/3+x^{4}/4+\cdots.$$ Motivated by the fact that hedge funds invest in different sectors of the economy, we make one further extension, particularly relevant for analyzing hedge fund performance. Namely, we measure market concerns about future rare disaster events associated with various economic sectors, instead of relying on the S&P 500 index exclusively. In particular, we employ liquid index options on six sectors: the KBW banking sector (BKX), the PHLX semiconductor sector (SOX), the PHLX gold and silver sector (XAU), the PHLX housing sector (HGX), the PHLX oil service sector (OSX), and the PHLX utility sector (UTY). This allows us to avoid the caveat of using options on a single market index under which the perceived disastrous outcome of one economic sector may be offset by a euphoric outlook in another sector. Specifically, we first use OTM puts on each sector index to calculate sector-level disaster concern indices, and then take a simple average across them to obtain a market-level $$\mathbb{RIX}$$. Such a construction is likely to incorporate disaster concerns about various economic sectors, which is particularly important for investigating hedge fund performance. 2.1.2 Option data and empirical estimation We obtain daily data on options from 1996 through 2010 from OptionMetrics. For both European calls and puts on the six sector indices we consider, the dataset includes the daily best closing bid and ask prices, in addition to implied volatility and option Greeks (delta, gamma, vega, and theta). Following the literature, we clean the data as follows: (1) we exclude options with nonstandard expiration dates, missing implied volatility, zero open interest, or either a zero bid price or a negative bid-ask spread; (2) we discard observations with a bid or ask price of less than 0.05 to mitigate the effect of price recording errors; and (3) we remove observations where option prices violate no-arbitrage bounds. Because there is no closing price in OptionMetrics, we use the mid-quote price (i.e., the average of the best bid and ask prices) as the option price.7 Finally, we consider only options with maturities longer than 7 days and shorter than 180 days for liquidity reasons. We focus on a 30-day horizon to illustrate the construction of $$\mathbb{RIX}$$, that is, $$\left. T-t=30\right.$$. On a daily basis, we choose options with exactly 30 days to expiration, if they are available. Otherwise, we choose two contracts with the nearest maturities to 30 days, with one longer and the other one shorter than 30 days. We keep only out-of-the-money put options and exclude days with fewer than two option quotes of different moneyness levels for each chosen maturity. As observed in Equation (4), the computation of $$\mathbb{RIX}$$ relies on a continuum of moneyness levels. Similar to Carr and Wu (2009), we interpolate implied volatilities across the range of observed moneyness levels. For moneyness levels outside the available range, we use the implied volatility of the lowest (highest) moneyness contract for moneyness levels below (above) it. In total, we generate 2,000 implied volatility points equally spaced over a strike range of zero to three times the current spot price for each chosen maturity on each date. We then obtain a 30-day implied volatility curve, either exactly or by interpolating the two implied volatility curves of