The Idiosyncratic Volatility Expected Return Relation: Reconciling the Conflicting Evidence

Transcription

1 The Idiosyncratic Volatility Expected Return Relation: Reconciling the Conflicting Evidence Doron Avramov and Scott Cederburg July 26, 2014 ABSTRACT This paper develops a simple dividend discount model to resolve the conflicting evidence of a large negative (Ang, Hodrick, Xing, and Zhang (AHXZ, 2006)) versus large positive (Fu (2009)) relation between idiosyncratic volatility (IVOL) and average returns. In the proposed model, IVOL strongly predicts the cross section of average returns, even when it is unpriced. That predictive ability is attributable to the relations of IVOL with dividend size and expected dividend growth, both of which are related to risk premiums. In particular, firms with small dividends exhibit high IVOL and high expected returns, while low dividend growth firms have high IVOL and low expected returns. Empirical evidence strongly supports the model s novel prediction of a negative relation between IVOL and firm growth. Moreover, consistent with model predictions, IVOL is positively related to returns in the dividend size dimension and negatively along the dividend growth dimension. Finally, the AHXZ and Fu measures are more closely aligned with dividend growth and dividend size, respectively, consistent with their opposing relations with IVOL. JEL Codes: G10, G12 We thank Fangjian Fu, Campbell Harvey, Ivalina Kalcheva, Eric Kelley, Michael O Doherty, Avanidhar Subrahmanyam, Jianfeng Yu, participants at the 2014 Jerusalem Finance Conference, and seminar participants at the University of Arizona for helpful comments. All errors are our own. Doron Avramov (davramov@huji.ac.il) is from the Hebrew University of Jerusalem, Israel, and Scott Cederburg (cederburg@ .arizona.edu) is from the University of Arizona.

2 1 Introduction The recent empirical literature on the relation between idiosyncratic volatility (IVOL) and average stock returns has produced conflicting evidence. While Ang, Hodrick, Xing, and Zhang (2006, 2009) (hereafter AHXZ) show a statistically significant and economically meaningful negative IVOL-return relation, Fu (2009) documents an equally prominent positive relation using an alternative IVOL proxy. The empirical evidence is often interpreted in the context of whether IVOL is priced as prescribed by economic theory. Indeed, the Capital Asset Pricing Model of Sharpe (1964) and Lintner (1965) as well as its consumption-based version (e.g., Breeden (1979)), among other prominent extensions, suggest that IVOL should be unrewarded. However, the equilibrium setups of Miller (1977) and Merton (1987) assert that IVOL does affect expected returns negatively and positively, respectively, due to market frictions. This paper studies that conflicting evidence from the perspective of firm cash flow characteristics, which ultimately govern the IVOL effect. Focusing on firm cash flows is motivated by recent work based on the return decomposition framework of Campbell and Shiller (1988), which implies that return volatility arises due to shocks to discount rates and/or cash flows. Extensions of this decomposition to firm-level returns by Vuolteenaho (2002) and Chen, Da, and Zhao (2013) provide empirical evidence that individual stock return volatility is predominantly driven by firm-specific cash-flow shocks. 1 These findings suggest firm-level linkages between IVOL and cash-flow characteristics. In particular, if the idiosyncratic component of cash-flow growth is iid, a firm-specific shock to current cash-flow growth produces a one-for-one effect on idiosyncratic returns as expectations of all future cash flows are revised proportionally. When expected cash-flow growth is mean reverting, however, expected growth is an additional and strong determinant of the effect of cash-flow volatility on return volatility. Then, returns respond less than one-for-one to cash-flow shocks, and stocks with higher expected growth display lower IVOL. Building upon this intuition, we develop and apply a simple dividend discount model in which 1 See also Irvine and Pontiff (2009), who find that the time-series pattern in average idiosyncratic cash-flow volatility closely match that of idiosyncratic return volatility which provides additional evidence that cash-flow risk is important in determining IVOL. 1

3 IVOL is a strong predictor of the cross section of average returns even when it is unpriced. Moreover, the IVOL-return relation can either be positive or negative. Opposing IVOL effects emerge due to the relations of IVOL with two firm characteristics dividend size and expected dividend growth both of which are related to risk premiums. In particular, a positive IVOL effect arises in the dividend size dimension, while a negative effect characterizes the dividend growth dimension. We show that inferences about the IVOL effect could be markedly different, depending on whether an IVOL proxy is aligned more closely with dividend size or with dividend growth. The conflicting IVOL effects thus do not really establish a puzzle and are well captured in our setup. 2 Three assumptions underly our model. First, the term structure of discount rates is upward sloping. This property is endogenously obtained in well-known models including Campbell and Cochrane (1999) and Bansal and Yaron (2004). Our inferences would indeed apply to such models after incorporating a formal cross section of firms with the proper aggregation, i.e., that the sum of firm dividends is equal to the aggregate dividend. To promise proper aggregation, we further assume that firms with smaller dividend face greater uncertainty about dividend growth, suggesting that smaller dividend firms exhibit higher IVOL. This assumption is in line with the notion that smaller firms exhibit greater cash flow uncertainty. Next, to ensure that no single firm eventually dominates the economy, our third assumption is mean reversion in firm expected dividend growth, while the aggregate dividend growth follows an iid process for parsimony. We then formulate stock prices, expected returns, and IVOL for the cross section of firms. In our setup, expected returns are highest for firms with dividend streams which are concentrated at a long horizon due to the upward-sloping term structure of discount rates. Even though IVOL is unpriced, it can be related to expected returns for two reasons. First, IVOL and dividend size are negatively related, and moreover firms with smaller current dividends tend to exhibit higher 2 Previous work offers potential explanations for a negative IVOL-return relation based on unpriced information risk (Johnson (2004)), preferences for skewness (Boyer, Mitton, and Vorkink (2010)), lottery-like payoffs (Bali, Cakici, and Whitelaw (2011)), growth options (Barinov (2013)), limits to arbitrage (Stambaugh, Yu, and Yuan (2013)), or the effect of idiosyncratic shocks on firm risk exposure (Babenko, Boguth, and Tserlukevich (2013)). Prior studies also motivate a positive IVOL effect due to investor underdiversification (e.g., Levy (1978), Merton (1987), and Liu (2012)) or loss aversion (Barberis and Huang (2001)). Our own focus is different as we investigate the simultaneous occurrence of positive and negative IVOL effects obtained by using different proxies for IVOL. 2

