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Scenarios are outcomes of events with am influence on the return of a portfolio. Examples might include an increase in interest rates, war in the Middle Ease, 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.

Variance is a measure of the fluctuation of return above and below its average. As a statistic, variance weights a scenario that returns 20percent above average the same as a scenario that returns 20 percent 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 other 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 that returns 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.

In this model, you are considering investing in three stocks (ATT, GM, and USX). You have determined there are 12 equally likely scenarios in the forthcoming period. You have come up with pedicted rates of return for each of the stocks under the twelve scenarios. You have a target growth rate of 15percent for your portfolio. Your goal is to construct optimal portfolios that minimize expected risk while meeting the expected level of return using three different risk measures--variance, downside, and semi-variance.