the two chosen maturities. Finally, we use the generated 30-day implied volatility curve to compute the OTM option prices based on the Black and Scholes (1973) formula and then $$\mathbb{RIX}$$ according to a discretization of Equation (4) for each day. After obtaining those daily estimates, we take the daily average over the month to deliver a monthly time series of $$\mathbb{RIX}$$, extending from January 1996 to June 2010. We further divide $$\mathbb{RIX}$$ by $$\mathbb{V}^{-}$$ as a normalization, to mitigate the effect of different volatility levels across different economy sectors. The sector-level OTM index puts we use are generally liquid, and thus the liquidity effect of these OTM puts on $$\mathbb{RIX}$$ is expected to be small.8 2.1.3 Descriptive statistics Table 1 presents descriptive statistics for the disaster concern indices. Panel A shows the monthly aggregated $$\mathbb{RIX}$$ has a mean of 0.063, with a standard deviation of 0.02. Among sector-level disaster concern indices, the semiconductor sector has the highest mean (0.076) and median (0.070), whereas the utility sector has the lowest mean (0.029) and median (0.027). Interestingly, the banking sector has the highest standard deviation, likely due to the 2007–2008 financial crisis. Figure 1 presents a time-series plot of the aggregated $$\mathbb{RIX}$$ that illustrates how the market’s perception of future disaster events varies over time. We observe that disaster concerns spike not only when disaster shocks hit the market, such as the LTCM collapse, the crash of Nasdaq, and the recent financial crisis, but also during bull markets, such as the peak of Nasdaq and the market rally in October 2011. Figure 1 View largeDownload slide Time series of RIX This figure plots the monthly time series of the rare disaster concern index (RIX) from January 1996 to December 2011. Figure 1 View largeDownload slide Time series of RIX This figure plots the monthly time series of the rare disaster concern index (RIX) from January 1996 to December 2011. Table 1 Descriptive statistics of rare disaster concern indices A. Summary statistics of monthly time series Mean Min P25 Median P75 Max SD N KBW Banking Sector (BKX) 0.057 0.017 0.037 0.054 0.068 0.165 0.029 192 PHLX Semiconductor Sector (SOX) 0.076 0.037 0.055 0.070 0.095 0.143 0.025 192 PHLX Gold Silver Sector (XAU) 0.065 0.036 0.051 0.063 0.073 0.140 0.018 192 PHLX Housing Sector (HGX) 0.063 0.030 0.046 0.054 0.073 0.139 0.023 114 PHLX Oil Service Sector (OSX) 0.072 0.039 0.053 0.066 0.087 0.165 0.025 179 PHLX Utility Sector (UTY) 0.029 0.012 0.023 0.027 0.033 0.071 0.010 165 Aggregated Factor ($$\mathbb{RIX}$$) 0.063 0.034 0.046 0.061 0.074 0.141 0.020 192 A. Summary statistics of monthly time series Mean Min P25 Median P75 Max SD N KBW Banking Sector (BKX) 0.057 0.017 0.037 0.054 0.068 0.165 0.029 192 PHLX Semiconductor Sector (SOX) 0.076 0.037 0.055 0.070 0.095 0.143 0.025 192 PHLX Gold Silver Sector (XAU) 0.065 0.036 0.051 0.063 0.073 0.140 0.018 192 PHLX Housing Sector (HGX) 0.063 0.030 0.046 0.054 0.073 0.139 0.023 114 PHLX Oil Service Sector (OSX) 0.072 0.039 0.053 0.066 0.087 0.165 0.025 179 PHLX Utility Sector (UTY) 0.029 0.012 0.023 0.027 0.033 0.071 0.010 165 Aggregated Factor ($$\mathbb{RIX}$$) 0.063 0.034 0.046 0.061 0.074 0.141 0.020 192 B. Correlations between rare disaster concern indices and other factors $$\mathbb{RIX}$$ BKX SOX XAU HGX OSX UTY MKTRF –0.102 –0.110 –0.013 –0.150 –0.172 –0.067 –0.177 SMB 0.002 –0.019 0.087 –0.019 –0.030 –0.032 –0.051 HML –0.165 –0.109 –0.112 –0.211 –0.209 –0.171 –0.032 UMD –0.121 –0.202 –0.055 0.017 –0.225 –0.020 –0.080 PTFSBD 0.248 0.194 0.259 0.226 0.277 0.239 0.172 PTFSFX 0.051 0.073 –0.024 0.130 0.102 0.009 –0.005 PTFSCOM –0.055 –0.031 –0.112 0.056 0.048 –0.083 –0.154 PTFSIR 0.177 0.242 –0.029 0.163 0.348 0.091 0.069 PTRSSTK –0.026 0.032 –0.114 0.017 0.139 –0.079 –0.015 Liquidity risk: PS –0.193 –0.214 –0.087 –0.226 –0.325 –0.096 –0.242 Liquidity risk: Sadka –0.310 –0.401 –0.130 –0.283 –0.408 –0.195 –0.220 Liquidity risk: Noise –0.009 –0.046 –0.055 0.092 –0.009 0.005 0.010 Change of term spread 0.222 0.251 0.169 0.167 0.280 0.141 0.259 Change of default spread 0.221 0.146 0.123 0.256 0.208 0.241 0.002 Change of VIX –0.094 –0.065 –0.112 –0.025 –0.048 –0.096 –0.057 B. Correlations between rare disaster concern indices and other factors $$\mathbb{RIX}$$ BKX SOX XAU HGX OSX UTY MKTRF –0.102 –0.110 –0.013 –0.150 –0.172 –0.067 –0.177 SMB 0.002 –0.019 0.087 –0.019 –0.030 –0.032 –0.051 HML –0.165 –0.109 –0.112 –0.211 –0.209 –0.171 –0.032 UMD –0.121 –0.202 –0.055 0.017 –0.225 –0.020 –0.080 PTFSBD 0.248 0.194 0.259 0.226 0.277 0.239 0.172 PTFSFX 0.051 0.073 –0.024 0.130 0.102 0.009 –0.005 PTFSCOM –0.055 –0.031 –0.112 0.056 0.048 –0.083 –0.154 PTFSIR 0.177 0.242 –0.029 0.163 0.348 0.091 0.069 PTRSSTK –0.026 0.032 –0.114 0.017 0.139 –0.079 –0.015 Liquidity risk: PS –0.193 –0.214 –0.087 –0.226 –0.325 –0.096 –0.242 Liquidity risk: Sadka –0.310 –0.401 –0.130 –0.283 –0.408 –0.195 –0.220 Liquidity risk: Noise –0.009 –0.046 –0.055 0.092 –0.009 0.005 0.010 Change of term spread 0.222 0.251 0.169 0.167 0.280 0.141 0.259 Change of default spread 0.221 0.146 0.123 0.256 0.208 0.241 0.002 Change of VIX –0.094 –0.065 –0.112 –0.025 –0.048 –0.096 –0.057 Rare disaster concern indices are constructed using the prices of 30-day out-of-the-money put options on 6 different sector indices from 1996 to 2011. The aggregated rare disaster concern index ($$\mathbb{RIX}$$) is a simple average of the 6 sector-level disaster concern indices. Panel A reports summary statistics for their monthly time series, and panel B reports the time-series correlations between one rare disaster concern index and a number of factors, including the Fama-French-Carhart four factors (MKTRF, SMB, HML, and UMD), the Fung-Hsieh 5 trend-following factors (PTFSBD, PTFSFX, PTFSCOM, PTFSIR, and PTFSSTK), the Pastor-Stambaugh (PS) liquidity risk factor, the Sadka liquidity risk factor, the Hu-Pan-Wang liquidity risk factor (Noise), change in term spread, change in default spread, and change in VIX. Table 1 Descriptive statistics of rare disaster concern indices A. Summary statistics of monthly time series Mean Min P25 Median P75 Max SD N KBW Banking Sector (BKX) 0.057 0.017 0.037 0.054 0.068 0.165 0.029 192 PHLX Semiconductor Sector (SOX) 0.076 0.037 0.055 0.070 0.095 0.143 0.025 192 PHLX Gold Silver Sector (XAU) 0.065 0.036 0.051 0.063 0.073 0.140 0.018 192 PHLX Housing Sector (HGX) 0.063 0.030 0.046 0.054 0.073 0.139 0.023 114 PHLX Oil Service Sector (OSX) 0.072 0.039 0.053 0.066 0.087 0.165 0.025 179 PHLX Utility Sector (UTY) 0.029 0.012 0.023 0.027 0.033 0.071 0.010 165 Aggregated Factor ($$\mathbb{RIX}$$) 0.063 0.034 0.046 0.061 0.074 0.141 0.020 192 A. Summary