Problem description

The single-period portfolio selection problem is about selecting assets from a given list in order to create a portfolio that has the greatest expected value. The assets are bought at the market price and their value is subject to variation. The investment budget is fixed, and each selected asset is bought using a percentage of the available budget. The market price for each asset is known at the time of buying and corresponds to the payment to be made by the trader to buy one unit. The future asset value is not known, but we assume the distribution characteristics of the random variable is known. The trader is risk averse and wants to have some guarantees about the worst case value of the portfolio. She considers that in the worst case the downward variation of an asset value reaches 1.5 times the variance of the asset. In other words, the trader assumes it is very unlikely that the value of an asset decreases by more than 1.5 times its variance.

Example: Let's assume that asset #20 has a market price of 100$ and its variability is +/- 10$. Then the considered worst case value of the asset is 85$.

In this context, the objective of the trader is to maximize the expected value of her portfolio, but without taking too much risk. A conservative trader would definitely spend all her budget in the asset with the highest worst case value in order to be highly protected against the worst possible outcome. Unfortunately, this strategy would dramatically reduce the expected value of the portfolio. Another extreme policy would be an opportunistic trader who spends all her budget in the asset with the highest expected value, without paying much attention to the worst case realization. Both approaches are extreme and do not take into account the fact that it is quite unlikely to see all asset values going down (or going up) to the same degree. In practice traders often aim for portfolios with a guaranteed probability of having a value greater than a minimum target value. This is also known as chance constrained optimization. In the 'Results' section (Results) we will show how robust optimization can be used to provide solutions with a large expected value without sacrificing protection against worst case realization.