A state-of-the-art fund performance index: Higher-order omega and its consistency with almost stochastic dominance

Lu, Hengzhen; Zhang, Yingying; Xiao, Ling; Dhesi, Gurjeet Article A state-of-the-art fund performance index: Higherorder omega and its consistency with almost stochastic dominance Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Lu, Hengzhen; Zhang, Yingying; Xiao, Ling; Dhesi, Gurjeet (2022) : A state-ofthe-art fund performance index: Higher-order omega and its consistency with almost stochastic dominance, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 15, Iss. 10, pp. 1-20, https://doi.org/10.3390/jrfm15100438 This Version is available at: https://hdl.handle.net/10419/274958 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Citation: Lu, Hengzhen, Yingying Zhang, Ling Xiao, and Gurjeet Dhesi. 2022. A State-of-the-Art Fund Performance Index: Higher-Order Omega and Its Consistency with Almost Stochastic Dominance. Journal of Risk and Financial Management 15: 438. https:// doi.org/10.3390/jrfm15100438 Academic Editors: Thanasis Stengos and Jong-Min Kim Received: 3 July 2022 Accepted: 22 September 2022 Published: 28 September 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Journal of Risk and Financial Management Article A State-of-the-Art Fund Performance Index: Higher-Order Omega and Its Consistency with Almost Stochastic Dominance Hengzhen Lu 1, Yingying Zhang 1,*, Ling Xiao 2,* and Gurjeet Dhesi 3 1College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China 2School of Business and Management, Royal Holloway, University of London, Egham TW20 0EX, UK 3Group of Researchers Applying Physics in Economy and Sociology (GRAPES), Beauvallon, Rue de la Belle Jardinière, 483, 0021 Sart Tilman, Angleur, B-4031 Liège, Belgium *Correspondence: [email protected] (Y.Z.); [email protected] (L.X.) Abstract: This paper provides a mathematical proof and theoretical analysis of the one-to-one consistency between higher-order Omega and Almost Stochastic Dominance rules when evaluating fund performance. The consistency between higher-order Omega and Almost Nth-degree Stochastic Dominance reinforces the effectiveness of applying the higher-order Omega function in fund performance measurement, as the Almost Stochastic Dominance rules are more likely to be observed in real life. This study also clarifies that the higher-order Omega decreases when threshold Lincreases. The ranking of funds based on higher-order Omega changes at different thresholds. Hence, it is critical to specify the Lso that the consistency holds. Through evaluating the performance of eleven U.S. funds between 2010 and 2020, we demonstrate the applications of the Nth-order Omega in the concept of Almost Stochastic Dominance rules. Furthermore, the empirical results also show the superiority of the Nth-order Omega over the traditional fund performance measure, i.e., Sharpe ratio and the lower-order Omega. The ranking of fund performance based on higher-order Omega is consistent with Almost Stochastic Dominance rules. Keywords: higher-order Omega; Almost Stochastic Dominance rules; fund performance evaluation 1. Introduction Academics and investors are still skeptical about the effectiveness of traditional fund performance measures. The performance indexes such as the Sharpe ratio and Treynor ratio (Sharpe 1966;Treynor 1965) work only if the evaluated funds follow a normal distribution and/or investors make an investment decision based on mean-variance decision making. Furthermore, it is well-documented that higher-order moments such as skewness and kurtosis also play an important role in investors’ preference decision making, but they are often neglected (Harvey and Siddique 2000;Guidolin and Timmermann 2008;Stutzer 2000). Keating and Shadwick (2002) first introduced the Omega function to measure fund performance to address the shortcomings of the traditional fund performance measures. The Omega function is defined as the probability-weighted ratio of potential gains over possible losses at a given level of expected return. It uses the complete information of assets and portfolio return distributions (Chan and Nadarajah 2019;Yang et al. 2021). van Dyk et al. (2014) reported that Omega ratio provides useful information to investors in addition to that provided by the Sharpe ratio alone. Hence, the Omega function could perform effectively even if the returns do not follow a normal distribution. Moreover, it is proved that the Omega function provides a consistent ranking as the First-order Stochastic Dominance (hereafter, FSD) (Benhamou et al. 2019). Bi et al. (2019) extend the Omega function to the Nth-order Omega based on the concept of Almost Acceptance Dominance (hereafter, AAD). The Nth-order Omega includes the Omega function as a special case when Nequals 1. The J. Risk Financial Manag. 