Third-order Stochastic Dominance Research Articles

Likelihood Ratio Tests for Lorenz Dominance

In testing hypotheses pertaining to Lorenz dominance (LD), researchers have examined second- and third-order stochastic dominance using empirical Lorenz processes and integrated stochastic processes with the aid of bootstrap analysis. Among these topics, analysis of third-order stochastic dominance (TSD) based on the notion of risk aversion has been examined using crossing (generalized) Lorenz curves. These facts motivated the present study to characterize distribution pairs displaying the TSD without second-order (generalized Lorenz) dominance. It further motivated the development of likelihood ratio (LR) goodness-of-fit tests for examining the respective hypotheses of the LD, crossing (generalized) Lorenz curves, and TSD through approximate Chi-squared distributions. The proposed LR tests were assessed using simulated distributions, and applied to examine the COVID-19 regional death counts of bivariate samples collected by the World Health Organization between March 2020 and February 2021.

Incomplete risk-preference information in portfolio decision analysis

Portfolio decision analysis models support decisions on the allocation of resources among assets with uncertain outcomes (e.g., investments, projects or stocks). These models require information on the decision maker’s risk-preferences which can be difficult to obtain in practice. Stochastic dominance criteria show promise in this regard as they can compare portfolios without exact specification of risk-preferences, but the current literature lacks practical approaches for generating the efficient frontier, i.e., the set of those portfolios that are not stochastically dominated by any other portfolio. We address this gap by developing models to identify sets of portfolios that are efficient in the sense of second- or third-order stochastic dominance (SSD, TSD). These models provide novel insights into the composition of portfolios belonging to the efficient frontier by, e.g., identifying those assets that are included in all efficient portfolios. Moreover, the identification of the efficient frontier makes it possible to utilize additional information on the decision maker’s risk-preferences to further reduce the set of admissible portfolio alternatives, and to analyze the implications this information has on the amount of capital that should be allocated to each individual asset. We illustrate the usefulness of these models with applications in project portfolio selection and financial portfolio diversification.

Dominance criteria on grids for measuring competitive balance in sports leagues

The paper proposes a dominance criterion that assesses whether a seasonal outcome of a sports league is more balanced than another. This criterion is based on a novel third-order stochastic dominance defined on finite sets of evenly spaced seasonal points (seasonal grids), called downward seasonal balance (DSB). The DSB criterion makes the same assessments as the well-known Lorenz criterion. However, the converse is not true: The DSB criterion makes assessments even in cases where the Lorenz criterion cannot. The former is then less incomplete than the latter. The assessments of the DSB criterion reflect the unanimity of a class of competitive balance indices. A seasonal outcome is more balanced than another according to the DSB criterion if and only if every index of the class agrees. Such a class is axiomatically characterized so that the indices place at least as much emphasis on the balance between leading competitors as on the balance occurring among the nonleading competitors. An empirical application provides comparisons of seasonal outcomes of the five most competitive soccer leagues in Europe from 2014–2015 to 2018–2019.

Interval-based stochastic dominance: theoretical framework and application to portfolio choices.

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 second- to 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.

The relative efficiency of option hedging strategies using the third-order stochastic dominance

The relative efficiency of option hedging strategies using the third-order stochastic dominance

A Stochastic Dominance-Based Approach for Hotel Selection under Probabilistic Linguistic Environment

Online reviews of hotels reflect tourist perception and evaluation, which are becoming an important perspective of studying hotel selection. In this paper, we prefer to use a probabilistic linguistic term set (PLTS) to fully reveal evaluation grades and the corresponding probability distribution in the online reviews of hotels. In this way, we propose a novel stochastic dominance-based approach based on stochastic dominance degrees of PLTSs and a stochastic multi-criteria acceptability analysis (SMAA) method that tolerates missing information. Among them, first-, second-, and third-order stochastic dominance degrees of PLTSs are calculated on the premise that the dominance relationships between PLTSs can be defined based on first-, second-, and third-order stochastic dominance rules of PLTSs. Based on these basic researches, five hotels are selected as alternatives in our case study to verify the validity and feasibility of the proposed approach. In the end, data analysis illustrates the influence of parameter and linguistic scale functions and how to choose appropriate parameter values. Furthermore, comparative analysis with other methods shows the stability of the proposed approach.

New development on the third-order stochastic dominance for risk-averse and risk-seeking investors with application in risk management

This paper develops new financial theory to link the third-order stochastic dominance (TSD) for risk-averse and risk-seeking investors and provide illustration of application in risk management. We present some interesting new properties of TSD for risk-averse and risk-seeking investors. We show that the means of the assets being compared should be included in the definition of TSD for both investor types. We also derive the conditions on the variance order of two assets with equal means for both investor types and extend the second-order SD reversal result of Levy and Levy (Manag Sci 48(10):1334–1349, 2002) to TSD. We apply our results to analyze the investment behaviors on traditional stocks and internet stocks for both risk averters and risk seekers.

