By investigating the behavior of robust portfolios, the authors explore how portfolio managers can better manage risk and uncertainty. They describe how robustness leads to increased dependence on factor movements and conclude that as the robustness of a portfolio increases, its optimal weights approach the portfolio variance that is maximally explained by the given factors.
The authors’ objective is to decipher robust portfolios for the purpose of analyzing any noticeable attributes. They analyze the behavior of stock portfolios formed from the robust formulation with an ellipsoidal uncertainty set for the expected returns and examine how robust portfolios tilt their exposure to market factors. Furthermore, they offer a mathematical framework and an analytic explanation, as well as empirical analyses, that help explain why increased robustness of portfolios owing to robust optimization leads to increased dependence on market factors.
How Is This Research Useful to Practitioners?
Much previous research has been performed regarding the creation of robust portfolios, but not many researchers have analyzed the characteristics of portfolios formed from robust optimization or deciphered robust portfolios to analyze noticeable trends. The authors’ research is intended to provide practitioners in portfolio management, algorithm equity traders, economists, and academics with tools to arrive at an optimal portfolio and to model its behavior as factors move.
The authors describe robust formulation as a quadratic programming representation with ellipsoidal uncertainty to analyze robust behavior. Furthermore, they use a stylized analytical approach that specifically details the assumptions used and includes three portfolio scenarios: a portfolio with maximum dependency on factors, a portfolio with maximum robustness, and a convergence of robust portfolios. Finally, the empirical approach includes a simulation with generated returns and an analysis with historical returns.
Three main conclusions are highlighted. First, regarding the robust portfolio formulation with an ellipsoidal uncertainty set for expected returns, the authors show that an increase in robustness results in the optimal portfolio being more dependent on factor movements. Second, they find a quadratic program with behavior equivalent to that of a second-order cone problem, and they provide mathematical proofs on the pattern of the relationship between the magnitude of the penalized matrix and the distance from the factors. Third, their results reveal the factor exposure of robust equity portfolios, and they provide evidence that the portfolios might be robust because they are betting more on market factors.
How Did the Authors Conduct This Research?
The authors present the risk–return of the portfolios by solving Problem 1 (a second-order cone programming problem) and Problem 2 (Lemma 1—the optimal solution of a quadratic program for analyzing robust behavior). Increasing the level of δ in Problem 1 influences the portfolios in the same manner as increasing the level of α in Problem 2. Regarding the annualized risk–return of the portfolios, both δ and α modify portfolios so that they shift to the lower-left region (lower risk and lower return), and the frontiers’ curves are similar.
Next, the distance between the factor portfolio and the optimal portfolio from simulations is graphically represented. In this exercise, the results are produced for 10 simulations with 100 stocks and four factors. The authors present the Euclidean distance between the optimal portfolio and the factor portfolio; the factor portfolio decreases as the value of α is increased, which also means that increasing the robustness moves portfolios closer to the factor portfolio even in the generic case.
The authors perform an analysis with historical equity market returns and present the distance between the factor portfolio and the industry-level optimal portfolio. The historical returns are obtained from daily US equity returns in 49 industries and three factors during the analysis period 1970–2012. The portfolios are analyzed during three-year periods, and the variable α ranges from 1 to 100 because it forms portfolios with annualized risk roughly between 10% and 20%. The optimal portfolio becomes more like the factor portfolio as the magnitude of penalization increases.
Finally, the authors reveal the distance between the factor portfolio and the industry-level robust optimal portfolio when varying the robustness (confidence level or uncertainty). Their results indicate that using a small confidence level results in much more dependence on factors than using a 0% confidence level.
The authors highlight how robust portfolios behave outside the common worst-case approach to portfolio selection. Although this research is geared for practitioners with a mathematical background, the authors provide clear definitions, figures that support the conclusions, and appendices for each of the mathematical proofs referenced within the research paper. It would be useful for portfolio managers seeking to increase their client returns using a robust portfolio optimization method rather than, for example, a portfolio created from a Markowitz mean–variance model, which is limited because of its high sensitivity to inputs.