Dividend Volatility and Asset Prices: A Loss Aversion/Narrow Framing Approach

Transcription

1 Dividend Volatility and Asset Prices: A Loss Aversion/Narrow Framing Approach Yan Li and Liyan Yang Abstract This paper documents that the aggregate dividend growth rate exhibits strong volatility clustering. We incorporate this feature into a theoretical model and study the role of dividend volatility in asset prices when investors are loss-averse over fluctuations in the value of their financial wealth. We find that our model explains many salient features of the stock market, including the high mean, excess volatility, and predictability of stock returns; the low correlation between consumption growth and stock returns; time-varying Sharpe ratios; the GARCH effect and the volatility feedback effect in stock returns; and the decline of equity premiums in the postwar period. JEL classification: G11, G12 Keywords: dividend volatility clustering, loss aversion, narrow framing, asset pricing Yan Li: Department of Finance, Fox School of Business, Temple University, Philadelphia, PA 19122; liyanlpl@temple.edu; Tel: Liyan Yang: Department of Finance, Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario M5S 3E6; liyan.yang@rotman.utoronto.ca; Tel: We thank Nick Barberis, David Easley, Ming Huang, Maureen O Hara and participants in 2009 China International Finance Conference and 2010 Northern Finance Association Meetings for helpful suggestions. All errors are our responsibility. 1

2 Dividend Volatility and Asset Prices: A Loss Aversion/Narrow Framing Approach Abstract This paper documents that the aggregate dividend growth rate exhibits strong volatility clustering. We incorporate this feature into a theoretical model and study the role of dividend volatility in asset prices when investors are loss-averse over uctuations in the value of their nancial wealth. We nd that our model explains many salient features of the stock market, including the high mean, excess volatility, and predictability of stock returns; the low correlation between consumption growth and stock returns; time-varying Sharpe ratios; the GARCH e ect and the volatility feedback e ect in stock returns; and the decline of equity premiums in the postwar period. JEL classi cation: G11, G12 Keywords: dividend volatility clustering, loss aversion, narrow framing, asset pricing 1

3 1 Introduction How does aggregate dividend volatility a ect asset prices? 1 Despite the fact that the dividend is one of the most important nancial variables, there has been little research on this subject largely because, in standard consumption-based models, the pricing kernel is entirely driven by consumption. Thus, it is the covariance between consumption and dividends, not the volatility of dividends, that determines stock prices. 2 By contrast, the role of dividend volatility has been highlighted by the loss aversion/narrow framing approach advocated in the recent behavioral nance literature. 3 As can be seen from its name, the key elements of this approach are loss aversion and narrow framing, two of the most important ndings from experimental literature on decisionmaking under uncertainty. In the context of nancial markets, loss aversion means that investors are more sensitive to losses in their nancial investments than to gains of the same magnitude. Narrow framing states that investors tend to isolate the stock market risk from their overall wealth risk (such as labor income risk and housing risk) and derive utility directly from changes in the value of their stock holdings. Since dividend volatility drives uctuations in the value of their nancial wealth, investors may perceive aggregate dividend volatility rather than consumption volatility, the commonly used measure in the 1 Throughout the paper, the term dividend volatility refers to the standard deviation of the growth rate of (not the level of) the aggregate dividends paid to all stocks. See equation (3) for a technical de nition. 2 To the best of our knowledge, the only exception is Longsta and Piazzesi (2004), who demonstrate that volatile and procyclical dividends can raise equity premiums in a representative agent model with power utility. However, they limit their discussion to the equity premium and do not explore whether dividend volatility can help explain other puzzling facts in the aggregate stock market, such as excess volatility and return predictability. 3 See Barberis and Huang (2007) for a comprehensive survey of models adopting the loss aversion/narrow framing approach. 2

4 literature, as a metric for representing risk. This immediately implies that dividend volatility has signi cant implications for asset prices. Barberis and Huang (2007, p200) summarize by stating that models adopting the loss aversion/narrow framing approach can generate a large equity premium and a low and stable risk-free rate, even when consumption growth is smooth and only weakly correlated with the stock market. To date, studies adopting the loss aversion/narrow framing approach have assumed that the dividend growth rate process is independent and identically distributed (i.i.d.) over time to re ect the fact that dividend growth is di cult to forecast at the mean level. But is the dividend growth rate truly an i.i.d. process? Is it possible that the dividend growth rate is forecastable at higher moments, such as volatility? If yes, what are the pricing implications of this forecastability in dividend volatility? In this paper, we empirically investigate the data and nd strong evidence that dividend volatility indeed exhibits strong persistence (usually called volatility clustering) indicating the tendency of a large (small) change today to be followed by a large (small) change tomorrow. We then incorporate dividend volatility clustering into a standard model featuring loss aversion/narrow framing and nd that this simple extension helps to explain many salient features of the aggregate market. Speci cally, in addition to the low and stable risk-free rate, high equity premium, and low correlation between consumption and stock returns that can be generated from a standard model adopting the loss aversion/narrow framing approach with an assumed i.i.d. dividend growth rate, 4 our extended model can explain the excess volatility and predictability of stock 4 For a review of a standard loss aversion/narrow framing model, see Barberis and Huang (2007). In fact, our model will be reduced to a standard model adopting the loss aversion/narrow framing approach if we remove the time varying feature in the dividend process, i.e., if we set u to be 0 in equation (4). 3

