Idiosyncratic Volatility, Growth Options, and the Cross-Section of Returns

Transcription

1 Idiosyncratic Volatility, Growth Options, and the Cross-Section of Returns This version: September 2013 Abstract The paper shows that the value effect and the idiosyncratic volatility discount (Ang et al., 2006) arise because growth firms and high idiosyncratic volatility firms beat the CAPM during the periods of increasing aggregate volatility, which makes their risk low. Growth options become less sensitive to the underlying asset value as idiosyncratic volatility goes up together with aggregate volatility. Hence, growth options betas decrease more and their value decreases less in volatile times, which are typically recessions. All else equal, growth options value also increases with volatility. The impact of both effects on the firm s value is naturally stronger for growth firms and high idiosyncratic volatility firms. The two-factor ICAPM with the market factor and the aggregate volatility risk factor completely explains the value effect and the idiosyncratic discount. The two-factor ICAPM also explains why those puzzles are stronger for the firms with high short sale constraints. JEL Classification: G12, G13, E44 Keywords: idiosyncratic volatility discount, growth options, aggregate volatility risk, value premium, real options

2 1 Introduction A recent paper by Ang, Hodrick, Xing, and Zhang (2006) (hereafter - AHXZ) finds that firms with high idiosyncratic volatility earn negative abnormal returns. The return differential between high and low volatility firms is around 13% per year. Meanwhile, the conventional wisdom says that the correlation between idiosyncratic volatility and future returns should be either zero or positive. In what follows, we call this puzzle the idiosyncratic volatility discount. Another recent paper by Ali, Hwang, and Trombley (2003) finds that the value effect is about 6% per year larger for high idiosyncratic volatility firms. It poses a challenge to any risk-based story for the value effect. Any such story has to explain why the value effect is related to something that is seemingly not risk - idiosyncratic volatility. The main contribution of this paper is showing that aggregate volatility risk is the common explanation for both the value effect and the idiosyncratic volatility discount, as well as the dependence of the value effect on idiosyncratic volatility. We treat aggregate volatility risk as a separate risk factor in the Merton s (1973) Intertemporal CAPM (henceforth - ICAPM). We show that firms with abundant growth options and high idiosyncratic volatility offer a natural hedge against aggregate volatility increases. The concept of aggregate volatility risk was developed in the ICAPM-type models of Campbell (1993) and Chen (2002). In the Campbell model, higher aggregate volatility implies higher future risk premium. Stocks that covary negatively with changes in aggregate volatility command a risk premium because they lose value when the future is also turning bleak. In the Chen model, investors care not only about future returns, but also about future volatility. Aggregate volatility increases imply the need to boost precautionary savings 1

3 and cut current consumption. The stocks that covary negatively with aggregate volatility changes again command a risk premium, but for a different reason. They lose value exactly when consumption is reduced to build up savings. Our paper contributes to the aggregate volatility risk literature by predicting which firms are less exposed to aggregate volatility risk. Using a simple model, we show that the more growth options and idiosyncratic volatility a firm has, the less it is exposed to aggregate volatility risk. The reason for that is that higher idiosyncratic volatility makes growth options less sensitive to the current value of the underlying asset. The beta of the underlying asset does not change with idiosyncratic volatility, so the response of the underlying asset value to a given market return stays the same. However, when idiosyncratic volatility is high, growth options are less sensitive to the value of the underlying asset. Hence, the response of the growth options value to the same market return decreases with idiosyncratic volatility. That is, higher idiosyncratic volatility means a lower beta of growth options. In recessions, both aggregate volatility and idiosyncratic volatility increase 1. The increase in idiosyncratic volatility makes growth options betas smaller and mutes the increase in their risk premiums. Because a lower expected return means a higher current price, the value of growth options drops less after an increase in aggregate volatility, especially if the underlying asset is volatile. The effect on the firm value is naturally stronger for the firms with abundant growth options 2. Higher volatility also means higher value of growth options. Hence, all else equal, growth options should react less negatively to aggregate volatility increases. Quite natu- 1 See, e.g., Campbell, Lettau, Malkiel, and Xu, 2001, and Section 5.2 in this paper. 2 Note that the argument would not go through for systematic or total volatility. While the growth options elasticity declines with both idiosyncratic and systematic volatility, higher systematic volatility of the underlying asset is equivalent to its higher beta. Hence, the overall effect of higher systematic/total volatility of the underlying asset on the growth options beta is ambiguous. 2

4 rally, this effect is also stronger for high volatility firms. Our main empirical prediction is that the value effect and the idiosyncratic volatility discount arise because growth firms and high volatility firms are a hedge against aggregate volatility risk. When aggregate volatility increases and aggregate consumption drops, their value drops less than the value of the firms with the same market beta, since their expected returns increase less and, all else equal, the value of growth options increases with volatility. Because this hedge is created through growth options, the second prediction states that the idiosyncratic volatility discount is stronger for growth firms, and this regularity is explained by aggregate volatility risk. The flip side of this prediction is that the hedging power of growth options against aggregate volatility risk increases with idiosyncratic volatility, thus explaining why the return differential between value and growth firms increases with idiosyncratic volatility. An important feature of our aggregate volatility risk story is that the story is true holding the market return constant. Because the market return is strongly negatively correlated with aggregate volatility (the correlation between the market factor and the change in the VIX index is ), any stock with a positive beta, including growth stocks and high idiosyncratic volatility stocks, will react negatively to increases in expected aggregate volatility. Our prediction is that growth stocks and high idiosyncratic volatility stocks react less negatively to aggregate volatility increases than what the CAPM predicts. This is the reason why these firms have negative CAPM alphas. We do not predict, however, that growth stocks and high idiosyncratic volatility stocks will go up in value when aggregate volatility increases. We use the ICAPM with the aggregate volatility risk factor to explain the idiosyncratic volatility discount and the value effect. Our aggregate volatility risk factor (hereafter - 3

