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

$dS^i_t= S^i_t \left\{ \sum_{j=1}^N \sigma^{ij} d B^j_t + \mu^i dt \right\} \, , \quad i=1,...,d$

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

Wealth now evolves according the SDE

$dW_t = r \left( W_t - \bm n_t \cdot \bm S_t \right) dt + \bm n_t \cdot d \bm S_t -c_t dt$

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

$\begin{aligned} V(w) = \max_{(\bm n_t,c_t)_{t\geq 0}\in\mathcal P ( w_0)} \mathbb{E}\left[\int_0^\infty e^{-\rho t} u(c_t) dt\right]. \end{aligned}$

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

$0 = \max_{c} \left\{ u(c) - c \partial_w V \right\} + \max_{\bm \theta }\left\{ \bm \theta \cdot (\bm \mu -\bm r ) \partial_w V + \frac{1}{2} | \sigma \bm \theta |^2\partial_{ww} V \right\} -\rho V + r w \partial_w V.$

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

$d V ( W_t ) = \partial _w V(W_t ) dW_t + \frac{1}{2} \partial_{ww} V(W_t) d [W]_t$

where

$\begin{gathered} dW_t = \Big( r W_t - \underbrace{r \bm n_t \cdot \bm S_t}_{\bm \theta \cdot \bm r} \Big) dt + \!\!\! \underbrace{ \bm n_t \cdot d \bm S_t }_{ \bm \theta^\top [ \sigma d \bm B_t + \bm \mu dt ] } \!\!\! -c_t dt = \left( r W_t +\bm \theta_t (\bm \mu - \bm r ) -c_t \right)dt + \bm \theta^\top_t \sigma d \bm B_t \\ d[W]_t = \sum_{ij} (\bm \theta^\top_t \sigma )_i (\bm \theta^\top_t \sigma )_j d[B^i_t,B^j_t] = | \bm \theta_t \sigma |^2 dt.\end{gathered}$ Thus

$d V(W_t) = \left[ \left( r W_t +\bm \theta_t (\bm \mu - \bm r ) -c_t \right) \partial_w V (W_t) + \frac{1}{2} | \bm \theta \sigma |^2 \partial_{ww} V(W_t) \right] dt + \bm \theta^\top_t \sigma d \bm B_t.$

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

$\begin{aligned} \bm \theta^* = - \frac{\partial_w V}{\partial_{ww} V } (\sigma \sigma^\top)^{-1} (\bm \mu -\bm r)\end{aligned}$

and

$\begin{aligned} \max_{\bm \theta }\left\{ \bm \theta \cdot (\bm \mu -\bm r ) \partial_w V + \frac{1}{2} ( \bm \theta^\top \sigma^\top \sigma \bm \theta) \partial_{ww} V \right\} = -\frac{1}{2} | \bm \kappa |^2 \frac{ (\partial_w V)^2 }{\partial_{ww} V}\end{aligned}$

Ans 2. Considering [1] we have that

$\max_{\bm \theta }\left\{ \bm \theta \cdot (\bm \mu -\bm r ) \partial_w V + \frac{1}{2} ( \bm \theta^\top \sigma^\top \sigma \bm \theta) \partial_{ww} V \right\} \implies (\bm \mu - \bm r) \partial_w V + \partial_{ww} V (\sigma^\top \sigma ) \bm \theta^* = 0.$

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

$\begin{aligned} \bm \theta^* = \frac{w}{R} (\sigma \sigma^\top)^{-1} ( \bm \mu - \bm r ) \, , \quad c^* =\gamma^{- \frac{1}{R}} w \end{aligned}$

where

$\gamma^{-\frac{1}{R}} = R^{-1} \left\{ \rho + (R-1) (r + \frac{1}{2} \frac{|\bm \kappa |^2}{R} ) \right\}\,\qquad \bm \kappa = \sigma^{-1} (\bm \mu - \bm r ).$

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

$V(w) = \gamma \frac{w^{1-R}}{1-R}$

for some constant $\gamma$ . Differentiating twice gives

$\begin{aligned} \bm \theta^* = - \frac{\partial_w V}{\partial_{ww} V } (\sigma \sigma^\top)^{-1} (\bm \mu -\bm r) = \frac{w}{R} (\sigma \sigma^\top)^{-1} ( \bm \mu - \bm r ).\end{aligned}$