5 returns, the time-varying Sharpe ratios, the GARCH and volatility feedbacks e ect in stock returns, and the declining equity premium in the postwar period. The intuition of our new results is as follows. Suppose that the dividend volatility, which is the state variable in our model, is high this period. Given the persistence in dividend volatility, the investor forecasts highly volatile dividends in the next period; this implies highly volatile returns in the next period. Since she is loss-averse, she nds this prospect scary and requires a higher equity premium. This pushes stock prices down, generating excess volatility. The same mechanism also produces predictability in stock returns: at moments when the investor forecasts especially high return volatility, she pushes current stock prices down, generating a low price-dividend ratio. But these are also moments when she requires a high equity premium; the price-dividend ratio therefore predicts returns. In addition, the conditional mean and conditional standard deviation of expected returns are driven di erently by dividend volatility; hence, the Sharpe ratio, as a measure of the price of risk, changes over time. Under the loss aversion/narrow framing approach, stock returns are ultimately driven by dividend news. Thus, they inherit its property of volatility clustering. This explains the well-documented volatility clustering e ect, also known as the GARCH e ect, in stock returns. 5 Moreover, our model provides a simple preference-based explanation for the volatility feedback e ect (e.g., French, Schwert and Stambaugh, 1987; Campbell and Hentschel, 1992): an increase in future stock return volatility leads to a decrease in current stock return. In our 5 Volatility clustering is called the GARCH e ect because it is typically modeled using generalized autoregressive conditional heteroscedasticity proposed by Bollerslev (1986). 4

6 model, stock volatility is priced because the loss-averse investor does not like uncertainty; therefore, an anticipated increase in volatility would raise the required rate of return and lower current prices. Last but not least, the declining dividend volatility in the postwar period also helps to explain the decline in ex-ante equity premium, which has been documented in many studies (e.g., Blanchard, 1993; Fama and French, 2002; Buranavityawut, Freeman and Freeman, 2006). Our model generates the three results predicted by a standard loss aversion/narrow framing model through the same mechanism. Speci cally, since the risk-free rate is determined by a smooth consumption process, it is stable and its level is also low for empirically plausible values of the intertemporal rate of substitution. Given that our model generates volatile stock prices, loss aversion is responsible for the high unconditional equity premium. In addition, unlike the consumption-based models, stock returns are ultimately driven by dividend news, which has a low correlation with consumption news; thus, the correlation between consumption and stock returns is low in our economy. Our paper is most closely related to the literature that relies on the loss aversion/narrow framing approach to explain aggregate stock market phenomena. Benartzi and Thaler (1995) are the rst to incorporate loss aversion and narrow framing in a theoretical model to study the equity premium puzzle. Barberis, Huang, and Santos (2001) study the joint properties of stock returns and consumption growth by incorporating dynamic features of loss aversion, usually labeled the house money e ect, meaning that people are more (less) willing to bear risks when they have had prior gains (losses). 6 Our study di ers from that of Barberis, 6 Unlike loss aversion itself, the idea that loss aversion changes over time is less well-supported by experi- 5

7 Huang, and Santos (2001) in three ways: (i) while their model makes two psychological assumptions, namely loss aversion and the house money e ect, we use loss aversion as the only psychological assumption; (ii) in terms of model mechanism, while Barberis, Huang, and Santos (2001) rely on stochastic risk aversion to explain the stylized facts in the stock market, our results are driven not by investors changing risk aversion, but by salient properties of dividend volatility for which we provide strong econometric evidence; and (iii) our model can also match more dimensions of data, such as the GARCH and volatility feedback e ects in stock returns as well as the decline in equity premium in the postwar period. McQueen and Vorkink (2004) introduce a new feature, namely, state-dependent sensitivity to news, into the model of Barberis, Huang, and Santos (2001) to explain conditional volatility clustering in stock returns. Our study di ers from theirs in that our primary goal is to systematically investigate the role of dividend volatility in asset prices. Our work also contributes to the literature examining how low-frequency risk can provide a justi cation for observed risk premiums (e.g., Bansal and Yaron, 2004; Hansen and Sargent, 2009). An important aspect of these models is that shocks to both the level and the volatility of consumption are persistent, as are shocks to the level and volatility of dividends. So far, this literature has been conducted within consumption-based models. Our work shows that under the loss aversion/narrow framing approach which has the advantage over consumptionbased models of matching the observed low correlation between consumption and stock returns, the presence of a low-frequency component in aggregate dividend volatility can also mental evidence. Moreover, Brunnermeier and Nagel s (2008) recent work has shown that time-varying risk aversion ts poorly with household-level data. 6

8 shed light on key observed nancial market phenomena. Since we attempt to provide a comprehensive study of the role of aggregate dividend volatility in asset prices, our work also touches on several other areas of the literature, including studies that relate dividend volatility to asset prices (Longsta and Piazzesi, 2004; Sheinkman and Xiong, 2003; Dumas, Kurshev and Uppal, 2009), studies that address the declining trend in equity premiums (Jagannathan, McGrattan and Scherbina, 2000; Pastor and Stambaugh, 2001; Kim, Morley and Nelson, 2004, 2005; Lettau, Ludvigson and Wachter, 2008), and studies that explore the GARCH and volatility feedback e ects in stock returns, to name a few, Engle (1982), Bollerslev (1986), French, Schwert and Stambaugh (1987), Campbell and Hentschel (1992) and Tauchen (2004). The rest of the paper is organized as follows. Section 2 summarizes the three key ingredients of our model. In particular, it provides empirical evidence that dividend volatility is persistent over time. Section 3 presents the model and characterizes the equilibrium asset prices. Section 4 calibrates the model, solves the price-dividend ratios, and analyzes model simulation results. It also provides a variety of sensitivity analyses. The nal section concludes the paper. 2 Key Ingredients of the Model In this section, we discuss the three key ingredients underlying our model: dividend volatility clustering, loss aversion and narrow framing. We rst provide extensive empirical evidence that dividend volatility is highly persistent, which provides a rationale for incorporating this 7

