# Computing statistics from historical returns

I'm reading age 35 of "Advances in Machine Learning" by de Prado.

Consider an IID multivariate Gaussian process characterized by a vector of means μ, of size Nx1, and a covariance matrix V, of size NxN. This stochastic process describes an invariant random variable, like the returns of stocks, the changes in yield of bonds, or changes in options’ volatilities, for a portfolio of N instruments. We would like to compute the vector of allocations ω that conforms to a particular distribution of risks across V’s principal components.

The only way to calculate these means or covariance matrices are from historical data for a particular stock. But this does not seem to be useful at all. The "risk" or variance of a stock will drastically change due to events. Tesla may be a low variance stock from 2008 - 2012, but then balloon like crazy upon news its profit is positive. So how is designing a portfolio using historical statistics of mean or covariance useful at all?

• That’s why it is a model - you have to carefully assess how wild your assumptions are and check their impact. The assumption of multivariate normality may have some deep theoretical foundation and its implications are very well studied and understood - but its practical relevance and the materiality of its ‘strictness’ must be asserted with each application - as you have to do with any other model, for that matter. – Kermittfrog Jan 31 at 9:41
• @Kermittfrog Point taken. I think the main erroneous assumption here is IID. I can imagine the returns of "all stocks" roughly following a Gaussian distribution at a certain time. But I cannot imagine fitting a Gaussian to a single company's historical time-series is at all reasonable. it seems completely incorrect. – user3180 Jan 31 at 23:35
• You could shorten the interval between observations, e.g. from 1 month to 1 minute, to reduce the impact of outlier intervals when material information was released. – Sergei Rodionov Feb 1 at 7:59