Inverse Problems in Portfolio Selection: Scenario Optimization Framework
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A number of researchers have proposed several Bayesian methods for portfolio selection, which combine statistical information from financial time series with the prior beliefs of the portfolio manager, in an attempt to reduce the impact of estimation errors in distribution parameters on the portfolio selection process and the effect of these errors on the performance of 'optimal' portfolios in out-of-sample-data. This thesis seeks to reverse the direction of this process, inferring portfolio managers’ probabilistic beliefs about future distributions based on the portfolios that they hold. We refer to the process of portfolio selection as the forward problem and the process of retrieving the implied probabilities, given an optimal portfolio, as the inverse problem. We attempt to solve the inverse problem in a general setting by using a finite set of scenarios. Using a discrete time framework, we can retrieve probabilities associated with each of the scenarios, which tells us the views of the portfolio manager implicit in the choice of a portfolio considered optimal. We conduct the implied views analysis for portfolios selected using expected utility maximization, where the investor's utility function is a globally non-optimal concave function, and in the mean-variance setting with the covariance matrix assumed to be given. We then use the models developed for inverse problem on empirical data to retrieve the implied views implicit in a given portfolio, and attempt to determine whether incorporating these views in portfolio selection improves portfolio performance out of sample.