4 expected returns as they often gain much of their value from higher long-horizon dividends. Hence, there is a positive IVOL-return relation in the dividend size dimension. Next, in the presence of mean reversion in expected dividend growth, expectations of long-term dividends are less affected by current dividend shocks than are short-term dividends. Thus, prices of high expected growth stocks are less responsive to firm-specific dividend shocks, which reduces their IVOL. Altogether, this leads to negative relation between IVOL and expected dividend growth. Firms with higher expected dividend growth also have higher expected returns. Thus, the IVOL-return relation is negative in the expected dividend growth dimension. While the IVOL effect along the dividend size dimension follows straightforwardly, let us build some intuition about the IVOL effect along the expected dividend growth dimension. In particular, consider two firms on opposite extremes of the cross section in our model. The first firm has a stock price of $1,000 and dividend of $1. The high price-dividend ratio (1,000) is supported by high expected dividend growth. If the firm s dividend increases by 100% today, which is considerably higher than the expected growth, the realized return would only be 3.3%, yielding a price-dividend ratio of (=$1,033/$2). The valuation ratio diminishes because part of the initial expected growth has already been realized, thus expected continued growth is downwardly revised. The potential impact of dividend shocks on realized returns is largely offset by opposing changes in valuation ratios for high growth stocks. Hence the firm s value displays lower sensitivity to dividend shocks. This firm exhibits low IVOL of 2.63% per month paired with a high expected return of 1.23%. On the other extreme, consider a second firm with a stock price of $3, dividend of $1, and a large negative expected growth rate. If its dividend drops unexpectedly to $0.50, the firm realizes a 46.4% return, almost a one-for-one response. Dividend shocks exert a large impact because firm value heavily depends on short-run dividends, and expectations of these dividends are strongly impacted by current shocks. This firm s reliance on short-run cash flows produces a relatively low expected return of 0.54% per month while its sensitivity to firm-specific shocks leads to high IVOL of 24.94% per month. These two extreme cases illustrate the negative IVOL-return relation along the dividend growth dimension. 3

5 To test whether our model does accommodate the conflicting evidence about the IVOL effect, we use both simulation studies and empirical analysis. To reconstruct the conflicting evidence and implement our empirical experiments, we use U.S. data spanning the July 1963 through December 2012 sample period. The AHXZ measure of IVOL is the volatility of residuals from a Fama French (1993) model regression using daily returns over the past month, while Fu s alternative measure uses expected IVOL from an exponential generalized autoregressive conditional heteroskedasticity (EGARCH) model fitted to monthly residuals from a Fama French model regression. We show that a value-weighted strategy that buys the high IVOL and sells the low IVOL portfolios sorted on the AHXZ measure delivers a significant 1.20% payoff per month. The IVOL effect is found to be stronger based on value-weighted portfolios and larger stocks, consistent with the findings of Bali and Cakici (2008) and Chen, Jiang, Xu, and Yao (2012). Sorting on the Fu measure, however, the evidence shows that an equal-weighted strategy that buys high IVOL stocks and sells low IVOL stocks earns 2.79% per month. The Fu measure is indeed more strongly related to returns based on equal-weighted portfolios, and moreover the effect only exists among microcap stocks while no relation prevails among bigger stocks. Consistent with past work, the two IVOL measures create economically large return spreads in opposing directions, and moreover the effects are more prominent among different subsets of stocks. Back to our proposed model, simulation studies of a cross section of 2,500 firms validate the accuracy of the model predictions. In particular, sorting stocks on the dividend size and expected dividend growth components of IVOL produces positive and negative IVOL effects, respectively. The negative IVOL effect for the dividend growth component is strongest among large cap stocks consistent with the findings using the AHXZ measure, while the positive effect based on the dividend size component is stronger for very small stocks, which is similar to the pattern obtained by sorting on the Fu measure. Empirically, we confirm our model s predictions that IVOL is negatively related to cash flow size and cash flow growth. In particular, a strong negative and monotonic relation between cash flow size and IVOL characterizes the data, as dividends paid by low IVOL firms are over 100 4

