Log-robust portfolio management

This dissertation investigates the 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 is implied in the Log-Normal model, although no one distribution has emerged as a better replacement. In this work, 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 captures stock price dynamics by incorporating randomness at the level of the true uncertainty drivers, rather than of the stock returns. We call our approach log-robust portfolio management.;Chapter 2 of this dissertation focuses on static portfolio management without short sales. In this setting, the amount invested in each asset must be non-negative. We derive tractable linear formulations, which can be solved efficiently for large numbers of assets, both when the assets are independent -- a situation that arises when managers invest in asset classes such as gold or real estate, rather than stocks -- and when they are correlated. We show that diversification arises naturally from the robust optimization approach. In particular, when assets are independent, we show that the optimal allocation has a simple structure, where the manager invests in the stocks with the highest nominal returns, in amounts inversely proportional to the standard deviation of the continuously compounded rate of return. The number of assets invested in increases with the decision-maker's aversion to ambiguity, until it becomes optimal to invest the whole budget in the asset with the highest worst-case return.;In Chapter 3 of the Dissertation, we extend the approach to the case with short sales, i.e., amounts invested in the assets can be negative. This requires the development of new solution techniques, as computing the worst-case value of the portfolio for a given allocation is no longer a convex problem. We derive tractable robust formulations, which involve solving linear programming problems. We provide insights into the structure of the optimal solution and highlight the phenomenon of diversification, for independent assets as well as correlated assets. We also present numerical experiments for the cases with and without short sales; results are very encouraging.;In Chapter 4, we present a scenario-based approach to log-robust portfolio management in which the model parameters have several possible estimates, for instance computed with different time horizons for the historical data. We study the independent and correlated assets models. In the case of independent assets, we derive a tractable convex problem which can be solved efficiently for a large number of assets. We then devise an algorithm that only requires solving linear problems and generates a series of progressively tightening upper and lower bounds. The algorithm terminates with a solution that is within epsilon from optimality. For the correlated assets model, we provide a tractable heuristic that builds upon the theoretical insights derived in the independent assets case.;Chapter 5 studies the risk-return tradeoffs in log-robust portfolio management. We minimize the portfolio shortfall at the end of the time horizon in a one period setting, i.e., we minimize the difference between the nominal targeted return and the worst-case portfolio value (over a set of allowable deviations of the uncertain parameters from their nominal values), subject to a budget constraint and a constraint on the nominal portfolio return at the end of the time horizon. We study both, the independent and correlated assets models. We are able to derive convex and linear formulations and obtain tractable solutions for both models.;The last part of the dissertation presents concluding remarks and directions for future work.;An analysis of the effect of the size of the portfolio n on the appropriate value of the parameter c to yield performance guarantees is studied in Appendix A. In Appendix B, we present a sensitivity analysis on the effect of the estimation of the covariance matrix Q.