Equilibrium Asset Pricing with Nonparametric Horizon Risk
26 Pages Posted: 19 Aug 2002
Date Written: March 2002
This paper presents an equilibrium asset pricing model where, unlike in the CAPM, returns distributions and investors' utility functions are both unspecified. Rather, only the centered moments up to order four of the unknown unconditional distribution of returns can be observed and investors unanismously associate a security's measure of riskiness to the time required for its mean return to converge around its expectation with a specified tolerance. The measurement of this horizon is obtained through the use of Chebyshev-type inequalities. The input parameters for the definition of risk horizon can be calibrated using the term structure of interest rates. Investors may let alone differ in their attitude vis-a-vis risk horizon for any level of expected return, but they may also exhibit various sensitivities with respect to each moment of the returns distributions. This model overcomes two important weaknesses of parametric multi-moments asset pricing models. First, it does not imply moment preferences, respecting Brockett and Kahane's (1992) theoretical finding. Furthermore, provided that the measurement of risk horizon largely emphasizes catastrophic events, investors may also not have any incentive to diversify their portfolio, which was already found by Simkowitz and Beedles (1978). At equilibrium, the horizon-based capital market line displays a nonlinear, but monotonically increasing relationship between the expected returns of portfolios held by rational investors and the risk horizon of their selected efficient portfolio. If two-funds separation does not obtain, idiosyncratic risk can even be priced. If diversification holds, the model yields a three-factor linear equation for the determination of individual securities returns, involving their covariance, coskewness and cokurtosis with the single market portfolio returns. In spite of its inclusion in the model, the riskless rate is not the intercept of this pricing equation, while all risk premia are nonlinear functions of the first four moments of the market porfolio returns distribution. By proper restrictions on the parameters characterizing investment horizon, the CAPM, the zero-beta CAPM and the Kraus-Litzenberger (1976) model are nested in this nonparametric model, with different degrees of freedom depending on whether they rest upon distribution-based or utility-based assumptions. Considering Value-at-Risk (VaR) as a market-wide measure of risk would also be compatible with this approach.
Keywords: equilibrium asset pricing, multi-moment, skewness, kurtosis, nonparametric risk, horizon, Chebyshev's inequality
JEL Classification: G11, G12, C14
Suggested Citation: Suggested Citation