The Sharpe Ratio and Preferences: A Parametric Approach

Abstract

We use a log-normal framework to examine the effect of preferences on the market price for risk, that is, the Sharpe ratio. In our framework, the Sharpe ratio can be calculated directly from the elasticity of the stochastic discount factor with respect to consumption innovations as well as the volatility of consumption innovations. This can be understood as an analytical shortcut to the calculation of the Hansen–Jagannathan volatility bounds, and therefore provides a convenient tool for theorists searching for models capable of explaining asset-pricing facts. To illustrate the usefulness of our approach, we examine several popular preference specifications, such as CRRA, various types of habit formation, and the recursive preferences of Epstein–Zin–Weil. Furthermore, we show how the models with idiosyncratic consumption shocks can be studied.

A new methodology for studying the equity premium

“…9 In the consumption CAPM literature, the equity premium and the resulting Sharpe ratio depend on the consumption based stochastic discount factor. Lettau and Uhlig (2002) show that one requires an implausibly large elasticity of this discount factor with respect to the consumption innovations. Similar problems are reported by Hansen and Jagannathan (1991), who derive an upper bound for the Sharpe ratio for asset returns.…”

“…Under reasonable assumptions, however, the standard asset pricing models predict an equity premium of only 0.25%. Subsequently, Hansen and Jagannathan (1991), as well as Lettau and Uhlig (2002), emphasize the volatility component of the equity premium puzzle. 3 Other possible explanations for the puzzle that were proposed in the literature include: market imperfections (Jagannathan et al 2000;Treynor 1994), crash states (Reitz 1988), dividend taxes and regulation (McGrattan and Prescott 2001) GDP growth and portfolio insurance against downside risk (Faugère and Van Erlach 2006).…”

“…SeeLettau and Uhlig (2002) for an excellent survey of various types of closed form solutions for equity premium functions and their limitations.…”

Habit formation and the equity–premium puzzle: a skeptical view

“…Lettau and Uhlig (2002), Boldrin et al (1997), or Chapman (2002. Indeed, Gruber (2004) utilizes the excess-volatility result in order to explain the high volatility of the current-account deficit in a consumption-based open-economy model.…”

C-CAPM Refinements and the Cross-Section of Returns

“…5 Lettau and Uhlig (2002) also provide simple (but different) analytical expressions for these models -mostly with the aim to discuss the equity premium and riskfree rate puzzles. 6 Otherwise, the relevant covariance would be between Z it and (C t /C t−1 ) −γ .…”