Fundamentals and Stock Returns in Japan

Working Paper Series _______________________________________________________________________________________________________________________ National Centre of Competence in Research Financial Valuation and Risk Management Working Paper No. 488 Understanding Stock Return Volatility Emilio Osambela First version: August 2008 Current version: August 2008 This research has been carried out within the NCCR FINRISK project on “Equilibrium Asset Pricing” ___________________________________________________________________________________________________________ Understanding Stock Return Volatility EMILIO OSAMBELAy August 5, 2008 Abstract This article studies the e⁄ect of limited commitment on stock return volatility in a dynamic general equilibrium economy populated by investors with heterogeneous beliefs. Due to het-erogeneity of beliefs investors disagree about the fundamentals, introducing an additional risk factor denoted sentiment risk. Limited commitment introduces an endogenous solvency con-straint which scales up sentiment risk every time it binds, which is labeled solvency risk. The equilibrium market price of risk which drives the short-run stock return volatility has three components: endowment risk, sentiment risk, and solvency risk. The three components are persistent, in line with volatility clustering or GARCH e⁄ects. The solvency risk component in the market price of risk is novel, and it is the main contribution of the paper. It is related to the optimal exercise boundary of an American-style contingent claim, and exhibits a markedly dif-ferent pattern with transient persistence according to the binding of solvency constraints. This is consistent with multifactor volatility models. Due to solvency risk, the correlation between stock return volatility and stock expected returns depends on the direction of disagreement of the population facing limited commitment, which may be relatively optimistic (positive correla-tion) or relatively pessimistic (negative correlation). This may explain the con‡icting empirical evidence in the correlation between stock return volatility and stock expected returns. A consump-tion CAPM with endowment, sentiment and solvency risk factors is obtained, and the model predicts that the di⁄erent factors which drive volatility should be priced in the cross section of stock returns, in line with recent empirical evidence. I thank Bernard Dumas for very helpful discussions and suggestions, as well as Jerome Detemple, Durrell Du¢ e and Julien Hugonnier for valuable comments. ySwiss Finance Institute and University of Lausanne (Institute of Banking and Finance). E-mail: emilio.osambela@s…-phd.ch 1 1 Introduction Over the past few decades, research in …nancial economics has taken a high e⁄ort to increase the understanding of the volatility patterns of stock market returns. Indeed, good knowledge of return volatility is crucial for portfolio choice, risk management and derivatives asset pricing. Perhaps the most robust empirical regularity of stock return volatility is volatility clustering. As …rst noted by Mandelbrot (1963) when referring to stock market returns, "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes". This persistence in volatility lead to the development of GARCH models by Engle (1982) and Bollerslev (1986) which have been very successful in modeling stock return volatility. A second feature of stock market return volatility which has received considerable attention is the fact that stock return volatility has a transitory (rapid decay) and a more permanent (slow decay) component. It was noted that real data on stock return has a longer memory in volatility than the one GARCH models suggest. This observation lead to the development of the component GARCH (CGARCH) models by Ding and Granger (1996) and Engle and Lee (1999). Following these papers, several studies have found that multi factor volatility models outperform single fac-tor speci…cations in explaining the behavior of stock market return volatility.1 Importantly, two component volatility models have also shown superior performance in the option pricing literature, as shown by Xu and Taylor (1994), Bates (2000) and Christophersen, Jacobs and Wang (2006). A third empirical issue which has been raised is the correlation between stock return volatility and stock expected returns. Merton (1980) postulated that expected stock market return should be positively related with the variance of market return (and proportional to it). However, the empirical evidence is not conclusive. French, Schwert and Stambaugh (1987) and Campbell and Hentschel (1992) …nd this correlation to be positive, while Turner, Starz and Nelson (1989), Glosten, Jangannathan and Runkle (1993) and Nelson (1991) …nd this correlation to be negative. Often the coe¢ cient linking stock return volatility and stock expected returns is statistically insigni…cant. The fourth empirical …nding is the fact that the di⁄erent volatility factors are due to systematic risk, and therefore are priced in the cross section of stock returns. In a recent contribution, Adrian and Rosenberg (2008) show that volatility factor models compare favorably to benchmark models in explaining the cross section of stock returns. While the statistical knowledge of stock return volatility is impressive, several questions remain regarding their economic explanation. For example, why does stock return volatility cluster? why is it that stock return volatility is composed of several factors? what do these factors represent? why do they matter in the cross section of stock returns? what are the reasons for the con‡icting results regarding the correlation between stock return volatility and stock expected returns? These questions have both a theoretical interest and far-reaching implications for portfolio choice, risk management and derivatives asset pricing. This article investigates these questions by introducing limited commitment in a general equilibrium model in which investors have heterogeneous beliefs about the fundamentals. I demonstrate that the interaction of heterogeneous beliefs with limited commitment introduces a new risk factor which I denote solvency risk. This new risk factor is able to reproduce many of the stylized …ndings about stock return volatility and provides with potential answers to the questions raised above. 