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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?

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  • $\begingroup$ I'm pretty sure it's because in order for covariance to be defined between two or more assets, these assets must follow Fractional or Geometric Brownian Motion whereby a change in one price results in the expected fractional/proportional change in another price. If comovement is not proportional, then instantaneous asset price changes are not normally distributed, and covariance cannnot exist even if a more general form of comovement is permitted. $\endgroup$ – David Addison Apr 23 '17 at 23:23
  • $\begingroup$ I am not convinced that the assumption of normality is required. As long as the means and (co)variances exist and the user's utility function depends only on these (not on the skewness for example) then Markowitz's results are valid. $\endgroup$ – Alex C Apr 24 '17 at 0:26
  • $\begingroup$ I'm not convinced either. @DavidAddison, why do assets must follow GBM for the covariance to be defined? $\endgroup$ – Chicoscience Apr 24 '17 at 1:27
  • $\begingroup$ It was my assumption that semi-normality is required in order for observed co-variances to be robust (i.e., anything but spurious). I do not question that securities prices behave in semi-normal ways (accounting higher level moments allow us to further relaxes these assumptions). However, consider the utility of MVA as a forward looking decision tool if any two or more processes have distributions which are not approximately normal (i.e., through a Cauchy or variance-Gamma process)... there would be no utility because any observed co-variance would be spurious (even if co-movement is real). $\endgroup$ – David Addison Apr 24 '17 at 3:04
  • $\begingroup$ @DavidAddison I don't think covariance as a general concept requires FBM or GBM. Perhaps you could write up your comments as an answer and more fully develop your point. $\endgroup$ – John Apr 24 '17 at 14:50
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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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Portfolio optimization techniques, such as those defined under Modern Portfolio Theory (MPT), are mildly predicated on the assumption of joint normality. Even though there will be a set of portfolio weights which minimizes variance regardless of the underlying distributions, correlation is only a complete measure of association if the joint multivariate distribution is normal; i.e., covariance is only an exhaustive measure of co-movement if the joint distributions are themselves normal. We can see this is true because the joint distribution of X and Y is defined by joint normality:

${\frac {1}{2\pi \sigma _{X}\sigma _{Y}{\sqrt {1-\rho ^{2}}}}}\iint _{X\,Y}\exp \left[-{\frac {1}{2(1-\rho ^{2})}}\left({\frac {X^{2}}{\sigma _{X}^{2}}}+{\frac {Y^{2}}{\sigma _{Y}^{2}}}-{\frac {2\rho XY}{\sigma _{X}\sigma _{Y}}}\right)\right]\,\mathrm {d} X\,\mathrm {d} Y$

Which through a proof can be show to produce:

$\sigma _{X+Y}={\sqrt {\sigma _{X}^{2}+\sigma _{Y}^{2}+2\rho \sigma _{X}\sigma _{Y}}},$

If now, we define $\omega_i \sigma^2_i=\sigma_X$, and $\omega_j \sigma^2_j=\sigma_Y$, then we get back the equation which is used as the basis of mean variance optimization of a two asset portfolio:

$\mathbb{E}[\sigma _{p}^{2}]=\omega_{i}^{2}\sigma _{i}^{2}+\omega_{j}^{2}\sigma _{j}^{2}+2\omega_{i}\omega_{j}\sigma _{i}\sigma _{j}\rho _{ij}$

So while the portfolio covariance matrix can always be computed, to the extent that underlying assets have returns which are not normal the optimization is likely to result in spuriously optimal weights.

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