5 FVIX factor) tracks the daily changes in the CBOE VIX index. The VIX index measures the implied volatility of S&P 100 options. AHXZ show that changes in VIX are a good proxy for changes in expected aggregate volatility. Using the data for the period of VIX availability (January December 2012), we regress daily changes in the VIX index on the daily excess returns to the six size and book-to-market portfolios (sorted in two groups on size and three groups on book-to-market). The FVIX factor is the fitted part of the regression less the constant cumulated to the monthly level. All results in our paper are robust to using the aggregate volatility sensitivity quintiles or the 10 industry portfolios from Fama and French (1997) instead of the six size and book-to-market portfolios. We start our empirical tests by sorting firms on market-to-book and idiosyncratic volatility. As our story predicts, the value effect (the idiosyncratic volatility discount) starts at zero for low volatility (value) firms and monotonically increases with idiosyncratic volatility (market-to-book). We also find that high volatility firms, growth firms, and especially high volatility growth firms have positive FVIX betas, which means that their value reacts to increases in expected aggregate volatility significantly less negatively than the value of the firms with comparable market betas. The ICAPM with the FVIX factor completely explains the idiosyncratic volatility discount and why it is stronger for growth firms. The FVIX factor also completely explains the returns to the HML factor and reduces the strong value effect for high volatility firms by up to a half. Nagel (2005) finds that the value effect and the idiosyncratic volatility discount are stronger for the firms with low institutional ownership, which is interpreted as a proxy for short sale constraints. Boehme, Danielsen, Kumar, and Sorescu (2009) find that the idiosyncratic volatility discount is higher when short sales are harder to perform. We show that these patterns can be explained by exposure to aggregate volatility risk. 4

6 In particular, buying low volatility firms and shorting high volatility firms or buying growth firms and shorting value firms exposes the investor to more severe losses during volatile periods if this strategy was performed in the low institutional ownership subsample or in the subsample of stocks with the highest estimated probability to be on special, where the value effect and the idiosyncratic volatility discount are the strongest. After we control for aggregate volatility risk, the value effect and the idiosyncratic volatility discount become insignificant even in the subsample of stocks with the lowest institutional ownership and the highest estimated probability to be on special. The probable reason is that institutional investors tend to avoid highly volatile stocks and stocks with extreme levels of market-to-book (which turn out to be the best hedges against aggregate volatility risk), and these stocks are likely to be short-sale constrained. We perform several robustness checks, which confirm that buying value (low volatility) firms and shorting growth (high volatility) firms results in inferior performance during hard times, especially if this strategy is followed in the high volatility (growth) subsample, low institutional ownership subsample, or high probability to be on special subsample. Value minus growth and low volatility minus high volatility portfolios witness the increase in market betas during recessions, especially if these portfolios were formed of the firms with high idiosyncratic volatility and growth firms, respectively, or of the firms with low institutional ownership or high probability to be on special. We also replace FVIX with daily changes in the VIX index it mimics and confirm directly that growth firms, high idiosyncratic volatility firms, and especially high idiosyncratic volatility growth firms beat the CAPM when expected aggregate volatility increases. The paper proceeds as follows. Section 2 provides a brief literature review. Section 3 lays out a simple model. Section 4 discusses the data sources. Section 5 shows that the value effect and the idiosyncratic volatility discount are explained by aggregate volatility 5

7 risk. Section 6 considers the competing behavioral stories. Section 7 summarizes the robustness checks. Section 8 offers the conclusion. 2 Literature Review The central idea of the paper is that higher idiosyncratic volatility of the underlying asset makes the systematic risk of growth options smaller. Our story is related to Veronesi (2000) and Johnson (2004). They show that parameter risk can negatively affect expected returns by lowering the covariance with the stochastic discount factor. Johnson (2004) also uses the idea that the beta of a call option is negatively related to volatility. In our paper, we take a broader definition of idiosyncratic risk. We argue that it can affect expected returns even if there is no parameter risk, but there is idiosyncratic volatility. We also focus on growth options instead of focusing on leverage, as Johnson (2004) does. It allows me to study the relation between idiosyncratic volatility and the value effect. Most importantly, we tie the effect the idiosyncratic risk of the underlying asset has on the risk of growth options to aggregate volatility risk, thus explaining why the lower risk of growth options on volatile assets is not captured by the existing asset-pricing models - these models miss out the aggregate volatility risk factor, and our story predicts that, all else equal, higher idiosyncratic volatility means less exposure to aggregate volatility risk, not just lower market beta. The Merton (1987) model predicts a positive relation between idiosyncratic volatility and expected returns for risky assets. It does not contradict our story that predicts the opposite relation for common stock. Rather, our story emphasizes the option-like nature of common stocks, which produces another effect in the opposite direction. Therefore, our story is consistent with the evidence supporting the Merton model for other risky assets 3. 3 Green and Rydqvist (1997) find a positive relation between idiosyncratic risk and expected returns 6