2022,15, 438. https://doi.org/10.3390/jrfm15100438 https://www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2022,15, 438 2 of 20 Nth-order Omega not only retains the best properties of Omega, that no assumption of return distribution is required, but also takes higher moments into account. Performance indexes including Sharpe ratio, Omega function, and higher-order Omega, are based on evaluation of the trade-off between return and risk (Heufer 2014). Another popular framework of fund performance evaluation consists of Stochastic Dominance, Almost Stochastic Dominance, and a series of dominance rules that provide a ranking based on two assets return distributions. According to the dominance rules, investors will choose one risky asset over the other according to the ranking order. However, it is well-known that the Sharpe ratio is not consistent with the FSD (Klar and Müller 2018). The investors could make an irrational decision, i.e., choose the investment with a smaller return by maximizing the Sharpe ratio. Therefore, it is important to examine if there is one-to-one consistency between the Omega type of measure and the (Almost) Stochastic Dominance rules. No conclusion has been drawn from existing literature. Some studies support the consistency between the Omega ratio and Stochastic Dominance. For example, Fong (2016) showed the consistency between the Omega ratio and Second-order Stochastic Dominance (thereafter, SSD) by using a simple deformation of Omega. Their results suggest that all loss-averse and risk-averse investors could adopt the Omega ratio to make the same investment decision which is also suggested by the SSD rule. Guo et al. (2017) find that the Omega ratio is consistent with SSD rules when the threshold of the Omega function is less than the average return of a higher-return portfolio. Their findings were applied to test the efficiency of the Hongkong estate market. Similar conclusions are drawn by Guo et al. (2018) that the Omega ratio criterion is consistent with the Stochastic Dominance. Their findings were supported by testing fund data. However, Klar and Müller (2018) claimed that there was a consistent relationship between Omega function and FSD, but no consistency with SSD. In addition, they also used expectiles (Bellini et al. 2016) to prove that the Omega index is consistent with fractional stochastic order (1 + γ ), which was proposed by Muller et al. (2017). Balder and Schweizer (2017) clarified when Omega ratio was consistent with Second-order Stochastic Dominance and when it was not. They also pointed out that in order to avoid the consistency problem, any Omega ratio below one should be discarded like a negative Sharpe ratio. Caporin et al. (2018) critiqued that the Omega function has been used excessively and inappropriately. They claimed that the relationship between threshold and mean return would affect the effectiveness of the Omega index, that is, the smaller threshold accounts for less variations between the Omega and SSD rankings. However, most of the extant literature is restricted to lower-order Stochastic Dominance, i.e., FSD or SSD. Bi et al. (2019) extended Klar and Müller’s (2018) work to a higher-order framework and proved the one-to-one consistency between higher-order Omega and its corresponding higher-order Stochastic Dominance. However, there are very limited discussions on the consistency of higher-order Omega with Stochastic Dominance criteria since the development of higher-order Omega. Why is it critical to prove a consistent relationship between higher-order Omega and the series rules of