Assume in (7.16) that logR˜ and log(˜c1/c0) are joint normally distributed.

Specifically, let logR˜ = ˜y and log(˜c1/c0) = ˜z with E[˜y] = μy, var(y˜) = σ2 y , E[˜z] = μ, var(z˜) = σ2, and corr(y˜, z˜) = γ .

(a) Show that μ = −logδ +ργσσy + ρμ− 1 2 ρ2 σ2 − 1 2 σ2 y .

(b) Let r = logRf denote the continuously compounded risk-free rate.
Using (7.22
), show that μ = r +ργσσy − 1 2 σ2 y . (7.37)
Note: γσσy is the covariance of the continuously compounded rate of return y˜
withthe continuously compounded consumption growth rate z˜, so (7.37) hasthe usual form Expected Return = Risk-Free Return+ψ ×Covariance , with ψ = ρ, except for the extra term −σ2 y /2. The extra term, which involves the total and hence idiosyncratic risk of the return, is usually called a Jensen’s inequality term, because it arises from the fact that E[ey˜
] = eμy+σ 2 y /2 > eμy .