8 Several empirical studies (e.g., Malkiel and Xu, 2003) find a positive relation between idiosyncratic volatility and future stock returns at the portfolio level. This evidence is not inconsistent with our story that studies the same relation at the firm level. Firm-level idiosyncratic volatility is diversified away at the portfolio level. The remaining portfoliolevel idiosyncratic volatility is more likely to result from omitted common factors. Hence, the two idiosyncratic volatility measures are likely to be poor proxies for each other. A large strand of literature, starting with Berk, Green, and Naik (1999) and Carlson, Fisher, and Giammarino (2004) uses the real options framework to explain the value effect and related puzzles. The common feature of these papers is that they assume that growth options, as levered claims, are riskier than assets in place, and then produce different stories of why market-to-book or other related firm characteristics are still negatively related to expected returns, despite growth options being riskier. Our paper looks at a completely different mechanism - the interaction of growth options and idiosyncratic volatility. We use the fact that growth options, all else equal, react positively to increases in idiosyncratic volatility, and conclude that growth options should be hedges against aggregate volatility risk, since aggregate volatility and idiosyncratic volatility tend to comove. We take no stand on whether growth options per se (i.e., without the presence of significant idiosyncratic volatility) are more risky or less risky than assets in place and make no prediction about the sign of the value premium in the low idiosyncratic volatility subsample. Another important difference of our model from the real options literature is that the real option models are usually set up in the CAPM world. While this is also true about our model due to the technical difficulties of the setup with a two-factor pricing kernel, in for lottery bonds. Bessembinder (1992) and Mansi, Maxwell, and Miller (2007) find a similar relation for currency and commodity futures and corporate bonds, respectively. 7

9 Section 3.2 we outline the reasons to bring a second risk factor into the analysis and use the ICAPM instead of the (conditional) CAPM in explaining the value effect. The empirical study closest to our paper is AHXZ, which is the first to establish the idiosyncratic volatility discount and the pricing of aggregate volatility risk. Our paper extends AHXZ by showing that the idiosyncratic volatility discount is explained by aggregate volatility risk. We also extend AHXZ by linking the idiosyncratic volatility discount and aggregate volatility risk to growth options. In their paper, AHXZ come to a different conclusion about the link between the idiosyncratic volatility discount and aggregate volatility risk. They show that making the sorts on idiosyncratic volatility conditional on FVIX betas does not eliminate the idiosyncratic volatility discount, and conclude therefore that aggregate volatility risk cannot explain the idiosyncratic volatility discount. In our paper, we perform a more direct test - we use the two-factor ICAPM with the market factor and FVIX - and find that the idiosyncratic volatility discount is completely explained by aggregate volatility risk. We also use an improved factor-mimicking procedure to form FVIX. Our version of FVIX is less noisy and has a significant factor premium (see Section 5.1 and footnote 7 for more details). 3 The Model 4 Data Sources Our data span the period between January 1986 and December 2012 due to the availability of the VIX index. Following AHXZ, we measure idiosyncratic volatility as the standard deviation of the Fama-French (1993) model residuals, which is fitted to daily data. We estimate the model separately for each firm-month, and compute the residuals in the same month. We require at least 15 daily returns to estimate the model and idiosyncratic 8

10 volatility. We compute market-to-book from Compustat data as the market value at the fiscal year end (CSHO times PRCC from the new Compustat files) divided by the book value of equity (CEQ plus TXDB from the new Compustat files). To compute the market-to-book, we use the current year book value for firms with the fiscal year end in June or earlier or the previous year book value for firms with later fiscal year end, to ensure that the book value is available before the date of portfolio formation. We use monthly cum-dividend returns from CRSP and complement them by the delisting returns from the CRSP events file. Following Shumway (1997) and Shumway and Warther (1999), we set delisting returns to -30% for NYSE and AMEX firms and to -55% for NASDAQ firms if CRSP reports missing or zero delisting returns and delisting is for performance reasons. Our results are robust to setting missing delisting returns to -100% or using no correction for the delisting bias. We obtain the daily and monthly values of the three Fama-French factors and the riskfree rate from Kenneth French web site at We define FVIX, the aggregate volatility risk factor, as a factor-mimicking portfolio that tracks the daily changes in the VIX index. Following Ang, Hodrick, Xing, and Zhang (2006), we regress the daily changes in VIX on the daily excess returns to the five portfolios sorted on past sensitivity to VIX changes: V IX t = γ 0 + γ 1 (V IX1 t RF t ) + γ 2 (V IX2 t RF t ) + (1) + γ 3 (V IX3 t RF t ) + γ 4 (V IX4 t RF t ) + γ 5 (V IX5 t RF t ), where V IX1 t,..., V IX5 t are the VIX sensitivity quintiles described above, with V IX1 t being the quintile with the most negative sensitivity. The VIX sensitivity is estimated by regressing daily stock returns in the previous month on the market factor and the change 9