Stochastic Dominance? Stochastic Dominance suffers the well-known partial order problem, only employing partial information on the investor’s preferences such as risk aversion. With Omega types of measures, including higher-order Omega (Bi et al. 2019), Omega (Keating and Shadwick 2002) can generate a complete order through analytical calculations. If the consistency between the two is proved, it does not only imply the use of Stochastic Dominance among portfolios but also confirms the decision making based on higher-order Omega. However, Stochastic Dominance rarely could be observed in real life compared to Almost Stochastic Dominance (Tzeng et al. 2013). To address this limitation, we turn to the Almost Stochastic Dominance rule. There is always an Almost Stochastic Dominance relation existing between the two portfolios corresponding to the degree no matter what the value of Nis (Tzeng et al. 2013). If there is consistency found between higher-order Omega and Almost Stochastic Dominance, then the application of higher-order Omega would become more feasible. J. Risk Financial Manag. 2022,15, 438 3 of 20 To our best knowledge, there is no research examining consistency between higherorder Omega and Almost Stochastic Dominance yet. This paper contributes to knowledge by exploring whether the higher-order Omega is consistent with Almost Stochastic Dominance. We also demonstrate the application of the higher-order Omega using U.S. fund data and provide empirical evidence to show the consistency between higher-order Omega and Almost Stochastic Dominance. The remainder of the paper is structured as follows. Section 2describes the relevant theoretical foundations, including Stochastic Dominance theory, Almost Stochastic Dominance theory, and higher-order Omega’s theoretical foundation. Section 3focuses on the conjecture and mathematical proof of the relationship between higher-order Omega and Almost Stochastic Dominance. In this section, we prove that the First-order Omega is consistent with Almost First Stochastic Dominance (thereafter, AFSD) when the threshold is in [L1,L2] . L1 and L2 are arithmetic expressions of the thresholds Lof Nth-order Omega and statistical value of Nth-degree Almost Stochastic Dominance. Furthermore, we also find the consistency for higher-order when Nis greater than 1 under certain conditions. Section 4applies the results from Section 3to evaluate the performance of real fund data based on the First-, Second-, and Third-order Omega index and the corresponding First-, Second-, and Third-order Almost Stochastic Dominance. Finally, conclusions are drawn, and we summarize that the consistency of higher-order Omega and Almost Nth-order Stochastic Dominance proves the superiority of this approach when evaluating and ranking more than two portfolios. 2. Definitions and Theories Bi et al. (2019) proposed a higher-order Omega index, which is free of the utility function form or/and distribution assumptions. This index is rooted in the framework of acceptance dominance. Bi et al. (2019) provide a theoretical foundation by showing the consistency between Stochastic Dominance and higher-order Omega. In this section, we examine the consistency between higher-order Omega and Almost Stochastic Dominance. In the subsequent sub-section, we provide a detailed description of the relevant theories including Stochastic Dominance theory, Almost Stochastic Dominance theory, and describe the higher-order Omega. 2.1. (Almost) Stochastic Dominance Assume variables X,Y, and their second-order integral are, F(2) Z(ϕ) = Zϕ −∞F(1) Z(t)dt,F(1) Z(ϕ) = FZ(ϕ) = P(Z<ϕ),Z=X,Y. According to Levy (2015), First-order Stochastic Dominance (FSD) is defined as: Definition 1. FSD: X ≥FSD Y if and only if F(1) X(ϕ)≤F(1) Y(ϕ)f or any ϕ∈R,F(1) X(b)≤F(1) Y(b). FSD correlates with the case where the first derivative of the utility function is nonnegative. Increasing utility means that the investor is always willing to have more wealth or return when all other things are equal. Regardless of investors’ risk attitude, i.e., riskneutral, risk-loving, or risk-averse, FSD is the valid criterion as long as the utility function has a non-negative first derivative. Similarly, assume the Nth-order integral of X,Yas follows, F(N) Z(ϕ) = Zϕ −∞F(N−1) Z(t)dt,Z=X,Y. Thus, Nth-degree Stochastic Dominance (thereafter, NSD), N≥2, could be defined. J. Risk Financial Manag. 