Assume the investor has constant relative risk aversion . Define optimal consumption C and terminal wealth WT

Assume the investor has constant relative risk aversion ρ. Define optimal consumption C and terminal wealth WT from the first-order conditions (14.7), and define Wt from (14.5).

(a) Show that Wt = M−1/ρ

t f(t,Xt)for some function f .

(b) Derive a PDE for f .

(c) Explain why the optimal portfolio is π(t,Xt), where π satisfies

π(t,x)

 =



1

ρ

λ(x)



+



i=1

∂ logf(t,x)

∂xi

νi(x)





σ (x)

−1 .

How does this compare to the optimal portfolio derived in Exercise 14.6 for a constant investment opportunity set? How does it compare to the optimal portfolio (14.24) derived from dynamic programming?