11 in VIX. The fitted part of the regression above less the constant is my aggregate volatility risk factor (FVIX factor): F V IX t = ˆγ 1 (V IX1 t RF t ) + ˆγ 2 (V IX2 t RF t ) + ˆγ 3 (V IX3 t RF t ) + (2) + ˆγ 4 (V IX4 t RF t ) + ˆγ 5 (V IX5 t RF t ). The returns are then cumulated to the monthly level to get the monthly return to FVIX. All results in the paper are robust to changing the base assets from the five portfolios sorted on past sensitivity to VIX changes to the ten industry portfolios (Fama and French (1997)) or the six size and book-to-market portfolios (Fama and French (1993)). Likewise, the idiosyncratic volatility risk factor, FIVol, is the factor-mimicking portfolio that tracks monthly innovations to average idiosyncratic volatility, which is the simple average of the idiosyncratic volatilities (as defined above) of all firms traded during the given month. The innovation is the residual from an ARMA(1,1) model fitted to the average idiosyncratic volatility. The base assets used to create FIVol are IVol sensitivity quintiles, where IVol sensitivity is measured in the previous month by regressing monthly stock returns in the most recent 36 months on the market factor and innovations to average idiosyncratic volatility. In the tests of our story against behavioral stories we use two measures of limits to arbitrage - residual institutional ownership, RInst, and the estimated probability to be on special, Short, which proxy for the severity of short sale constraints. We define institutional ownership of each stock as the sum of institutional holdings from Thompson Financial 13F database, divided by the shares outstanding from CRSP. If the stock is on CRSP, but not on Thompson Financial 13F database, it is assumed to have zero institutional ownership. Following Nagel (2005), we drop all stocks below the 20th NYSE/AMEX 10

12 size percentile and measure residual institutional ownership for the remaining stocks as the residual from Inst log( 1 Inst ) = γ 0 + γ 1 log(size) + γ 2 log 2 (Size) + ɛ (3) The estimated probability to be on special is defined as in D Avolio (2002) and Ali and Trombley (2006) Short = ey 1 + e y, (4) y = 0.46 log(size) 2.8 Inst T urn 0.09 CF IP O Glam (5) T A Equation (20) uses the coefficients estimated by D Avolio (2002) for a short 18-month sample of short sale data. Ali and Trombley (2006) use the same formula to estimate the probability to be on special for the intersection of Compustat, CRSP, and Thompson Financial populations. They show that the estimated probability is closely tied to other short sale constraint measures, including the real shorting fees, in different sub-periods. In (20) Size is defined as shares outstanding times the price per share (both from CRSP) and measured in millions, Inst is institutional ownership, T urn is turnover, defined as the trading volume over shares outstanding, CF is cash flow 4, T A are total assets (Compustat item AT), IP O is the dummy variable equal to 1 if the stock first appeared on CRSP 12 or less months ago, and Glam is the dummy variable equal to 1 for three top market-to-book deciles. 4 Following D Avolio (2002) and Ali and Trombley (2006), we define cash flow as operating income before depreciation (Compustat item OIADP plus Compustat item DP) less non-depreciation accruals, which are change in current assets (Compustat item ACT) less change in current liabilities (Compustat item LCT) plus change in short-term debt (Compustat item DLC) less change in cash (Compustat item CHE). 11

13 5 Explaining the Puzzles 5.1 Is Aggregate Volatility Risk Priced? AHXZ show that stocks with positive return sensitivity to the innovations in the VIX index earn about 1% per month less than stocks with negative sensitivity. The VIX index measures the implied volatility of the S&P100 options and behaves like a random walk. The change in VIX is therefore a good proxy for the innovation in expected aggregate volatility, and, according to the evidence in AHXZ, it should be priced. We form the aggregate volatility risk factor, the FVIX factor, as the zero-investment portfolio that tracks daily changes in expected aggregate volatility. AHXZ use a very similar factor-mimicking portfolio. The only difference is that they perform the factormimicking regression of VIX changes on the excess returns to the base assets separately for each month, and we perform a single factor-mimicking regression using all available data. Clearly, the estimates of six or seven parameters using 22 data points are not too precise, and it is especially true about the constant, which varies considerably month to month. This variation adds noise to their version of FVIX, and the imprecise estimation of the constant makes the FVIX factor premium small and insignificant. In unreported results we find that the AHXZ version of FVIX is significantly correlated with our version of FVIX (the correlation is 0.56, t-statistic 10.7) and produces the betas of the same sign if used instead of our FVIX. However, the use of the the AHXZ version of FVIX in asset-pricing tests is problematic because of the noise in it and the small factor premium 5. 5 The choice between the month-by-month factor-mimicking regressions and the full-sample factormimicking regressions is the choice between precision and the potential bias from ignoring the fact that the coefficients in the factor-mimicking regression may be time-varying. In untabulated results, we look at the coefficients from the month-by-month regressions and find that the coefficients are very volatile, are not autocorrelated, and their values are unrelated to a long list of possible state variables. The evidence suggests that the true coefficients are likely to be constant and the estimated coefficients from the month-by-month regression are very noisy, which makes the full-sample factor-mimicking regression a better choice. 12

