Univariate vs multivariate distributions

Hi, in one of the questions it says that for modeling returns of a portfolio of technology stocks :

Statistical interrelationships must be considered, resulting in the need to use a multivariate normal distribution. Why do we need the statistical interrelationships (Covariances/Correlations?) to model the returns? Shouldn’t all we need are the different expected returns and the weights?

So by this saying, as soon as we have a distribution with different means, this must be a multivariate distribution?

Thanks in advance

Hi @Alexandre_B

I think it is important to clarify some concepts here:

No. We need each asset’s expected return, individual variance and covariance (which will be discussed further ahead). These are the parameters that fully describe a multivariate normal distribution for modeling a portfolio’s return. Weights are not included in this definition.

We need statistical interrelationships because it affects the portfolio’s variance (risk). It is impossible to disregard the importance of co-movements in asset’s pairwise. This concept is a building block of the importance of diversification.

I got a little confused here, so if I didn’t understand what you said, forgive me. A distribution of returns doesn’t have different means, unless we are talking about arithmetic x geometric x harmonic, for example. Now, if we are dealing with multiple assets, each one have a distribution of returns with its respective mean and variance, therefore, the portfolio is based on a multivariate distribution.

I hope this answer is helpful. If not, please let me know. Kind regards.

Thank you

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