9 feature of dividend volatility into our theoretical model in Section 2.2. We then review two of the most important ideas emerging from the experimental literature on decision-making under risk: loss aversion and narrow framing. 2.1 Dividend Volatility Clustering Volatility clustering, which characterizes the persistence in volatility, has been documented as a standard feature of many nancial series. For instance, Bollerslev, Engle and Wooldridge (1988) show that conditional variance of market returns uctuates across time and is very persistent. For high-frequency return data, the ARCH literature nds a very high coe cient in the correlation of conditional standard deviations of returns. In this subsection, we document that the dividend growth rate also exihibits strong volatility clustering. Moreover, even though our data are at a low-frequency, the estimated coe cient is very similar to those found in high-frequency data. Following Campbell (2000) and Bansal, Khatchatrian and Yaron (2005), we back out the aggregate quarterly dividend time series from CRSP stock return data from 1926.Q Q3. 7 The detailed data construction is provided in Appendix A. This imputed dividend series has accounted for stock repurchases as an increasingly signi cant component of dividends since the 1980s. We rst ran two standard diagnostic tests, the Box-Pierce-Ljung test and the ARCH test, to determine whether the dividend growth rate can be predicted at the second moment. 7 In this section, we provide empirical evidence of dividend series at a quarterly frequency to match our benchmark model in section 4. In unreported results, we also examine dividend series at monthly and annual frequencies, which also exhibit strong volatility clustering. 8

10 The results are reported in Panel A of Table 1; means that the estimates are signi cantly di erent from zero at the 1% level. The statistics from both tests signi cantly reject the null hypothesis of no volatility clustering in the dividend growth rate, contradicting the standard i.i.d. assumptions in existing models. The results of the preliminary tests suggest that use of the exponential GARCH (EGARCH) model to identify the persistent component in dividend volatility is appropriate. We choose EGARCH instead of other types of GARCH models (such as (Fractionally) Integrated GARCH) because it matches best with our theoretical dividend volatility speci cation in section 3. Moreover, the EGARCH speci cation can capture potential asymmetric behaviors in volatility. Speci cally, we consider the following regression: g D;t+1 = g D;t + t Z t+1, log 2 t = + 1 log 2 t 1 + A1 [jz t j E jz t j] + L 1 Z t, Z t+1 i:i:d: N (0; 1), where g D;t+1 is the dividend growth rate, 2 t is the conditional variance of g D;t+1, and 0 ; 1 ; ; G 1 ; A 1 and L 1 are coe cients. In what follows, we report results based on this AR(1)-EGARCH(1,1) speci cation. We also applied AR(2)-EGARCH(1,1) and other speci cations, and the main results remained unchanged. In addition to the EGARCH speci - cation, we tried the speci cations described in Bansal, Khatchatrian and Yaron (2005) and obtained similar results not reported here. 9

11 Panel B of Table 1 reports the estimation result. The standard errors of the estimated parameters are reported in parentheses; and mean that the estimates are signi cantly di erent from zero at the 1% and 5% levels, respectively. The coe cient for dividend volatility is ^ 1 = 0:968, indicating that dividend volatility is highly persistent, which is consistent with the standard ndings in the ARCH literature. This estimated persistence parameter ^ 1 represents an important input parameter for our model. 8 Note that the coe cient that measures the persistence in the dividend growth rate per se is much smaller (^ 1 = 0:451). In the long-run risk literature (e.g., Bansal and Yaron, 2004; Bansal, Kiku and Yaron, 2007), it is crucial to have persistence in both the mean and the volatility of the dividend growth rate to explain the high equity premium; in other words, both ^ 1 and ^ 1 are assumed to be close to one. In our model, we require the persistent component only in the volatility of dividend growth rate and not in the level of dividend growth rate; that is, we only require that ^ 1 is su ciently high, which is supported by the data. To further con rm that the persistence in dividend volatility is indeed very high, we resorted to the augmented Dicky-Fuller unit root test by running the following regression: log (^ t ) = log (^ t 1 ) + 2 log (^ t 1 ) + e t, where ^ t is conditional dividend volatility obtained from the EGARCH estimation, and 0 ; 1 and 2 are coe cients. Panel C of Table 1 reports the test statistics together with the critical values at 1%, 5%, and 10% levels. We can hardly reject the null hypothesis 8 That is, in subsection 4.3, we use 0:968 for the value of in our model (see equation [4]). 10