14 Our model implies that the part of idiosyncratic volatility that is related to business cycle should also be priced. Therefore, we introduce another ICAPM factor, FIVol, that mimics innovations to average idiosyncratic volatility in the economy. In order to be valid ICAPM factors, FVIX and FIVol have to satisfy two necessary conditions. First, FVIX and FIVol have to be significantly correlated with the variables they mimics - the change in the VIX index and the innovation to aggregate idiosyncratic volatility. The larger is the correlation, the better are FVIX and FIVol at mimicking the innovations in the state variables. Second, FVIX and FIVol have to earn a significant risk premium, controlling for other sources of risk. Since FVIX and FIVol, by construction, are positively correlated with the innovations investors try to hedge against, their risk premiums have to be negative: FVIX (FIVol) tends to earn positive returns when expected aggregate (idiosyncratic) volatility increases and consumption drops, thereby providing a valuable hedge. In Table 1, we look at whether these two conditions are satisfied for FVIX and the two alternative aggregate volatility risk factors - FVIX5 and FVIXind. The difference between the aggregate volatility risk factors is the base assets. FVIX mimics the daily VIX changes by regressing them on the returns to the six portfolios sorted two-by-three on size and market-to-book, respectively. In the same factor-mimicking regression, FVIXind uses the ten industry portfolios from Fama and French (1997) as the independent variables. FVIX5 replaces the industry portfolios by the quintile portfolios sorted on the return sensitivity to the changes in VIX. The return sensitivity is estimated separately for each firm-month from the regression of the daily excess returns on the excess returns to the market factor and the VIX change. In Panel A of Table 1, we look at the correlations between the factors and the state variables. First, we find that FVIX and FIVol are closely related to the innovations they 13

15 mimic: the correlation between FVIX and V IX is 0.653, and the correlation between FIVol and IV ol U is The relation between FVIX and FIVol and the levels of the respective state variables is expectedly weaker, but still significantly positive. Second, we find that while FVIX and FIVol, as well as the respective state variables, are highly correlated, the correlation is far from perfect. For example, the correlation between the change in VIX ( V IX) and the innovation to average idiosyncratic volatility (IV ol U ) is 0.417, and the correlation between FVIX and FIVol is 0.635, meaning that FVIX and FIVol are related, but different factors. In Panels B and C, we look at the raw returns, the CAPM alphas, and the Fama-French (1993) alphas of the aggregate and idiosyncratic volatility risk factors. We find that FVIX loses 1.24% per month, or bp per month on risk-adjusted basis, with all t-statistics comfortably exceeding 3. Likewise, FIVol has average return of -105 bp per month, the CAPM alpha of -72 bp per month, and the Fama-French alpha is bp per month, all statistically significant. We conclude that FVIX and FIVol are priced factors. Panels B and C also report the CAPM betas and the Fama-French betas of FVIX and FIVol, as well as their betas from the ICAPM with the market and the other factor. We find that both factors tend to have very negative market betas. This is to be expected, since the market factor is strongly negatively correlated with both changes in the VIX index and innovations to average idiosyncratic volatility, and FVIX and FIVol are positively correlated with these variables by construction. We also find that FVIX is unrelated to HML, but positively related to SMB (suggesting that FVIX will be more useful in explaining the idiosyncratic volatility discount than the value effect) and FIVol is negatively related to both HML and SMB (suggesting that FIVol will be the main force in explaining the value effect, but probably not the idiosyncratic volatility discount). Lastly, we find that, controlling for the market factor, which is negatively related to both FVIX 14

16 and FIVol, the relation between FVIX and FIVol is absent, strengthening our view of FVIX and FIVol as two completely different factors. 5.2 Idiosyncratic Volatility, Aggregate Volatility, and the FVIX Factor The necessary condition the validity of our idiosyncratic volatility factor and our explanation of the idiosyncratic volatility discount and the value effect is that aggregate volatility and average idiosyncratic volatility comove. In Panel A of Table 2, we verify if average idiosyncratic volatility indeed tends to increase in recessions and to comove with aggregate volatility (the results are robust to using the median idiosyncratic volatility instead). In Panel A, we run pairwise regressions of average idiosyncratic volatility on the NBER recession dummy (one between peak and trough, zero otherwise) and three measures of market volatility. For each business cycle variable we run regressions with it lagged up to four quarters and leaded up to four quarters, and report the slopes in the respective columns of Panel A. For example, in the column labeled -3 we report γ 2 from log(iv ol t ) = γ 0 + γ 1 t + γ 2 log(x t 3 ) (6) where X t 3 is one of the business cycle variables lagged by three months. To account for the fact that idiosyncratic volatility has trended up in our sample period, we also add the linear trend into the regressions. Also, to make the slopes on the business cycle variables easier to interpret, we take the log of the average idiosyncratic volatility and the log of the market volatility. The numbers in the first row, where we report the slopes from the regression of average idiosyncratic volatility on the NBER recession dummy, represent the percentage increase in idiosyncratic volatility during recessions. We find that idiosyncratic volatility is on average by 25-30% higher in recessions than in expansions. This is a significant change, 15

