Orderings and Risk Probability Functionals in Portfolio Theory

2005

This paper discusses and analyzes risk measure properties in order to understand how a risk measure has to be used to optimize the investor's portfolio choices. In particular, we distinguish between two admissible classes of risk measures proposed in the portfolio literature: safety risk measures and dispersion measures. We study and describe how the risk could depend on other distributional parameters. Then, we examine and discuss the differences between statistical parametric models and linear fund separation ones. Finally, we propose an empirical comparison among three different portfolio choice models which depend on the mean, on a risk measure, and on a skewness parameter. Thus, we assess and value the impact on the investor's preferences of three different risk measures even considering some derivative assets among the possible choices.

2011

There is a duality theory connecting certain stochastic orderings between cumulative distribution functions F 1 , F 2 and stochastic orderings between their inverses F −1 1 , F −1 2 . This underlies some theories of utility in the case of the cdf and deprivation indices in the case of the inverse. Under certain conditions there is an equivalence between the two theories. An example is the equivalence between second order stochastic dominance and the Lorenz ordering. This duality is generalised to include the case where there is "distortion" of the cdf of the form v(F ) and also of the inverse. A comprehensive duality theorem is presented in a form which includes the distortions and links the duality to the parallel theories of risk and deprivation indices. It is shown that some wellknown examples are special cases of the results, including some from the Yaari social welfare theory and the theory of majorization.

2015

The paper proposes a multivariate comparison among different financial markets, using risk/variability measures consistent with investors’ preferences. First of all, we recall a recent classification of multivariate stochastic orderings and properly define the selection problem among different financial markets. Then, we propose an empirical financial application, using multivariate stochastic orderings consistent with the non-satiable and risk averse investors’ preferences. For the empirical analysis we examine two different approaches; first, we assume that the return are normally distributed; second, we deal with the more generalassumption that the returns’ distribution follow a stable sub-Gaussian law.

2010

Abstract: In the paper, we consider the application of the theory of probability metrics in several areas in the eld of nance. First, we argue that specially structured probability metrics can be used to quantify stochastic dominance relations. Second, the methods of the theory ...

Insurance: Mathematics and Economics, 2001

This paper discusses a class of risk measures developed from a risk measure recently proposed for insurance pricing. This paper reviews the distortion function approach developed in the actuarial literature for insurance risk. The proportional hazards transform is a particular case. The relationship between this approach to risk and other approaches including the dual theory of choice under risk is discussed. A new class of risk measures with suitable properties for asset allocation based on the distortion function approach to insurance risk is developed. This measure treats upside and downside risk differently. Properties of special cases of the risk measure and links to conventional portfolio selection risk measures are discussed.

Statistics: A Series of Textbooks and Monographs, 2005

In this paper we review and extend some key results on the stochastic ordering of risks and on bounding the influence of stochastic dependence on risk functionals. The first part of the paper is concerned with a.s. constructions of random vectors and with diffusion kernel type comparisons which are of importance for various comparison results. In the second part we consider generalizations of the classical Fréchet-bounds, in particular for the distribution of sums and maxima and for more general monotonic functionals of the risk vector. In the final part we discuss three important orderings of risks which arise from ∆-monotone, supermodular, and directionally convex functions. We give some new criteria for these orderings. For the basic results we also take care to give references to "original sources" of these results.

2018

Utility and risk are two often competing measurements on the investment success. We show that efficient trade-off between these two measurements for investment portfolios happens, in general, on a convex curve in the two dimensional space of utility and risk. This is a rather general pattern. The modern portfolio theory of [Markowitz (1959)] and its natural generalization, the capital market pricing model [Sharpe (1964)], are special cases of our general framework when the risk measure is taken to be the standard deviation and the utility function is the identity mapping. Using our general framework we also recover and extend the results in [Rockafellar, Uryasev & Zabarankin (2006)] which were already an extension of the capital market pricing model to allow for the use of more general deviation measures. This generalized capital asset pricing model also applies to e.g. when an approximation of the maximum drawdown is considered as a risk measure. Furthermore, the consideration of a...

SSRN Electronic Journal, 2007

This paper characterizes higher order risk e¤ects, such as prudence and temperance, via preferences that partially order a set of simple 50-50 lotteries. In particular, consider the random variables e X N ; e Y N ; e X M and e Y M , and assume that e X i dominates e Y i via i th-order stochastic dominance for i = M; N. We show that the 50-50 lottery [ e X N + e Y M ; e Y N + e X M ] dominates the lottery [ e X N + e X M ; e Y N + e Y M ] via (N +M) th-order stochastic dominance. A preference ranking over these lotteries is shown to generalize the concept of risk apportionment, as introduced by Eeckhoudt and Schlesinger (2006). We apply our results in several examples of decision making under risk.

Journal of Mathematical Economics, 2004

This article presents various notions of risk generated by the intuitively appealing single-crossing operations between distribution functions. These stochastic orders, Bickel & Lehmann dispersion or (its equal-mean version) Quiggin's monotone mean-preserving increase in risk and Jewitt's location-independent risk, have proved to be useful in the study of Pareto allocations, ordering of insurance premia and other applications in the Expected Utility setup. These notions of risk are also relevant to the Quiggin-Yaari Rank-dependent Expected Utility (RDEU) model of choice among lotteries. Risk aversion is modeled in the vNM Expected Utility model by Rothschild & Stiglitz's Mean Preserving Increase in Risk (MPIR). Realizing that in the broader rank-dependent set-up this order is too weak to classify choice, Quiggin developed the stronger monotone MPIR for this purpose.