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.
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.
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.