6 times the size of those paid by high IVOL firms. Moreover, the novel prediction of our model that IVOL is negatively related to cash flow growth is strongly supported, as higher IVOL firms have substantially lower growth rates over a three-year period around portfolio formation. Using the AHXZ measure, high IVOL firms have average dividend growth and asset growth rates which are about 19% and 13% lower per year, respectively. The same figures corresponding to the Fu measure are 20% and 11%. Further, based on the prescription of the model, we hypothesize that the observed positive and negative relations between IVOL and returns arise due to the relations of IVOL with dividend size and dividend growth. The empirical evidence supports this hypothesis. In particular, portfolios sorted on dividend size display a positive relation between IVOL and average returns. The high dividend portfolio earns 0.85% per month with average AHXZ (Fu) IVOL of 6.00% (6.14%), while the low dividend portfolio returns 1.41% per month with IVOL measures of 13.00% and 11.44%. Thus, as the model suggests, a positive IVOL effect appears along the dividend size dimension. Next, a strategy that buys high dividend growth stocks and sells low dividend growth stocks earns 0.14% per month using value-weighted and 0.34% per month using equal-weighted portfolios. The AHXZ and Fu measures are 1.73% and 1.22% lower for high dividend growth stocks, respectively. Hence, a negative relation appears in this dimension of past realized dividend growth, which is admittedly a noisy proxy of expected dividend growth. Double-sorted portfolios on dividend size and dividend growth provide further evidence of the simultaneous existence of a positive IVOLreturn relation along the dividend size dimension and a negative relation along the dividend growth dimension. We next hypothesize that the AHXZ measure is more closely related to dividend growth, while the Fu measure is more closely aligned with dividend size. The empirical evidence conclusively supports this joint hypothesis. First, double-sorting firms into portfolios based on the two IVOL measures allows one to isolate variation in each measure while holding the other measure relatively constant. While both measures are negatively associated with cash flow size, the Fu measure is more strongly and nearly monotonically related to both dividends and earnings. Examining div- 5

7 idend growth, earnings growth, and asset growth reveals that the AHXZ measure is consistently and strongly negatively related to such growth rates. In contrast, controlling for AHXZ, the Fu measure is often either positively related or unrelated to growth rates. Second, a complementary panel regression approach further validates the model predictions along the joint hypothesis described here. In particular, implementing panel regressions of both IVOL measures on dividend growth, log dividend size, the interaction of these two variables, as well as other control variables, the evidence shows that the AHXZ measure has a negative slope which is nearly three times the magnitude ( 4.06 versus 1.43) with respect to dividend growth, while the Fu measure displays greater dependence on dividend size ( 1.18 versus 0.75). All these slope coefficients are highly significant. To summarize, we develop a dividend discount model in which different proxies of IVOL display strong relations with average returns even when IVOL is unpriced. Different outcomes on the direction and magnitude of the IVOL effect could emerge depending on whether the IVOL proxy is more titled towards dividend growth (which produces a negative IVOL effect) or dividend size (which invokes a positive effect). In the data, we find evidence consistent with our model s predictions for the IVOL effects produced by both the AHXZ and Fu measures. The remainder of the paper proceeds as follows. Section 2 reviews the related literature on the IVOL effect and establishes several empirical regularities. Section 3 presents a simple model for investigating the IVOL effect and demonstrates the model s ability to match several crosssectional features of the data. Section 4 presents empirical evidence on the predictions of the model. Section 5 concludes. 2 The Idiosyncratic Volatility Effect Before formulating our model of asset prices, we establish several empirical highlights to gain some thorough perspective about the IVOL effect. We first describe the data that will also serve to test our own model implications and then report evidence on the IVOL effect using the two alternative IVOL measures as well as using both portfolio analysis and cross section regressions. 6

8 2.1 Data The sample spans the July 1963 through December 2012 period and it consists of U.S. firms with shares of ordinary common stock listed on the NYSE, Amex, and NASDAQ. Included are all firms with necessary stock return and financial information, as elaborated below. Stock return data is from CRSP, while financial data is from Compustat. If a stock gets delisted, any missing returns are replaced with their corresponding delisting returns. 3 The AHXZ measure is calculated as the standard deviation of residuals from a regression of excess stock returns on the Fama French (1993) factors. 4 The regression is run each month using daily return data. The resulting standard deviation is then multiplied by the square root of the number of trading days in the month to convert it into monthly IVOL. The IVOL measure is finally lagged one month so it is available to investors at the beginning of the month for portfolio formation. To calculate the AHXZ measure, it is required that there are no more than five missing daily returns during the month. The Fu measure is calculated as follows. For each firm that has at least 30 monthly return observations, we fit an EGARCH model to the residuals of Fama French monthly regressions. Nine versions of the EGARCH model are considered with lags on past returns and past volatility that range from one to three. For each firm-month, the model with the highest Akaike Information Criterion (AIC) is chosen. The expected IVOL implied by the EGARCH model serves as the IVOL measure. 5 Table I shows equity characteristics for portfolios sorted on the IVOL measures of AHXZ (2006) (Panel A) and Fu (2009) (Panel B). IVOL portfolios are rebalanced on a monthly basis. Firm market capitalization, total assets, and the book-to-market ratio are calculated following 3 Shumway (1997) discusses potential biases related to stock delistings. 4 Factor returns are available at library.html. We thank Kenneth French for making this data available. 5 Fu (2009) uses an expanding window estimation approach by estimating the EGARCH model using historical data in each month. This approach avoids potential look-ahead bias but it requires the estimation of over 25 million EGARCH models, including many periods with sample sizes that are small for accurate estimation of the EGARCH model. We perform a full-period estimation of the EGARCH model parameters and obtain qualitatively similar results as Fu (2009). 7