2022,15, 438 4 of 20 Definition 2. NSD: X ≥NSD Y if and only if F(N) X(ϕ)≤F(N) Y(ϕ)f or any ϕ∈R,F(N) X(b)≤F(N) Y(b),n=1, 2, . . . , N. When N= 2, it is the definition of Second-order Stochastic Dominance (SSD). The SSD requires the second derivative of the utility function to be less than zero, that is, SSD is only effective for risk-averse investors. If the utility function’s second derivative is positive, Risk-seeker Stochastic Dominance (SRSD) is its effectiveness criterion (Bai et al. 2015;Levy 2015). When N= 3, Third-order Stochastic Dominance (TSD) requires the third derivative of investors’ utility function to be positive, namely, risk prudential. The theory of NSD has strict ordering. In other words, a higher-order SD such as SSD and TSD will be found provided the FSD is established. There is also a so-called pathological preference, based on which the utility function is limited to construct Almost Stochastic Dominance (ANSD). Adopting the interpretation of ANSD by Tzeng et al. (2013), define as follows, UN(εN)=(u (−1)n+1u(n)(x)>0, n=1, 2, . . . , N,and (−1)N+1u(N)≤in f n(−1)N+1uN(x)o1 εN−1,∀x) ˆ SN(FX,FY)=nx∈[x,x]:F(N) X>F(N) Yo, kF(N) X−F(N) Yk=Rx xF(N) X−F(N) Ydx. Definition 3. εN−ANSD : 0 <εN<0.5, N≥1, XεN−ANSDY, if and only if Rˆ SN(FX,FY)F(N) X(x)−F(N) Y(x)dx ≤εNkF(N) X−F(N) Yk, F(n) Y(x)−F(n) X(x)≥0, n=1, 2, . . . , N. Compared with Nth-degree Stochastic Dominance, Almost Nth-degree Stochastic Dominance eliminates unusual pathological preferences. It can be also seen from the mathematical expression as the critical condition to satisfy Stochastic Dominance. The condition specifies that one cumulative distribution is ‘absolutely greater’ than another for all distributions, while Almost Stochastic Dominance only requires ‘the relatively greater’ as a whole distribution. This also explains why the Almost Stochastic Dominance relationship is always found between any two portfolios, but not the Stochastic Dominance. This motivates us to prove the consistency between higher-order Omega and Almost Stochastic Dominance. 2.2. Higher-Order Omega Similarly, consider a random variable x∈[a,b] , where a< 0 <b , cumulative distribution function (CDF) and Nth-order integral of xis defined as: F(1)(x) = F(x) = P(X≤x),x∈[a,b], FN(x) = Rx aF(N−1)(t)dt. Let Obe a CDF of threshold L, similarly defined, O(N)(x) = Zx aO(N−1)(t)dt,and O(x)=0, x<L 1, x≥L. J. Risk Financial Manag. 2022,15, 438 5 of 20 Definition 4. The Nth-order Omega is defined as ΩN F(L) = RF(N)≤O(N)O(N)(x)−F(N)(x)dx RF(N)>O(N)F(N)(x)−O(N)(x)dx (1) where the denominator is the violating area of F that dominates O in terms of Nth-degree Stochastic Dominance, and it represents risk. The numerator is the part satisfying Nth-degree Stochastic Dominance, and it measures returns. When N = 1, First-order Omega equals the standard Omega function: Ω1 F(L) = RF(1)≤O(1)O(1)(x)−F(1)(x)dx RF(1)>O(1)F(1)(x)−O(1)(x)dx =Rb L[1−F(x)]dx RL aF(x)dx (2) As shown above, 1 −F(x) represents the probability that the return is higher than the threshold L, and F(x) represents the probability of being lower than the threshold L. It has been proven that Ω1 F(L) is a smooth monotone decreasing function for Land embodies all information of the CDF. Some equivalent representations are derived in the literature (as seen in Bi et al. 2019). The Nth-order Omega is equal to the following form: ΩN F(L) = (b−L)N−EF(b−e x)N EF[(S∗ N−e x)+]N−(S∗ N−L)N+1 (3) where (b−L)N calculates the higher-order moment of maximum return and threshold L, which represents the higher-order moment of the maximum excess return. Accordingly, EF(b−e x)N is processed in the same way. The numerator is positively related to the overall difference between O(N)(x) and F(N)(x) suggesting a difference in the probability of obtaining a higher return. It measures the return of F(x) under the threshold L. S∗ N is the intersection of O(N)(x) and F(N)(x) , and S∗ N≥L , EF[(S∗ N−e x)+]N is the generalized lower partial moment. It is the area of F(N)(x)≤O(N)(x) on behalf of the downside risk. The denominator shows the difference between the Nth-order lower partial moment (LPM) of O(N)(x) and F(N)(x) . The higher the denominator is, the larger the difference will be. In other words, Fhas a greater downward risk compared with the threshold L. In summary, the higher-order Omega measures returns by the upper part of the distribution and measures the risk by the lower part, namely, the LPM. It is an extension of Omega’s framework, a ratio of potential gains out of possible losses. Moreover, different-order Omega can be selected for investors with different preferences to reflect their diversified investment requirements. 