Uncovering the Risk-Return Relation in the Stock Market
Hui Guo, Robert Whitelaw
There is an ongoing debate in the literature about the apparent weak or negative relation between risk (conditional variance) and return (expected returns) in the aggregate stock market. We develop and estimate an empirical model based on the ICAPM to investigate this relation. Our primary innovation is to model and identify empirically the two components of expected returns--the risk component and the component due to the desire to hedge changes in investment opportunities. We also explicitly
... odel the effect of shocks to expected returns on ex post returns and use implied volatility from traded options to increase estimation efficiency. As a result, the coefficient of relative risk aversion is estimated more precisely, and we find it to be positive and reasonable in magnitude. Although volatility risk is priced, as theory dictates, it contributes only a small amount to the timevariation in expected returns. Expected returns are driven primarily by the desire to hedge changes in investment opportunities. It is the omission of this hedge component that is responsible for the contradictory and counter-intuitive results in the existing literature. The return on the market portfolio plays a central role in the capital asset pricing model (CAPM), the financial theory widely used by both academics and practitioners. However, the intertemporal properties of stock market returns are not yet fully understood. 1 In particular, there is an ongoing debate in the literature about the relationship between stock market risk and return and the extent to which stock market volatility moves stock prices. This paper provides new evidence on the risk-return relation by estimating a variant of Merton's (1973) intertemporal capital asset pricing model (ICAPM). In his seminal paper, Merton (1973) shows that the conditional excess market return, E t−1 r M,t − r f,t , is a linear function of its conditional variance, σ 2 M,t−1 , (the risk component) and its covariance with investment opportunities, σ MF,t−1 , (the hedge component), i.e., where J(W (t), F (t), t) is the indirect utility function with subscripts denoting partial derivatives, W (t) is wealth, and F (t) is a vector of state variables that describe investment opportunities. is a measure of relative risk aversion, which is usually assumed to be constant over time. If people are risk averse, then this quantity should be positive. Under certain conditions, Merton (1980) argues that the hedge component is negligible and the conditional excess market return is proportional to its conditional variance. Since Merton's work, this specification has been subject to dozens of empirical investigations, but these papers have drawn conflicting conclusions on the sign of the coefficient of relative risk aversion. In general, however, despite widely differing specifications and estimation techniques, most studies find a weak or negative relation. Examples include French, Stambaugh (1987), Campbell (1987), Glosten, Jagannathan and Runkle (1993), Whitelaw (1994), and more recent papers, including Goyal and Santa-Clara (2003) and Lettau and Ludvigson (2003) . 1 The expected stock market return was long considered to be constant until relatively recent work documenting the predictability of market returns (e.g., Fama and French (1989) ). It is now well understood that time-varying expected returns are consistent with rational expectations. See Campbell and Cochrane (1999) and Guo (2003) for recent examples of this literature. 2 Strictly speaking, equation (1) is the discrete time version of Merton's ICAPM (see Long  ). In addition, the equation holds for the aggregate wealth portfolio for which we use the market portfolio as a proxy.