Abstract: Farinelli and Tibiletti (F-T) ratio, a general risk-reward performance measurement ratio, is popular due to its simplicity and yet generality that both Omega ratio and upside potential ratio are its special cases. The F-T ratios are ratios of average gains to average losses with respect to a target, each raised by a power index, p and q. In this paper, we establish the consistency of F-T ratios with any nonnegative values p and q with respect to first-order **stochastic** **dominance**. Second-order **stochastic** **dominance** does not lead to F-T ratios with any nonnegative values p and q, but can lead to F-T **dominance** with any p < 1 and q ≥ 1. Furthermore, higher-order **stochastic** **dominance** (n ≥ 3) leads to F-T **dominance** with any p < 1 and q ≥ n − 1. We also find that when the variables being compared belong to the same location-scale family or the same linear combination of location-scale families, we can get the necessary relationship between the **stochastic** **dominance** with the F-T ratio after imposing some conditions on the means. Our findings enable academics and practitioners to draw better decision in their analysis.

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This paper extends the theory between Kappa ratio and **stochastic** **dominance** (SD) and risk-seeking SD (RSD) by establishing several relationships between first- and higher- order risk measures and (higher-order) SD and RSD. We first show the sufficient rela- tionship between the (n + 1)-order SD and the n-order Kappa ratio. We then find that, in general, the necessary relationship between SD/RSD and the Kappa ratio cannot be established. Thereafter, we find that when the variables being compared belong to the same location-scale family or the same linear combination of location-scale families, we can get the necessary relationships between the (n + 1)-order SD with the n-order Kappa ratio when we impose some conditions on the means. Our findings enable academics and practitioners to draw better decision in their analysis.

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According to the von Neuman and Morgenstern (1944) expected utility theory, the func- tions for risk averters and risk seekers are concave and convex respectively, and both are increasing functions. In this context **stochastic** **dominance** (SD) theory has generated a rich and growing academic literature. Linking SD theory to the selection rules for risk averters under different restrictions on the utility functions include Quirk and Saposnik (1962), Fishburn (1964), Hanoch and Levy (1969), Whitmore (1970), Hammond (1974) and Tesfatsion (1976). Linking SD theory to the selection rules for risk seekers include Hammond (1974), Meyer (1977), Stoyan (1983), Wong and Li (1999), Anderson (2004), and Wong (2007).

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In this paper we tighten the Nth (N ≥ 3) **stochastic** **dominance** (hereafter NSD) option bounds by using the observed prices of concurrently expiring op- tions. We show that given the prices of a unit bond, underlying stock, and n option prices, the kth order **stochastic** **dominance** option bounds are given by a pricing kernel whose (N − 2)th derivative is (n/2)-segmented and piecewise constant if n is even or ((n + 1)/2)-segmented and piecewise constant if n is odd.

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successfully generalized the notion of **stochastic** **dominance**. The results that follow differ from that of Meyer in several respects. First, all families of utility functions with prudence measures that are pointwise bounded from below by a similar looking ratio for the transformation of the background risk space will experience increased risk aversion whenever the distribution undergoes a second degree spread in the transformed background risk. Secondly, all preferences with prudence measures that exceed the aforementioned bound will experience an increase in their expected marginal utilities under the deterioration in background risk. Finally, all **stochastic** dominating spreads will be occurring for transformations of background risks rather than background risks directly.

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domination relationship. Nonetheless, the almost SD rule developed by Leshno and Levy (2002) and Tzeng et al. (2012) could draw preference of A over B. However, though the almost SD rule developed by Leshno and Levy (2002) and others could draw preference for risk averters, but not risk seekers. To complete the theory of almost SD, we will develop the almost **stochastic** **dominance** concept for risk seekers. We modify from Example 2.1 to get the following example to illustrate the motivation:

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There are two major types of persons: risk averters and risk seekers. Markowitz (1952) and Tobin (1958) propose the mean-variance (MV) selection rules for risk averters and risk seekers. **Stochastic** **Dominance** (SD) is first introduced in mathematics by Mann and Whitney (1947) and Lehmann (1955). Quirk and Saposnik (1962), Hanoch and Levy (1969), and many others develop the theory of SD related to economics and develop the **stochastic** **dominance** rules for risk averters. On the other hand, Meyer (1977), Stoyan (1983), Wong (2007), and many others develop the **stochastic** **dominance** rules for risk seekers.