3. Mathematical Proof of the Consistency between the Nth-Omega and Almost Stochastic Dominance Bi et al. (2019) show that there is one-to-one consistency between higher-order Omega and Nth-degree Stochastic Dominance. To ensure the rigor of the study, we will first discuss the relationship between NSD and higher-order Omega using a novel approach. As this study presents a new way of proving, this is spelt out as propositions in Section 3.1, with a new proof of propositions in Appendices Aand B. We further support the previous proposition of Bi et al. (2019), but using a different approach. This new mathematical proposition leads to our original contribution to proving Theorems 3 and 4. Section 3.2 presents the theoretical foundation to enhance the state-of-the-art performance index and its consistency with Almost Stochastic Dominance. See Theorems 1 and 2 in Section 3.2 and their proofs in Appendices Cand D. J. Risk Financial Manag. 2022,15, 438 6 of 20 3.1. Nth-Order Omega and Nth-Degree Stochastic Dominance There are some attempts to examine the relationship between the First-order Omega and FSD (as seen in Guo et al. 2018;Klar and Müller 2018;Fong 2016). And it is reported in Bi et al. (2019) that the Nth-order Omega is monotonic with Nth-degree Stochastic Dominance. Especially, if two portfolios could be ranked by NSD, the order would be kept by the Nth-order Omega. We start with proving their conclusions using a new approach. Our proofs could strengthen conclusions in prior literature. The relationship between First-order Omega and FSD can be expressed as follows: Proposition 1. Assume portfolio X and Y, mean return is EX and EY, when the threshold is L, First-order Omega is Ω1 X(L),Ω1 Y(L). If X ≥FSD Y,Ω1 X(L)≥Ω1 Y(L)for any L ∈R. This is also true in Second-order Omega and SSD. Proposition 2. Assume portfolio X and Y, Second-order Omega is Ω2 X(L) , Ω2 Y(L) . If X≥SSD Y,Ω2 X(L)≥Ω2 Y(L)for any L ∈R. Proof is as in Appendix A. This consistency relationship can be further extended to the higher-order Omega. Proposition 3. Assume portfolios X and Y, the Nth-order Omega are ΩN X(L) , ΩN Y(L) . If X≥NSD Y,ΩN X(L)≥ΩN Y(L)for any L ∈R. Proof is as in Appendix B. The relationship between NSD and Nth-order Omega could be established unconditionally. If portfolio Xdominates portfolio Yby NSD, Y’s Nth-order Omega must be lower than portfolio X’s. When N= 1, this relationship could be used by all investors, for that first-order requires investors to hold, the more, the better. When N= 2, investors are riskaverse, and Proposition 2 holds. Proposition 3 is established to meet various investment needs, for example, when N= 3, investors are risk-prudential, and when N= 4 they are risk-temperance. However, it is not set up that if ΩN X(L)≥ΩN Y(L) , X≥NSD Y . Take a counterexample. When N=1, L=0, O(x) = 1(x≥0). If Ω1 X(0)≥Ω1 Y(0), Rb 00−F(1) X(ϕ)dϕ R0 aF(1) X(ϕ)dϕ ≥Rb 00−F(1) Y(ϕ)dϕ R0 aF(1) Y(ϕ)dϕ It has no relationship with X≥FSD Y,Rϕ aF(1) X(t)dt ≤Rϕ aF(1) Y(t)dt. As Propositions 2 and 3 show that NSD is a sufficient condition of Nth-order Omega, we turn to Almost Nth-degree Stochastic Dominance (ANSD) in the subsequent sub-section. That is, replacing NSD with ANSD is to relax sufficient conditions in order to be closer to Nth-order Omega. 