Risk measures, robust portfolios and other minimax models

The classical mean-variance model treats the upside and downside equally as risks. This feature is undesirable, in the eyes of a profit-making investor. In this regard, the downside Lower Partial Moments (LPM) are more attractive as alternative risk measures, since they only penalize the downside. This thesis is mainly concerned with the issues related to downside risk measures. We consider two different environments, under which our investigations shall proceed. The first one is the world of Q-radial distributions. The Q-radial distributions generalize the normal distribution and uniform distribution, among many other useful classes of probability distributions. The second type of setting that we will investigate assumes that the distribution of the assets' returns is ambiguous, and the only available (and reliable) knowledge that we have is the first few moments of the distribution. In the first setting, we show that if the investment return rates follow a Q-radial distribution, then the LPM related Risk Adjusted Performance Measures (RAPM), such as the Sortino ratio, the Omega Statistic, the upside potential ratio, and the normalized LPM, are all equivalent to the ordinary Sharpe ratio, which is easy to compute and optimize. Conversely, if all normalized LPM's are equivalent to the Sharpe ratio, then the underlying distribution must be Q-radial. Therefore, this property characterizes the class of Q-radial distributions in which the Sharpe ratio is essentially the only risk adjusted performance measure. If the distribution is unspecified, and only the first few moments (first, second, and/or fourth) are known, we develop tight upper bounds on the lower partial moment E[(r -- X+m], where r ∈ reals and X is stochastic. Based on such tight bounds we then consider the corresponding robust portfolio selection problem, in which the distribution of the investment return is ambiguous, but its first few moments are assumed to be known. We show that if the first two moments are known and the risk measures are either the lower partial moments or the Conditional Value-at-Risk (CVaR), then the optimal portfolio is mean-variance efficient. Moreover, one can formulate the (adjustable) two-stage robust portfolio selection problem as a convex program with finite representations. If more than two moments are known, then the problem is NP-hard in general. In that case we consider approximative models instead. We then proceed to consider the problem of how to alleviate regrets in a decision problem when the parameters are ambiguous, or part of the information will only become known in a dynamic fashion. Since the models we consider in this thesis are mostly in the minimax format, we also consider a general minimax model and study a progressive finite representation approach, which can be used to prove the minimax theorem constructively without any fixed-point theorem or hyperplane separation theorems.