# On a source for a mean-variance portfolio optimization result

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.

• 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.
– John
Feb 3, 2015 at 15:44
• Feb 3, 2015 at 15:51

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 $$\frac{dZ}{d\omega}=\mu-\gamma V\omega\overset{!}{=}0\\ \Leftrightarrow \frac{1}{\gamma}\mu=V\omega^*\\ \Leftrightarrow\omega^* = \frac{1}{\gamma}V^{-1}\mu$$
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