statistics of monthly time series Mean Min P25 Median P75 Max SD N KBW Banking Sector (BKX) 0.057 0.017 0.037 0.054 0.068 0.165 0.029 192 PHLX Semiconductor Sector (SOX) 0.076 0.037 0.055 0.070 0.095 0.143 0.025 192 PHLX Gold Silver Sector (XAU) 0.065 0.036 0.051 0.063 0.073 0.140 0.018 192 PHLX Housing Sector (HGX) 0.063 0.030 0.046 0.054 0.073 0.139 0.023 114 PHLX Oil Service Sector (OSX) 0.072 0.039 0.053 0.066 0.087 0.165 0.025 179 PHLX Utility Sector (UTY) 0.029 0.012 0.023 0.027 0.033 0.071 0.010 165 Aggregated Factor ($$\mathbb{RIX}$$) 0.063 0.034 0.046 0.061 0.074 0.141 0.020 192 B. Correlations between rare disaster concern indices and other factors $$\mathbb{RIX}$$ BKX SOX XAU HGX OSX UTY MKTRF –0.102 –0.110 –0.013 –0.150 –0.172 –0.067 –0.177 SMB 0.002 –0.019 0.087 –0.019 –0.030 –0.032 –0.051 HML –0.165 –0.109 –0.112 –0.211 –0.209 –0.171 –0.032 UMD –0.121 –0.202 –0.055 0.017 –0.225 –0.020 –0.080 PTFSBD 0.248 0.194 0.259 0.226 0.277 0.239 0.172 PTFSFX 0.051 0.073 –0.024 0.130 0.102 0.009 –0.005 PTFSCOM –0.055 –0.031 –0.112 0.056 0.048 –0.083 –0.154 PTFSIR 0.177 0.242 –0.029 0.163 0.348 0.091 0.069 PTRSSTK –0.026 0.032 –0.114 0.017 0.139 –0.079 –0.015 Liquidity risk: PS –0.193 –0.214 –0.087 –0.226 –0.325 –0.096 –0.242 Liquidity risk: Sadka –0.310 –0.401 –0.130 –0.283 –0.408 –0.195 –0.220 Liquidity risk: Noise –0.009 –0.046 –0.055 0.092 –0.009 0.005 0.010 Change of term spread 0.222 0.251 0.169 0.167 0.280 0.141 0.259 Change of default spread 0.221 0.146 0.123 0.256 0.208 0.241 0.002 Change of VIX –0.094 –0.065 –0.112 –0.025 –0.048 –0.096 –0.057 B. Correlations between rare disaster concern indices and other factors $$\mathbb{RIX}$$ BKX SOX XAU HGX OSX UTY MKTRF –0.102 –0.110 –0.013 –0.150 –0.172 –0.067 –0.177 SMB 0.002 –0.019 0.087 –0.019 –0.030 –0.032 –0.051 HML –0.165 –0.109 –0.112 –0.211 –0.209 –0.171 –0.032 UMD –0.121 –0.202 –0.055 0.017 –0.225 –0.020 –0.080 PTFSBD 0.248 0.194 0.259 0.226 0.277 0.239 0.172 PTFSFX 0.051 0.073 –0.024 0.130 0.102 0.009 –0.005 PTFSCOM –0.055 –0.031 –0.112 0.056 0.048 –0.083 –0.154 PTFSIR 0.177 0.242 –0.029 0.163 0.348 0.091 0.069 PTRSSTK –0.026 0.032 –0.114 0.017 0.139 –0.079 –0.015 Liquidity risk: PS –0.193 –0.214 –0.087 –0.226 –0.325 –0.096 –0.242 Liquidity risk: Sadka –0.310 –0.401 –0.130 –0.283 –0.408 –0.195 –0.220 Liquidity risk: Noise –0.009 –0.046 –0.055 0.092 –0.009 0.005 0.010 Change of term spread 0.222 0.251 0.169 0.167 0.280 0.141 0.259 Change of default spread 0.221 0.146 0.123 0.256 0.208 0.241 0.002 Change of VIX –0.094 –0.065 –0.112 –0.025 –0.048 –0.096 –0.057 Rare disaster concern indices are constructed using the prices of 30-day out-of-the-money put options on 6 different sector indices from 1996 to 2011. The aggregated rare disaster concern index ($$\mathbb{RIX}$$) is a simple average of the 6 sector-level disaster concern indices. Panel A reports summary statistics for their monthly time series, and panel B reports the time-series correlations between one rare disaster concern index and a number of factors, including the Fama-French-Carhart four factors (MKTRF, SMB, HML, and UMD), the Fung-Hsieh 5 trend-following factors (PTFSBD, PTFSFX, PTFSCOM, PTFSIR, and PTFSSTK), the Pastor-Stambaugh (PS) liquidity risk factor, the Sadka liquidity risk factor, the Hu-Pan-Wang liquidity risk factor (Noise), change in term spread, change in default spread, and change in VIX. Panel