17 given that the spread between the calmest period in the expansion and the most volatile period in the recession is likely to be much wider. We also notice from looking at the leads and lags that the switch from expansion to recession predicts higher idiosyncratic volatility for a year ahead and probably longer, while the increase in idiosyncratic volatility can potentially forecast recessions only one quarter ahead. This evidence suggest that the increase in average idiosyncratic volatility during recessions is not short-lived. In the next rows of Panel A, we look at the slopes from the regressions of average volatility on the log of the VIX index, on the TARCH(1,1) forecast of market volatility, and on the log of realized market volatility. The TARCH model (see Glosten, Jagannathan, and Runkle, 1993) is estimated for the whole sample by fitting the following model to the value-weighted CRSP index returns: Ret CRSP t = γ 0 + γ 1 Ret CRSP t 1 + ɛ t, σ 2 t = c 0 + c 1 σ 2 t 1 + c 2 ɛ 2 t 1 + c 3 I(ɛ t 1 < 0) Since our idiosyncratic volatility measure is standard deviation, we use ˆσ from the above regression, not ˆσ 2. Realized market volatility is defined as the square root of the average daily return to the CRSP market index in each month. We find that an increase in market volatility (expected or realized) by 1% triggers the increase in average idiosyncratic volatility by 0.25% to 0.45%. The volatility measures have the ratio of the standard deviation to the mean close to 1, hence, a two-standard deviation change in aggregate volatility (from one standard deviation below the mean to one standard deviation above the mean) can trigger the increase in average idiosyncratic volatility by 50-90%. We also find that higher market volatility predicts higher idiosyncratic volatility for up to a year ahead, and higher idiosyncratic volatility predicts higher market volatility for up to three quarters ahead. 16

18 Chen (2002) shows that investors appreciate the hedge against volatility increases if volatility increases mean higher future volatility (otherwise there is no need for them to cut current consumption and increase precautionary savings in response to an increase in volatility). If FVIX provides such a hedge, its returns should predict recessions and market volatility. We test this prediction in Panel B of Table 2. In the first row, we perform the probit regression of the NBER recession dummy on FVIX. We find that the increase in the FVIX return by 1% increases the probability of recession in the current or the next quarters by 0.4% to 0.7%. We also find that FVIX return can predict recessions for up to a year ahead, but the recession dummy cannot predict the FVIX return, consistent with market efficiency. In the next three rows of Panel B, we use FVIX returns to predict aggregate volatility and vice versa. Consistent with Chen (2002) and the hypothesis that FVIX is a valid ICAPM factor, we find that FVIX returns can predict aggregate volatility for up to two quarters ahead, while FVIX returns are unpredictable using the aggregate volatility measures. A 1% FVIX return corresponds to the increase in the current aggregate volatility by 1.2% to 2.1% and the increase in the future aggregate volatility by 0.7% to 1%. We conclude therefore that FVIX is a valid hedge against aggregate volatility increases and the corresponding need to cut current consumption and to increase precautionary spending. In the last row of Panel B, we look at whether FVIX returns are related to average idiosyncratic volatility, and find that FVIX does not post noticeably higher returns when average idiosyncratic returns, though it does post higher one (and probably two) quarter(s) before the increase. We conclude that FVIX has limited power as a hedge against average idiosyncratic volatility. In Panel C, we perform similar analysis with FIVol. We find that FIVol returns increase prior to recessions and prior to increases in average idiosyncratic volatility. The 17

19 relation between FIVol returns and aggregate volatility is weaker and is visible only as a contemporaneous relation. We conclude that FIVol is a good hedge against recessions in general and increases in average idiosyncratic volatility in particular, as it should be, and that FIVol also has some ability to hedge against aggregate volatility risk. 5.3 Idiosyncratic Volatility, Market-to-Book, and Volatility Risk Our model predicts that the idiosyncratic volatility discount increases with market-tobook and is absent for value firms. The prediction about the value effect is symmetric and implies that the value effect increases with idiosyncratic volatility and is absent for high idiosyncratic volatility firms. In Panel A of Table 3, we look at the value-weighted CAPM alphas in the five-by-five independent portfolio sorts on market-to-book and idiosyncratic volatility. The sorts are performed using NYSE (exchcd=1) breakpoints. The results are robust to using conditional sorting and/or CRSP breakpoints, as well as to using raw returns or the Fama-French alphas instead, and/or using equal-weighted returns. Panel A shows that our predictions are strongly supported by the data. The magnitude of the idiosyncratic volatility discount monotonically increases with market-to-book from 3.4 bp per month, t-statistic 0.1, in the extreme value portfolio to 92.3 bp per month, t-statistic 2.83, in the extreme growth portfolio. The difference is highly significant with t-statistic In terms of statistical significance, the idiosyncratic volatility discount is confined to the two highest market-to-book quintiles. A similar pattern emerges for the value effect. The value minus growth strategy starts with the CAPM alpha of 28.3 bp per month, t-statistic 0.99, in the lowest idiosyncratic volatility quintile and ends up with the CAPM alpha of bp per month, t-statistic 3.3, in the highest idiosyncratic volatility quintile. The portfolio that seems to generate the majority of the value effect and the idiosyn- 18