12 of 1 = 1 at the 10% critical level, which implies that dividend volatility is indeed very persistent. For comparison, we also ran the unit root test in the dividend growth rate; the unreported result strongly rejects the unit root hypothesis at any critical level, which is not surprising given that ^ 1 is only 0:451 in Panel B of Table 1. Given the strong econometric evidence, we believe that dividend volatility clustering is an important feature of the actual dividend data. Our theoretical model incorporates this feature when we specify the dividend growth rate process. 2.2 Loss Aversion and Narrow Framing Loss aversion is a central feature of Kahneman and Tversky s (1979) prospect theory, a prominent descriptive theory of how people evaluate risk. 9 According to this theory, people derive utility not from absolute wealth levels but rather from gains and losses measured relative to a reference point. Loss aversion is the nding that people have a greater sensitivity to losses than to gains of the same magnitude; this is represented by a kink in the utility function. The most basic evidence for loss aversion is that people tend to reject the following 9 To improve upon Kahneman and Tversky (1979), Tversky and Kahneman (1992) propose a generalization of prospect theory, the cumulative prospect theory. Loss aversion remains a central feature of the cumulative prospect theory. In addition to loss aversion, this theory has two other main features: diminishing sensitivity and probability weighting. In the present paper, we study the implication of loss aversion alone. We would not expect our results to be very di erent even if it were possible to solve a dynamic asset pricing model that did incorporate all three features of prospect theory. The asset pricing implications of other features of prospect theory have been studied, for example, by Gomes (2005), Barberis and Huang (2008), Barberis and Xiong (2009), Hens and Vlcek (2009), and Li and Yang (2009). 11

13 gamble: (110; 50%; 100; 50%), which says that there is a 50% probability of winning 110 dollars and a 50% probability of losing 100 dollars. It is very di cult to reconcile this evidence with di erentiable utility functions because the very high local risk aversion required to reject this gamble will also lead people to reject large-scale gambles (Epstein and Zin, 1990; Rabin, 2000; Barberis, Huang and Thaler, 2006). Under traditional utility functions de ned over consumption or total wealth, investors evaluate a new gamble by rst merging it with the other risks they already face and then checking whether the combination is attractive. Experimental evidence, however, has shown that when o ered a new gamble, people often engage in narrow framing in the sense that they tend to evaluate the new gamble in isolation. In other words, they act as if they obtain direct utility from the outcome of this new gamble rather than indirect utility via its contribution to their total wealth as predicted by traditional utility functions. Several explanations have been proposed to explain why investors might frame stock market risk narrowly. One view is that narrow framing stems from non-consumption utility, such as regret. Regret is the pain that we feel when we realize that we could be better o today had we taken a di erent action in the past. Investing in the stock market exposes the investor to possible future regret: if the stock market performs badly, she might regret having allocated some fraction of her wealth to the stock market. Another interpretation of narrow framing is proposed by Kahneman (2003), who argues that narrow framing arises 12

14 due to accessibility: the return distribution of a new gamble itself is much more accessible than the distribution of the investor s overall wealth once the new gamble has been merged with other risks; as a result, it is easier for the investor to evaluate the stock market risk in isolation than it is for her to evaluate the wealth risk as a whole. 10 Although we describe loss aversion and narrow framing separately, recent studies have shown that these two attitudes may form a natural pair (e.g., Kahneman, 2003; Barberis, Huang and Thaler, 2006). Kahneman (2003) provides an explanation for this, arguing that since both prospect theory and narrow framing capture the way people act when they make decisions intuitively, it is natural for loss aversion and narrow framing to appear in combination. In our context, this combination manifests as follows: even though stock market risk is just one of the many risks investors face, they still obtain direct utility from stock market uctuations (narrow framing) and are more sensitive to gains than to losses (loss aversion). 3 The Model 3.1 Setup Consider an economy populated by a continuum of identical, in nitely-lived, investors who exhibit narrow framing and loss aversion. Two assets are available to trade: a risk-free asset in zero net supply paying a gross interest rate R f;t, and one unit of a risky asset paying a gross return R t+1 between time t and t People can frame the stock market risk at di erent horizons, see Thaler, Tversky, Kahneman, and Schwartz (1997). For more discussions on narrow framing, please refer to Barberis and Huang (2007, 2009). 13

15 The loss-averse investor chooses consumption C t and risky asset holdings S t to maximize the utility function E " 1 X t=0 t C1 t 1 + b 0 C t t+1 v (X t+1 ) # ; (1) subject to the standard budget constraint, and X t+1 = S t (R t+1 R f;t ) ; 8 >< X t+1 ; if X t+1 0; v (X t+1 ) = >: X t+1 ; if X t+1 < 0: : The rst term in the objective function is the standard utility over consumption, where 2 (0; 1) is the time discount factor and > 0 measures the curvature of the investor s utility over consumption. 11 For ease of expression, we call this term consumption utility. The second term deserves more attention than the rst because it captures the direct utility that the investor derives from uctuations in the value of her nancial wealth. For simplicity, we call this additional utility the prospect theory utility; this description captures the idea that the investor exhibits narrow framing and loss aversion. Ct is the aggregate per capita consumption at time t, which is exogenous to the investor. The exogenous scalar C t is introduced to ensure that consumption utility and prospect theory utility are of the same order as aggregate wealth increases over time. 12 Depending on the return of the risky asset, the investor s total portfolio excess return 11 For = 1, we replace C 1 t = (1 ) with log (C t ). 12 Another tractable preference speci cation that incorporates loss aversion and narrow framing but does not rely on a scaling to ensure stationarity can be found in Barberis and Huang (2007, 2009). 14

