Prove Proposition 6.38 Su' (es) u(-1)' Proposition 6.38 Under the assumptions of the present section, let m

Prove Proposition 6.38

Transcribed Image Text:

Su' (es) u(-1)' Proposition 6.38 Under the assumptions of the present section, let m = for all t = 1,..., T, and suppose that (log m,, log r) conditionally on F-1 follows a bivariate normal distribution with mean " R and covariance " = R22, for all n = 1,..., N and t = 1,..., T. Then the following holds: Elog log ry: == 222-212 2 Var (log) Cov (log m+1, log+1). 2 (6.78) for all n = 1,..., N and t = 0,1,..., T-1. A similar result has been obtained in Hansen & Singleton [896] (see also Singleton [1549]) by assuming that the representative agent's utility function is of the power form u(x) = x/y and that the aggregate consumption growth rate and the return of each asset are jointly distributed according to a log-normal distribution. Indeed, if (log(e/e-1), log) conditionally on F-1 has a bivariate normal distribution with mean " R2 and covariance "R22, for all n = 1,..., N and t = 1,..., T, then it holds that Elog log ri =-- 2 - (1 - 1) 2 Var (log+1) 2 -(y-1) Cov (log(e+1/e), log+1),