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Third-order **stochastic** **dominance** (TSD) is becoming an important area of research in finance. For example, Post, et al. (2015) developed and implemented linear formula- tions of convex **stochastic** **dominance** relations based on decreasing absolute risk aversion (DARA) for discrete and polyhedral choice sets in which DARA is related to the TSD. In addition, Post and Milos (2016) developed an optimization method for constructing investment portfolios that dominate a given benchmark portfolio in terms of TSD. Al- though risk averse investor behavior is the conventional assumption in most financial research, risk seeking behavior has long been recognized as an important element that should also be considered. For example, Friedman and Savage (1948) observed investors’ risk-seeking behavior to buy both insurance and lottery tickets. To circumvent this prob- lem, Markowitz (1952) suggested including convex functions in both the positive and the negative domains. Williams (1966) found evidence to get a translation of outcomes that produces a dramatic shift from risk aversion to risk seeking. Tobin (1958) proposed the mean-variance rules for risk seekers and Kahneman and Tversky (1979) observed risk- seeking behavior in the negative domain. In this paper we include risk-seeking behavior in our study of TSD. More specifically, we analyze TSD in the context of both risk-averse and risk-seeking investors and present some interesting new properties of TSD for both types of investor. Before we discuss our contribution to the literature, we first discuss the literature.

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In order to examine the change in the diversification level and benefit, the data is divided into two groups. Each group contains 100 monthly returns from each stock market. Next, the six portfolios are built for the study’s analysis. The first five portfolios are the stock index portfolios from the examined countries. Their monthly returns are directly from their stock index returns. The sixth portfolio is the equally weighted stock index portfolio built by investing the equally amount of money in each of the previous five portfolios. Hence, the monthly returns of the sixth portfolio are the mathematic average of the first five portfolios’ monthly returns. The mean-variance and the **stochastic** **dominance** technique are employed for the analysis as follow.

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Markowitz (1952b) proposes that decisions are based on change in wealth and individuals are risk averse for losses and risk seeking for gains as long as the possible outcomes are not very extreme, i.e., value functions are inverse S-shaped (concave in the losses and convex in the gains). For extreme outcomes, individ- uals are risk seeking for losses and risk averse for gains. Levy and Levy (2002) extend the work of Markowitz (1952b) to propose (second-degree) Markowitz **Stochastic** **Dominance** (MSD) for all inverse S-shaped functions.Wong and Chan (2008) further generalize second-degree MSD to the third-degree MSD, or TMSD, conditions: (i) R a b [G(z) − F (z)]dz ≥ 0, and (ii) R a x R a z [G(t) − F (t)]dtdz ≥ 0 for all x ∈ [a, 0], and R y b R z b [G(t)− F (t)]dtdz ≥ 0 for all y ∈ [0, b]. It is notable that Wong and Chan (2008) implicitly restrict that the second derivatives of the inverse S- shaped value functions are equal to zero at the reference point (zero). We next provide alternative SD conditions for increases in downside risk and downside risk aversion without requiring the second derivatives of the inverse S-shaped value functions to be zero at the reference point. Following Levy and Levy (2002), we omit PWFs.

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significant SD tests have recently been developed. For example, Barrett and Donald (2003) exploit a Kolmogorov-Smirnov-type test and Linton et al. (2005) further relax the i.i.d. assumption. In addition, Anderson (1996) and Davidson and Duclos (2000) develop SD tests that examine the underlying distributions at a finite number of grid points. Armed with these powerful tests, the SD approach becomes widely applicable. Most critically, and as detailed in the Technical Appendix, first-order **stochastic** **dominance** (FSD) of one investment prospect over another implies that regardless of one’s risk preferences, the dominant prospect is the preferred investment. We apply the Davidson and Duclos (2000) DD test to make this determination.

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This paper applies **stochastic** **dominance** (SD) tests to examine the **dominance** relationships between the futures and spot markets in Hong Kong. We also analyze the preferences for the risk averters, risk seekers, prospect investors, and Markowitz investors with further in dept of their positive and negative domains in these markets. We find that for the risk averters, spot dominates futures while for the risk seekers, futures dominate spot. This implies that the risk averters prefer to buy indexed stocks, while risk seekers are attracted to long index futures to maximize their expected utilities, but not necessary their wealth. We also conclude that in general, the prospect investors prefer spot in the positive domain and prefer futures in the negative domain while the Markowitz investors prefer spot in the negative domain and prefer futures in the positive domain.

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Abstract: This paper presents a decision making approach for mid-term scheduling of large industrial consumers based on the recently introduced class of **Stochastic** **Dominance** (SD)- constrained **stochastic** programming. In this study, the electricity price in the pool as well as the rate of availability (unavailability) of the generating unit (forced outage rate) is considered as uncertain parameters. The self-scheduling problem is formulated as a **stochastic** programming problem with SSD constraints by generating appropriate scenarios for pool price and self-generation unit's forced outage rate. Furthermore, while most approaches optimize the cost subject to an assumed demand profile, our method enforces the electricity consumption to follow an optimum profile for mid-term time scheduling, i.e. three months (12 weeks), so that the total production will remain constant.