9 Fama and French (2008). 6 Prior month return is the lagged monthly return for each of the stocks included in our sample. Reported are time-series averages, throughout the entire sample period, of monthly cross-sectional averages. The evidence shows substantial cross sectional dispersion in each of the IVOL measures. The AHXZ (Fu) measure ranges from 3.3% (4.7%) to 38.5% (30.4%) among the low and high portfolios. Both IVOL measures exhibit strong negative relations with market capitalization and total assets but there is no obvious relation with the book-to-market ratio. While the AHXZ measure is strongly positively related to past one-month return, the Fu measure shows a strong negative relation. Overall, the IVOL effect seems to be related to stock price reversals. Nevertheless, as we show below, controlling for price reversals does not capture the IVOL effect. We report portfolio transition probabilities in Table A.I of the Appendix A. Portfolio membership is reasonably persistent. Using the IVOL measure of AHXZ, nearly half of the firms in the high and low IVOL portfolios carry over from the same portfolio in a typical month. At an annual horizon, a firm that was initially in the high (low) IVOL decile portfolio and meets current data requirements is still included in the high (low) portfolio about 38% (39%) of the time. The Fu measure is even more persistent, such that 65% (73%) of the high (low) IVOL firms are still included in the portfolio one month later and 49% (61%) appear in the same portfolio after one year. 2.2 Idiosyncratic Volatility and Average Returns We confirm the AHXZ findings in the first column of Panel A of Table II. The return spread between the high and low IVOL portfolios is economically large at 1.20% per month (t-stat of 3.40). Past work shows that the IVOL effect could also depend on the sorting mechanism. In particular, Bali and Cakici (2008) consider the AHXZ measure and examine various sorting 6 Specifically, market capitalization is the product of shares outstanding and stock price from CRSP. The book value of equity is defined as Total Assets (AT) minus Total Liabilities (LT) plus Deferred Taxes and Investment Tax Credit (TXDITC), if available, minus the value of preferred stock, if available. Preferred stock is defined as the first available of Preferred Stock Liquidating Value (PSTKL), Preferred Stock Redemption Value (PSTKRV), and Preferred Stock at Carrying Value (UPSTK). Book-to-market is the ratio of the book value of equity to market capitalization, and we postpone use of book-to-market for six months following the fiscal year end following the convention of Fama and French (1992). 8

10 techniques upon forming the IVOL portfolios. They find that the effect is much smaller and even insignificant among equal-weighted portfolios. Indeed, the second column in Panel A of Table I shows lower average return spread of 0.33% per month among equal-weighted portfolios, which is statistically insignificant recording a t-stat of We next investigate the IVOL effect within size subgroups. Following Fama and French (2008), Micro stocks are those with market capitalization less than the 20th percentile of NYSE size, and All but Micro stocks consists of all other stocks. This classification implies that about 60% of the firms are Micro stocks in a typical month during the sample period. Panel A of Table II reports average returns of portfolios formed based on Micro stocks and All but Micro stocks. Among Micro stocks, the value-weighted return spread is 1.46% per month and highly significant (t-stat of 4.17) while the equal-weighted high-low return is only 0.39% and statistically insignificant. In contrast, when Micro stocks are excluded in the last two columns of Panel A, both the value- and equal-weighted return spreads are large and highly significant. The value-weighted strategy achieves an average monthly return of 1.41% (t-stat of 3.59) while the equal-weighted portfolios earns 1.71% (t-stat of 4.26). A common link between these findings is that the IVOL effect using the AHXZ measure is stronger when larger stocks are emphasized (in value-weighted portfolios and when excluding Micro stocks) and weaker when smaller stocks are more prominently featured (in equal-weighted portfolios and among Micro stocks). Panel B reports the same statistics, but focusing on the measure of Fu. Interestingly, the opposite pattern emerges. In the full sample, an equal-weighted strategy records a positive average return of 2.79% (t-stat of 5.33) while the value-weighted portfolio earns a statistically insignificant 0.52%. When only Micro stocks are considered, both the value-weighted (1.22% with t-stat of 2.50) and equal-weighted (3.16% with t-stat of 5.95) portfolios earn large returns. In stark contrast, each of the two strategies earn just 0.02% when the sample accounts for all but Micro stocks. The 7 Bali and Cakici (2008) also find that sorting stocks using NYSE breakpoints of the AHXZ measure, which consider more small stocks into the high IVOL portfolio, weakens the IVOL effect. Chen, Jiang, Xu, and Yao (2012) find that the IVOL effect with the AHXZ measure is stronger in bigger firms than among microcap stocks, and the effect strengthens when penny stocks are eliminated from the sample. These findings are particularly interesting given that anomalies are typically stronger among microcaps and are weaker or insignificant among big stocks (e.g. Fama and French (2008)). 9

