In this talk, we investigate models, insights and algorithms arising in portfolio management when the uncertain rates of return and the investor’s attitude towards ambiguity are quantified using robust optimization techniques based on uncertainty sets. Portfolio management traditionally assumes the precise knowledge of the probabilities of asset price movements; in particular, stock returns are generally assumed to be Log-Normal. There is, however, substantial evidence that stock returns have fatter tails than what the Log-Normal model assumes, although no one distribution has emerged as a better replacement. In this talk, we consider a robust optimization approach to address these issues. Specifically, we model the continuously compounded rates of return as uncertain parameters belonging to a polyhedral uncertainty set, and maximize the worst-case portfolio value, where the worst case is measured over that set. To the best of our knowledge, this is the first work applying robust optimization to finance that incorporates randomness at the level of these uncertainty drivers, rather than of the stock returns. This leads us to analyze nonlinear convex and non-convex optimization models, for which we develop tractable exact algorithms or heuristics; we also provide insights into the optimal solution. We call our approach log-robust portfolio management.
The first part of the talk will focus on the robust optimization approach without short sales. In this setting, the amount invested in each asset must be non-negative. An example of insight we derive is that, when assets are independent, the manager invests in the stocks with the highest nominal return, in an amount inversely proportional to the standard deviation of the continuously compounded rate of return. Furthermore, we show that diversification arises naturally from our approach. The second part of the talk will describe extensions to (1) additional allocation constraints to model risk-return tradeoffs, (2) short sales, i.e., a setting where amounts invested can be negative, and (3) parameter ambiguity, when the decision-maker has the choice between different time horizons to compute the parameters using historical data. Numerical results are encouraging.
Joint work with Dr. Ban Kawas, IBM Research Lab, Zurich, Switzerland.
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