<p>Does anyone know how to solve the following problem?</p>

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

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

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

<p>$R_{f}$ 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 $F$.</p>

<p>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.</p>

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

CC BY-SA 3.0
2016-09-13T21:38:09.743Z
Have you given it an attempt?
VCG
CC BY-SA 3.0
2016-09-13T21:41:35.790Z
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
CC BY-SA 3.0
2016-09-13T22:15:10.590Z
Shouldn't you get an Euler equation for $c_t$?
VCG
CC BY-SA 3.0
2016-09-13T22:27:07.777Z
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
CC BY-SA 3.0
2016-09-13T22:34:11.607Z
I meant the coefficient G from the guess...
Slakeon
CC BY-SA 3.0
2016-09-13T22:48:49.870Z
Oh ya duh. Okay I just replicated your answer for C. Attempting the other.
VCG
CC BY-SA 3.0
2016-09-13T23:03:39.950Z
I get something like this for the second part where I'm stuck: $0=\frac{\partial V(W)}{\partial \phi}=\beta G E_{t}[\frac{R_{+1}-R_{f}}{\phi R_{+1}-\phi R_{f}+R_{f}}]$.
Slakeon
CC BY-SA 3.0
2016-09-13T23:27:53.417Z
I got that too. That's a concave function in $R_{t+1}$ but  I see your point on not sure where to go next.
VCG
CC BY-SA 3.0
2016-09-14T00:03:10.897Z
I was thinking that maybe if I assume that the distribution of the random variable is normal, I can get this through the moment generating function $e^{E[ln(R_{+1}-R_{f})-ln(\phi R_{+1}-\phi R_{f}+R_{f})]+\frac{1}{2}V[ln(R_{+1}-R_{f})-ln(\phi R_{+1}-\phi R_{f}+R_{f})]}$ but I don't see if it gets me anywhere, can't still isolate the $\phi$.
Slakeon
CC BY-SA 3.0
2016-09-14T02:21:47.977Z
Here's a thought: the only variables in the derivative for $\phi$ are constants and $R_{t+1}$ which you are taking an expectation of, so that will also be a constant. C and W are not present.
VCG
CC BY-SA 3.0
2016-09-14T04:00:49.927Z
I think I'm not following, isn't $\phi$ is still a variable?
Slakeon
CC BY-SA 3.0
2016-09-14T11:14:13.787Z
If we could solve for $\phi$ it would be a function of only constants.
VCG
<macroeconomics><finance><dynamic-programming><stochastic-calculus>
How to solve a variation of Merton's optimal portfolio problem?
Slakeon
# How to solve a variation of Merton's optimal portfolio problem?

(Asked by: username_0 on 2016-09-13)

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 $V(W)=F+Gln(W)$.

**Tags:** macroeconomics, finance, dynamic-programming, stochastic-calculus

> **Comments:**
> **username_1**: Have you given it an attempt?  
> **username_0**: 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.  
> **username_1**: Shouldn't you get an Euler equation for $c_t$?  
> **username_0**: 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.  
> **username_0**: I meant the coefficient G from the guess...  
> **username_1**: Oh ya duh. Okay I just replicated your answer for C. Attempting the other.  
> **username_0**: I get something like this for the second part where I'm stuck: $0=\frac{\partial V(W)}{\partial \phi}=\beta G E_{t}[\frac{R_{+1}-R_{f}}{\phi R_{+1}-\phi R_{f}+R_{f}}]$.  
> **username_1**: I got that too. That's a concave function in $R_{t+1}$ but  I see your point on not sure where to go next.  
> **username_0**: I was thinking that maybe if I assume that the distribution of the random variable is normal, I can get this through the moment generating function $e^{E[ln(R_{+1}-R_{f})-ln(\phi R_{+1}-\phi R_{f}+R_{f})]+\frac{1}{2}V[ln(R_{+1}-R_{f})-ln(\phi R_{+1}-\phi R_{f}+R_{f})]}$ but I don't see if it gets me anywhere, can't still isolate the $\phi$.  
> **username_1**: Here's a thought: the only variables in the derivative for $\phi$ are constants and $R_{t+1}$ which you are taking an expectation of, so that will also be a constant. C and W are not present.  
> **username_0**: I think I'm not following, isn't $\phi$ is still a variable?  
> **username_1**: If we could solve for $\phi$ it would be a function of only constants.  

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## 0 answers

