نوسانات بازار سهام نامتقارن و رفتار دوره ای از بازده مورد انتظار

Recent explanations of aggregate stock market fluctuations suggest that countercyclical stock market volatility is consistent with rational asset evaluations. In this paper, I develop a framework to study the causes of countercyclical stock market volatility. I find that countercyclical risk premia do not imply countercyclical return volatility. Instead, countercyclical stock volatility occurs if risk premia increase more in bad times than they decrease in good times, thereby inducing price–dividend ratios to fluctuate more in bad times than in good. The business cycle asymmetry in the investors’ attitude toward discounting future cash flows plays a novel and critical role in many rational explanations of asset price fluctuations.

Why does stock market volatility vary over time? Economists have been intrigued by this issue for decades. For example, Schwert (1989b) finds that the volatility of no single macroeconomic variable could help explain low frequency movements of aggregate stock market volatility. Yet stock market volatility is related to the business cycle. A number of empirical studies confirm further findings from Schwert, 1989a and Schwert, 1989b that the volatility of stock returns is higher in bad times than in good times (see, e.g., Brandt and Kang, 2004, and the additional evidence provided here). This paper addresses an important but still unanswered question: Why is stock market volatility asymmetric over the business cycle? My central result is that, in economies with rational expectations, return volatility is countercyclical because risk premia (i.e., the compensation investors require to invest in the stock market) change asymmetrically in response to variations in economic conditions. That risk premia are countercyclical has been a widely known empirical fact since the seminal contributions of Fama and French (1989) and Ferson and Harvey (1991). However, the main message of this paper is not a simple statement that risk premia must be countercyclical to generate countercyclical return volatility. Instead, the crucial point is that, to induce countercyclical return volatility, risk premia must increase more in bad times than they decrease in good times, a new hypothesis that I support with substantial empirical evidence. So why do asymmetric risk premia fluctuations translate into countercyclical return volatility? Consider Fig. 1, in which I assume that the investors’ risk-adjusted discount rates are inversely and asymmetrically related to some variable y that tracks the state of the economy. This asymmetry implies that in good times investors do not significantly alter the discount rates used to evaluate future dividends. Consequently, price–dividend ratios do not fluctuate widely in good times. In bad times, however, the investors’ discount rates are extremely sensitive to changes in economic conditions. Therefore, variations in the price–dividend ratios become increasingly volatile as economic conditions deteriorate. The main result of this paper is that these asymmetric movements of the price–dividend ratios occur when the asymmetry in discounting is sufficiently pronounced. I calculate a theoretical lower bound for the asymmetric movements of the risk premia that triggers the previous asymmetric variations in the price–dividend ratios. This bound can be tight. For example, economies exist in which risk premia are countercyclical but do not satisfy this bound and, consequently, induce price–dividend ratios to fluctuate more in good times than in bad. Countercyclical return volatility. If price–dividend ratios are concave in some state variable y tracking the state of the economy, then return volatility increases on the downside and is consequently countercyclical. According to the theory in this article, price–dividend ratios are concave in y if the risk-adjusted discount rates are decreasing and sufficiently convex in y. Figure options Naturally, countercyclical return volatility could also arise because the volatility of the state variables in the economy is inherently countercyclical. Alternatively, the conditions developed here highlight the mechanism through which countercyclical return volatility is endogenously induced by rational fluctuations of the price–dividend ratio. Moreover, empirical evidence suggests that price–dividend ratios exhibit the pattern predicted in this paper. I find that, over the last 50 years, price–dividend ratios movements in the US have been asymmetric over the business cycle: Downward changes occurring in recessions have been far more severe than upward changes during expansions. In the economy I study, dividend growth is independent and identically distributed, while interest rates and risk premia are driven by a state variable that is interpreted as an index of the state of the economy. This economy is rich enough to include many model examples in the literature. The distinctive feature of this article is the way I deal with interest rates and risk premia. The standard approach is to link interest rates and risk premia to markets, preferences, and technology (e.g., Basak and Cuoco, 1998, Campbell and Cochrane, 1999 and Jermann, 2005) or in general to make use of higher