Publication: Parameter uncertainty in portfolio optimization
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2013-07
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2013-09-27
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Abstract
La modelización de decisiones reales supone la interacción de dos elementos: un problema de optimización y un procedimiento para estimar los parámetros que definen dicho modelo. Cualquier técnica de estimación requiere de la utilización de información muestral disponible, la cual es aleatoriamente dada. Dependiendo de dicha muestra, los estimadores pueden variar ampliamente, y en consecuencia uno puede obtener soluciones muy distintas del modelo. Concretamente, la incertidumbre de los estimadores que definen el modelo resulta en decisiones inciertas.
El análisis del impacto de la incertidumbre de los parámetros en la optimización de carteras es un área muy activo en estadística e investigación operativa. En esta tesis tratamos el impacto de la incertidumbre de los parámetros en la optimización de carteras. En concreto, estudiamos y caracterizamos la pérdida esperada de los inversores que usan información muestral para construir sus carteras optimas, y además proponemos nuevas técnicas para aliviar dicha incertidumbre. Primero estudiamos diferentes criterios de calibración para estimadores shrinkage en
el contexto de la optimización de carteras. En concreto consideramos diferentes métodos de calibración para estimadores shrinkage del vector de medias, la matriz de covarianzas y el vector de pesos. Para cada método de calibración damos expresiones explícitas de la intensidad optima del shrinkage y además proponemos un nuevo enfoque no-paramétrico para el cálculo de la intensidad de shrinkage de cada criterio de calibración. Finalmente evaluamos el comportamiento de cada método de calibración con datos simulados y empíricos. En segundo lugar analizamos el impacto de la incertidumbre de los parámetros para un inversor multiperíodo que se enfrenta a costes de transacción. Caracterizamos la pérdida
esperada del inversor multiperíodo y encontramos que dicha pérdida es igual al producto de la perdida de un solo periodo y otro término que recoge los efectos multiperíodo en la perdida de utilidad. Además proponemos dos carteras multiperíodo de tipo shrinkage que ayudan a mitigar la incertidumbre de los parámetros. Finalmente analizamos el comportamiento de las carteras multiperíodo que proponemos y encontramos que el inversor puede sufrir grandes pérdidas si ignora los costes de transacción, la incertidumbre de los parámetros o ambos elementos.
Modeling every real-world decision involves two elements: an optimization problem and a procedure to estimate the parameters of the model. Any estimation technique requires the utilization of available sample information, which is random. Depending on the given sample, the estimates may vary widely, and in turn, one may obtain very different solutions from the model. Precisely, the uncertainty of the estimates that define the parameters of the model results into uncertain decisions. Analyzing the impact of parameter uncertainty in optimization models is an active area of study in statistics and operations research. In this dissertation, we address the impact of parameter uncertainty within the context of portfolio optimization. In particular, we study and characterize the expected loss for investors that use sample estimators to construct their optimal portfolios, and we propose several techniques to mitigate the impact of parameter uncertainty. First, we study different calibration criteria for shrinkage estimators in the context of portfolio optimization. Precisely, we study shrinkage estimators for both the inputs and the output of the portfolio model. In particular, we consider a set of dffierent calibration criteria to construct shrinkage estimators for the vector of means, the covariance matrix, and the vector of portfolio weights. We provide analytical expressions for the optimal shrinkage intensity of each calibration criteria, and in addition, we propose a novel non-parametric approach to compute the optimal shrinkage intensity. We characterize the out-of-sample performance of shrinkage estimators for portfolio selection with simulatedand empirical datasets. Second, we study the impact of parameter uncertainty in multiperiod portfolio selection with transaction costs. We characterize the expected loss of a multiperiod investor, and we find that it is equal to the product between the single-period utility loss and a second term that captures the multiperiod effects on the overall utility loss. In addition, we propose two multiperiod shrinkage portfolios to mitigate the impact of parameter uncertainty. We test the out-of-sample performance of these novel multiperiod shrinkage portfolios with simulated and empirical datasets, and we find that ignoring transaction costs, parameter uncertainty, or both, results into large losses in the investor's performance.
Modeling every real-world decision involves two elements: an optimization problem and a procedure to estimate the parameters of the model. Any estimation technique requires the utilization of available sample information, which is random. Depending on the given sample, the estimates may vary widely, and in turn, one may obtain very different solutions from the model. Precisely, the uncertainty of the estimates that define the parameters of the model results into uncertain decisions. Analyzing the impact of parameter uncertainty in optimization models is an active area of study in statistics and operations research. In this dissertation, we address the impact of parameter uncertainty within the context of portfolio optimization. In particular, we study and characterize the expected loss for investors that use sample estimators to construct their optimal portfolios, and we propose several techniques to mitigate the impact of parameter uncertainty. First, we study different calibration criteria for shrinkage estimators in the context of portfolio optimization. Precisely, we study shrinkage estimators for both the inputs and the output of the portfolio model. In particular, we consider a set of dffierent calibration criteria to construct shrinkage estimators for the vector of means, the covariance matrix, and the vector of portfolio weights. We provide analytical expressions for the optimal shrinkage intensity of each calibration criteria, and in addition, we propose a novel non-parametric approach to compute the optimal shrinkage intensity. We characterize the out-of-sample performance of shrinkage estimators for portfolio selection with simulatedand empirical datasets. Second, we study the impact of parameter uncertainty in multiperiod portfolio selection with transaction costs. We characterize the expected loss of a multiperiod investor, and we find that it is equal to the product between the single-period utility loss and a second term that captures the multiperiod effects on the overall utility loss. In addition, we propose two multiperiod shrinkage portfolios to mitigate the impact of parameter uncertainty. We test the out-of-sample performance of these novel multiperiod shrinkage portfolios with simulated and empirical datasets, and we find that ignoring transaction costs, parameter uncertainty, or both, results into large losses in the investor's performance.
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Keywords
Gestión de cartera, Incertidumbre, Modelo matemático