Scenarios are outcomes of events with an influence on the return of a portfolio. Examples might include an increase in interest rates, war in the Middle East, etc. In the scenario based approach to portfolio selection, the modeler comes up with a set of scenarios, each with a certain probability of occurring during the period of interest. Given this set of scenarios and their probabilities, the goal is to select a portfolio that minimizes risk, while meeting a target return level.

In the Markowitz portfolio model, presented above, we used a portfolio's variance as a measure of risk. As one might imagine, variance is not the only possible measure of risk. Variance is a measure of the fluctuation of return above and below its average. As a statistic, variance weights a scenario that returns 20% above average the same as a scenario that returns 20% below average. If you're like most investors, you’re probably more worried about the risk that return will be below average. In our scenario based model, we will expand our options by including two new measures of risk that focus on returns below the target level—downside risk and semi-variance risk.

Both semi-variance and downside risk only consider the option return will be below the target. Downside risk is a measure of the expected difference between the target and returns below the target, while semi-variance is a measure of the squared expected difference between the target and returns below the target. Therefore, semi-variance puts a relatively higher weight on larger shortfalls.