Some remarks on stochastic dominance

The present work proposes a definition of dominance (dominance in the strict sense), which is weaker than first order stochastic dominance, stating precisely that the r.v.Y dominatesX (XY) if Pr(YX)=1.Such a dominance in the strict sense is then compared with first and second order stochastic dominance and with dominance between descisions of the same decision problem summarised in a table of results, arriving at certain general remarks about decision problems and the choice between r.v.'s. Indications are also given about how it is possible to obtain simple and useful bounds for Pr(YX).

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Riassunto Nel presente lavoro si propone una definizione di dominanza (dominanza in senso stretto) più debole della dominanza stocastica del prim'ordine, precisamente dicendo che la v.a.Y dominaX (XY) se Pr(YX)=1.Si confronta poi tale dominanza in senso stretto con le dominanze stocastiche del primo e del secondo ordine e con la dominanza tra decisioni di uno stesso problema di decisione sintetizzato in una tabella dei risultati giungendo ad alcune precisazioni generali sui problemi di decisione e di scelta tra v.a. Si danno anche indicazioni su come sia possibile ottenere limitazioni per la Pr(YX).

     

Stochastic dominance with pair-wise risk aversion

A class of stochastic orders is defined on the set of bivariate distribution functions. This class of orders is linearly orderable by inclusion. A family of utility functions, coherent with each of the stochastic orders previously defined, is determined. These utility functions represent pair-wise risk aversion. The relations with univariate stochastic orders are examined.     

A note on portfolio selection and stochastic dominance

This note provides new and simpler conditions ensuring that, when one portfolio dominates another via stochastic dominance, a decision maker prefers the first one. The conditions are derived for the case of third-order stochastic dominance and for the general case of Nth-order stochastic dominance.     

An improved bootstrap test of stochastic dominance

We propose a new method of testing stochastic dominance that improves on existing tests based on the standard bootstrap or subsampling. The method admits prospects involving infinite as well as finite dimensional unknown parameters, so that the variables are allowed to be residuals from nonparametric and semiparametric models. The proposed bootstrap tests have asymptotic sizes that are less than or equal to the nominal level uniformly over probabilities in the null hypothesis under regularity conditions. This paper also characterizes the set of probabilities so that the asymptotic size is exactly equal to the nominal level uniformly. As our simulation results show, these characteristics of our tests lead to an improved power property in general. The improvement stems from the design of the bootstrap test whose limiting behavior mimics the discontinuity of the original test’s limiting distribution.     

Stochastic dominance and statistical comparisons of income distribution functions

This paper assesses the economic well being of the U.S. elderly population by applying asymptotically distribution-free statistical tests of stochastic dominance relations characterizing income distribution functions. The procedure is new and overcomes serious difficulties encountered in naive comparisons of summary statistics which are widely used in assessing economic well being. An important advantage of this approach is that it exploits all the information contained in the income distribution function. Application of the method to compare the elderly population to the entire U.S. population in the 1980's reveals evidence of a higher per capita level of economic well being among older Americans. The method can be applied to other socioeconomic groups and in other countries provided that appropriate micro data are available.     

Probabilistic sophistication,second order stochastic dominance and uncertainty aversion

We study the interplay of probabilistic sophistication, second order stochastic dominance and uncertainty aversion, three fundamental notions in choice under uncertainty. In particular, our main result, Theorem 2, characterizes uncertainty averse preferences that are probabilistically sophisticated, as well as uncertainty averse preferences that satisfy second order stochastic dominance. As a byproduct, Proposition 2 highlights a fundamental tension between probabilistic sophistication/second order stochastic dominance and uncertainty aversion in the presence of nontrivial unambiguous events.     

Testing for stochastic dominance using the weighted McFadden-type statistic

We demonstrate that when testing for stochastic dominance of order three and above, using a weighted version of the Kolmogorov–Smirnov-type statistic proposed by McFadden [1989. In: Fomby, T.B., Seo, T.K. (Eds.), Studies in the Economics of Uncertainty. Springer, New York, pp. 113–134] is necessary for obtaining a non-degenerate asymptotic distribution. Since the asymptotic distribution is complex, we discuss a bootstrap approximation for it in the context of a real application.     