16 X t+1 can be either positive or negative, with a positive value indicating a nancial gain and a negative one indicating a nancial loss. The function v (X t+1 ) describes how she feels about her investment performance. Since she is loss-averse, the pain she receives from nancial losses outweighs the happiness she derives from nancial gains of the same magnitude. Therefore, v (X t+1 ) takes di erent functional forms with respect to the values of X t+1 : when X t+1 is positive, showing that she makes money, v (X t+1 ) is linear in X t+1 with slope one; by contrast, when X t+1 is negative, meaning that she loses money, v (X t+1 ) ampli es her utility loss by a magnitude of ; with being greater than one. Figure 1 plots the function v (X t+1 ). Parameter b 0 determines how much the second utility factors into the investor s total utility. If b 0 = 0, narrow framing and loss aversion do not play a role in the overall utility and the model is reduced to a traditional asset pricing setting with time-varying dividend volatilities. In this case, as we will show in subsection 4.2, higher dividend volatility leads to a higher dividend growth rate, resulting in a higher price-dividend ratio and a lower equity premium. However, as the value of b 0 increases, the e ect of prospect theory utility increases such that the investor su ers more disutility from her nancial loss and demands a higher risk premium in holding stocks. Given that we are considering a pure exchange economy, the technology is fully charac- 15

17 terized by the consumption and dividend growth processes, which are speci ed as follows: g C;t+1 = log C t+1 = C t = gc + C t+1 ; (2) g D;t+1 = log (D t+1 =D t ) = g D + t " t+1 ; (3) log ( t+1 ) log () = [log ( t ) log ()] + u u t+1 ; (4) where g C;t+1 is the growth rate of aggregate consumption C t, g C and C are the mean and standard deviation of the consumption growth rate, and D t is the dividend, the growth rate of which is denoted as g D;t+1 ; with mean g D and standard deviation t. Equation (3) is a standard assumption in asset pricing literature that is used to capture the idea that the mean level of dividend is di cult to predict. Similar to other models that adopt the loss aversion/narrow framing approach, we model consumption and dividend as two distinct processes: besides the dividend stream, the agent also receives a stream of non- nancial income, such as labor income, to nance her aggregate consumption. We draw special attention to equation (4), which characterizes the evolution of dividend volatility. At this point, we depart from existing models such as that of Barberis, Huang and Santos (2001). That is, we incorporate the realistic volatility-clustering feature of the dividend growth rate documented in Section 2.1. To ensure that t is positive, we model log ( t+1 ) instead of t as an AR(1) process. In this sense, the dividend volatility equation (4) is very similar to the EGARCH speci cation in subsection 2.1. u captures the magnitude of the innovation to the conditional volatility t : a larger u will increase divi- 16

18 dend volatility. A particularly interesting parameter is the coe cient, which controls the strength of dependence on past volatilities. A larger implies that the impact of a shock to dividend volatility is more persistent. As has been shown in subsection 2.1, this persistence parameter is very high in actual dividend data. The innovations to the consumption growth t, the dividend growth " t, and the dividend volatility u t are jointly normally distributed as t " t u t i.i.d. N C 6B A 4@ ; C B 1! 0! (5) C7 A5 Parameter! represents the correlation between consumption shocks and dividend shocks. Note that when there is persistence in dividend volatility, the unconditional correlation between g C;t+1 and g D;t+1 is!e 0:52 u=(1 2 ) <!. For simplicity, we assume that consumption growth shocks are independent of dividend volatility shocks, i.e., Cov ( t ; u t ) = 0. However, we do allow for interaction between shocks to the dividend growth rate " t and shocks to the dividend volatility u t ; the interaction of these two shocks is denoted by. As will be shown later, also plays a role in generating certain model results, including the excess volatility and negative autocorrelation of stock returns. Under equations (2)-(5), the consumption growth rate is, in our setting, an i.i.d. process, while the dividend growth rate is a white noise process featuring volatility clustering. 17

19 3.2 Equilibrium Prices This subsection derives the equilibrium asset prices. We rst construct a one-factor Markov equilibrium in which the risk-free rate is a constant and the state variable t (dividend volatility) determines the distribution of future stock returns. Assume that the price-dividend ratio is a function of t : f t P t =D t = f( t ). We will verify that there is indeed an equilibrium satisfying this assumption. Given the one-factor assumption, the stock returns R t+1 can be determined as R t+1 = P t+1 + D t+1 P t = 1 + P t+1=d t+1 P t =D t D t+1 D t = 1 + f ( t+1) e g D+ t" t+1. (6) f ( t ) Intuitively, the change in stock returns can be attributed to either the news about dividend growth " t+1 ; the nancial market uncertainty t ; or changes in the price-dividend ratio f: Given that the dividend process is exogenous, the key to solving R t+1 is to solve the pricedividend ratio f: In equilibrium, the dynamics of the economy are fully captured by the Euler equations: 13 1 = R f;t E t h Ct+1 = C t i (7) 13 The Euler equations are both necessary and su cient to characterize the equilibrium. See Barberis, Huang and Santos (2001) for a proof. 18

20 1 = E t h R t+1 Ct+1 = C t i + b 0 E t [^v (R t+1 )] (8) where 8 >< R t+1 R f;t ; if R t+1 R f;t ; ^v (R t+1 ) = >: (R t+1 R f;t ) ; if R t+1 < R f;t : Equation (7) and the i.i.d. assumption regarding the consumption growth together imply a constant risk-free rate, R f;t = 1 e g C 2 2 C =2 : (9) After substituting in the respective consumption and dividend processes, equation (8) becomes 1 + f (t+1 ) 1 = E t e g D+ t" t+1 e (g C+ C t+1 ) 1 + f (t+1 ) + b 0 E t ^v e g D+ t" t+1, 8 t. f ( t ) f ( t ) In equilibrium, the function f must evolve according to equation (10), which also veri es the conjectured one-factor Markov equilibrium price function. (10) Methodology of Numerical Computation We solve f numerically on a grid search of the state variable t. We begin by guessing a solution to (10), say, f (0). According to equation (4), the distribution of t+1 is completely determined by t and u t+1. We then 19

