\( \newcommand{\bm}[1]{\boldsymbol{#1}} \newcommand{\textm}[1]{\textsf{#1}} \def\T{{\mkern-2mu\raise-1mu\mathsf{T}}} \newcommand{\R}{\mathbb{R}} % real numbers \newcommand{\E}{{\rm I\kern-.2em E}} \newcommand{\w}{\bm{w}} % bold w \newcommand{\bmu}{\bm{\mu}} % bold mu \newcommand{\bSigma}{\bm{\Sigma}} % bold mu \newcommand{\bigO}{O} %\mathcal{O} \renewcommand{\d}[1]{\operatorname{d}\!{#1}} \)

3.7 Summary

Countless of models have been put forth in the literature for financial data. The i.i.d. model may be a rough approximation of reality, but it is functional and widely used by academics and practitioners. Some key points of the i.i.d. model for financial data include:

  • Sample estimators perform poorly: This is not unexpected since the sample mean and sample covariance matrix are optimal estimators under the assumption of Gaussian distributed data, which does not hold in practice.

  • Robust estimators are necessary: The spatial median and Tyler estimator are examples of robust estimators against outliers for the mean vector and covariance matrix, respectively.

  • Heavy-tailed estimators are well suited for financial data: Estimators derived under the assumption of heavy-tailed distributed data are naturally robust and fit well financial data. In addition, simple iterative algorithms can be used to compute them in practice.

  • Estimating the mean vector from historical data is extremely noisy: Practitioners typically obtain factors from data providers (at a high premium) and then use them for regression; using just historical data is the “poor man’s” substitute and it is not without its risks.

  • Prior information should be used when available: This can be, among others, in the form of shrinkage target, factor modeling, or information fusion via the Black-Litterman model (or similar).