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

14.4 Summary

An optimal solution to a portfolio optimization problem typically produces unacceptable results in practice, which may seem counterintuitive. Why is that and how can we address it?

• Under ideal conditions, the solution to a portfolio formulation should indeed achieve the desired optimal objective subject to the constraints.

• In reality, however, it may fail miserably. The reason is that the formulation relies on some parameters, such as the mean vector and covariance matrix, but these parameters have to be estimated from noisy and scarce data and will inevitably contain estimation errors.

• The end result of naively ignoring the parameter estimation errors in a portfolio formulation can be catastrophic. For this reason, such portfolio optimization problems have been called “estimation-error maximizers” with solutions that are financially meaningless.

• Some effective approaches to avoid naive solutions include:

  • Robust portfolios: Robust optimization is a mature approach in operations research that is able to incorporate the fact that the parameters in the formulation will contain some unknown errors (rather than naively assuming no errors). This has been widely developed in the context of portfolio optimization.

  • Resampled portfolios: Bootstrapping and resampling are mature techniques in statistics that allow the aggregation of multiple naive solutions into a more stable and reliable solution. These techniques can be easily applied to portfolio design.