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



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


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


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


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.

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