Good deals and benchmarks in robust portfolio selection

Abstract

This paper deals with portfolio selection problems under risk and ambiguity. The investor may be ambiguouswith respect to the set of states of nature and their probabilities. Both static and discrete or continuous timedynamic pricing models are included in the analysis. Risk and ambiguity are measured in general settings. Theconsidered risk measures contain, as particular cases, the usual deviations and the coherent and expectationbounded measures of risk.Four contributions seem to be reached. Firstly, necessary and sufficient optimality conditions are given. Sec-ondly, the portfolio selection problem may be frequently solved by linear programming linked methods, de-spite the fact that risk and ambiguity cannot be given by linear expressions. Thirdly, if there is a market priceof risk then there exists a benchmark that creates a robust capital market line when combined with the risk-less asset. The global risk of every portfolio may be divided into systemic and specific. Moreover, if there is noambiguity with respect to the states of nature (only their probabilities are uncertain), then classical CAPM-formulae may be found. Fourthly, some recent pathological findings for ambiguity-free analyses also apply inambiguous frameworks. In particular, there may exist arbitrage free markets such that the ambiguous agentcan guarantee every expected return with a maximum risk bounded from above by zero,i.e., the capitalmarket line (risk, return) becomes vertical. For instance, in the (non-ambiguous) Black and Scholes modelthis property holds for important risk measures such as the absolute deviation or the CVaR. Nevertheless, inambiguous settings, adequate increments of the ambiguity level will allow us to recover capital market linesconsistent with the empirical evidence. The introduction of ambiguity may overcome several caveats of manyimportant pricing models.