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priced [1] [2] [3] [4]. Most debates have focused on expensive out-of-the-money (hereafter, OTM) put options [5] [6]. However, the **stochastic** **dominance** (he- reafter, SD) literature has documented that index call options are also too ex- pensive. Specifically, [7] and [8] argued that Standard and Poor’s 500 (hereafter, S & P 500) Index call options are frequently overpriced in the sense that every rational agent can improve her expected utility by writing these call options that violate the SD upper bound. Hence, expensive index call options are also an un- solved puzzle in the finance literature.

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Ma, C., Wong, W.K. (2010) **Stochastic** **Dominance** and Risk Measure: A Decision-Theoretic Foundation for VaR and C-VaR, European Journal of Operational Research 207(2), 927-935. Markowitz, H.M. (1952). The utility of wealth. Journal of Political Economy 60, 151-156. Markowitz, H.M. (1959). Portfolio Selection. Wiley, New York.

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In the present paper, we have compared the CPPI and OBPI strategies, mainly with respect to the third stochas- tic **dominance** (TSD). We find that the CPPI method third order stochastically dominates the OBPI one for high implied volatility relatively to the empirical volatility. We have checked the TSD of the CPPI method compared to the OBPI method for low values of the drift weighted by high values of the multiplier. We have shown that the relation of SDT is rejected for the low values of the implicit volatility with respect to the statistical one. Fur- ther extensions could be based on the use of almost **stochastic** **dominance** as defined by Leshno and Levy [19], in order to extend the range of the multiple for which the CPPI dominates the OBPI.

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range, and employs Almost **Stochastic** **Dominance** to develop a simple **stochastic** model that can explain the empirically observed power-law wealth distribution. Benitez, Kuosmanen, Olschewski, and van Kooten [24] use Almost **Stochastic** **Dominance** to determine the minimal con- servation payments required to guarantee that the environmentally- preferred use of land dominates other less environmentally-preferred alternatives. Gasbarro, Wong, and Zumwalt [13] employ Almost Sto- chastic Dominating in ranking 18 country market indices. Huang [25] employs Almost **Stochastic** **Dominance** to derive bounds on the prices of various options. Levy, Leshno and Leibovich [26] investigate ASD experimentally.

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This paper first extends some well-known univariate **stochastic** **dominance** results to multivari- ate **stochastic** dominances (MSD) for both risk averters and risk seekers, respectively, to n order for any n ≥ 1 when the attributes are assumed to be independent and the utility is assumed to be additively and separable. Under these assumptions, we develop some properties for MSD for both risk averters and risk seekers. For example, we prove that MSD are equivalent to the expected-utility maximization for both risk averters and risk seekers, respectively. We show that the hierarchical relationship exists for MSD. We establish some dual relationships between the MSD for risk averters and risk seekers. We develop some properties for non-negative com- binations and convex combinations random variables of MSD and develop the theory of MSD for the preferences of both risk averters and risk seekers on diversification. At last, we discuss some MSD relationships when attributes are dependent and discuss the importance and the use of the results developed in this paper.

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In this paper, we explore the circumstances under which a terrorist decision-maker might be expected to choose the more risky and—generally potentially more damaging—of two attack methods. We use two related approaches: (1) cumulative prospect theory (CPT); and (2) **stochastic** **dominance**. The two approaches are linked. Any preference ordering obtained from CPT must not—and will not—violate first- order **stochastic** **dominance** (FSD). It can be shown that there are points within both frameworks that are associated with the decision-maker switching to the more risky alternative in a given pair of risky prospects. We call these points ‘trigger points’. Within CPT these trigger points are associated with reference points. When the decision-maker’s reference point is high enough, the more risky prospect is accorded the higher prospect value. Within **stochastic** **dominance** these trigger points are associated with the intersection of the cumulative distributions for each prospect. Convex incentives schedules applied at the intersection of the cumulative distributions will activate the trigger point. Trigger points emerge without violating FSD. In the case of CPT, trigger points emerge for prospects where no clear **dominance** exists to be violated. In the case of **stochastic** **dominance**, second-order **dominance** (SSD) has embedded within it the property that past the point of intersection of two cumulative distributions, the prospect dominated by SSD dominates its alternative by FSD 6 .

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Our three methods work with **stochastic** **dominance** techniques for ordinal and dichoto- mous variables in order to document whether autonomy comparisons across Indian states are robust to broad classes of individual welfare functions (each, embedding large varieties of composite indices). When the **dominance** conditions hold, the ensuing robust ordering (i.e. comparison) has an interpretation in terms of preference over lotteries based on these individual welfare ("utility") functions. Moreover, if our purpose is to rank Indian states in terms of the relative desirability of their autonomy distributions, then when the conditions hold, we do not need to choose, and justify the choice, among several potentially suitable composite indices. More specifically, our second and third methods relate to the counting approach (Townsend, 1979) now popular in the measurement of multidimensional poverty (e.g. in the UNDP’s "Multidimensional Poverty Index", see (Alkire and Santos, 2014)), but also being used in different areas of social science, including female autonomy itself (e.g. Alkire et al., 2013).

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