Portfolio Optimization I: An Introduction to Markowitz-related Approaches

The topic of the selection and weighting of assets (or projects) associated with an optimal portfolio is a large and complex one.  For example in general business applications, the determination of the optimal set of companies or subsidiaries that a holding company or conglomerate should own may depend on a large number of factors (including basic risk-return measures, but perhaps also the timing and risk of future cash flows from each business, exchange rate effects, the choice of reporting currency and so on).

In general, the need to reflect specific aspects of such situations (as well as the risk profile of decision-makers) may require the use of tools which readily allow many risk measures and heuristic methods to be easily implemented. Palisade’s Evolver (statistical analysis using genetic algorithm optimization in Microsoft Excel) and RISKOptimizer (Monte Carlo software with genetic algorithm optimization) belong to this set.

However, there is a special class of optimization problem which is standard in financial theory. This problem consists of the determination of an optimal portfolio of a set of financial assets, where the expected return and volatility, and correlation between the assets’ returns are known (or assumed).

As a reminder, the graphs that result for the return of a two asset portfolio with various asset weightings and correlation coefficients between them are shown in the chart below. In a multi-asset situation (more than two assets), theses lines cannot be directly calculated, but are determined as the result of an optimization calculation.

Note that a key (often unstated) assumption in this approach is that the decision-makers’ risk profile can be described by these and no other metrics, so that such lines do indeed describe what a decision-maker would consider to be optimal. (As far as I am aware, there is as such no requirement for an assumption that asset returns are normally distributed. Such an assumption is more restrictive, but which would then imply that the portfolio’s returns are also normally distributed, and so the mean and standard deviation of the portfolio can be used to determine any other relevant risk metrics).

In a subsequent entry, we provide an overview of methods to find this efficient frontier.

Dr. Michael Rees
Director of Training and Consulting

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