1
$\begingroup$

In the context of a mean_variance framework consider an optimizing investor who chooses at time $T$ portfolio weights $w$ so as to maximize the quadratic objective function:

$$U(w) = E[R_p] - \frac{\gamma}{2}Var[R_p]= w'\mu - \frac{\gamma}{2}w'Vw$$

Where $E$ and $Var$ denote the mean and variance of the uncertain portfolio rate of return $R_p = w'R_{T+1}$ to be realized in time $T + 1$ and $\gamma$ is the relative risk aversion coefficient. The optimal portfolio weights will be:

$$w^* = \frac{1}{\gamma}V^{-1}\mu $$

Could I have a reference that proves this result? preferably a textbook that builds up to it.

$\endgroup$
  • 1
    $\begingroup$ I think Markowitz' 1959 book does, but it's a straightforward optimization that is easy if you look up the relevant matrix derivatives. I think I went through the math in another question here, but can't find it now. $\endgroup$ – John Feb 3 '15 at 15:44
  • 1
    $\begingroup$ Here: quant.stackexchange.com/questions/8594/… $\endgroup$ – Monolite Feb 3 '15 at 15:51
2
$\begingroup$

You do note require a sum up constraint that gives you that the weights sum up to 1? Then the problem is equivalent to a maximization without constraints: $$Z(\omega)=w'\mu - \frac{\gamma}{2}w'Vw$$ then it holds that $$\frac{dZ}{d\omega}=\mu-\gamma V\omega\overset{!}{=}0\\ \Leftrightarrow \frac{1}{\gamma}\mu=V\omega^*\\ \Leftrightarrow\omega^* = \frac{1}{\gamma}V^{-1}\mu $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.