21 obtain a new candidate solution f (1) by the following recursion: 1 = e g D g C C(1! 2 ) 1 + f (i) ( t+1 ) Et e (t f (i+1) ( t ) 1 + f +b 0 E t ^v (i) ( t+1 ) e g D+ t" t+1, 8 f (i+1) t. ( t )! C)" t+1 We continue this process until f (i)! f. 4 Model Results 4.1 Benchmark Calibration of Parameter Values There has been no consensus regarding the decision interval of loss-averse investors. Kahneman (2003) provides evidence that the frequency at which information is most salient or available is likely to be the frequency the investor uses to evaluate performance. Since much nancial information is provided at an annual frequency, previous studies, such as that of Barberis, Huang and Santos (2001), assume an annual evaluation of performance by investors. By contrast, in this paper, we interpret one period in our model as one quarter; thus, the investor re-evaluates her investment performance each quarter. We choose to calibrate the model at a quarterly frequency because much information is also provided on a quarterly basis. In particular, rms routinely report their earnings/dividends on a quarterly basis. Since the dividend is the main focus of the present paper, quarterly calibration 20

22 might be more relevant. 14 For a robustness check, we also calibrate the model at an annual frequency and report the results in subsection 4.4. Table 2 summarizes our choice of parameter values for the model calibrated at a quarterly frequency. We choose similar values to those of Barberis, Huang and Santos (2001) for the consumption growth parameters and the preference parameters. More speci cally, for g C and C, the mean and standard deviation of log consumption growth, we follow Cecchetti, Lam and Mark (1990) and set g C = 0:46% and C = 1:90%; this corresponds to an annual growth rate of 1:84% with a volatility of 3:79%. The curvature of utility over consumption and the time discount factor are set as 1:0 and 0:995, respectively, bringing the net annual risk-free rate close to 3:86 percent by equation (9) and the values of g C and C. The loss aversion parameter is chosen to be 2:25, because many independent experimental studies have estimated it as being around this level. Because b 0 does not have an empirical counterpart, following Barberis, Huang and Santos (2001), we present results for a range of values of b 0. Using NYSE/AMEX data and Fama risk-free rate data from 1926.Q3 to 2006.Q4 from CRSP, we calibrate the unconditional mean of the quarterly dividend growth rate as its empirical mean, g D = 0:39%. By matching the rst moment of Equation (3), i.e., E [log (jg D;t+1 g D j)] = log () + E [log (j" t+1 j)], we calibrate log () as 3:91. The parameter, which governs the persistence in dividend volatility, takes the value of 0:968, the estimated value of ^ 1 in the 14 In reality, since each rm chooses to issue its dividend/earning report on a di erent date, we could have modeled the aggregate dividend process as a continuous process with time-varying arrival rates. However, we believe our speci cation is a good approximation. First, as shown in Lamont and Frazzini (2007), most rms concentrate their reports in the rst month of the quarter. Second, the decision interval of agents can be di erent from the interval at which information arrives. Thus, investors might aggregate all the information within the quarter to evaluate their investment performance. 21

23 EGARCH model in subsection 2.1. Another important parameter is u, which measures the magnitude of dividend volatility. We calibrate this parameter as 0:25, such that the model-implied annual dividend growth rate has a volatility equal to that of its empirical counterpart. Compared to Bansal and Yaron (2004), the value of u appears large. However, this is an artifact of our speci cation of the volatility process in equation (4), where the logarithm of dividend volatility, rather than its square as in Bansal and Yaron (2004), follows an AR(1) process. Indeed, given 1 and taking a rst-order approximation of (4), we have 2 t+1 2 t + w u t+1, where w = 2E ( 2 t ) u = 1:5 10 3, which is closer to the value found in Bansal and Yaron s (2004) speci cation. Two more model parameters remain to be calibrated:, which captures the interaction between innovations in dividend growth rate and in dividend volatility, and!, the correlation between consumption and dividend. By equations (3) and (4), we calibrate at 0:38; the calibration details are given in Appendix B. Following Campbell (2000), we set! = 0:16, which implies an unconditional correlation of 0:1 between consumption and dividend growth processes. 4.2 Price-dividend Ratio Function f Figure 2 plots the price-dividend ratio f as a function of log ( t ) for b 0 = 0, b 0 = 0:7, and b 0 = 100. The choices of b 0 = 0 and b 0 = 100 in Panels (a) and (b) correspond to two extreme cases: when b 0 = 0, the investor only cares about consumption utility and does not 22

