<p>The problem is equivalent to maximizing</p>

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

<p>subject to</p>

<p>$$
\mathbf{1}' \phi = w_0.
$$</p>

<p>(boldface 1 is column vector of ones, ' stands for transposition).</p>

<p>The lagrangian is</p>

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

<p>and its jacobian wrt. $\phi$ is</p>

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

<p>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 $(\phi,\lambda)$:</p>

<p>$$
\begin{split}
\Sigma \phi + \mathbf{1} \lambda &amp;= \frac{1}{\alpha}\mu \\
\mathbf{1}' \phi &amp;= w_0
\end{split}
$$</p>

<p>We can solve for $\phi$ as a function of multiplier $\lambda$:</p>

<p>$$
\phi = \frac{1}{\alpha} \Sigma^{-1} \mu - \lambda \Sigma^{-1} \mathbf{1} 
$$</p>

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

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

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

CC BY-SA 3.0
2015-02-05T04:05:43.527Z
2015-02-05T04:05:43.527Z
ivansml
<p>Ok, I am working on a problem that consists of the following:</p>

<p>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.</p>

<hr>

<p><strong>Problem:</strong></p>

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

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

<p>Show that the optimal portfolio is:</p>

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

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

<hr>

<p><strong>Work/Attempt</strong></p>

<p>Ok, these are the things I know:</p>

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

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

<p>$w$~$N(\mu,\sigma^2)$ 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.</p>

<p>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:</p>

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

<p>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.</p>

<p>All that to say, I need:</p>

<p>$\frac{\partial}{\partial\phi}\phi'\mu+\frac{\alpha\phi'\Sigma\phi}{2}=0$</p>

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

<p>$1'\mu+\alpha\Sigma\phi=0$</p>

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

<p>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.</p>

CC BY-SA 3.0
2015-02-05T01:26:24.613Z
Where is the $\theta$ in $\frac{\partial}{\partial\theta}\phi'\mu+\frac{\alpha\phi'\Sigma\phi}{2}=0$ coming from?
BKay
CC BY-SA 3.0
2015-02-05T02:03:28.100Z
@BKay My mistake, I believe that should be a ϕ
user2034
CC BY-SA 3.0
2015-02-05T02:18:33.400Z
You should edit the question to fix the typo. Chris Carroll has some nice notes on the single risky asset version of this problem.
http://www.econ2.jhu.edu/people/ccarroll/public/lecturenotes/AssetPricing/CARAPortfolio/
BKay
CC BY-SA 3.0
2015-02-05T02:19:16.917Z
You are not considering the constraint $\sum \phi_i=w_0$ (you need to form a Lagrangian, or substitute out one of phis).
ivansml
CC BY-SA 3.0
2015-02-05T03:02:54.587Z
@ivansml could you give a more detailed explanation? I'm not sure I'm following.
user2034
CC BY-SA 3.0
2015-02-05T03:04:24.253Z
@BKay Thanks I'll edit it. My reference (Asset pricing and portfolio choice theory by Back) covers the single risky asset case and even the multivariate case with a risk free asset quite well. However I'm stuck on this one
user2034
<asset-pricing><optimization><risk-aversion><portfolio-theory>
Portfolio choice problem of a CARA investor with n risky assets
user2034
# Portfolio choice problem of a CARA investor with n risky assets

(Asked by: username_0 on 2015-02-04)

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:

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

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.

**Tags:** asset-pricing, optimization, risk-aversion, portfolio-theory
> **Comments:**
> **username_1**: Where is the $\theta$ in $\frac{\partial}{\partial\theta}\phi'\mu+\frac{\alpha\phi'\Sigma\phi}{2}=0$ coming from?  
> **username_0**: @BKay My mistake, I believe that should be a ϕ  
> **username_1**: You should edit the question to fix the typo. Chris Carroll has some nice notes on the single risky asset version of this problem. http://www.econ2.jhu.edu/people/ccarroll/public/lecturenotes/AssetPricing/CARAPortfolio/  
> **username_2**: You are not considering the constraint $\sum \phi_i=w_0$ (you need to form a Lagrangian, or substitute out one of phis).  
> **username_0**: @ivansml could you give a more detailed explanation? I'm not sure I'm following.  
> **username_0**: @BKay Thanks I'll edit it. My reference (Asset pricing and portfolio choice theory by Back) covers the single risky asset case and even the multivariate case with a risk free asset quite well. However I'm stuck on this one  

---

## 1 answers

### Answer by username_2 on 2015-02-05. Score: 3 | Accepted answer

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.


---