Stable sets in matching problems with coalitional sovereignty and path dominance

We study von Neumann Morgenstern stable sets for one-to-one matching problems under the assumption of coalitional sovereignty (C), meaning that a deviating coalition of players does not have the power to arrange the matches of agents outside the coalition. We study both the case of pairwise and coalitional deviations. We argue further that dominance has to be replaced by path dominance (P) along the lines of van Deemen (1991) and Page and Wooders (2009). This results in the pairwise CP vNM set in the case of pairwise deviations and the CP vNM set in the case of coalitional deviations. We obtain a unique prediction for both types of stable sets: the set of matchings that belong to the core.     

Continua of stochastic dominance relations for unbounded probability distributions

Let P = {F,G,…} be the set of all probability distribution functions with support (0, ∞). An unrestricted stochastic dominance relation>_∞ is defined on P for each real ∞ 1, where F >_∞ G means that ^x_y = 0 (x - y)^{∞ - 1} dG(y) ^x_{n = 0}(x − y)^∞−1 dG(y) for all ∞ 0, with < for some x. These relations are partial orders that increase as ∞ increases with limit relation>_∞. A class U_∞ of utility functions u on (0, ∞) is defined in such a way that F >_∞ G iff udF > udG for all u ε U_∞. The U_∞ decrease as ∞ increases and have a non-empty intersection U_∞. Each u ε U_∞ is an increasing function that has derivatives of all orders that alternate in sign. Criteria which tell when F eventually dominates G in the sense of F >_∞ G are noted. Comparisons with bounded stochastic dominance results are made in several places.     

Roots and effects of financial misperception in a stochastic dominance framework

This work deals with the issue of investors’ irrational behavior and financial products’ misperception. The theoretical analysis of the mechanisms driving erroneous assessment of investment performances is explored. The study is supported by the application of Monte Carlo simulations to the remarkable case of structured financial products. Some motivations explaining the popularity of these complex financial instruments among retail investors are also provided. In particular, investors are assumed to compare the performances of different projects through stochastic dominance rules. Unreasonably and in contrast with results obtained by the application of the selected criteria, investors prefer complex securities to standard ones. In this paper, introducing a new definition for stochastic dominance which presents asymmetric property, we provide theoretical and numerical results showing how investors distort stochastic returns and make questionable investment choices. Results are explained in terms of framing and representative effects, which are behavioral finance type arguments showing how decisions may depend on the way the available alternatives are presented to investors.     

Stochastic volatility with leverage: Fast and efficient likelihood inference

This paper is concerned with the Bayesian analysis of stochastic volatility (SV) models with leverage. Specifically, the paper shows how the often used Kim et al. [1998. Stochastic volatility: likelihood inference and comparison with ARCH models. Review of Economic Studies 65, 361–393] method that was developed for SV models without leverage can be extended to models with leverage. The approach relies on the novel idea of approximating the joint distribution of the outcome and volatility innovations by a suitably constructed ten-component mixture of bivariate normal distributions. The resulting posterior distribution is summarized by MCMC methods and the small approximation error in working with the mixture approximation is corrected by a reweighting procedure. The overall procedure is fast and highly efficient. We illustrate the ideas on daily returns of the Tokyo Stock Price Index. Finally, extensions of the method are described for superposition models (where the log-volatility is made up of a linear combination of heterogenous and independent autoregressions) and heavy-tailed error distributions (student and log-normal).     

Testing for infinite order stochastic dominance with applications to finance, risk and income inequality

In this paper we develop a test of infinite order degree stochastic dominance based on the use of the empirical Laplace transform function. Two applications are considered. One uses the income data of Anderson (Econometrica 64:1183–1193, 1996) and derives results consistent with his. In the other application we examine the dominance between the U.S. and U.K. stock markets. Using data on the S&P 500 and the FTALL-Share we show that the U.S. displays infinite order degree stochastic dominance of the U.K.