B of Table 1 reports correlations between $$\mathbb{RIX}$$ and a set of factors related to market, size, book-to-market equity, momentum, trend following, market liquidity, funding liquidity, term spread, default spread, and volatility. We find that $$\mathbb{RIX}$$ is only mildly correlated with the usual equity risk factors ($$-$$0.17 and $$-$$0.12 for book-to-market and momentum factors, respectively) and hedge fund risk factors (0.25 and 0.18 for the Fung-Hsieh trend-following factors PTFSBD for bonds, and PTFSIR for short-term interest rates, respectively). More importantly, $$\mathbb{RIX}$$ is weakly correlated with risk factors related to market disaster shocks, for example, between 0.20 and 0.31 with market liquidity (Pastor and Stambaugh 2003; Sadka 2006), around 0.22 with the change of the default spread, and only $$-$$0.10 with the change of VIX for volatility risk. These low correlations further indicate that ex ante disaster concerns are quite distinct from realized disaster shocks ex post, even though they often spike simultaneously. 2.2 Hedge fund data The data on hedge fund monthly returns are obtained from the Lipper TASS database. In addition to the returns, the database also provides fund characteristics, including assets under management (AUM), net asset value (NAV), management fees, and incentive fees, among others.9 Because our measure of rare disaster concerns begins in 1996 when the OptionMetrics data become available, the full sample period of hedge funds we use is from January 1996 through July 2010. We further require funds to report returns net of fees in US dollars, and to have at least 18 months of return history in the TASS database. Moreover, from an institutional investment and market impact perspective, funds with low AUM are of less economic importance; we hence include only funds with at least $${\$}$$10 million AUM at the time of portfolio formation (but not after) in our baseline analysis, following Cao et al. (2013) and Hu, Pan, and Wang (2013). Overall, we exclude 3674 funds from the TASS database, and retain 5864 funds in total over our time period of 01/1996-07/2010. An equal-weight hedge fund portfolio on average earns $$\left. 0.8\%\right.$$ per month with a standard deviation of $$\left. 1.9\%\right.$$; it earned the highest (lowest) mean return of $$\left. 2.2\%\right.$$ ($$\left. -1.4\%\right.$$) per month in 1999 (2008). Table IA-2 of the Internet Appendix provides more detailed descriptive statistics for our hedge fund sample. 2.3 The SED estimates of hedge funds We now explain our measure of hedge fund manager skill at exploiting rare disaster concerns and present various properties of SED-sorted hedge fund portfolios. 2.3.1 The SED estimates We measure hedge fund managers’ skill at exploiting rare disaster concerns through the covariation between fund returns and our measure of ex ante rare disaster concerns ($$\mathbb{RIX}$$). At the end of each month from June 1997 to June 2010, for each hedge fund, we first perform 24-month rolling-window regressions of a fund’s monthly excess return on the CRSP value-weighted market excess return and $$\mathbb{RIX}$$. Then, we measure the fund’s SED using the estimated regression coefficient on $$\mathbb{RIX}$$. To ensure that we have a reasonable number of observations in the estimation, we require funds to have at least 18 months of