Portfolio choice problem of a CARA investor with n risky assets

Question

Ok, I am working on a problem that consists of the following:

I am looking to solve the portfolio choice optimization problem (maximizing utility with a known utility function) in the case where all of the underlying random variables are multivariate normal.

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Problem:

define $\phi$ as the amount invested in each of $n$ risky assets, such that the budget constraint is:

$\Sigma_{i=1}^{n}\phi_i=w_0$ for some initial wealth, $w_0$

Show that the optimal portfolio is:

$\phi=\frac{1}{\alpha}\Sigma^{-1}\mu+[\frac{\alpha w_0-1'\Sigma^{-1}\mu}{\alpha 1'\Sigma^{-1}1}]\Sigma^{-1}1$

where each of the 1's is an $n$-dimensional column vector of 1's.

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Work/Attempt

Ok, these are the things I know:

I am dealing with CARA utility, which gives me a utility function of the form:

$u(w)=-e^{-\alpha w}$ where $w$ is my random end-of-period wealth which I believe to be distributed as

$w$~$N(\mu,\sigma^2)$ with $\mu=\phi'\mu$ (a vector of expected returns scaled by the amount invested in each), and $\sigma^2=\phi'\Sigma\phi$ where $\Sigma$ is the covariance matrix of the $n$ risky assets.

So, to find the expected utility of this function, I use the fact that the expectation of an exponential of normals is the exponential of the mean plus half the variance, to arrive at:

$E(u(w))=-e^{-\alpha\phi'\mu+\frac{\alpha^2\phi'\Sigma\phi}{2}}$

Factoring out a negative alpha, and equating the remaining part of the exponential as the certainty equivalent of a random wealth (I might not be explaining that well, but I am almost certain this is the correct path), I can maximize utility by maximizing the utility of the certainty equivalent, which is done by maximizing the certainty equivalent itself.

All that to say, I need:

$\frac{\partial}{\partial\theta}\phi'\mu+\frac{\alpha\phi'\Sigma\phi}{2}=0$

From there I can't seem to get anything even remotely close to the result I am supposed to show. I have

$1'\mu+\alpha\Sigma\phi=0$

which seems to mirror the first term in the result, but I am lost as to where the rest comes from.

Any help would be appreciated. I'm not sure if my mistake is in the multi-dimensional partial derivative, or if it is in obtaining the function that needs to be maximized. The book I am using has a similar problem for a single risky asset which I can work through just fine, but the exclusion of a risk-free asset (which would seem to simplify the wealth constraint) makes it more confusing to me.

Answer 1

This problem is from the exercise for Chapter 2 of Kerry Back's Asset Pricing Book. The setup of the problem is rather simple. You want to \begin{equation*} \begin{aligned} & \underset{\phi}{\text{maximize}} & & \phi'\mu + \frac{1}{2} \alpha \phi' \Sigma \phi\\ & \text{subject to} & & 1'\phi = w_0 \end{aligned} \end{equation*}

The general method to solve this is to setup Lagrangian, and take derivative w.r.t $\phi$,

\begin{equation*} \begin{aligned} & L = \phi'\mu + \frac{1}{2} \alpha \phi' \Sigma \phi + \lambda(1'\phi - w_0) \end{aligned} \end{equation*}

\begin{equation*} \begin{aligned} & \frac{\partial L}{\partial \phi} = \mu + \alpha \Sigma \phi + \lambda 1 = 0 \end{aligned} \end{equation*}

Solve this for $\phi$ in terms of $\lambda$, you will have $$ \phi = \frac{1}{\alpha} \Sigma^{-1} \mu - \lambda \Sigma^{-1} 1 $$ Now substitute this $\phi$ back into the constraint, and solve for $\lambda$, you will have: $$ \lambda = (1'\Sigma^{-1}\mu - \alpha w_0)(1'\Sigma^{-1}1)^{-1} $$

Finally substitute this $\lambda$ into the expression for $\phi$, you will have what you are looking for.