¹Universidad Nacional de Colombia, Facultad de Ciencias Humanas y Económicas, Departamento de Economía, Medellín, Colombia. Universidad de Antioquia, Facultad de Ciencias Económicas, Grupo de Econometría Aplicada, Medellín, Colombia. Assistant professor. Email: kgomezp@unal.edu.co
²Universidad de Antioquia, Facultad de Ciencias Económicas, Departamento de Estadística y Matemáticas - Departamento de Economía, Medellín, Colombia. Universidad de Antioquia, Facultad de Ciencias Económicas, Grupo de Econometría Aplicada, Medellín, Colombia. Assistant professor. Email: santiagog@udea.edu.co

Accurate measures of the volatility matrix and its inverse play a central role in risk and portfolio management problems. Due to the accumulation of errors in the estimation of expected returns and covariance matrix, the solution to these problems is very sensitive, particularly when the number of assets (p) exceeds the sample size (T). Recent research has focused on developing different methods to estimate high dimensional covariance matrixes under small sample size. The aim of this paper is to examine and compare the minimum variance optimal portfolio constructed using five different estimation methods for the covariance matrix: the sample covariance, RiskMetrics, factor model, shrinkage and mixed frequency factor model. Using the Monte Carlo simulation we provide evidence that the mixed frequency factor model and the factor model provide a high accuracy when there are portfolios with p closer or larger than T.

Key words: Covariance matrix, High dimensional data, Penalized least squares, Portfolio optimization, Shrinkage.

Medidas precisas para la matriz de volatilidad y su inversa son herramientas fundamentales en problemas de administración del riesgo y portafolio. Debido a la acumulación de errores en la estimación de los retornos esperados y la matriz de covarianza la solución de estos problemas son muy sensibles, en particular cuando el número de activos (p) excede el tamaño muestral (T). La investigación reciente se ha centrado en desarrollar diferentes métodos para estimar matrices de alta dimensión bajo tamaños muestrales pequeños. El objetivo de este artículo consiste en examinar y comparar el portafolio óptimo de mínima varianza construido usando cinco diferentes métodos de estimación para la matriz de covarianza: la covarianza muestral, el RiskMetrics, el modelo de factores, el shrinkage y el modelo de factores de frecuencia mixta. Usando simulación Monte Carlo hallamos evidencia de que el modelo de factores de frecuencia mixta y el modelo de factores tienen una alta precisión cuando existen portafolios con p cercano o mayor que T.

Palabras clave: matrix de covarianza, datos de alta dimension, mínimos cuadrados penalizados, optimización de portafolio, shrinkage.

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