On Extending the LP Computable Risk Measures to Account Downside Risk

Abstract

Abstract.A mathematical model of portfolio optimization is usually quantified with mean-risk models offering a lucid form of two criteria with possible trade-off analysis. In the classical Markowitz model the risk is measured by a variance, thus resulting in a quadratic programming model. Following Sharpe's work on linear approximation to the mean-variance model, many attempts have been made to linearize the portfolio optimization problem. There were introduced several alternative risk measures which are computationally attractive as (for discrete random variables) they result in solving linear programming (LP) problems. Typical LP computable risk measures, like the mean absolute deviation (MAD) or the Gini's mean absolute difference (GMD) are symmetric with respect to the below-mean and over-mean performances. The paper shows how the measures can be further combined to extend their modeling capabilities with respect to enhancement of the below-mean downside risk aversion. The relations of the below-mean downside stochastic dominance are formally introduced and the corresponding techniques to enhance risk measures are derived. The resulting mean-risk models generate efficient solutions with respect to second degree stochastic dominance, while at the same time preserving simplicity and LP computability of the original models. The models are tested on real-life historical data.

Efficient optimization of the reward-risk ratio with polyhedral risk measures

“…Similar computationally efficient dual reformulations may be also achieved for more complex polyhedral risk measures. In particular for the measures with enhanced downside risk focus (Michalowski and Ogryczak 2001;Krzemienowski and Ogryczak 2005;Mansini et al 2007). …”

“…On the other hand, measures like MAD or GMD are rather symmetric. Nevertheless, all they can be extended to enhance downside risk focus (Michalowski and Ogryczak 2001;Krzemienowski and Ogryczak 2005) while preserving their polyhedrality.…”

On Fuzzy Driven Support for SD-Efficient Portfolio Selection

“…Following the seminal work by Markowitz [6], the portfolio optimization problem is modeled as a mean-risk bicriteria optimization problem where z(x) is maximized and some risk measure (x) is minimized. In the original Markowitz model [6] the risk is measured by the standard deviation or variance while several other risk measures have been later considered thus creating the entire family of mean-risk (Markowitz-type) models [2,5].…”

“…is used to define the first degree stochastic dominance (FSD). The second function is derived from the cdf as F (2) R(x) (η) = η −∞ F R(x) (ξ) dξ and it defines the second degree stochastic dominance (SSD). Function F (2) R(x) , used to define the SSD relation, can also be presented [10] as F (2) R(x) (η) = E{(η − R(x)) + } where (.)…”

“…+ denotes the nonnegative part. Hence, while function F R(x) expresses the probability of a shortfall to target η, function F (2) R(x) measures the mean shortfall to the target. The weak relations of stochastic dominance (FSD or SSD) are defined by pointwise inequalities for all real targets:…”