# More on Merton Portfolio Optimization

## Merton Portfolio Optimization with Multiple Assets

We now note how the above results extend to the case where there aren’t many assets. Now suppose that there are $d$ assets that can be in invested in. These obey the Stochastic Differential Equation

where $B^j_t$ is an independent Brownian motion for each $j=1,...,N$.

Wealth now evolves according the SDE

where $n_t = ( n^i_t : i=1,...,d )$ gives the amount of each asset $S_t = (S^i_t : i =1,...,d)$ held in the portfolio at time $t$. Also we define $\theta_t = ( n^i_t S^i_t : i=1,...,d)$ as the wealth in each asset. As given in Def [Merton:Objective], our task is the maximize the objective

We now proceed through exercises that are very similar to the case with a single risky asset.

Ex 1. Show that the HJB equation for the Merton Problem can be written as

where $r= (r: i=1,...,d)$.

Ans 1. The answer follows more-or-less identically to Merton Portfolio Optimization [3]. Note that in this case Ito’s formula applied to $V(W_t)$ gives

where

Thus

This is the drift term applied in the HJB equation, similar to [[cDP:MertonHJB]] and gives the require HJB equation.

Ex 2. Show the optimal asset portfolio in the HJB equation is given by

and

Ans 2. Considering [1] we have that

Solving for $\theta^*$ and substituting back into the maximization gives the answer.

Ex 3. Show that for a CRRA utility the optimal solution to the HJB equation is given by

where

Ans 3. [Merton:Multi-CRRA] By (Merton Portfolio Optimization [7])

for some constant $\gamma$. Differentiating twice gives

By (Merton Portfolio Optimization [8]), $c^* =\gamma^{- \frac{1}{R}} w$. Substituting these solutions into the HJB equation using [2] and (Merton Portfolio Optimization [9]) gives

Rearranging and solving for $\gamma$ gives the required solution for $\gamma^*$.

Def 1. [Merton Portfolio and Market Price Risk Vector] As given in [3],

is called the Merton Portfolio and

is called the Market Price Risk Vector.

## Dual value function approach

We could solve the CRRA utility case because it had a special structure. We now give a method for solving for general utilities $u(t,c)$.

Here we assume that $u(t,c)$ is continuous in $t$ and $c$, concave in $c$ and satisfies

The HJB equation for the Merton problem is

We take the LF transform of $u$,

Further we define

where $w$ is such that $z=\partial_w V(t,w)$.

Ex 4. Show that

Ans 4. We can ignore the dependence of $t$ for the first two expressions i.e. take $J(t,z) = J(z)$. Now $J(z) = V( (V')^{-1} (z) ) - z (V')^{-1} (z)$, so

and

Now reintroducing dependence on $t$,

as required.

Ex 5. Show that the HJB equation becomes

[Noticed that we have got rid of the maximisation are we now have a linear PDE.]

Ans 5. The HJB equation is

Ex 6. Now suppose that $u(t,x)=e^{-\rho t}u(x)$, for $u(x)$ concave and increasing. Argue that

Ans 6. This the same as  (Merton Portfolio Optimization [2]).

Ex 7. [Continued] Further defining $j(z) =V(w) -wz$ where $w$ is such that $z=\partial_w V(t,w)$, show that

where $y=z e^{\rho t}$.

Ans 7.  This is a straightforward calculation.

Ex 8. [Continued] Show that the HJB equation becomes

[noticed that we have reduced the PDE in [5] to an ordinary differential equation.]

Ans 8.

which implies the stated answer.

Ex 9. Show that the second order ODE given the above has a solution of the form

where $\alpha$ and $\beta$ solve the quadratic equation

Ans 9. This is standard ODE theory.