24 exhibit loss aversion and narrow framing in her preference; when b 0 = 100, her preference is mostly determined by loss aversion and narrow framing. The choice of b 0 = 0:7 in Panel (c) corresponds to the situation in which both consumption utility and prospect theory utility are important. We also try a variety of other intermediate values for b 0, for example, b 0 = 2, b 0 = 6, b 0 = 20, etc. The essential pattern, however, is fully depicted by Figure 2. When b 0 = 0, the investor s preference corresponds to the standard consumption utility. In this case, the price-dividend ratio f( t ) increases with the dividend volatility t. Now that the model is reduced to a standard consumption-based model, the standard consumption growth rate becomes the stochastic discount factor (SDF). Because the dividend process is only weakly correlated with SDF, the investor does not care much about the stock risk and, thus, behaves as if she is risk-neutral toward the stock. A higher dividend volatility implies higher expected cash ows from holding stocks and makes the stock more attractive to the risk-neutral alike investor. 15 Therefore, when b 0 = 0, stock prices rise with dividend volatility, resulting in a positive relationship between f( t ) and t. When b 0 = 100, the investor derives utility primarily from uctuations in the value of her stock holdings (loss aversion and narrow framing) and derives very little consumption utility. This is similar to the situation studied in Benartzi and Thaler (1995). In this scenario, the overall relationship between t and the price-dividend ratio f ( t ) is negative. The intuition is as follows. A higher value of t corresponds to a more volatile dividend process and, hence, a more volatile return process, which implies that the investor is more likely to su er 15 In mathematical terms, by equation (3), the conditional expected dividend at date t + 1 is E t (D t+1 ) = D t E t (e g D;t+1 ) = D t e g D+ 2 t =2, which is increasing in the state variable t. 23

25 nancial losses. This causes her great pain and makes stocks less desirable. As a result, she requires more compensation when faced with more volatile dividend processes, which, in turn, results in lower stock prices or higher equity premiums. In fact, this negative slope of f function is consistent with Bansal and Yaron (2004) and Bansal, Khatchatrian and Yaron (2005), who document that asset prices drop as economic uncertainty rises, although their measure of economic uncertainty is conditional consumption volatility rather than conditional dividend volatility. It is rather di cult to justify this negative relationship within the standard power utility framework, where, as we have shown above, a higher dividend volatility is associated with higher expected dividend growth and, thus, price-dividend ratios always vary positively with dividend volatility. However, it can be easily understood with the introduction of loss aversion/narrow framing preferences. When b 0 = 0:7, both consumption utility and prospect theory utility will matter. Now the investor cares not only about consumption, the standard expected power utility term in (1), but also about uctuations in the value of her investments, the additional prospect theory utility term in (1). As the standard consumption utility contributes to a positive relationship between t and the price-dividend ratio f ( t ), and because prospect theory utility contributes to a negative relationship between t and the price-dividend ratio f ( t ), the overall shape of the function f is U-shaped. For low values of t, the impact of loss aversion and narrow framing is dominant; hence, the function f is downward-sloping. As t becomes larger, the impact of consumption utility catches up, and the function f eventually becomes upward-sloping. 24

26 Examining Panels (a) through (c) of Figure 2, we nd that as b 0 increases, the function f will move downward. This is because the larger the value of b 0, the more the investor cares about her wealth uctuations, which, in turn, means that she requires a higher risk premium to hold stocks. What does f look like in the data? We performed an AR(1)-EGARCH(1,1) estimation on the quarterly dividend growth for 1926.Q Q4; the resulting price-dividend ratios are plotted against the estimated conditional dividend volatilities ^ t in Figure 3. The data display a U-shaped pattern, as predicted by Panel (c) of Figure 2. On the one hand, 80 percent of the observed data display a negative relationship between the empirical price-dividend ratios and the estimated dividend volatility. On the other hand, for certain extremely high realizations of ^ t (for instance, when the logarithm of dividend volatility is larger than 0:05) price-dividend ratios begin to rise with dividend volatility. 4.3 Simulation Results In this subsection, we generate arti cial data under the parameter con guration in Table 2 and show that the model-simulated data replicate the interesting patterns found in the stock market. In our benchmark results here, we report the results for three values of b 0 : b 0 = 0:7, b 0 = 2, b 0 = 6. In subsection 4.4, we will investigate how robust our results are with respect to other parameter con gurations. To facilitate a comparison with historical data, we simulate the model at a quarterly frequency and time-aggregate results to obtain annual data. We do 10,000 simulations each 25

27 with 320 quarterly observations. We then calculate the statistics of interest and report their sample moments. Given that the simulation number is large enough, the sample moments should serve as good approximations to population moments Stock Returns and Stock Volatility Two important features of the aggregate market include the historically signi cantly higher average return that stocks have earned compared to T-bills (Mehra and Prescott, 1985) and the greater volatility of stock returns compared to the underlying dividends (Shiller, 1981). We examine whether our model is able to match these two dimensions of the data. Table 3 reports a variety of statistics calculated from model-simulated data and the corresponding statistics from historical data. It is noteworthy that the model can match the mean and standard deviations of excess stock returns fairly well. When b 0 = 6 ; the model generates a sizable premium of 5:59% per annum, which is slightly lower than the empirical value 5:90%; the model also generates a standard deviation of 17:49%, which is close to the corresponding value of 19:17% in the data. We also report the mean and standard deviation of the simulated annual price-dividend ratios, E (P a t =D a t ) and (P a t =D a t ), where the superscript a indicates annualized variables. The empirical value (P a t =D a t ) = 12:43% is relatively high compared to that found in other papers (Campbell and Cochrane, 1999; Barberis, Huang and Santos, 2001; Bansal and Yaron, 2004). This is due to the relatively high price-dividend ratios from 1996 to 2006, a period that includes the high-tech bubble. 26