3.2. Nth-Order Omega and Almost Nth-Degree Stochastic Dominance As discussed above, when the Nth-order Omega value ordering is the same as NSD, there exists NSD between two portfolios. However, we note that vice versa is not true. If the Nth-order Omega of one portfolio is greater than that of another, it does not necessarily mean that there is NSD between them. We hope to remove this prerequisite for a wider application of higher-order Omega. The possible solution could be Almost Stochastic Dominance. Almost Nth-degree Stochastic Dominance is less restricted than Nth-degree Stochastic Dominance. Compared to ANSD, it is hard to find that there exists a lower-order SD relationship between two portfolios. The lower-order SD is significantly perfect, and a large number of portfolios tend to be similar. Thus, we could not compare the dominance J. Risk Financial Manag. 2022,15, 438 7 of 20 rules just by FSD, SSD, or even TSD. The higher-order process cannot solve this problem because we usually use only N ≤ 4. The higher-order process means more requirements of preferences, and they are more difficult to calculate. By exploring whether the higherorder Omega is consistent with ANSD, we not only solve the partial order problem when comparing ANSD relationships among more than two portfolios, but also broaden the use of Omega function. Theorem 1. Assume portfolios X and Y, when the threshold is L, the First-order Omega is Ω1 X(L) , Ω1 Y(L) . If X≥AFSD Y , Ω1 X(L)≥Ω1 Y(L) when the threshold is valued at a special interval. Proof is as seen in Appendix C. According to Theorem 1, the First-order Omega ranking can be deduced from the Almost First Stochastic Dominance (AFSD) relationship between them, provided that the threshold we choose is not significantly small or large. This reflects investors’ behavior in reality, i.e., we usually select by referring to bank deposits, treasury bonds, etc. Namely, the relation between AFSD and First-order Omega in our daily investment always holds. Theorem 2. Assume portfolios X and Y, the Nth-order Omega are ΩN X(L) , ΩN Y(L) . If X≥ANSD Y,ΩN X(L)≥ΩN Y(L)for when L is valued at a special interval. For the flow of the text, the full proofs of Theorem 2 are supplied in Appendix D. Theorem 2 is an extension of Theorem 1 from a higher-order perspective. The higherorder relationship satisfies more specific requirements while the first-order is more suitable for the common cases. In real life, we choose the most appropriate order for different types of investors. For instance, Second-order Omega and Almost Second Stochastic Dominance (ASSD) suits risk-averse investors, whereas risk prudence corresponds to Third-order Omega and Almost Third Stochastic Dominance (ATSD). The higher-order Omega satisfies more detailed preferences, and therefore, Theorem 2 describing ANSD and higher-order Omega is required. In the next section, we use U.S. fund data and provide empirical evidence to show the consistency between higher-order Omega and ANSD. 4. Fund Performance Evaluation 4.1. Data and Summary Statistics This sub-section demonstrates the application of the higher-order Omega, and its comparison with the traditional Sharpe ratio. We select eleven established U.S. funds following the previous literature (Sharpe 1966;Keating and Shadwick 2002;Kaplan and Knowles 2004). These funds were ranked highest based on their net asset value during the sample period of 120 months between September 2010 and August 2020. The monthly return rate of each fund is defined as the growth percentage of the fund’s price on the last trading day of the month and the previous month. The data source is Refinitiv Lipper. The risk-free interest rate is computed from the monthly expected return rate of the 10-year treasury bond during the same period. Table 1presents descriptive statistics of the monthly returns of the eleven funds. Overall, the mean values are mainly between 1 and 2, and the standard deviation is between 4 and 6. Sharp peaks and thick tails characterize their distribution. Except for the three funds of Invesco QQQ Trust, American Century Ultra, and JPMorgan Large Cap, the skewness of the other funds is negative. The kurtosis coefficients of the selected funds are all positive, and the returns of the funds are relatively concentrated. The skewness and kurtosis coefficients indicate that the selected funds do not follow the normal distribution. J. Risk Financial Manag. 