New Development on the Third Order Stochastic Dominance for Risk-Averse and Risk-Seeking Investors with Application in Risk Management

This paper develops new financial theory to link the third order stochastic dominance for risk-averse and risk-seeking investors and provide illustration of application in risk management. We present some interesting new properties of third order stochastic dominance (TSD) for risk-averse and risk-seeking investors. We show that the means of the assets being compared should be included in the definition of TSD for both investor types. We also derive the conditions on the variance order of two assets with equal means for both investor types and extend the second order SD (SSD) reversal result of Levy and Levy (2002) to TSD. We apply our results to analyze the investment behaviors on traditional stocks and internet stocks for both risk averters and risk seekers.

Linear Algorithm for Portfolio Optimization with Third-Order Stochastic Dominance

We propose a novel linear approximation of expected utility. The approximation guides us as we transfer the traditional quadratic dependence of third-order stochastic dominance (TSD) into an equivalent linear system. The finding also shows a dual relationship between traditional low partial moment condition and the efficient condition of Post (2003). Based on the transformation, we develop a linear algorithm of TSD. Furthermore, we refine the superconvex TSD of Post and Kopa (2017) and introduce a corresponding linear system. The portfolio optimization performances of various criteria are also investigated.

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.

Strong Increases in Downside Risk Aversion

A ranking of risk preferences is of economic interest insofar as it leads to unambiguous comparative statics predictions, and for this to be the case, the ranking must be a strict partial ordering. The ranking by greater risk aversion meets this demand at the second order, and yields a variety of well-known predictions concerning the effect of greater risk aversion on demands for insurance and risky assets, among many other applications. There has been less success at the third order, where ranking preferences by aversion to downside risk has not produced a strict partial ordering. The problem is that account has not been taken of the fact that an increase in downside risk aversion must induce changes in risk aversion as well. We propose a definition of stronger downside risk aversion that does yield a strict partial ordering by requiring a nested increase in both second- and third-order risk aversion, so that v is more strongly downside risk averse than u if v is more risk averse and more downside risk averse than u. We demonstrate that v being more strongly downside risk averse than u is characterized by v never liking any change in the probability distribution for y that induces a third-order stochastic dominance deterioration in the distribution for u(y). We apply the definition to obtain intuitive comparative statics predictions in the precautionary saving problem, and relate the definition to alternatives proposed in the literature.

Mean-Variance and Stochastic Dominance Analysis of Global Exchange-Traded Funds

We employ the stochastic dominance (SD) approach that utilizes the entire return distribution to rank the performance of exchange-traded funds as traditional mean-variance and CAPM approaches may be inappropriate given the nature of non-normal returns. We find second and third-order stochastic dominance relationships amongst the assets and conclude that by investing in higher-order dominant funds, risk-averse investors can maximize their expected utilities but not their wealth. In addition, we find the common characteristic for most pairs of funds is that one fund is preferred to another in the negative domain whereas the preference reverses in the positive domain. We conclude that the stochastic dominance approach is more appropriate compared with traditional approaches as a filter in exchange-traded fund selection. Compared with traditional approaches, the SD approach, not only is assumption free, but also provides greater insights to the performance and risk inherent in an exchange-traded fund’s track record.

Stochastic dominance relationships between stock and stock index futures markets: International evidence

In this paper, we first modify the stochastic dominance (SD) test for risk averters proposed by Davidson and Duclos (2000) to be the SD test for risk seekers. We then adopt both tests to examine the SD relationships between stock indices and their corresponding index futures for 10 countries. The sample contains data from 6 developed countries and 4 developing countries. The study proposes that there should be no SD relationship between spot and futures markets in developed financial markets in which arbitrage opportunities (both pure and quasi) are rare and short-lived. However, we expect that SD relationships could be found in emerging financial markets that have more impediments to arbitrage. Consistent with this conjecture, our study finds that there are no SD relationships between spot and futures markets in the mature market sample, implying that these markets could be efficient. However, for the emerging markets, spot dominates futures for risk averters, while futures dominate spot for risk seekers in the second- and third-order SD. These results indicate that there are potential gains in expected utilities for risk averters (seekers) if they switch their investment from futures (spot) to spot (futures) in the emerging markets.

Modeling heterogeneous risk-taking behavior in route choice: A stochastic dominance approach

This paper proposes a unified approach to modeling heterogonous risk-taking behavior in route choice based on the theory of stochastic dominance (SD). Specifically, the first-, second-, and third-order stochastic dominance (FSD, SSD, TSD) are respectively linked to insatiability, risk-aversion and ruin-aversion within the framework of utility maximization. The paths that may be selected by travelers of different risk-taking preferences can be obtained from the corresponding SD-admissible paths, which can be generated using general dynamic programming. This paper also analyzes the relationship between the SD-based approach and other route choice models that consider risk-taking behavior. These route choice models employ a variety of reliability indexes, which often make the problem of finding optimal paths intractable. We show that the optimal paths with respect to these reliability indexes often belong to one of the three SD-admissible path sets. This finding offers not only an interpretation of risk-taking behavior consistent with the SD theory for these route choice models, but also a unified and computationally viable solution approach through SD-admissible path sets, which are usually small and can be generated without having to enumerate all paths. A generic label-correcting algorithm is proposed to generate FSD-, SSD-, and TSD-admissible paths, and numerical experiments are conducted to test the algorithm and to verify the analytical results.

