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

Chapter 14 Robust Portfolios

“Nobody’s easier to fool […] than the person who is convinced that he is right.”

— Haruki Murakami, 1Q84

“In theory there is no difference between theory and practice, while in practice there is.”

— Benjamin Brewster

“By failing to prepare, you are preparing to fail.”

— Benjamin Franklin

Markowitz’s mean–variance portfolio optimizes a trade-off between expected return and risk measured by the variance. This formulation requires a prior estimation of some parameters: the mean vector and covariance matrix of the assets. In ideal conditions, this would be a perfectly fine approach. In practice, however, this fails miserably due to estimation errors in these parameters, which is one of the reasons why Markowitz’s portfolio has not been widely adopted by practitioners. This has been referred to as “Markowitz optimization enigma” and portfolio optimization problems have been called “estimation-error maximizers.”

The parameters of an optimization problem have to be estimated and they will inevitably contain estimation errors. The naive approach simply ignores the existence of such estimation errors and proceeds as if the parameters were perfectly known. This leads to totally unacceptable solutions due to their instability and sensitivity to errors. Fortunately, several approaches have been proposed in the literature to mitigate such sensitivity.

This chapter explores two main approaches to deal with the error sensitivity:

  • Robust optimization is a way to formulate optimization problems such that they are aware of the possible errors in the parameters. This approach goes back to the 1990s in the operations research literature and can be effectively applied to portfolio optimization.

  • Resampling and bootstrapping methods constitute the bread and butter in the statistics literature. They rely on a smart resampling of the data to obtain multiple solutions that are then aggregated to form the final stable solution.

The good news is that these two philosophies are very mature and do not destroy the convexity (if any) of the original portfolio formulation.

This material will be published by Cambridge University Press as Portfolio Optimization: Theory and Application by Daniel P. Palomar. This pre-publication version is free to view and download for personal use only; not for re-distribution, re-sale, or use in derivative works. © Daniel P. Palomar 2024.