20 cratic volatility discount and represents the worst failure of the CAPM in the double sorts on market-to-book and idiosyncratic volatility, is the highest volatility growth portfolio. In Panel A, this portfolio witnesses the negative alpha of bp, t-statistic Such a low return to high volatility growth firms is consistent with our model. We predict that the firms with the highest volatility and the highest market-to-book should be the best hedges against aggregate volatility increases and should therefore earn the lowest expected return. 6 The bottom line of Panel A is that the idiosyncratic volatility discount increases with market-to-book and the value effect increases with idiosyncratic volatility. These regularities prove the importance of the interaction between idiosyncratic volatility and growth options, which is the focus of our model. Panels B to D present the alphas and FVIX and FIVol betas from the three-factor ICAPM with the market and the two volatility risk factors. The ICAPM alphas in Panel B suggest that there is no idiosyncratic volatility discount and no value effect after we control for volatility risk. The idiosyncratic volatility discount turns insignificantly negative in all market-to-book quintiles and no longer depends on market-to-book. The value effect in the ICAPM alphas fluctuates between -49 bp per month and 44 bp per month, but is never statistically significant and does not depend on idiosyncratic volatility. Panel C of Table 3 shows that the FVIX betas are closely aligned with the CAPM alphas in Panel A. High idiosyncratic volatility firms have positive FVIX betas that are significantly greater than the FVIX betas of low volatility firms. In all market-to-book quintiles the FVIX betas increase (almost) monotonically as idiosyncratic volatility increases. Growth firms also have significantly higher FVIX betas than value firms in all 6 Two more notable portfolios are the low volatility value portfolio and the high volatility value portfolio, with alpha is positive at 60.4 bp per month, t-statistic 2.37 and 57 bp per month, t-statistic 1.9. While the first alpha is consistent with our model (low volatility value firms are the worst hedges against aggregate volatility risk, the second alpha, while marginally significant, is puzzling. 19

21 idiosyncratic volatility quintiles except for the lowest one. The FVIX beta of the lowminus-high volatility portfolio increases strongly and monotonically with market-to-book, and the FVIX beta of the low-minus-high market-to-book portfolio also increases with idiosyncratic volatility. The FVIX beta of the highest volatility growth portfolio is 1.725, t-statistic 6.1, by far the largest number in the table. This shows that the highest volatility growth portfolio is a very good hedge against aggregate volatility increases, just as our model predicts. Comparing the FVIX betas of the low-minus-high volatility portfolios and the lowminus-high market-to-book portfolios, we find that FVIX is more likely to help in explaining the idiosyncratic volatility than the value effect. The FVIX beta of the low-minus-high volatility portfolio is always strongly negative, even in the value quintile, where the idiosyncratic volatility discount is virtually absent. The FVIX beta of the low-minus-high market-to-book portfolio is only negative in the highest idiosyncratic volatility quintile, though this quintile is the quintile in which the value effect is by far the strongest. Panel D reports the FIVol betas and finds that FIVol is the factor that explains the value effect, most particularly the positive alphas of value stocks (the FIVol betas of these stocks are large and negative), but does not contribute much to explaining the idiosyncratic volatility. I also find that the FIVol betas of the low-minus-high market-to-book portfolio are equally strong in all idiosyncratic volatility quintiles (though in equal-weighted returns I do see significantly more negative FIVol betas of the low-minus-high market-to-book portfolio in the high idiosyncratic volatility quintile). The conclusion from Panels C and D is that, consistent with our theory, high volatility firms, growth firms, and especially high volatility growth firms are hedges against volatility risk. Their prices tend to react much less negatively to increases in expected aggregate volatility and average idiosyncratic volatility compared to what the CAPM predicts. The 20

22 reverse is generally true about low volatility, value, and low volatility value firms. 5.4 Explaining the Value Effect and the Idiosyncratic Volatility Discount In Table 4, we test the ability of a variety of asset-pricing models to explain the value effect and the idiosyncratic volatility discount, as well as the dependence of these anomalies on idiosyncratic volatility and market-to-book, respectively. For brevity, we focus on five arbitrage portfolios. The first portfolio is the HML factor, which is our measure of the value effect. We contrast it with the second portfolio - HMLh, which looks at the value effect for high volatility firms. HMLh buys the firms from the intersection of the lowest marketto-book quintile and the highest volatility quintile and shorts the firms the intersection of the highest market-to-book quintile and the highest volatility quintile. The third portfolio - IVol - captures the idiosyncratic volatility discount. It goes long in low volatility firms and short in high volatility firms. The IVolh portfolio does the same for growth firms only to capture the stronger idiosyncratic volatility discount for growth firms. The IVol55 portfolio is long in the highest volatility growth firms and short in the one-month Treasury bill. In the first three columns of Table 4, we present the alphas from the unconditional CAPM, the unconditional Fama-French model, and the conditional CAPM. The conditional CAPM assumes that the market beta is the linear function of the dividend yield, the default premium, the one-month Treasury bill rate, and the term spread 7 and estimates Ret it = α i +(β 0i +β 1i DIV t 1 +β 2i DEF t 1 +β 3i T B t 1 +β 4i T ERM t 1 ) MKT t +ɛ it (7) The unconditional CAPM turns out to be incapable of explaining either the valueweighted or the equal-weighted returns to any of the portfolios, except for the HML factor 7 The detailed definitions of the four variables can be found in, e.g., Petkova and Zhang (2005). 21