2022,15, 438 8 of 20 Table 1. Descriptive Statistics of the eleven funds’ monthly return. Fund Name Mean Return (%) Standard Deviation Skewness Kurtosis American Century Ultra 1.6111 4.4748 0.0048 1.1547 American Funds Growth 1.4790 4.1604 −0.1610 1.3951 Fidelity Blue Chip Growth 1.7089 4.6816 −0.0649 1.0033 Fidelity Contrafund 1.4121 3.9989 −0.0879 1.1835 Fidelity OTC Portfolio 1.7774 5.0226 −0.2042 0.5733 Harbor Capital Appreciation 1.6457 4.6096 −0.0895 0.6735 Invesco QQQ Trust 1.7948 4.4728 0.0079 0.3949 JNL/T Rowe Price Established 1.5354 4.4444 −0.0933 1.1473 JPMorgan Large Cap Growth 1.6673 4.6173 0.0976 1.2822 T Rowe Price Blue Chip Growth 1.6300 4.4352 −0.0242 0.9554 Vanguard US Growth 1.6277 4.4806 −0.0023 1.2025 4.2. Consistency with Almost Stochastic Dominance Table 2shows the First-order Omega, Second-order Omega, and Third-order Omega, as well as the Sharpe Ratio of the selected funds. Comparing the results of First-order Omega and Sharpe ratios, it is found that the ranking of different indexes varies. Among them, the First-order Omega value of Harbor Capital and Fidelity Blue is lower than JNL/T Rowe, but their Sharpe ratios are greater than JNL/T Rowe. Table 2. The 1st-, 2nd-,3rd-order Omega and Sharpe Ratio. Fund Name Ω1 F(Riskless) Ω2 F(Riskless) Ω3 F(Riskless) Sharpe Ratio Fidelity Blue Chip Growth 3.9885(5) 11.2800(2) 11.9774(2) 0.5382(3) Fidelity OTC Portfolio 3.6151(6) 7.8487(6) 5.9280(6) 0.5152(6) Harbor Capital Appreciation 3.9998(4) 9.6957(4) 8.7164(4) 0.5328(4) Invesco QQQ Trust 4.3541(1) 13.0826(1) 14.6950(1) 0.5825(1) JNL/T Rowe Price Established 4.0157(3) 9.5737(5) 8.5982(5) 0.5278(5) T Rowe Price Blue Chip Growth 4.2241(2) 10.9485(3) 10.9183(3) 0.5502(2) In Table 3, Almost first-order Stochastic Dominance (AFSD) selected test results are presented following the existing literature (as seen in Bali et al. 2013;Davidson and Duclos 2000; Leshno and Levy 2010). There is no clear Stochastic Dominance relationship between the two funds, but as expected there exist Almost Stochastic Dominance relationships. JNL/T Rowe ranks AFSD better than Harbor Capital and Fidelity Blue, which is consistent with the ranking results of First-order Omega; the performance between First-order Omega and AFSD is consistent. There is no such consistency found with Sharpe ratio. Table 3. Almost First Stochastic Dominance (AFSD) test results. JNL/T Rowe Price Establish Harbor Capital Appreciation Fidelity Blue Chip Growth JNL/T Rowe Price Establish Harbor Capital Appreciation 0.0021 < ε∗ 1 Fidelity Blue Chip Growth 0.0065 < ε∗ 10.0000 < ε∗ 1 Note: If the value ε<ε∗ N , it is shown that the fund in a row dominated the funds in a column. Refer to Leshno and Levy (2010), ε∗ 1= 5.9%. In this AFSD test, the AFSD ranking is JNL/T Rowe Price Establish > Harbor Capital Appreciation > Fidelity Blue Chip Growth, which is consistent with the First-order Omega ranking. Not all funds have Second-Order Stochastic Dominance (SSD). SSD relationship with each other, so we further examine the Almost Second Stochastic Dominance (ASSD) among some of them. Selected second-order tests are reported in Table 4. Invesco QQQ Trust is significantly better ASSD over the other three funds. The Invesco QQQ Trust is also ranked first by Second-order Omega, and T Rowe Price Blue Chip is superior to Harbor Capital Appreciation and Fidelity OTC Portfolio. J. Risk Financial Manag. 2022,15, 438 15 of 20 Appendix B Proof of Proposition 3. If X≥NSD Y,ΩN X(L)≥ΩN Y(L). x,y∈[a,b],FX,FYare CDFs and F(N) X,F(N) Yare second-order integrals. CDF of L,O(x) = 1(x≥L),O(N)(x) = Rx LO(N−1)(t)dt =x−L,f or x ≥L. The proof is similar to N= 2. We make a simple description as, ∀x , y∈[a,b] , 0 ≤FX , FY , O(x)≤ 1, F(N) X , F(N) Y , O(N)(x) are non-negative and nondecreasing monotonical functions. F(N) X , F(N) Y intersect O(N) from above once at SN X , SN Y when F(N) X(b) , F(N) Y(b)≤O(N)(b) (from Bi et al. 2019): ZF(N) X≤O(N)O(N)(ϕ)−F(N) X(ϕ)dϕ≥ZF(N) Y≤O(N)O(N)(ϕ)−F(N) Y(ϕ)dϕ, and RF(N) X≥O(N)F(N) X(ϕ)−O(N)(ϕ)dϕ≤RF(N) Y≥O(N)F(N) Y(ϕ)−O(N)(ϕ)dϕ. Therefore, ΩN X(L)≥ΩN Y(L). Appendix C Proof of Theorem 1. When L∈[L1,L2], if X≥AFSD Y,Ω1 X(L)≥Ω1 Y(L). x , y∈[a,b] , FX , FY are CDFs and F(1) X , F(1) Y are first-order integrals. According to properties of CDF: ∀x , y∈[a,b] , 0 ≤F(1) X , F(1) Y≤ 1, and both are monotonical and nondecreasing functions. CDF of L, O(1)(x) = 1(x≥L). F(1) X , F(1) Y intersect O(1)(x) once at s∗ X , s∗ Y and we know from Bi et al. (2019), s∗ X= L,s∗ Y=L. Define UL=[a,L],UH=[L,b]. U1=nx∈[a,b]F(1) X(ϕ)≤F(1) Y(ϕ)o,U2=nx∈[a,b]F(1) X(ϕ)>F(1) Y(ϕ)o, M=RU1F(1) Y(ϕ)−F(1) X(ϕ)dϕ,N=RU2F(1) X(ϕ)−F(1) Y(ϕ)dϕ. M1=RU1∩ULF(1) Y(ϕ)−F(1) X(ϕ)dϕ,M2=RU1∩UHF(1) Y(ϕ)−F(1) X(ϕ)dϕ, N1=RU2∩ULF(1) X(ϕ)−F(1) X(ϕ)dϕ,N2=RU2∩UHF(1) X(ϕ)−F(1) Y(ϕ)dϕ, M1+M2=M, and similarly, N1+N2=N. If X≥AFSD Y: RU2F(1) X(ϕ)−F(1) Y(ϕ)dϕ RF(1) Y(ϕ)−F(1) X(ϕ)dϕ =N M+N≤ε1 Therefore, there exists a constant δ1=ε1 1−ε1, satisfying N≤δ1∗M(A2) J. Risk Financial Manag. 