The Generalized Stein/Rubinstein Covariance Formula and Its Application to Estimate Real Systematic Risk

This paper generalizes Stein's (Stein, C. 1973. Estimation of the mean of a multivariate normal distribution. Proc. Prague Sympos. Asymptotic Statistics, September 1973.), Rubinstein's (Rubinstein, M. 1973b. A comparative static analysis of risk premiums. J. Bus. 46(October) 604--615; Rubinstein, M. 1976. The valuation of uncertain income streams and pricing of options. Bell J. Econom. Management Sci. 7(Autumn) 407--425.), and Losq and Chateau's (Losq, E., J. P. D. Chateau. 1982. A generalization of the CAPM based on a property of covariance operator. J. Financial and Quant. Anal. 17(December) 783--797.) covariance formula to the case where both variables are functions of multivariate normal random variables. The resulting formula is extremely useful for either implicit functions of, or nonpolynomials of, multivariate normal random variables, such as exponential functions. An application of the use of the generalized Stein/Rubinstein covariance formula to the estimation of real systemic risk is provided to illustrate the results.