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 [6], 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 [8] and [10]

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 [15] 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 [16] and [17], we see that

Applying this to [14] 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.

 

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