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

6.6 Summary

  • A portfolio over \(N\) assets is conveniently represented by the \(N\)-dimensional vector \(\w\). The portfolio return is given by \(\w^\T\bm{r}\), where \(\bm{r}\) contains the returns of the assets.

  • In practice, portfolios have to be periodically rebalanced, incurring transaction costs which erode the potential return.

  • Portfolios typically need to satisfy many constraints, some imposed by the regulators or brokers (such as shorting, leverage, and margin requirements), while others depend on the investor’s views (such as market neutrality, portfolio diversity, sparsity level, and turnover control). Most of these constraints are expressed as convex functions that can be easily handled in the optimization process (a notable exception is sparsity control).

  • The performance of portfolios can be measured and monitored in a multitude of ways (such as Sharpe ratio, downside risk measures, drawdown, and CVaR). Most interesting is when such performance measures are included in the optimization process, leading to challenging formulations.

  • Some heuristic portfolios have endured the test of time and can be easily calculated without the need for complicated optimization methods; notable examples are the \(1/N\) portfolio and the family of quintile portfolios.

  • Risk-based portfolios are focused on reducing the variability in the returns without attempting to ride the market trend; they are typically easy to compute and exhibit good results in practice.