returns. To understand the cross-sectional variations of hedge fund managers’ skill at exploiting disaster concerns, Table 2 reports panel regressions of the SED estimates on fund characteristics as of June for each year from 1997 to 2010. We find that higher-SED funds tend to have smaller AUM as well as positive return skewness over the past two years. We also observe a strong negative relation between the Fung-Hsieh alpha and SED. This is not surprising because funds with a high Fung-Hsieh alpha behave like they are purchasing disaster insurance by loading highly on the Fung and Hsieh (2001) trend-following factors that are constructed through lookback straddles and hence earn negative mean returns.10 They are consequently less likely to sell disaster insurance and more likely to be low-SED funds. Finally, the variations of SED seem to be more significant across funds than across time. For instance, the adjusted R-squared increases from 3.5% to 21.1% when fund fixed effects are included, while it only increases from 3.5% to 9.2% when year fixed effects are included.11 Table 2 Determinants of hedge fund skills in exploiting rare disaster concerns (SED) (1) (2) (3) (4) (5) (6) (7) (8) Minimal investment (log) 0.029 0.020 0.0187 0.010 (2.25) (1.58) (1.49) (0.80) Management fee (%) $$-$$5.7253 $$-$$3.604 $$-$$5.1803 $$-$$3.146 ($$-$$1.99) ($$-$$1.27) ($$-$$1.82) ($$-$$1.13) Incentive fee (%) 0.0077 $$-$$0.067 $$-$$0.1361 $$-$$0.184 (0.03) ($$-$$0.27) ($$-$$0.56) ($$-$$0.75) Redemption notice 0.0002 0.000 $$-$$0.0002 0.000 $$\quad$$ period (month) (0.40) (0.80) ($$-$$0.38) (0.01) Lockup period (month) $$-$$0.0022 $$-$$0.003 $$-$$0.0011 $$-$$0.002 ($$-$$1.29) ($$-$$1.67) ($$-$$0.63) ($$-$$1.02) High water mark (dummy) 0.0541 0.052 0.0571 0.056 (1.62) (1.58) (1.72) (1.70) Personal capital $$-$$0.0157 $$-$$0.008 $$-$$0.0087 $$-$$0.002 $$\quad$$ invested (dummy) ($$-$$0.51) ($$-$$0.25) ($$-$$0.28) ($$-$$0.05) Leverage (dummy) 0.0509 0.031 0.0476 0.029 (1.85) (1.14) (1.75) (1.07) AUM (log) $$-$$0.0291 $$-$$0.023 $$-$$0.0225 $$-$$0.017 $$-$$0.092 $$-$$0.087 $$-$$0.0514 $$-$$0.049 ($$-$$2.49) ($$-$$2.03) ($$-$$1.99) ($$-$$1.56) ($$-$$3.13) ($$-$$2.87) ($$-$$1.84) ($$-$$1.68) AGE (log) 0.0195 0.015 0.0066 0.005 0.0044 0.007 $$-$$0.0912 $$-$$0.075 (0.64) (0.50) (0.22) (0.17) (0.07) (0.11) ($$-$$0.81) ($$-$$0.68) Fund flow (past 1 year) 0.0164 0.011 0.0059 0.001 $$-$$0.0111 $$-$$0.012 $$-$$0.0368 $$-$$0.037 (1.18) (0.83) (0.40) (0.07) ($$-$$0.23) ($$-$$0.27) ($$-$$0.69) ($$-$$0.73) Return volatility 1.7971 2.737 0.6537 1.541 6.5084 8.006 2.9691 4.251 $$\quad$$ (past 2 years) (0.69) (1.04) (0.25) (0.57) (2.00) (2.58) (0.81) (1.23) Return skewness 0.1032 0.079 0.0791 0.059 0.1703 0.140 0.1358 0.110 $$\quad$$ (past 2 years) (4.66) (3.47) (3.54) (2.57) (6.96) (5.44) (5.22) (4.03) Return kurtosis 0.0206 0.024 0.0069 0.010 0.0043 0.008 $$-$$0.0010 0.003 $$\quad$$ (past 2 years) (2.47) (2.83) (0.81) (1.13) (0.44) (0.86) ($$-$$0.11) (0.31) Alpha (F-H $$-$$11.1859 $$-$$10.656 $$-$$10.2262 $$-$$9.833 $$-$$14.922 $$-$$14.227 $$-$$13.8962 $$-$$13.325 $$\quad$$ factor model) ($$-$$3.95) ($$-$$3.81) ($$-$$3.61) ($$-$$3.50) ($$-$$4.20) ($$-$$4.04) ($$-$$3.85) ($$-$$3.70) R-squared (F-H 0.5228 0.409 0.4406 0.340 0.4021 0.341 0.3107 0.253 $$\quad$$ factor model) (5.67) (4.56) (5.00) (3.94) (3.27) (2.86) (2.73) (2.27) SDI 0.458 0.362 0.3408 0.251 0.3079 0.268 0.2646 