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:

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

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

The problem is equivalent to maximizing

$$ \max_\phi \left( \phi ' \mu - \frac{1}{2} \alpha \phi ' \Sigma \phi \right) $$

subject to

$$ \mathbf{1}' \phi = w_0. $$

(boldface 1 is column vector of ones, ' stands for transposition).

The lagrangian is

$$ L(\phi, \lambda) = \phi ' \mu - \frac{1}{2} \alpha \phi ' \Sigma \phi - \lambda \left( \mathbf{1}' \phi - w_0 \right), $$

and its jacobian wrt. $\phi$ is

$$ \mathrm{D}_\phi L(\phi, \lambda) = \mu' - \alpha \phi ' \Sigma - \lambda \mathbf{1}'. $$

First order conditions require that the jacobian is equal to zero (in each element), plus the original budget constraint, which together (after transposing the jacobian) yields a system of linear equations in $(\phi,\lambda)$:

$$ \begin{split} \Sigma \phi + \mathbf{1} \lambda &= \frac{1}{\alpha}\mu \\ \mathbf{1}' \phi &= w_0 \end{split} $$

We can solve for $\phi$ as a function of multiplier $\lambda$:

$$ \phi = \frac{1}{\alpha} \Sigma^{-1} \mu - \lambda \Sigma^{-1} \mathbf{1} $$

Plugging this into the budget constraint and some algebra yields an expression for $\lambda$:

$$ \lambda = \frac{\frac{1}{\alpha}\mathbf{1}' \Sigma^{-1} \mu - w_0}{\mathbf{1}' \Sigma^{-1} \mathbf{1}}, $$

and substituting this into the expression for $\phi$ should yield the result as given in the question.