7.6 Summary
In 1952, Markowitz published a seminal paper that initiated the era referred to as modern portfolio theory, which postulates the design of a portfolio in terms of expected return and variance as a measure of risk.
This mean–variance formulation is in the form of a convex problem that can be efficiently solved and has remained central in portfolio optimization. Rather than a unique solution, it produces an efficient frontier of portfolios with different risk profiles.
Unfortunately, it performs poorly in practice due to a multitude of reasons, such as the sensitivity to errors in the parameters that characterize the market (i.e., the expected return vector and covariance matrix) or the simplistic characterization of risk via the variance or volatility.
To overcome these drawbacks, practitioners have come up with a variety of tricks and improvements, namely, adding heuristic constraints to control the solution, improving the estimators of the market parameters (such as shrinkage estimators or robust estimators), using alternative measures of risk, characterizing the risk with a more refined risk-profile vector, and so on.
One particular solution of interest that lies on the efficient frontier is the portfolio that maximizes the Sharpe ratio. Its formulation leads to a nonconvex problem with potentially difficult resolution. Fortunately, a number of practical numerical methods exist that produce the optimal solution.
The Kelly criterion portfolio and, more generally, expected utility portfolios are generalizations of the formulation of the trade-off between expected return and risk. In practice, however, they are closely approximated by the mean–variance framework and efficient numerical algorithms are available.