28 We are able to match the volatility of stock returns even though the volatility of pricedividend ratios is lower than its empirical counterparts, which is a common problem with one-factor models. The reason behind the achievement of excess volatility in stock returns is the positive relationship between price-dividend ratios and dividend innovations. To see this more clearly, consider the following approximate relationship (Campbell, Lo and MacKinlay, 1997): r t+1 A + log [f ( t+1 ) =f ( t )] + t " t+1 ; where A is a constant. The excess volatility of market return relative to that of the dividend growth (or the fundamental ), V ar (r t+1 ) V ar ( t " t+1 ), comes from two sources: the volatility of price-dividend ratios, V ar (log (f t+1 =f t )), and the covariance between the pricedividend ratios and the news regarding the dividends, Cov (log (f t+1 =f t ) ; t " t+1 ). In actual data, because < 0 in (5), good dividend news (positive " t+1 ) tends to be associated with negative dividend volatility shock (negative u t+1 ). This implies that price-dividend ratios will increase next period when b 0 takes the values in Table 3 (see Panel (c) of Figure 2). Therefore, the covariance term Cov (log (f t+1 =f t ) ; t " t+1 ) is positive. We notice that as b 0 grows, both the means and standard deviations of model-simulated excess returns increase. When b 0 increases, loss aversion becomes a more important feature of the investor s preference. Thus, she becomes more fearful of the risky asset and requires a higher premium to hold it. A higher b 0 also implies a larger covariance term Cov (log (f t+1 =f t ) ; t " t+1 ), resulting in a larger excess volatility of stock returns. The model is also able to generate the low correlation between stock returns and con- 27

29 sumption growth, Corr r a t+1; g a C;t+1 = 0:1. This occurs because the variation in stock returns is completely driven by the innovations in the dividend process, which are only weakly correlated with the consumption process Autocorrelation of Returns and Price-Dividend Ratios Table 4 presents autocorrelation in returns and price-dividend ratios. Our model predicts a negative autocorrelation in stock returns, as documented by Poterba and Summers (1988) and Fama and French (1988a). This negative correlation comes from the negative correlation between shocks to the dividend growth rate " t and shocks to the dividend volatility u t. Suppose the stock return R t is high in period t. This corresponds to good dividend news. That is, " t is high, which, in turn, implies a low realized dividend volatility, t, as " t and u t are negatively correlated. Recall that, when b 0 takes the values in Table 4, the price-dividend ratios f ( t ) decrease with the dividend volatility t (see Panel (c) of Figure 2). Therefore, a high R t leads to a high f ( t ) through the low t, and the high f ( t ), in turn, leads to a low expected future stock return, E t (R t+1 ) = 1 f( t) E t [(f ( t+1 ) + 1) e g D;t+1 ]. Moreover, our model closely matches the highly positively correlated price-dividend ratios in the data. This is because the price-dividend ratios depend solely on the persistent AR(1) dividend volatility process. 28

30 4.3.3 Return Predictability To analyze the predictability pattern of returns, we run the following regression on both simulated and historical data: r a t+1 + r a t+2 + ::: + r a t+j = j + j (D a t =P a t ) + j;t, where r a t+j refers to the annual cumulative log returns from year t + j 1 to t + j. Table 5 presents the regression result for di erent values of b 0. This estimation result from modelsimulated data resembles the classic pattern documented by Campbell and Shiller (1988) and Fama and French (1988b). The coe cients are signi cant and negative, indicating that high prices tend to predict low expected returns. Moreover, the forecasting power increases with forecasting horizons, as re ected by the increasing coe cients and R 20 s. The pattern of return predictability generated by our model can be understood through the volatility test in Cochrane (1992). Starting from the accounting identity 1 = R 1 t+1r t+1 with R t+1 = (P t+1 +D t+1 )=P t, the log-linearization around the average price-dividend ratios, P=D, implies that, in the absence of rational asset price bubbles, V ar(p t d t ) 1X h j Cov(p t d t ; g D;t+j ) j=1 1X h j Cov(p t d t ; r t+j ), j=1 where the lower case indicates log values and h = P=D=(1 + P=D). This suggests that the variation in the price-dividend ratio will forecast either the change in expected dividend growth rate, that in the discount rate, or both. 29

31 In our model, even though dividend volatility is time-varying, the dividend growth rate per se is still a white noise process, meaning that Cov(p t d t ; g D;t+j ) = 0. Given that the risk-free rate is maintained as a constant, the only thing remaining for the price-dividend ratio to predict is the excess return. A high price-dividend ratio is associated with a decline in dividend volatility, and, thus, the required expected return will be lower. Therefore, our model implies an extreme version of the volatility test results. As shown in Cochrane (2005), both the estimated coe cients and R 20 s increase with the persistence of the price-dividend ratio, which, in our model, depends on dividend volatility. Therefore, it is the persistence in dividend volatility that drives the predictability of returns Time-varying Sharpe Ratios Empirical evidence suggests that estimates of both conditional means and conditional standard deviations of returns change through time but do not move one-for-one. Hence, the Sharpe ratios are time-varying. Figure 4 presents the conditional means and conditional standard deviations as functions of the state variable log( t ) for the case of b 0 = 6. Those conditional moments are computed numerically from the price-dividend ratio function and equations (3)-(6) as follows: E t (R t+1 ) = e g D+ 1 2(1 2 ) 2 t 1 + f (t+1 ) Et f ( t ) v t (R t+1 ) = u t e tu t+1 ; 2 e 2g D+2(1 2 ) 2 t 1+f(t+1 ) Et f( t) e 2 tu t+1 e 2g D+(1 2 ) 2 t h E t 1+f(t+1 ) f( t) e tu t+1 i 2 : 30