11 general takeaways from Table II is that the IVOL effect displays opposing signs using the AHXZ and Fu measures, and that the IVOL effect is strongest among bigger firms using the AHXZ measure but among smaller firms using the Fu measure. Another stream of the literature relates the IVOL effect to the credit risk anomaly. In particular, Avramov, Chordia, Jostova, and Philipov (2013) show that the IVOL effect is strongest among financially distressed firms. While we do not explicitly account for leverage, our model implies that the negative relation between returns and IVOL is driven largely by small firms with poor growth prospects. Such firms are more likely to display high credit risk. In Table III, we use Fama MacBeth tests to further examine the marginal influence of IVOL on average returns. We control for size and book-to-market in each specification following Fama and French (2008), yet inferences are not sensitive to excluding these variables. The benchmark model (Model 1) shows a negative size effect and a positive book-to-market effect, consistent with prior literature. Models 2 and 3 include the IVOL measures of Ang, Hodrick, Xing, and Zhang (2006) and Fu (2009), respectively. Notice that the AHXZ measure is significantly negatively related to returns (Model 2) with a coefficient of 0.04 (t-stat of 4.60), while Fu s (2009) measure is significantly positively related to returns (Model 3) recording slope coefficient of 0.11 and t-stat of The effect of Fu s (2009) measure of IVOL appears to be partially related to the size effect, as the coefficient on size changes from 0.16 (t-stat of 3.25) in Model 1 to 0.08 (t-stat of 2.44) in Model 3. When both IVOL measures are included in Model 4, their explanatory abilities actually sharpen with the AHXZ measure s coefficient becoming more negative at 0.08 (t-stat of 13.59) and the coefficient of the Fu measure increasing to 0.15 (t-stat of 10.45). Thus, based on cross-sectional regressions, the evidence shows that both IVOL measures are useful in predicting average returns, with each measure capturing a distinct effect on the cross section of average returns. Huang, Liu, Rhee, and Zhang (2010) claim that the effect shown by Ang, Hodrick, Xing, and Zhang (2006) is an artifact of the short-term reversal anomaly. In response, we consider lagged one-month returns in the tests reported in Table III. We find that the AHXZ measure of IVOL 10

12 is robustly negatively related to returns even after controlling for short-term reversal, with a coefficient of 0.02 with a t-stat of 2.18 in Model 5 and a coefficient of 0.06 (t-stat of 9.55) in Model 7. 8 In sum, each of the IVOL measures appears to be useful in forecasting returns even after controlling for size, book-to-market, short-term reversal, and the alternative IVOL measure. However, the inferences about the IVOL effect are quite dependent on the measure used for portfolio formation as well as the portfolio formation technique. The measure of Ang, Hodrick, Xing, and Zhang (2006) is strongly negatively related to returns, particularly when larger stocks are emphasized. In contrast, the measure of Fu (2009) is positively related to returns, especially when focusing on smaller stocks. These conflicting results have led many researchers, formally or informally, to question whether IVOL is indeed related to expected returns. However, as we show below, the conflicting results do not really establish a puzzle as they are fully consistent with the predictions of a simple model that proposes an economic link between IVOL and expected returns. 3 Model This section formulates a simple dividend discount model that captures all the documented crosssectional features of IVOL and its relation with expected stock returns, and further imposes new testable restrictions on the data. The model is built upon three primary assumptions: (i) an upward-sloping term structure of discount rates, (ii) a negative relation between the size and the volatility of dividend growth, and (iii) mean reverting firm-level dividend growth. We do not intend to provide a full description of reality with our model. Rather, our goal is to demonstrate that a market which exhibits these three (plausible) features will produce the IVOL effects we observe in the data. 8 Chen, Jiang, Xu, and Yao (2012) also find that the AHXZ IVOL effect survives controlling for return reversal except among penny stocks. In unreported results, we further investigate whether the low returns of high IVOL stocks are attributable to short-term reversal from good prior-month performance. We run the value-weighted AHXZ strategy in subsamples in which stocks have lagged one-month returns below or above the median lagged return from the same period. Among stocks with low past return the AHXZ strategy earns 1.37% (t-stat of 3.48), while it earns 1.23% (t-stat of 3.85) among high past return stocks. Despite the strong relation between IVOL and past return shown in Table I, the IVOL effect does not appear to be attributable to return reversal. 11

13 3.1 Model Setup Discount Rates For purposes of parsimony, we exogenously specify a term structure of discount rates for dividends as r t,τ = a + bτ, a 0, b > 0 (1) where r t,τ is the annualized continuously compounded rate at time t used to discount dividends to be paid at a horizon of τ years (i.e., at time t + τ). The joint conditions a 0 and b > 0 imply a positive and upward-sloping term structure. The discount rate specification in equation (1) is a stylized model which is purposely simplistic in order to isolate the key feature of the aggregate economy: an upward-sloping term structure of discount rates. Of course, r t,τ is the sum of the riskless interest rate and the equity premium for horizon τ. Indeed, empirical evidence suggests that the term structure of riskless interest rates is, on average, upward sloping (e.g., the average spread between 10-year and 1-year Treasury yields is 0.98% over the July 1963 to December 2012 period), contributing to the likelihood of an upward-sloping term structure of the overall discount rates. 9 The implications of this model for the IVOL effect extend to other, more complex models, in which the term structure of discount rates is endogenously derived to be upward sloping. Such models include those of Jermann (1998), Abel (1999), Campbell and Cochrane (1999), Gomes, Kogan, and Zhang (2003), Bansal and Yaron (2004), Wachter (2006), Kaltenbrunner and Lochstoer (2010), and Gârleanu, Panageas, and Yu (2012), among others. While our results do not depend on the term structure being determined within a rational framework, we will often refer to long-run cash flows with high discount rates as riskier and short-run cash flows as safer. 9 A recent literature empirically investigates the slope of the term structure of the equity premium. Binsbergen, Brandt, and Koijen (2012) find evidence that synthetic short-term dividend strips on the S&P 500 earn higher returns than the index over the period 1996 to They suggest based on this evidence that the term structure may be downward sloping. In response, Boguth, Carlson, Fisher, and Simutin (2012) show that pricing frictions can account for this evidence as a result of the amplification of pricing errors in the levered strategy used to construct the dividend strips, while Schulz (2013) attributes the higher short-term returns to compensation for the tax burden of dividends on the ex-dividend date. Binsbergen, Hueskes, Koijen, and Vrugt (2013) present evidence that the term structure may be upward sloping except during the financial crisis period, which is consistent with the theoretical predictions of Muir (2013). 12

