# Merton Portfolio Optimization

We consider a specific diffusion control problem. We focus on setting where there is one risky asset and one riskless asset, though we will see that much of the analysis passes over to multiple assets.

Def. [The Merton Problem – Plant Equation] In the Merton problem you wish to optimise your long run consumption. You may invest your wealth in a bank account receiving riskless interest $r$, or in a risky asset with value $S_t$ obeying the following SDE where each $B=(B_t:t\geq 0)$ is an independent standard Brownian motion.

Wealth $(W_t : t\geq 0)$ obeys the SDE You can control $c_t$ your rate of consumption at time $t$ and $n_t$ the number of stocks the risky asset at time $t$. Also, we define $\theta_t = n_t S_t$ to be the wealth in the risky asset at time $t$.

Def. [The Merton Problem – Objective] Given the above plant equation, , the objective is to maximize the long-term utility of consumption Here $\rho$ is a positive constant and $u(c)$ is a concave increasing utility function. The set $\mathcal P(w_0)$ is the set of policies given initial wealth $w_0$. Further, let $V(w,t)$ be the optimal objective with the integral starting for time $t$ with $w_t=w$.

Let’s write the wealth respect to integrals in $dt$ and $dB_t$:

Ex 1. Show that Ans 1. Ex 2. Show that Ans 2. Notice if we shift time by $\tau$, a factor $e^{-\rho t}$ comes out, Ex 3.  Show that the HJB equation for the Merton Problem can be written as or, alternatively, as Ans 3. Recall that informally the HJB equation is Notice that if we apply Ito’s formula to $V(W_t)$ we get that Applying this to the above term gives as required Ex 4. [Continued] Optimizing the HJB over $\theta$, show that Ans 4. Differentiating the HJB equation in [[cDP:MertonHJB]] wrt $\theta$ gives Now rearrange for $\theta^*$.

Ex 5. [Continued] Given $\theta^*$, show that the HJB equation becomes Ans 5. Substituting gives ## Merton for CRRA Utility

We focus on the case of CRRA utility, that is: for $R >0$ (Recall the discussion on utility functions, Section [Util]). Thus we wish to solve for Ex 6. Show that Ans 6.  Here we note that having a policy for initial wealth $\lambda w_0$ is the same as having a policy of wealth $w_0$ and then multiplying each amount invested by $\lambda$: Ex 7. Show that for some position constant $\gamma >0$.

Ans 7. In , let $\lambda=w^{-1}$ and $\gamma = (1-R)V(1)$.

Ex 8. Show that Ans 8. Trival.

Ex 9. Show that if we define then for a CARA utility Ans 9. Differentiating gives $c^{-R} = z$ now rearrange and substitute.

Ex 10. Show that, when optimizing over $c$, the HJB equation is optimized by Ans 10. which since $u'(c)=c^{-R}$, gives the required formed.

Ex 11. [Merton:CARA4] Show that Ex 12. [Continued]Show that Ans 12. By  and Ex 13. Given the optimal choices of $c$ and $\theta$, Show that the HJB equation to be satisfied it must be that where To summarize: we notice we have show that the parameters given by in exercises [4, 10, 13] give a solution to the HJB equation for the Merton problem. (Although we have not yet proven them to be optimal.)

We now give rigourous argument for the optimality of parameters $c^*$, $\theta^*$ and $\gamma^*$ for the Merton problem with CRRA utility. (This section can be skipped if preferred.)

Ex 14. Show that where (Hint: $u(y)$ is concave.)

Ans 14. Since $u(y)$ is concave we have that $u(y) \leq u(x) + (y-x) u'(x)$. Thus Ex 15. Verify that is a positive local martingale. [Hint: apply Ito’s formula to ]

Ex 16. [Continued] Show that [Hint: Doob’s Martingale Convergence Theorem.]

Ans 16. Recall from stochastic integration theory that every positive local martingale is a supermartingale. Doob’s Martingale Convergence Theorem applied to  gives Ex 17. [Continued] By direct calculation show that [Hint: apply Fubini’s Theorem and note that ]

Ex 18. Now show that Ans 18. Combining  and , we see that Applying this to  we see that as required. Thus $c^*_t$ is optimal.

The last exercise shows that the portfolio $\theta^*,c^*$ is optimal for the Merton problem with CRRA utility.