level assumptions about the exact relations among interest rates, risk premia, and the primitives of the economy (e.g., Brennan, Wang, and Xia, 2004; Lettau and Wachter, 2007). In this paper, I take an opposite approach. Instead of making assumptions on interest rates and risk premia, I look for pricing kernels that make return volatility countercyclical. It is this search process that leads to the predictions summarized in Fig. 1. One additional contribution of the paper is to use these new predictions to understand when, why, and how models with time-varying discount rates could predict countercyclical volatility. For example, in a seminal contribution Campbell and Cochrane (1999) find that models with external habit formation might lead to countercyclical volatility. This paper explains the rationale behind this important result. At the same time, the predictions developed here go well beyond the case of habit formation. Countercyclical stock volatility is an empirical observation related to the so-called feedback effect; i.e., the effect by which asset returns and return volatility are negatively correlated. Indeed, this paper shows that a pronounced asymmetric behavior of the risk premia leads return volatility to be higher in bad times (when ex post returns are low) than in good (when ex post returns are high). Moreover, the asymmetric behavior of the risk premia could help explain why return volatility increases after prices fall. According to the explanations summarized in Fig. 1, return volatility increases after a price drop, i.e., when the price–dividend ratios enter the volatile region in Fig. 1. Campbell and Hentschel (1992) develop the first partial equilibrium explanation for the feedback effect. But, their explanation relies on a different channel. In the Campbell and Hentschel economy, the negative correlation between return volatility and returns arises through the combination of two inextricable effects: first, risk premia rise (and hence prices fall) with the volatility of dividend news; second, return volatility increases with the volatility of dividend news. Thus, in the Campbell and Hentschel economy the feedback effect arises because there is fluctuating economic uncertainty (i.e., dividend volatility is random) and investors fear this uncertainty. Wu (2001), Bansal and Yaron (2004), and Tauchen (2005) reconsider this channel of fluctuating economic uncertainty. Bansal and Yaron as well as Tauchen show that, in general equilibrium, investors with a preference for early resolution of uncertainty require compensation for economic uncertainty, thereby inducing negative co-movements between ex post returns and return volatility. This explanation of the feedback effect is not inconsistent with my explanation based on an asymmetric behavior of the risk premia. In fact, the last contribution of this paper is an extension of my previous analysis to economies in which the fundamentals are surrounded by fluctuating uncertainty. I consider two sources of volatility for the fundamentals of the economy. One is related to uncertain consumption growth volatility while the other, suggested by Tauchen (2005), relates to higher order uncertainty about consumption growth (the volatility of volatility). I show when and how the risk premia for these sources of uncertainty make prices fall after a rise in economic uncertainty. I provide a new role for the price–dividend ratio. For example, the relation between prices and the volatility of volatility is not uniquely tied down by the level of the volatility risk premia. It also depends on how asymmetrically the price–dividend ratio reacts to changes in consumption growth volatility. Moreover, I show that if investors have a preference for early resolution of uncertainty, an increase in the economic uncertainty can lower the risk-free rate, thereby producing a positive relation between asset prices and economic uncertainty. In particular, I show that the feedback effect arises when the volatility of volatility is not too responsive to changes in volatility, thereby dampening the effects associated with the preference for early resolution of uncertainty. I use these novel insights to shed new light on previous models of fluctuating economic uncertainty. The main scope of this paper is to isolate the business cycle determinants of return volatility. Its focus is on channels of asymmetric volatility that are markedly distinct from the leverage effects (the effects by which an increase in the debt-to-equity ratio boosts firms’ volatility). Instead, the general equilibrium analysis of leverage effects is in Aydemir, Gallmeyer, and Hollifield (2005), who conclude that these effects have marginal quantitative implications at the market level. The paper is organized as follows. In Section 2, I develop the core analysis. Section 3 hinges upon this analysis and provides examples of economies with countercyclical stock volatility. It also contains a calibration experiment to illustrate the key quantitative implications of the paper. Section 4 develops extensions and identifies conditions under which fluctuating economic uncertainty induces asset returns and volatility to co-move negatively. Section 5 concludes. The appendix contains technical details and proofs.