14 3.1.2 Dividend Characteristics and the Cross Section of Expected Returns We propose a formal cross section of firms that ultimately aggregate to form the economy as a whole. Starting with the economy level, the aggregate dividend is assumed to follow the diffusion process dd t D t = gdt + σdw D,t. (2) Indeed, time variation in the expected dividend growth at the firm level is key to our model s implications. Hence, we adopt this iid structure for the aggregate dividend to isolate the impact of cross-sectional differences in time-varying expected dividend growth rates. The aggregate economy is composed of a cross section of N firms. Firm-level dividends take the form D i,t = D t θ i,t (3) where θ i,t is the firm dividend share. That is, θ i,t is the proportion of the aggregate dividend paid by firm i at time t. 10 We assume that θ i,t follows a Wright Fisher process, 11 dθ i,t = α( θ i θ i,t )dt + δ θ i,t (1 θ i,t )dw θi,t (4) where θ i is the long-run dividend share, α > 0 is a mean reversion parameter, δ > 0 is a volatility parameter, dw θi,t is uncorrelated with dw D,t, and inter-firm correlation in share processes is given by θ i,t θ j,t ρ t (dw θi,t, dw θj,t) = (1 θ i,t )(1 θ j,t ). (5) Further, N i=1 θ i = 1. Under this set of assumptions, Proposition 1 below establishes the proper aggregation of the cross section of firms into the aggregate market. Before we formulate the proposition let us note 10 Several studies use dividend or consumption shares to model formal cross sections of firms (e.g., Menzly, Santos, and Veronesi (2004), Santos and Veronesi (2006, 2010), Croce, Lettau, and Ludvigson (2012), and Avramov, Cederburg, and Hore (2013)). 11 Wright Fisher processes originate in the genetics literature in studies by Fisher (1930) and Wright (1931). See Crow and Kimura (1970) and Karlin and Taylor (1981) for further discussion of Wright Fisher processes. 13

15 that Cochrane, Longstaff, and Santa-Clara (2008) consider equilibrium prices in the presence of multiple Lucas trees and iid processes for dividends. They show that, in this setup, a Wright Fisher process for dividend share arises endogenously. Proposition 1. Given N i=1 θ i,0 = 1 at time 0, then N i=1 θ i,t = 1 and N i=1 D i,t = D t for all t 0. Proof. See Appendix B.1. The instantaneous firm dividend growth then follows by applying Ito s lemma to equation (3): ( ( )) dd i,t θi 1 θ i,t = g + α 1 dt + σdw D,t + δ dw θi,t. (6) D i,t θ i,t θ i,t Expected firm dividend growth is increasing in the firm s share ratio, which is defined as θ i /θ i,t the ratio of the long-run dividend share to the current dividend share. 12 Thus the expected growth rates of the firm dividend and aggregate dividend are equal when the share ratio equals one. Otherwise, expected firm dividend growth is higher (lower) than expected aggregate dividend growth when θ i /θ i,t > 1 ( θi /θ i,t < 1 ). Due to the mean reverting nature of dividend share, the firm share ratio tends to revert to one over time. Two aspects of the firm dividend growth from equation (6) are important to the model s implications. First, due to the mean-reverting structure of the drift term in equation (6), expectations of future dividends are impacted less than one-for-one given a current dividend shock. That is, if a firm s dividend unexpectedly doubles in the current year, the expectation of the dividend paid by a firm 25 years from now is revised upward but by less than 100%. Formally, the expected firm dividend paid at a horizon of τ is ( ) θi E t (D i,t+τ ) = D i,t e gτ (1 e ατ ) + e ατ. (7) θ i,t 12 Share ratio is analogous to the relative share characteristic of Menzly, Santos, and Veronesi (2004). 14

16 The elasticity of this expectation with respect to θ i,t is then given by E t (D i,t+τ ) θ i,t θ i,t E t (D i,t+τ ) = e ατ ( θi θ i,t (1 e ατ ) + e ατ ). (8) Equation (8) illustrates the proportional impact of a shock to θ i,t on dividends at different horizons. For example, this elasticity is one when τ = 0 such that the current dividend is impacted fully by the change to the dividend share. As τ approaches infinity, the elasticity converges to zero so expectations of very long-horizon dividends are unaffected by a current return shock. For intermediate dividends, the revision in expected dividends lies between zero and one, meaning that expectations respond less than one-for-one to the shock. Further, the elasticity is decreasing in share ratio, so expected dividends are revised less (more) for high (low) share ratio stocks after a current dividend shock. Intuitively, expectations of continued dividend growth are downwardly revised when high expected growth firms experience a positive current shock. The second important aspect of the firm dividend growth is that the diffusion term in equation (6) implies higher dividend growth uncertainty for firms with low dividend shares. This assumption is necessary for proper aggregation of the cross section of stocks. It is also in line with the intuitive notion that smaller firms exhibit greater cash flow uncertainty Expected Returns and Idiosyncratic Volatility Expected Returns Given the above formulations, the firm value, expected stock return, realized return, and volatility are all determined by the dividend discount model. The firm value is given by P i,t = τ=0 E t D i,t+τ e rt,τ τ dτ (9) 13 The model assumes that term structure and dividend share parameters in equations (1) and (4), respectively, apply to all firms. While allowing these parameters to vary across firms may exert cross-sectional effects, the economic forces described here establish important determinants of the IVOL-return relation. We keep constant parameters for simplicity. 15

