Another way in which uncertainty about the inputs can be modeled is by incorporating it directly into the optimization process. Robust optimization is an intuitive and efficient way to deal with uncertainty. Robust portfolio optimization does not use the traditional forecasts, such as expected returns and stock covariances, but rather uncertainty sets containing these point estimates. An example of such an uncertainty set is a confidence interval around the forecast for each expected return (alpha). This uncertainty shape looks like a “box” in the space of the input parameters. (See Exhibit 18.2(A).) We can also formulate advanced uncertainty sets that incorporate more knowledge about the estimation error. For instance, a widely used uncertainty set is the ellipsoidal uncertainty set, which takes into consideration the covariance structure of the estimation errors. (See Exhibit 18.2(B).) We will see examples of both uncertainty sets in this section.

The robust optimization procedure for portfolio allocation is as follows. First, we specify the uncertainty sets around the input parameters in the problem. Then, we ask what the optimal portfolio allocation is when the input parameters take the worst possible value inside these uncertainty sets. In effect, we solve an inner problem which determines the worst possible realization of the uncertain parameters over the uncertainty set before we solve the original problem of optimal portfolio allocation.

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