Metrization of Stochastic Dominance Rules

2016

Farinelli and Tibiletti (2008) propose a general risk-reward performance measurement ratio. Due to its simplicity and generality, the F-T ratios have gained much attentions. F-T ratios are ratios of average gains to average losses with respect to a target, each raised by some power index. Omega ratio and Upside Potential ratio are both special cases of F-T ratios. In this paper, we establish the consistency of F-T ratios with respect to first-order stochastic dominance. It is shown that second-order stochastic dominance is not consistent to the F-T ratios. This point is illustrated by a simple example.

SSRN Electronic Journal, 2000

We examine the use of second-order stochastic dominance as both a technique for constructing portfolios and also as a way to measure performance. As a preference-free technique second-order stochastic dominance will suit any risk-averse investor, and it does not require normally distributed returns. Using in-sample data, we construct portfolios such that their second-order stochastic dominance over a benchmark is most probable. The empirical results based on 21 years of daily data suggest that this portfolio choice technique significantly outperforms the benchmark out-of-sample. Moreover, its performance is typically better -and frequently much better -than several alternative portfolio-choice approaches using equal weights, mean-variance optimization, or minimum-variance methods.

European Journal of Operational Research, 2012

LL-Almost Stochastic Dominance (LL-ASD) is a relaxation of the Stochastic Dominance (SD) concept proposed by Leshno and Levy that explains more of realistic preferences observed in practice than SD alone does. Unfortunately, numerical applications of this concept, such as identifying if a given portfolio is efficient or determining a marketed portfolio that dominates a given benchmark, are computationally prohibitive due to the structure of LL-ASD. We propose a new Almost Stochastic Dominance (ASD) concept that is computationally tractable. For instance, a marketed dominating portfolio can be identified by solving a simple linear programming problem. Moreover, the new concept performs well on all the intuitive examples from the literature, and in some cases leads to more realistic predictions than the earlier concept. We develop some properties of ASD, formulate efficient optimization models, and apply the concept to analyzing investors' preferences between bonds and stocks for the long run.

Wiley-Blackwell eBooks, 2011

The goals of this chapter are the following: • To explore the relationship between preference relations and quasi-semidistances. • To introduce a universal description of probability quasisemidistances in terms of a Hausdorff structure. • To provide examples with first-, second-, and higher-order stochastic dominance and to introduce primary, simple, and compound stochastic orders. • To explore new stochastic dominance rules based on a popular risk measure. • To provide a utility-type representation of probability quasisemidistances and to describe the degree of violation utilized in almost stochastic orders in terms of quasi-semidistances.

Annals of Operations Research

We introduce a new stochastic dominance relationship, the interval-based stochastic dominance (ISD). By choosing different reference points, we show that ISD may span a continuum of preferences between kth and (k + 1)th order stochastic dominance (SD). We distinguish accordingly between interval-based (or shortly just interval) SD of order 1 and of order 2: the former spanning from first-to second-order stochastic dominance, the latter from secondto third-order stochastic dominance. By examining the relationships between interval-based SD and SD, as well as between ISD and risk measures or utility functions, we frame the concept within decision theory and clarify its implications when applied to an optimal financial allocation problem. The formulation of ISD-constrained problems in the presence of discrete random variables is discussed in detail and applied to a portfolio selection problem.

SSRN Electronic Journal, 2017

Omega ratios have been introduced in [1] as a performance measure to compare the performance of different investment opportunities. It does not have some of the drawbacks of the famous Sharpe ratio. In particular, it is consistent with first order stochastic dominance. Omega ratios also have an interesting relation to expectiles, which found increasing interest recently as risk measures. There is some confusion in the literature about consistency with respect to second order stochastic dominance. In this paper, we clarify this and extend it to a consistency result with respect to stochastic dominance of order 1 + γ recently introduced in [2] and generalizing the classical concepts of stochastic dominance of first and second order. Several examples illustrate the usefulness of this result. Finally, some consistency results for even more general stochastic dominance rules are shown, including the concept of-almost stochastic dominance introduced in [3].

Applied Stochastic Models in Business and Industry

Actuarial risks and nancial asset returns are tipically heavy tailed. In this paper, we introduce two criteria, called the right tail order and the left tail order, to compare stochastically these variables. The criteria are based on comparisons of expected utilities, for two classes of utility functions that give more weight to the right or the left tail (depending on the context) of the distributions. We study their properties, applications and connections with other classical orders, including the increasing convex and increasing concave orders. Finally, we provide empirical evidence of these orders with an example using real data. I This is a previous version of the article published in Applied Stochastic Models in

RAIRO - Operations Research, 1999

In this paper, we develop some stochastic dominance theorems for the location and scale family and linear combinations of random variables and for risk lovers as well as risk averters that extend results in Hadar and Russell (1971) and Tesfatsion (1976). The results are discussed and applied to decision-making.

The Geneva Papers on Risk and Insurance Issues and Practice, 1981

Journal of Business & Economic Statistics, 2010

We consider consistent tests for stochastic dominance efficiency at any order of a given portfolio with respect to all possible portfolios constructed from a set of assets.