1Recent contributions which …nd that multifactor volatility speci…cations outperform single factor models for stock return volatility include Engle and Rosenberg (2000), Alizadeh, Brandt and Diebold (2002), Bollerslev and Zhou (2002), Chacko and Viceira (2003) and Chernov et. al. (2003) 2 Two populations of investors receive speci…c endowments which are commonly observable, but have incomplete (yet symmetric) information on their dynamics. In the continuous-time setting presented, both populations of investors deduce the volatility of the endowments, but must estimate their drifts via its conditional expectation. Due to heterogeneity of beliefs, each population of investors perform di⁄erent inferences about the unobserved drifts of speci…c endowments. This leads to an additional risk factor denoted sentiment risk. In every period, each population of investors receives its speci…c endowment and risk sharing is implemented through trade of contracts which specify future transfers of endowments between the two populations of investors.2 These contracts represent …nancial securities which pay the speci…c endowments as dividends. In the standard case where all investors can commit not to default on any prescribed endowment transfer, the optimal contracts achieve an e¢ cient risk-sharing allocation. If, however, one population of investors faces limited commitment in the sense that it cannot commit not to default, then e¢ cient risk sharing may be impeded as the optimal contract is constrained by the possibility of ex-post default. In this context, contracts should be self-enforcing, such that for any period and state a solvency constraint which prevents default is imposed. In general, solvency constraints limit endowment transfers and therefore reduce the scope for risk sharing. The possibility of solvency constraints binding leads to an additional risk factor which scales up sentiment risk, and is denoted solvency risk. The model yields the following results. First, the model is consistent with volatility clustering or GARCH e⁄ects. The market price of risk, which is the instantaneous component of stock return volatility, has three components: endowment risk, sentiment risk and solvency risk. These three components are persistent, hence the model reproduces volatility clustering or GARCH e⁄ects. Endowment risk is persistent because the "two-trees" feature of the model imply that endowment risk is proportional to the shares of aggregate endowment, which ‡uctuate randomly between zero and one. Sentiment risk is persistent because it is proportional to the optimal share of aggregate consumption which ‡uctuate randomly between zero and one. Finally, solvency risk is persistent because the binding of solvency constraints is also persistent: periods of binding solvency constraints tend to be followed by periods of binding of solvency constraints, and periods of non-binding solvency constraints tend to be followed by periods of non-binding of solvency constraints. Second, the model is consistent with multifactor volatility models or CGARCH e⁄ects. Both endowment risk and sentiment risk are associated with instantaneous shocks associated with the idiosyncratic risk embedded in the Brownian motions present in the investors endowments. In contrast, solvency risk is associated with the binding of solvency constraints, and therefore it occurs at a lower frequency. Because the shadow price of solvency constraints alternates behavior between endogenous regimes of binding constraints and endogenous regimes of non-binding constraints, solvency risk clusters with a lower decay rate and exhibits a transient persistence. To my knowledge, this is the …rst paper which …nds a theoretical foundation for the di⁄erent components of stock return volatility, and identi…es potential sources of the di⁄erent types of shocks occurring at di⁄erent frequencies. Third, the sign of the correlation between stock return volatility and stock expected return depends on the direction of disagreement of the population facing limited commitment. In peri-ods when solvency constraints bind the constrained investors experience a permanent increase in 2In the current setup, risk sharing refers not only to speci…c-endowment risk sharing, but also to sentiment risk sharing. 3 consumption, which is required in order to preclude default and switching to autarky. In order to restore the equilibrium in the …nancial markets, the cumulative interest rate must decrease. Due to no-arbitrage, the stock return must decrease by the same amount, implying that the binding of solvency constraints generates decreases in stock returns. In addition, solvency risk generates a positive premium in the market price of risk if the population of investors facing limited commit-ment is relatively optimistic, so in this case the relationship between the instantaneous price of risk and stock return is negative. Conversely, solvency risk generates a negative premium in the market price of risk if the population of investors facing limited commitment is relatively pessimistic, so in this case the relationship between the instantaneous price of risk and stock return is positive. Fourth, the model provides a rational to the empirical …nding that volatility factors are priced in the cross section of stock returns. In the current model, stock return volatility factors are driven by systematic risk factors: sentiment and solvency risks. These factors, which drive volatility factors, are shown to be priced across di⁄erent stocks in an equilibrium consumption CAPM. On the technical side, I use the martingale approach to solve for the equilibrium for a general shadow price of solvency constraints, and then solve for the equilibrium shadow price of solvency constraints which is consistent with optimal consumption and market clearing. To my knowledge, this is the …rst paper which solves a model with solvency constraints in a continuous time setting. The next section of the paper reviews additional literature that is related to this work. Section 3 presents the main setup of the model, including heterogeneity of beliefs and limited commitment. Section 4 determines the equilibrium in the economy. In section 5 I present the method used in order to solve for the equilibrium shadow price of solvency constraints. Section 6 implements the equilibrium in a complete …nancial market, characterizing stock return volatility and stock expected returns. Section 7 contains the conclusion, while all the mathematical derivations are contained in the appendix. 2 Review of the literature There have been some attempts to explain theoretically the behavior of the stock return volatility. Veronesi (1999) constructs a model with regime shifts in the endowments in which investors will-ingness to hedge against their own uncertainty on the true regime generates overreaction to good news in bad times and volatility clustering. In contrast to that paper, I assume that the exogenous state variables are not subject to regimes, neither do they exhibit mean-reversion. For instance, the current paper obtains volatility clustering in equilibrium even when all the exogenous state variables are geometric Brownian motions. In addition McQueen and Vorkink (2004) are able to reproduce GARCH volatility using a preference-based model with time-varying sensitivity to news. In a related paper, Vanden (2005) develops a model where the representative agent exhibits a utility function with several risk aversion regimes, which in equilibrium leads to volatility regimes and volatility clustering. In contrast to these models, I assume standard constant relative risk aversion preferences and volatility clustering occurs for di⁄erent possible levels of risk aversion, as long as risk aversion is higher than the case of logarithmic preferences. The model presented in this paper is close to a strand in the literature in which investors have incomplete (but symmetric) information about a relevant fundamental, and due to heterogeneous 4 ... - tailieumienphi.vn