# How to solve a variation of Merton's optimal portfolio problem?

Does anyone know how to solve the following problem?

I have tried to solve this but I'm lost since I have never dealt with a Stochastic Dynamic Programming problem with many variables.

$max_{c_{t},\phi_{t}}\ E_{0}[\sum_{{t=0}}^{\infty}\beta^t\ln(C_{t})]$

$s.t. \ W_{t+1}=(W_t-C_t)[\phi_tR_{t+1}+(1-\phi_t)R_f]$ with $W_{0}$ given

$R_{f}$ represents the gross return of a riskless asset and $R_{f}=(1+r)$ and $R_{t}$ is the return of a risky asset. $R_{t+1}$ is an $i.i.d.$ random variable with unkown cumulative distribution $F$.

I am supposed to solve the exersice using dynamic programming (bellman equation) by finding the optimal policy functions. I need to prove that the optimal share of the risky asset, $\phi_t$, is constant.

I am also given a guess for the value function which is $V(W)=F+Gln(W)$.

• Have you given it an attempt? – VCG Sep 13 '16 at 21:38
• Sure, I found $C=W/(\beta G+1)$ but I'm stuck when it comes to finding $\phi$. I'm not sure how to operate with with the expectation. – Slakeon Sep 13 '16 at 21:41
• Shouldn't you get an Euler equation for $c_t$? – VCG Sep 13 '16 at 22:15
• I think that If you get the FOC from the value function using the guess and replace $W_{t+1}$ with the rhs of the dynamic contraint you can avoid the Euler equation and get the policy for C, as a function of the unkown coefficient F and the state variable W. – Slakeon Sep 13 '16 at 22:27
• I meant the coefficient G from the guess... – Slakeon Sep 13 '16 at 22:34