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

$\begin{aligned} 0 & = \max_{c} \left\{ u(c) - c \partial_w V \right\} + \max_{\bm \theta }\left\{ \bm \theta \cdot (\bm \mu -\bm r ) \partial_w V + \frac{1}{2} | \sigma \bm \theta |^2\partial_{ww} V \right\} -\rho V + r w \partial_w V\\ & = \frac{R}{1-R} \gamma^{1-\frac{1}{R}} w^{1-R} +\frac{1}{2} | \bm \kappa |^2 \cdot \frac{\gamma}{R} w^{1-R} - \rho \gamma \frac{w^{1-R}}{1-R} + r \gamma w^{1-R}\end{aligned}$

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

$\theta^* = wR^{-1} \sigma^{-2} (\mu-r)$

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

$\bm \theta^* = \frac{w}{R} (\sigma \sigma^\top)^{-1} ( \bm \mu - \bm r ) \, ,$

is called the Merton Portfolio and

$\bm \kappa = \sigma^{-1} (\bm \mu - \bm r ).$

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

$\lim_{c \rightarrow \infty} u'(t,c) =0$

The HJB equation for the Merton problem is

$0 = \max_{\theta, c }\left\{ u(t,c) + \partial_{t} V + \left( rw + \theta \cdot (\mu-r)-c \right) \partial_w V + \frac{1}{2} |\sigma^\T \theta |^2\partial_{ww} V\right\}.$

We take the LF transform of $u$ ,

$u^*(t,z) = \max_{c} \{ u(t,c) - z c\}$

Further we define

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

Ex 4. Show that

$\partial_{z} J =-w, \qquad \partial_{zz}J = - \frac{1}{\partial_{ww} V} \, , \qquad \partial_{t} J = \partial_{t} V$

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

$J'(z) = \frac{d }{dz} (V')^{-1} (z) \cdot \underbrace{V'( (V')^{-1} (z) )}_{=z} - z \frac{d }{dz } (V')^{-1} (z) - (V')^{-1} (z) = - (V')^{-1} (z) = -w$ and

$J''(z) = - \frac{d }{dz} (V')^{-1} (z) = - \frac{1}{V''((V')^{-1}(z))} = - \frac{1}{V''(w)}\, .$

Now reintroducing dependence on $t$ ,

$\frac{\partial J}{\partial t} = \partial_t V(t,w) + \frac{d w}{dt} \underbrace{\partial_w V}_{=z} - \frac{d w}{dt} z = \partial_t V$

as required.

Ex 5. Show that the HJB equation becomes

$0= u^*(t,c) + \partial_t J - r z \partial_z J + \frac{1}{2} |\kappa|^2 z^2 \partial_{zz} J$

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

Ans 5. The HJB equation is

$\begin{aligned} 0 & = \max_{c} \left\{ u(t,z) - c \partial_w V \right\} + \max_{\bm \theta }\left\{ \bm \theta \cdot (\bm \mu -\bm r ) \partial_w V + \frac{1}{2} | \sigma \bm \theta |^2\partial_{ww} V \right\} +\partial_t V + r w \partial_w V \\ & = u^*(t,\partial_w V) -\frac{1}{2} | \bm \kappa |^2 \frac{ (\partial_w V)^2 }{\partial_{ww} V} + \partial_t V + rw \partial_w V \\ & = u^*(t,z) + \frac{1}{2} | \bm \kappa |^2 z^2 \partial_{zz} J + \partial_t J - r z \partial_z J \, .\end{aligned}$

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

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

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

$\begin{aligned} & J(t,z) = e^{-\rho t} j(y), && \partial_{t} J = -\rho e^{-\rho t} j(y) + \rho e^{-\rho t} y j'(y) \\ & \partial_{z} J = j'(y), && \partial_{zz}J =e^{\rho t}j''(y) \end{aligned}$

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

Ans 7. This is a straightforward calculation.

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

$0 = u^*(y) - \rho j(y) + (\rho -r) y j'(y) + \frac{1}{2} |\kappa |^{2} y^{2} j''(y)$

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

Ans 8.

$0= \underbrace{u^*(t,z)}_{=e^{-\rho t}u^*(y)} + \underbrace { \partial_t J }_{ = -\rho e^{-\rho t} j(y) + \rho e^{-\rho t} y j'(y) } - \underbrace{ r z \partial_z J }_{ r e^{-\rho t} yj'(y) } + \underbrace { \frac{1}{2} |\kappa|^2 z^2 \partial_{zz} J }_{ \frac{1}{2} |\kappa|^2 y^2 e^{-\rho t} j''(y) }$ which implies the stated answer.