0.224 (4.96) (4.05) (3.70) (2.80) (2.28) (2.08) (1.99) (1.75) Downside return 7.2535 7.440 9.451 9.631 4.7459 5.544 6.7865 7.461 (1.93) (1.98) (2.43) (2.50) (1.53) (1.85) (2.13) (2.42) Liquidity timing $$-$$0.0073 $$-$$0.008 $$-$$0.0178 $$-$$0.017 $$-$$0.0122 $$-$$0.013 $$-$$0.0186 $$-$$0.018 ($$-$$0.62) ($$-$$0.73) ($$-$$1.44) ($$-$$1.45) ($$-$$0.76) ($$-$$0.86) ($$-$$1.15) ($$-$$1.18) Volatility timing $$-$$0.2435 0.500 0.0319 0.697 $$-$$0.4652 0.557 $$-$$0.3144 0.628 ($$-$$0.82) (1.30) (0.10) (1.81) ($$-$$1.38) (1.37) ($$-$$0.88) (1.53) Market timing 0.0118 0.0104 0.0116 0.0113 (2.53) (2.24) (1.75) (1.70) Extreme market 0.009 0.010 0.002 0.004 $$\quad$$ timing (Bullish) (2.57) (2.84) (0.52) (0.96) Extreme market 0.027 0.026 0.030 0.029 $$\quad$$ timing (Bearish) (7.21) (6.91) (6.99) (6.93) Constant Included Included Included Included Included Included Included Included Year FEs No No Yes Yes No No Yes Yes Fund FEs No No No No Yes Yes Yes Yes Observations 10,330 10,245 10,330 10,245 10,330 10,313 10,330 10,313 Adjusted R-squared 0.0346 0.079 0.0921 0.134 0.2114 0.246 0.2550 0.286 (1) (2) (3) (4) (5) (6) (7) (8) Minimal investment (log) 0.029 0.020 0.0187 0.010 (2.25) (1.58) (1.49) (0.80) Management fee (%) $$-$$5.7253 $$-$$3.604 $$-$$5.1803 $$-$$3.146 ($$-$$1.99) ($$-$$1.27) ($$-$$1.82) ($$-$$1.13) Incentive fee (%) 0.0077 $$-$$0.067 $$-$$0.1361 $$-$$0.184 (0.03) ($$-$$0.27) ($$-$$0.56) ($$-$$0.75) Redemption notice 0.0002 0.000 $$-$$0.0002 0.000 $$\quad$$ period (month) (0.40) (0.80) ($$-$$0.38) (0.01) Lockup period (month) $$-$$0.0022 $$-$$0.003 $$-$$0.0011 $$-$$0.002 ($$-$$1.29) ($$-$$1.67) ($$-$$0.63) ($$-$$1.02) High water mark (dummy) 0.0541 0.052 0.0571 0.056 (1.62) (1.58) (1.72) (1.70) Personal capital $$-$$0.0157 $$-$$0.008 $$-$$0.0087 $$-$$0.002 $$\quad$$ invested (dummy) ($$-$$0.51) ($$-$$0.25) ($$-$$0.28) ($$-$$0.05) Leverage (dummy) 0.0509 0.031 0.0476 0.029 (1.85) (1.14) (1.75) (1.07) AUM (log) $$-$$0.0291 $$-$$0.023 $$-$$0.0225 $$-$$0.017 $$-$$0.092 $$-$$0.087 $$-$$0.0514 $$-$$0.049 ($$-$$2.49) ($$-$$2.03) ($$-$$1.99) ($$-$$1.56) ($$-$$3.13) ($$-$$2.87) ($$-$$1.84) ($$-$$1.68) AGE (log) 0.0195 0.015 0.0066 0.005 0.0044 0.007 $$-$$0.0912 $$-$$0.075 (0.64) (0.50) (0.22) (0.17) (0.07) (0.11) ($$-$$0.81) ($$-$$0.68) Fund flow (past 1 year) 0.0164 0.011 0.0059 0.001 $$-$$0.0111 $$-$$0.012 $$-$$0.0368 $$-$$0.037 (1.18) (0.83) (0.40) (0.07) ($$-$$0.23) ($$-$$0.27) ($$-$$0.69) ($$-$$0.73) Return volatility 1.7971 2.737 0.6537 1.541 6.5084 8.006 2.9691 4.251 $$\quad$$ (past 2 years) (0.69) (1.04) (0.25) (0.57) (2.00) (2.58) (0.81) (1.23) Return skewness 0.1032 0.079 0.0791 0.059 0.1703 0.140 0.1358 0.110 $$\quad$$ (past 2 years) (4.66) (3.47) (3.54) (2.57) (6.96) (5.44) (5.22) (4.03) Return kurtosis 0.0206 0.024 0.0069 0.010 0.0043 0.008 $$-$$0.0010 0.003 $$\quad$$ (past 2 years) (2.47) (2.83) (0.81) (1.13) (0.44) (0.86) ($$-$$0.11) (0.31) Alpha (F-H $$-$$11.1859 $$-$$10.656 $$-$$10.2262 $$-$$9.833 $$-$$14.922 $$-$$14.227 $$-$$13.8962 $$-$$13.325 $$\quad$$ factor model) ($$-$$3.95) ($$-$$3.81) ($$-$$3.61) ($$-$$3.50) ($$-$$4.20) ($$-$$4.04) ($$-$$3.85) ($$-$$3.70) R-squared (F-H 0.5228 0.409 0.4406 0.340 0.4021 0.341 0.3107 0.253 $$\quad$$ factor model) (5.67) (4.56) (5.00) (3.94) (3.27) (2.86) (2.73) (2.27) SDI 0.458 0.362 0.3408 0.251 0.3079 0.268 0.2646 0.224 (4.96) (4.05) (3.70) (2.80) (2.28) (2.08) (1.99) (1.75) Do