Robust Markowitz mean‐variance portfolio selection under ambiguous covariance matrix

Abstract

This paper studies a robust continuous-time Markowitz portfolio selection problem where the model uncertainty affects the covariance matrix of multiple risky assets. This problem is formulated into a min-max mean-variance problem over a set of nondominated probability measures that is solved by a McKean-Vlasov dynamic programming approach, which allows us to characterize the solution in terms of a Bellman-Isaacs equation in the Wasserstein space of probability measures. We provide explicit solutions for the optimal robust portfolio strategies and illustrate our results in the case of uncertain volatilities and ambiguous correlation between two risky assets. We then derive the robust efficient frontier in closed form, and obtain a lower bound for the Sharpe ratio of any robust efficient portfolio strategy. Finally, we compare the performance of Sharpe ratios for a robust investor and for an investor with a misspecified model. K E Y W O R D S ambiguous correlation, continuous-time Markowitz problem, covariance matrix uncertainty, dynamic programming, McKean-Vlasov, Wasserstein space wileyonlinelibrary.com/journal/mafi Mathematical Finance. 2019;29:174-207.

Mean-Variance Portfolio Selection with Tracking Error Penalization

“…Compared to the approaches of robust and bayesian optimization ( [9,21]), the regularization of the mean-variance allocation by a tracking-error penalization offers a more intuitive and simple approach from the financial point of view as the specification of the reference portfolio has a clear operationnal meaning. The benchmark tracking with improvement of the Sharpe ratio is intrinsically linked to the method described in this article and is, by definition, absent of the two other approaches.…”

“…In a continuous-time setting, ref. [9] have developed a robust approach by studying the mean-variance allocation with a market model where the model uncertainty affects the covariance matrix of multiple risky assets. In [10], the authors study the problem of utility maximization under uncertain parameters in a model where the parameters of the model do not evolve freely within a given range, but are constrained via a penalty function.…”

Modelo del portafolio eficiente para la toma de decisiones en la producción agrícola

“…Ante estas circunstancias, las investigaciones realizadas por Ismail & Pham (2018) lograron aplicar el criterio de la variación media formulado por Markowitz (1952) para dar mayor robustez al modelo en la selección del portafolio en condiciones de incertidumbre, permitiendo con esto, el análisis de la volatilidad de los activos incluidos y las correlaciones ambiguas entre los riesgos de cada uno de ellos, aplicando el índice del Modelo de Sharpe (1964), a pesar de que cuestiona la robustez de la frontera eficiente, dado que se obtiene un límite más bajo para carteras eficientes independientemente de los resultados que arroje la matriz de covarianza. Es por esto que, Georgalos et al (2018) combinaron el modelo de Markowitz con los modelos realizados por Stott (2006) ; Bouchouicha & Vieider (2017), utilizando las funciones alternativas de ponderación de probabilidades de Prelec (1998); Tversky & Kahneman (1992), llegando a concluir que a través del modelo de Markowitz realmente si se puede en gran medida obtener una explicación racional en función del riesgo a la elección de activos en un portafolio (Vitt et al, 2003).…”

Adaptive Robust Control in Continuous Time

“…Moreover, in the work of Ismail and Pham (2019) and Pham et al (2018), an agent maximises a mean-variance criterion (which is not time consistent) in a one-step optimal asset allocation problem. In both works, the authors assume that the unknown parameters lie in a fixed interval and estimates of the parameters are not updated as time evolves.…”