Load and no-load mutual fund dynamics during the 1987 market crash: A stochastic dominance analysis

Using a large sample of equity mutual fund returns, we compare performance of load and no-load funds during the 1987 crash. Differences in return distributions, particularly in the higher moments when the market was under stress, suggest a greater use of portfolio insurance by no-load fund managers. Using stochastic dominance, we find that load and no-load funds performed equally well before the crash. No-load returns dominated load fund returns during the crash. Load fund returns dominate after the crash. Over the entire month, no-load funds dominate. We attribute this to investor behavior motivated by the lack of a front-end load.     

Performance measurement and best-practice benchmarking of mutual funds: combining stochastic dominance criteria with data envelopment analysis

We propose a method for mutual fund performance measurement and best-practice benchmarking, which endogenously identifies a dominating benchmark portfolio for each evaluated mutual fund. Dominating benchmarks provide information about efficiency improvement potential as well as portfolio strategies for achieving them. Portfolio diversification possibilities are accounts for by using Data Envelopment Analysis (DEA). Portfolio risk is accounted for in terms of the full return distribution by utilizing Stochastic Dominance (SD) criteria. The approach is illustrated by an application to US based environmentally responsible mutual funds.     

Exploring efficient sets and equilibria on risky assets with APABO DSS prototype version 2.1

ABAPO 2.1 (Assets/Business(es) Analyser and Portfolio Optimizer) is a DSS prototype for portfolio managers. It assists the decision maker in two important stages of his task. First, it provides an integrated synthesis of the returns scenario in order to support the decision maker in the selection of the assets to retain, in accordance with his strategy. Second, on the retained assets it computes and shows the E- efficient solutions. For each efficient portfolio ABAPO supplies immediately a lot of information that should help the decision maker to single out the portfolio which fits his goals best.ABAPO works on an integration of elements derived from the Portfolio Selection Theory, the Capital Asset Pricing Model, the Utility Theory. ABAPO uses the principal component, the univariate, the bivariate regression and the correlation analysis, a parametric quadratic programming model and an algorithm based on the critical line method.ABAPO is implemented in C language under AIX operating system and runs on a IBM RT PC 6150. It is a ductile, interactive procedure that involves graphics in order to increase the efficiency and the effectiveness of the what if simulations.This version 2.1 improves upon the previous one by the evaluation of the Security Market Line, some statistics on the returns of the efficient portfolios, the evaluation of the quadratic indifference curves associated to the efficient portfolios.Research supported by the Italian National Research Council (CNR), grant n. 88.03065.10. ABAPO DSS 2.1 prototype has been developed with the support of the CNR (grant n. 87.1210.10), of the Banca Popolare di Verona, of the Cassa di Risparmio di Verona Vicenza e Belluno, of the Società Cattolica di Assicurazione, at the Istituto di Matematica, Università di Verona.     

Rearrangements and sequential rank order dominance

Distributive analysis typically involves comparisons of distributions where individuals differ in more than just one attribute. In the particular case where there are two attributes and where the distribution of one of these two attributes is fixed, one can appeal to sequential rank order dominance for comparing distributions. We show that sequential rank order domination of one distribution over another implies that the dominating distribution can be obtained from the dominated one by means of a finite sequence of favourable permutations, and conversely. We provide two examples where favourable permutations prove to have interesting implications from a normative point of view.     

Linear operators,time dominance and IRR

This paper studies the duality underlying the notion of time dominance among cash flows, by providing a dual characterization which is based on linear structures. This approach is prone to meaningful interpretations of both the same notion of time dominance and the well known results concerning it. In particular, we provide a general outline of the links interconnecting time dominance and the internal rate of return by following an alternative course as against other authors.

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Sommario Questo lavoro studia la dualità che sottende alla nozione di dominanza temporale per flussi di cassa, fornendo una caratterizzazione duale di tale relazione d'ordine. L'approccio proposto, basato sull'uso di strutture lineari, permette interpretazioni significative sia del concetto stesso di dominanza temporale, sia di alcuni noti risultati attinenti ad essa. In particolare, con riferimento a quest'ultimo aspetto, si fornisce, seguendo una via alternativa rispetto alla letteratura, un quadro generale dei legami intercorrenti tra dominanza temporale e tasso interno di rendimento.

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Financial support from M.U.R.S.T. and Bocconi University is gratefully acknowledged.