Modeling Heterogeneous Risk-Taking Behavior in Route Choice: A Stochastic Dominance Approach

This paper proposes a unified approach to modeling heterogonous risk-taking behavior in route choice based on the theory of stochastic dominance (SD). Specifically, the first-, second-, and third-order stochastic dominance (FSD, SSD, TSD) are respectively linked to insatiability, risk-aversion and ruin-aversion within the framework of utility maximization. The paths that may be selected by travelers of different risk-taking preferences can be obtained from the corresponding SD-admissible paths, which can be generated using general dynamic programming. This paper also analyzes the relationship between the SD-based approach and other route choice models that consider risk-taking behavior. These route choice models employ a variety of reliability indexes, which often make the problem of finding optimal paths intractable. We show that the optimal paths with respect to these reliability indexes often belong to one of the three SD-admissible path sets. This finding offers not only an interpretation of risk-taking behavior consistent with the SD theory for these route choice models, but also a unified and computationally viable solution approach through SD-admissible path sets, which are usually small and can be generated without having to enumerate all paths. A generic label-correcting algorithm is proposed to generate FSD-, SSD-, and TSD-admissible paths, and numerical experiments are conducted to test the algorithm and to verify the analytical results.

Stochastic dominance of portfolio insurance strategies

The purpose of this article is to analyze and compare two standard portfolio insurance methods: Option-based Portfolio Insurance (OBPI) and Constant Proportion Portfolio Insurance (CPPI). Various stochastic dominance criteria up to third order are considered. We derive parameter conditions implying the second- and third-order stochastic dominance of the CPPI strategy. In particular, restrictions on the CPPI multiplier resulting from the spread between the implied volatility and the empirical volatility are analyzed.

Stochastic Dominance of Portfolio Insurance Strategies - OBPI versus CPPI

The purpose of this article is to analyze and compare two standard portfolio insurance methods: Option-based Portfolio Insurance (OBPI) and Constant Proportion Portfolio Insurance (CPPI). Various stochastic dominance criteria up to third order are considered. We derive parameter conditions implying the second- and third-order stochastic dominance of the CPPI strategy. In particular, restrictions on the CPPI multiplier resulting from the spread between the implied volatility and the empirical volatility are analyzed.

TSD-consistent performance assessment of mutual funds

This paper presents three new data envelopment analysis-based approaches to assess the relative efficiency of mutual funds (MFs). Each model considers an appropriate risk measure as input and an appropriate return measure as output. The risk and return measures have been chosen so that the proposed models are consistent with third-order stochastic dominance (TSD) rules. This means that the MFs found efficient by the proposed models are also, in a necessity condition sense, TSD efficient and therefore of highest consideration for all non-satiated, risk averse investors that also have decreasing absolute risk aversion. The proposed approach is illustrated with real data on a set of Spanish MFs and compared with existing approaches from the literature based on Mean–Variance and Mean–Variance–Skewness models.

More on the measurement of inequality

The article, written in 1973, examines what comparisons of income distributions can be made when Lorenz curves cross, employing the concept of third-order stochastic dominance.

Tony Atkinson's 1973 manuscript “More on the measurement of inequality”

When Tony Atkinson submitted his paper “On the Measurement of Inequality” to the newly-introduced Journal of Economic Theory in 1969, he cannot have imagined that it would become one of the most widely-cited papers in the economics research literature. As the Social Science Citations Index reveals, Atkinson [1] has been cited some 1,013 times by other authors in the last 37 years (50 times in 1996 alone, evidently a good year). Tony wrote a follow-up piece, “More on the Measurement of Inequality” in 1973, but somehow this never was published. We are delighted to be able to reproduce Tony's 1973 manuscript here, with his kind permission. “More on the Measurement of Inequality” is replete with intuition and applicability. We can now see how Tony was thinking, when pushing forward the implications of his measurement system. What is more remarkable, though, is that his follow-up manuscript is also replete with developments which only appeared in print, in the work of other authors, years later. Consider Tony's treatment of third-order stochastic dominance in the 1973 manuscript. He based its welfare rationale on Kolm's Principle of Diminishing Transfers, which itself did not enter the research literature until 1976 (in the first of Kolm's two Journal of Economic Theory papers that year, see [6, 7]). This kind of dominance—for utilities with a positive third derivative—permits prescriptions to be given for distributions whose Lorenz curves cross, and brings in the coefficient of variation as a sometimes-decisive statistic—yet this empirically significant development took many more years to see the light of day in any journal (see Shorrocks and Foster [9], Dardanoni and Lambert [2], Davies and Hoy [3, 4]). Tony refers to supplementary work on higher orders of dominance still, with some percipience. For published treatments of this topic, we had to wait until the papers of Menezes et al. [8] and Ekern [5] appeared. One of Tony's closest collaborators, Francois Bourguignon, upon learning of this project, commented: “I think it is an excellent idea to publish this piece, even though it makes some publications by some of us somewhat trivial. I remember very well how I was struck some J Econ Inequal (2008) 6:275–276 DOI 10.1007/s10888-007-9074-8