Given $N$ assets, the Markowitz mean-variance model requires expected returns, expected variances and a $N \times N$ covariance matrix. The joint distribution is fully defined by these measures.
However I often read that assets are required to be normally distributed for consideration in the mean-variance model. While I understand that a normal joint distribution is fully defined by the statistics described above, I can't really see why normality is required.
Can't we simply assume that the distribution is fully described by $\mu$, $\sigma^2$ and $\Sigma$, and not necessarily imply normality? That is, an obvious drawback is not considering higher moments which influence assets, such as skewness and kurtosis, but why is normality an assumption?
Answer
it doesn't require normality. What it requires is that the investor's decisions are determined by mean and variance.
A normal distribution is determined by mean and variance, so if you assume joint normality then there is no point in the investor being interested in anything else.
(we try to discuss assumptions thoroughly in our book, Introduction to Mathematical Portfolio Theory.)
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