Merton Portfolio Optimization

• HJB equation for Merton Problem; CRRA utility solution; Proof of Optimality.
• Multiple Assets; Dual Value function Approach.

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 [cDP:MertonPlant]

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 $w_t=w$.

Prop 1. The HJB equation for the Merton Problem can be written as Here the optimal $\theta$ is given by

Proof. First we note that we can rewrite the SDE for $W_t$ as follows:

Further note that if we shift the value function time by $\tau$, a factor $e^{-\rho t}$ comes out,

So $V(w,t) = e^{-\rho t} V(x)$.

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

Differentiating the HJB equation w.r.t. $\theta$ gives

Now rearrange for $\theta^*$. $\square$

Merton for CRRA Utility

We focus on the case of CRRA utility, that is:

for $R >0$. (See the discussion on utility functions, Section [Util].) Thus we wish to solve for

Prop 2. For a CRRA utility it holds that:

a) The Value function takes the form

for some position constant $\gamma >0$.

b) The HJB equation is optimized by

c) The HJB equation is satisfied by parameters

where

Proof. a) 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 $\lambda$:

Letting $\lambda=w^{-1}$ and $\gamma = (1-R)V(1)$ gives the result.

b) By part a), $\partial_w V(w) = \gamma w^{-R}$ and $\partial_{ww} V(w) = - R \gamma w^{-xR-1}$. So, by Prop [Mert:Prop],

Also,

which since $u'(c)=c^{-R}$, gives that $c^*= \gamma^{-\frac{1}{R}}w$. Further,

as required.

c) Applying a) and b) to the HJB equation in Prop 1 gives

Cancelling $\gamma w^{1-R}$ and rearranging gives the required for for $\gamma$. $\square$

To summarize: we notice we have shown that the parameters

give a solution to the HJB equation for the Merton problem. (Although we have not yet proven them to be optimal.)

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

Thrm 1. The parameters

are optimal for the Merton problem.

Proof. Since $u(y)$ is concave, $u(y) \leq u(x) + (y-x) u'(x)$. Thus for $\zeta_t = e^{-\rho t} u'(c^*_t) \propto e^{-\kappa B_t -(r+\frac{1}{2} |\kappa|^2)t}$ we have that

Next we show that

is a positive local martingale. It is clear that the function $Y_t$ is positive. Note that

Define function

and note that $\zeta_t w_t = D f_t(W_t,B_t)$. Now lets apply Ito’s formula to $f_t(W_t,B_t)$. By Ito’s formula:

Now lets check terms.

Substituting these into Ito’s formula above gives,

Cancelling and rearrganging we get

So

is a local-Martingale. Recall from stochastic integration theory that every positive local martingale is a supermartingale.

Doob’s Martingale Convergence Theorem applied to $Y_t$ gives

Since $\zeta_t= e^{-\rho t} u'(c_t^*) =e^{-\rho t} (c_t^*)^{-R}$ and by the definition of $V(w_0)$:

The last equality holds since $\zeta_0 = (c^*_0)^{-R}$ and $c^*_s= \gamma^{1/R} w^*_s$.

Combining the last equality and the inequality before that, we see that

Applying this to we see that

and, as required, $c^*_t$ is optimal. $\square$

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

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

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

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

Proof. The proof follows more-or-less identically to Prop 1. 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. Thus recalling that

This gives the required HJB equation. $\square$

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

and

Proof. Considering Lemma 1 we have that

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

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

where

Proof. (In this proof when we refer to Prop 2 we mean that the argument which was applied in the single-asset setting is identical in the multiple asset setting.)

By Prop 2a)

for some constant $\gamma$. Differentiating twice gives

By Prop 2b), $c^* =\gamma^{- \frac{1}{R}} w$. Substituting these solutions into the HJB equation gives

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

Def [Merton Portfolio and Market Price Risk Vector] As given above,

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 $z=\partial_w V(t,w)$.

The HJB equation can be written as

Moreover if we suppose that $u(t,x)=e^{-\rho t}u(x)$, for $u(x)$ concave and increasing, the HJB equation becomes

Noticed in the first HJB equation above that we have got rid of the maximization and in the second we have a linear ODE, which can be solved using standard methods.

First we will show that

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 $t$,

This gives the required derivatives in .

Substituting the expressions in , the HJB equation is

Now is we suppose that $u(t,x)=e^{-\rho t}u(x)$, for $u(x)$ concave and increasing, then by the same argument as Prop 1a) we have that

Defining $j(z) =V(w) -wz$ where $w$ is such that $z=\partial_w V(t,w)$, the following are straightforward calculations:

where $y=z e^{\rho t}$. Now substituting these terms into the HJB equation gives the result:

$\square$