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

1.2 The Big Picture

The two main components for portfolio design are data modeling and portfolio optimization. Figure 1.3 illustrates these two building blocks for the case of mean–variance portfolios (i.e., based on the mean vector \(\bmu\) and covariance matrix \(\bSigma\)) to produce the optimal portfolio weights \(\w\).

Block diagram of data modeling and portfolio optimization.

Figure 1.3: Block diagram of data modeling and portfolio optimization.

Part I of this book examines the data modeling component in Figure 1.3. The main purpose of this block is to characterize the statistical distribution of future returns, primarily in terms of the first- and second-order moments, \(\bmu\) and \(\bSigma\), which will be utilized by the portfolio optimization block later on.

Part II fully explores a wide variety of formulations for the portfolio optimization component in Figure 1.3. These portfolio formulations can be classified according to different criteria leading to a diverse taxonomy of portfolios as follows.

  • Taxonomy according to the data used:

    • second-order portfolios: portfolios based on the mean and the variance, such as Markowitz mean–variance portfolio, maximum Sharpe ratio portfolio, index tracking portfolios, and volatility-based risk parity portfolios;
    • high-order portfolios: portfolios based directly on high-order moments as well as approximations of utility-based portfolios; and
    • raw-data portfolios: these include portfolios that require the raw data, such as downside risk portfolios, semi-variance portfolios, conditional variance-at-risk (CVaR) portfolios, drawdown portfolios, graph-based portfolios, and deep learning portfolios.

  • Taxonomy according to the view on the efficient-market hypothesis:2

    • active portfolios: most of the portfolio formulations that attempt to beat the market; and
    • passive portfolios: index tracking portfolios which simply track the market, avoiding frequent portfolio rebalancing.

  • Taxonomy according to the myopic nature of the portfolio formulation:

    • single period portfolios: most of the formulations considered here are based on a single step into the future; and
    • multi-period portfolios: more involved formulations that consider several steps into the future so that the long-term effect of current actions is better taken into account; this is not covered in this book, see (Boyd et al., 2017) for a monograph on multi-period portfolio optimization.

References

Boyd, S., Busseti, E., Diamond, S., Kahn, R., Koh, K., Nystrup, P., and Speth, J. (2017). Multi-period trading via convex optimization. Foundations and Trends in Optimization, Now Publishers, 3(1), 1–76.

  1. The efficient-market hypothesis states that asset prices reflect all information and, therefore, it should be impossible to outperform the overall market through expert stock selection or market timing.↩︎