17 and one-period expected returns are given as the value-weighted average of one-period forward rates applicable to firm dividends, E t (R i,t+1 ) = τ=0 E t D i,t+τ e r t,τ τ P i,t e a b+2bτ dτ. (10) Given that the forward rate curve inherits the upward slope of the term structure of discount rates, higher dividends at longer horizons contribute to larger expected returns in equation (10). A relation therefore emerges between expected dividend growth and expected returns. Intuitively, firms with high expected growth gain more value from long-horizon dividends, so their expected returns are high because they are weighted toward the higher long-run forward rates. Shorthorizon firms with low expected dividend growth have relatively more weight on low short-run discount rates. We now formally demonstrate the positive relation between expected dividend growth and expected returns. Equation (7) shows that expected dividend growth depends on firm-level parameters only through share ratio, and expected dividend growth is increasing in share ratio. Proposition 2 establishes the connection between share ratio and expected return. Proposition 2. Expected return is increasing in share ratio and related to dividend share only through share ratio. Proof. See Appendix B.2. The cross section of expected returns in this economy is determined by aggregate- and firmlevel parameters. From Proposition 2, expected return is independent of all firm-level parameters other than share ratio. Thus, share ratio explains all the cross-sectional variation in expected returns, while IVOL remains unpriced. This point is illustrated in Figure 1 which shows expected return across a range of share ratio and dividend share. Firms with high share ratios earn average returns over 1.2% per month while low share ratio firms earn returns of about 0.5%, creating economically significant return spreads in the cross section. This pattern occurs because high 16

18 duration firms have high expected returns to compensate for their riskier cash flows. The values of low duration firms, on the other hand, are dependent primarily on safer short-run dividends. The relation between expected return and share ratio extends to value-weighted portfolios. We show in Appendix B.3 that the share ratio of a value-weighted portfolio is a dividend-weighted average of the share ratios of the portfolio holdings. Portfolio-level expected returns then follow from equation (10) Idiosyncratic Volatility The overall return volatility emerges from shocks to both aggregate dividend growth and firm dividend share. The idiosyncratic portion of volatility, which is attributable to the firm-specific dividend share shocks, can be written as σ i,t = δ 1 θ i,t θ i,t ( 1 + (P i,t/d i,t ) θ i,t θ i,t P i,t /D i,t ). (11) The term δ 1 θ i,t θ i,t (12) captures the volatility of shocks to θ i,t. The term in parentheses has two components. First, there is a one-for-one effect of the current firm dividend growth shock resulting from the change in dividend share on realized return. Second, there is an effect of the change in dividend share on the valuation of the firm. The (P i,t /D i,t ) θ i,t θ i,t P i,t /D i,t = τ=0 θ i θ i,t (1 e ατ )e (g r t,τ )τ dτ τ=0 ( ) θi θ i,t (1 e ατ ) + e ατ e (g rt,τ )τ dτ (13) term is the elasticity of the firm s price-dividend ratio with respect to θ i,t, so this elasticity multiplied by the dividend share volatility term captures the effect of repricing based on changes in dividend share. Thus, firms display different idiosyncratic volatilities for two reasons: (i) the firms have different cash-flow volatilities, as formulated in equation (12), and (ii) firm values respond differently to an equivalent cash-flow shock, as evident from equation (13). 17

19 While we are interested in studying the relation between IVOL and expected return, the relation between share ratio and IVOL is of direct interest. Proposition 3 relates IVOL to share ratio and dividend share. Proposition 3. Idiosyncratic volatility is a function of dividend share and share ratio. Controlling for dividend share, idiosyncratic volatility is decreasing in share ratio. Similarly, controlling for share ratio, dividend share has a negative effect on idiosyncratic volatility. Proof. See Appendix B.4. Figure 2 shows IVOL as a function of share ratio and dividend share. IVOL can reach very high levels for firms with small dividend share and low share ratio. Such small firms that already have low dividend share, which is expected to further shrink since θ i is much smaller than θ i,t, tend to have high IVOL levels. IVOL is sharply decreasing in both the share ratio and dividend share dimensions, and large firms with high dividend share and good growth prospects, as evident by high share ratio, will have the lowest levels of IVOL. Analytically, it is obvious that the component of IVOL in equation (12) is decreasing in dividend share. From an economic perspective, firms with low dividend share have higher levels of dividend growth uncertainty which contributes to higher IVOL. This term of IVOL tends to be positively related to share ratio in a cross section of firms because θ i,t and θ i /θ i,t are negatively correlated. More specifically, the cross section of the model accommodates many firms with low dividend shares and high share ratios. Low dividend share contributes to higher IVOL as the component of IVOL in equation (12) is greater, while high share ratio implies high returns. Thus, this component of IVOL tends to contribute to a positive IVOL effect in the cross section. Indeed, the model predicts that in the dimension of dividend size, a positive IVOL-return relation is expected. The second term of IVOL in equation (13) is the elasticity of the firm s price-dividend ratio with respect to dividend share. This term captures the change in the firm s valuation ratio which accompanies a firm-specific dividend shock. Repricing occurs upon realization of a shock 18

