# Markowitz portfolio optimization

Say that there are 5 assets with given mean values, standard deviations and correlations. Is it possible to find the expected return of a risk-seeking portfolio (maximum expected return) by using markowitz model ? I am guessing that since I have the means if I can somehow find the weights it should be fine but not sure..

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Yes. If your returns vector is $\mu$ and your covariance matrix is $\Sigma$ then the optimal allocation is

$$w = \frac{1}{\lambda}\Sigma^{-1}\mu$$

where $\lambda$ is your risk aversion. There are many variations on this, taking into account target return, position limits, fully invested constraints, trading costs, parameter uncertainty, model risk etc.

Of course, no one actually uses the formula above in practice.

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Thanks but I have to ask this since I am a bit new to these topics. How do I get the "risk aversion" ? Sorry if this is very basic question –  Cemre Feb 17 '12 at 18:07
or let me put it this way: how do you find risk aversion without knowing the weights ? –  Cemre Feb 17 '12 at 18:20
One approach is to think of the 'risk aversion' as a parameter that you tune, giving you a family of portfolios. Another (more sensible) approach is to notice that your variance with this portfolio is $(1/\lambda^2)\mu' \Sigma^{-1}\mu$. If your target variance is $\sigma^2$ you should therefore choose $\lambda = (1/\sigma) \sqrt{\mu'\Sigma^{-1}\mu}$ –  Chris Taylor Feb 17 '12 at 18:21