23 and the equal-weighted IVol, which still have the alphas in excess of 35 bp per month and t-statistic above 1.5. The magnitude of the CAPM alphas is about 1% per month 8. The Fama-French model makes insignificant the alpha of the value-weighted IVol55, though at bp per month, t-statistic it is hardly small. The significant Fama- French alphas are around 0.7% per month. The conditional CAPM falls between the unconditional CAPM and the Fama-French model, shaving on average bp per month off the CAPM alphas and making several alphas marginally significant, but leaving them numerically large. In the three columns on the right, we estimate the three-factor ICAPM with the market factor, FVIX, and FIVol and find that it more than explains the returns to the IVol, IVolh, and IVol55 portfolios, which capture the idiosyncratic volatility discount and its crosssection. For example, the IVolh portfolio measures the idiosyncratic volatility discount in the extreme growth quintile and possesses the value-weighted CAPM and Fama-French alphas of 92.3 bp per month, t-statistic 2.83, and 69.8 bp per month, t-statistic 2.09, respectively. Adding the volatility risk factors to the CAPM makes the alpha flip the sign and become mere bp per month, t-statistic The decrease in the alphas brought about by FVIX and FIVol is comparable for the other five alphas of IVol, IVolh, and IVol55. The ICAPM with FVIX and FIVol also explains the returns to the HML factor: its alpha drops from 36.7 bp per month, t-statistic 1.66, in the CAPM, to -6.1 bp, t-statistic -0.33, in the ICAPM. The ICAPM with FVIX also handles quite well the value-weighted HMLh, reducing its alpha from bp per month, t-statistic 3.3 in the CAPM, to 44.3 bp, t-statistic 1.46, and significantly reduces the huge alpha of the equal-weighted HMLh. 8 One difference between IVol and IVolh is that IVol includes the stocks with non-missing idiosyncratic volatility, but missing market-to-book. Therefore, IVol is not the average idiosyncratic volatility discount across market-to-book groups and the fact that in Panel A the alphas of IVol are close to the alphas of IVolh does not contradict the increase of the idiosyncratic volatility discount with market-to-book. 22

24 Overall, it seems that volatility risk comes close to explaining the value effect and its dependence on idiosyncratic volatility. In columns five and six, we look at FVIX beta and FIVol beta and find that all portfolios, except for IVol55, load negatively on both factors, while IVol55 loads on them positively, consistent with its negative alpha. However, it looks like the idiosyncratic volatility discount is explained primarily by FVIX, because FIVol betas of IVol, IVolh, and IVol55, are insignificant, even though they all have the right sign. Similarly, FIVol seems to contribute more to explaining the value effect: while FVIX betas of HML and HMLh are significant, they are of the same magnitude as the respective FIVol betas, and FIVol has twice higher factor risk premium. In untabulated results, we try making the ICAPM betas conditional on the same four macroeconomic variables we use for the conditional CAPM and/or on the VIX index. We find that making the market beta time-varying reduces the alphas by at most 5 bp per month, and leaves the FVIX betas unchanged. These results are consistent with our story that suggests that FVIX and FIVol should subsume the effects of the time-varying market beta. We also try to make the FVIX and FIVol betas time-varying, but do not find any consistent evidence that the FVIX and FIVol betas are not constant, let alone that it varies in the way that would be helpful in explaining the anomalies discussed in this paper. 6 Behavioral Explanations 6.1 Idiosyncratic Volatility Discount, Institutional Ownership, and the Probability to Be on Special Several recent empirical papers find evidence consistent with the behavioral explanation of the idiosyncratic volatility discount. The behavioral story is based on the Miller (1977) argument that under short sale constraints firms with greater divergence of opinion about 23

25 their value will be more overpriced. Miller (1977) argues that short sale constraints keep pessimistic investors out of the market, and the market price reflects the average valuation of the optimists. The average valuation of the optimists is naturally higher than the fair price and increases with disagreement. Therefore, the overpricing should increase in both short sale constraints and disagreement/volatility, and the negative relation between disagreement/volatility and future returns should be the strongest for the most short sale constrained firms. Consistent with this idea, Nagel (2005) and Boehme, Danielsen, Kumar, and Sorescu (2009) find that the idiosyncratic volatility discount is much stronger for the firms they perceive to be the costliest to short. Nagel (2005) uses low institutional ownership (low supply of shares for shorting) as a proxy for high shorting costs. Boehme et al. (2009) look at high short interest (high demand for shorting). In this subsection, we restrict our sample to the stocks above the 20th NYSE/AMEX size percentile, since the institutional holdings of the stocks below the 20th percentile tend to be very small and therefore are often unreported. After we drop the firms below the 20th size percentile, we can assume that missing ownership by the institution means no ownership by the institution. We follow Nagel (2005) in looking at residual institutional ownership, which is orthogonalized to size (see equation (18) in Section 4). We do not have access to the short interest data and use instead the estimated probability that the stock is on special 9. The exact formula for the probability to be on special is given in (19) and (20) (see Section 4). It uses the coefficients estimated by D Avolio (2002) for a short 18- month sample of the stocks with available data on shorting fees. Ali and Trombley (2006) use the same formula to estimate the probability to be on special for the intersection of 9 When the stock is sold short, the party that sells it short (the borrower) borrows the stock from the party that owns it (the lender) and immediately sells it. The proceeds are left with the lender, and the lender pays the borrower the risk-free rate less the short sale fee for using the proceeds. If the stock is on special, the fee is larger than the risk-free rate, and the sum left with the lender declines with time. 24