20 as investors incorporate changes in expectations about future growth into the new valuation ratio. We show analytically in Appendix B.4 that equilibrium pricing within the model implies a negative relation between share ratio and this elasticity. As share ratio is a positive determinant of expected returns and has a negative influence on IVOL, a negative IVOL effect is produced in the cross section. This negative IVOL effect arises because of differences in the sensitivity of firm value to firmspecific dividend shocks. We show in Appendix B.4 that the elasticity in equation (13) is bounded between 1 and 0 such that ( 0 < 1 + (P i,t/d i,t ) θ i,t ) θ i,t < 1. (14) P i,t /D i,t Notice that this term in parentheses is multiplied by the firm s dividend growth volatility in equation (11) to determine IVOL. Thus, more negative values of the elasticity serve to dampen the effect of cash flow shocks while an elasticity close to zero implies that nearly the full impact of dividend shocks transfers to prices and returns. As share ratio approaches zero, the elasticity term also approaches zero. In this case, the full term in parentheses approaches one and IVOL approaches dividend growth volatility. On the other hand, as share ratio approaches infinity, the full term approaches zero hence, IVOL is a small fraction of dividend growth volatility. In the end, we observe a negative relation between share ratio and IVOL. The above analysis shows analytically that revaluation of the firm following a dividend shock serves to diminish IVOL. This effect occurs because valuation ratios decrease (increase) when a firm experiences a good (bad) dividend shock. The change in valuation reflects changes in expectations about future dividends. In the absence of mean reversion in expected dividend growth, a current dividend shock would lead to a one-for-one revision of all future cash flows so the realized return would equal the size of the shock to dividend growth. In the presence of mean reversion, however, future dividends and, as a result, prices are impacted less by the shock. Recall from equation (8) that expectations of future dividends are positively affected by current dividend shocks. This effect depends on the horizon, as short-run dividends experience large 19

21 revisions while expectations of long-term dividends are less affected. Such dependence explains why the prices of low share ratio firms with short duration experience large impacts from firmspecific shocks while high share ratio firms with long durations have diminished effects. 3.3 Model Simulation In this section we use Monte Carlo experiments to simulate from the proposed model, while the next section further tests our model implications using real U.S. data. The simulation study is based on 1,000 draws of a cross section of 2,500 stocks. Given the process for dividend shares in equations (4) and (5), it can be shown that the invariant distribution of the N-vector θ t is Dirichlet with the parameter vector 2α θ/δ 2. We can then calculate prices, expected returns, and IVOL for each stock from equations (9), (10), and (11), respectively, using numerical integration. When size categories are needed for portfolio formation, we designate the smallest 60% of firms as Micro stocks and the remaining 40% as All but Micro stocks to match the empirical allocation. Further details on the simulation appear in Appendix C. Table IV shows model-implied expected returns for Low, High, and High-Low IVOL portfolios. We examine both value-weighted and equal-weighted portfolios within the All Stocks, Micro Stocks, and All but Micro Stocks categories. Portfolio expected returns are on a monthly basis. Sorting by the model total implied IVOL, a negative expected return IVOL relation appears. Among all stocks, the value-weighted and equal-weighted High-Low portfolios earn 0.47% and 0.37% per month, respectively. Results are somewhat weaker among micro stocks with expected payoffs of 0.22% and 0.20% for the two strategies, while expected payoffs become 0.50% and 0.44% per month when micro stocks are excluded. As discussed in the previous section, IVOL can be decomposed into two separate effects. Component one from equation (12) captures the effect of dividend size on IVOL, and this term has a positive influence on the IVOL-return relation. Specifically, sorting firms into portfolios based on this first component, the High-Low portfolio earns large positive returns across all specifications. The value-weighted and equal-weighted strategies earn 0.59% and 0.51% among all 20

22 stocks, and performance is strongest among Micro stocks with both strategies returning 0.62%. The second component in equation (13) captures the impact of expected dividend growth on IVOL. This term contributes heavily to the negative IVOL-return relation. The value-weighted portfolio earns 0.61% per month among all stocks while the equal-weighted strategy returns 0.53%. The effects are strongest in the All but Micro stocks with value-weighted and equalweighted expected returns of 0.61% and 0.54%, respectively. In sum, we demonstrate that IVOL has components related to dividend size and expected dividend growth. The size-based component has a positive influence on the IVOL-return relation, while the growth-based component generates a negative relation. The model produces economically large spreads in returns across these two dimensions. 4 Testing Model Implications The model formulated in Section 3 has several implications for the relation between IVOL, expected returns, and other firm characteristics, which we examine here. Our roadmap is as follows. We first establish the negative relations between IVOL and both firm cash flow size and cash flow growth. We then present evidence that sorting firms by cash flow size produces a positive relation between IVOL and returns, while sorting firms by a measure of cash flow growth produces a negative IVOL effect. We finally demonstrate that while both measures are robustly related to cash flow size and cash flow growth, the AHXZ (2006) measure better captures the firm cash flow growth effect and the Fu (2009) measure better captures the cash flow size effect. These findings provide a potential explanation for simultaneously observing a negative relation between IVOL and returns using the AHXZ (2006) measure while a positive relation exists using the Fu (2009) measure. 4.1 Relations of Idiosyncratic Volatility with Cash Flow Growth and Shares Our model implies that share ratio has a negative influence on IVOL. Share ratio is directly related to expected dividend growth, as high (low) share ratio translates into high (low) expected growth. 21