2022,15, 438 16 of 20 Denote ω1=Rb LO(1)(ϕ)−F(1) X(ϕ)dϕand ω2=RL aF(1) X(ϕ)−O(1)(ϕ)dϕ, Ω1 X(L) = RF(1) X≤O(2)O(1)(ϕ)−F(1) X(ϕ)dϕ RF(1) X>O(1)F(1) X(ϕ)−O(1)(ϕ)dϕ =RUHO(1)(ϕ)−F(1) X(ϕ)dϕ RULF(1) X(ϕ)−O(1)(ϕ)dϕ =ω1 ω2 Ω1 Y(L) = RF(1) Y≤O(2)O(1)(ϕ)−F(1) Y(ϕ)dϕ RF(1) Y>O(1)F(1) Y(ϕ)−O(1)(ϕ)dϕ =RUHO(1)(ϕ)−F(1) Y(ϕ)dϕ RULF(1) Y(ϕ)−O(1)(ϕ)dϕ =RUHO(1)(ϕ)−F(1) X(ϕ)dϕ+RUHF(1) X(ϕ)−F(1) Y(ϕ)dϕ RULF(1) X(ϕ)−O(1)(ϕ)dϕ+RULF(1) Y(ϕ)−F(1) X(ϕ)dϕ =ω1+RU2∩UHF(1) X(ϕ)−F(1) Y(ϕ)dϕ−RU1∩UHF(1) Y(ϕ)−F(1) X(ϕ)dϕ ω2+RU1∩ULF(1) Y(ϕ)−F(1) X(ϕ)dϕ−RU1∩ULF(1) X(ϕ)−F(1) Y(ϕ)dϕ =ω1+N2−M2 ω2+M1−N1 Ω1 X(L)−Ω1 Y(L) = ω1 ω2−ω1+N2−M2 ω2+M1−N1 =ω1(ω2+M1−N1)−ω2(ω1+N2−M2) ω2(ω2+M1−N1) =ω1(M1−N1)−ω2(N2−M2) ω2(ω2+M1−N1) Due to the definition of Nth-order Omega, ω2(ω2+M1−N1)>0. The Theorem 1 to be proved is equivalent to this, Ω1 X(L)−Ω1 Y(L)≥0, ω1(M1−N1)−ω2(N2−M2)≥0(B) Then in three following cases: i. If M1>N1, N2>M2, When ω1≥ω2, we can conclude M1−N1>N2−M2>0 from (A1), so, ω1(M1−N1)>ω2(N2−M2). (B) holds. When ω1<ω2, ω1(M1−N1)−ω2(N2−M2)>ω1M1−ω2N1−ω2N2+ω1M2=ω1M−ω2N, J. Risk Financial Manag. 2022,15, 438 17 of 20 Furthermore, N≤δ1∗M, ω1M−ω2N≥ω1M−ω2δ1M≥0,ω1 ω2 ≥δ1(A3) There exists a constant L1such that, Rb L1O(1)(ϕ)−F(1) X(ϕ)dϕ RL1 aF(1) X(ϕ)−O(1)(ϕ)dϕ =δ1, Omega is a monotone function of return threshold L, if L≥L1 , (*) holds.. Therefore, if N2>M2, (*) holds when L≥L1, (*) holds. N2≤M2, ω1(M1−N1)>0, −ω2(N2−M2)>0. (B) holds. ii. If M1=N1, ω1(M1−N1)−ω2(N2−M2) =−ω2(N2−M2) =ω2[(N−N1)−(M−M1)] =−ω2(N−M)>0. (B) holds. iii. If M1<N1 , then M2>N2 , When ω1≤ω2 , we can conclude M2−N2>N1− M1>0 from (A1), so, ω2(M2−N2)>ω1(N1−M1), (B) holds. When ω1>ω2, ω1(M1−N1)−ω2(N2−M2) >ω2M1−ω1N1−ω1N2+ω2M2 =ω2M−ω1N Furthermore, N≤δ1∗M, ω2M−ω1N≥ω2M−ω1δ1M≥0,ω1 ω2 ≤1 δ1 (A4) There exists a constant L2such that, Rb L2O(1)(ϕ)−F(1) X(ϕ)dϕ RL2 aF(1) X(ϕ)−O(1)(ϕ)dϕ =1 δ1 , Omega is a monotone function of return threshold L, if L≤L2, (*) holds. Therefore, if M1<N1,M2>N2, (*) holds when L≤L2. For all three possible cases, we obtain that there exists constant L1,L2such that, If X≥AFSD Y,Ω1 X(L)≥Ω1 Y(L)when L∈[L1,L2]. Appendix D Proof of Theorem 2. When LN∈LN 1,LN 2, If X≥ANSD Y,ΩN X(L)≥ΩN Y(L). J. Risk Financial Manag. 2022,15, 438 18 of 20 (1) ∀x , y∈[a,b] , 0 ≤FX , FY , O(x)≤ 1, F(N) X , F(N) Y , O(N)(x) are non-negative and non-decreasing monotonical function. (2) F(N) X(b),F(N) Y(b)≤O(N), intersects only once at sN X,sN Y≥L. Assume sN X<sN Yas an example. J. Risk Financial Manag. 2022, 15 FOR PEER REVIEW 17 =𝜔𝑀−𝜔𝑁 Furthermore, 𝑁≤𝛿∗𝑀 , 𝜔𝑀−𝜔𝑁≥𝜔𝑀−𝜔𝛿𝑀≥0 , 𝜔 𝜔≤1 𝛿 (A4) There exists a constant 𝐿 such that, 󰇡𝑂()(𝜑)−𝐹()(𝜑)󰇢𝑑𝜑    󰇡𝐹()(𝜑)−𝑂()(𝜑)󰇢𝑑𝜑   =1 𝛿, Omega is a monotone function of return threshold L, if 𝐿≤𝐿., (*) holds. Therefore, if 𝑀<𝑁, 𝑀>𝑁, (*) holds when 𝐿≤𝐿. □ For all three possible cases, we obtain that there exists constant 𝐿,𝐿 such that, If 𝑋≥ 𝑌，𝛺 (𝐿)≥𝛺 (𝐿) when 𝐿∈[𝐿,𝐿]. Appendix D Proof of Theorem 2. When 𝐿∈[𝐿 ,𝐿 ], If 𝑋≥ 𝑌，Ω (𝐿)≥Ω (𝐿). (1) ∀𝑥,𝑦∈[𝑎,𝑏]，0≤𝐹,𝐹,𝑂(𝑥)≤1，𝐹(),𝐹(), 𝑂()(𝑥) are non-negative and nondecreasing monotonical function. (2) 𝐹()(𝑏),𝐹()(𝑏)≤ 𝑂()，intersects only once at 𝑠 ,𝑠 ≥𝐿 . Assume 𝑠 <𝑠  as an example. Figure A1. O(x) , 𝐹  ,𝐹  are Cumulative Distribution Functions (CDFS). Note: X-axis shows the threshold. Y-axis shows the value of CDFs. 𝐹  ,𝐹  intersect O(x) at 𝑆  ,𝑆  , and 𝐿≤𝑆  ,𝑆  ≤𝑏. Define 𝑈=[𝑎,𝑆 ], 𝑈 =[𝑆 ,𝑆 ], 𝑈 =[𝑆 ,𝑏]. 𝑈=󰇥𝑥∈[𝑎,𝑏]𝐹 ()(𝜑)≤𝐹 ()(𝜑)󰇦, 𝑈=󰇥𝑥∈[𝑎,𝑏]𝐹()(𝜑)>𝐹 ()(𝜑)󰇦. 𝐶=󰇡𝐹()(𝜑)−𝐹()(𝜑)󰇢𝑑𝜑    , 𝐶=󰇡𝐹()(𝜑)−𝐹()(𝜑)󰇢𝑑𝜑    ∩   +󰇡𝐹()(𝜑)−𝑂()(𝜑)󰇢𝑑𝜑    ∩   , 𝐶=󰇡𝐹()(𝜑)−𝐹()(𝜑)󰇢𝑑𝜑    ∩   +󰇡𝑂()(𝜑)−𝐹()(𝜑)󰇢𝑑𝜑    ∩   , 𝐶+𝐶=𝐶; 𝑉=󰇡𝐹()(𝜑)−𝐹()(𝜑)󰇢𝑑𝜑    , 𝑉=󰇡𝐹()(𝜑)−𝐹()(𝜑)󰇢𝑑𝜑    ∩   ,𝑉=󰇡𝐹()(𝜑)−𝐹()(𝜑)󰇢𝑑𝜑    ∩   , And +𝑉=𝑉. Figure A1. O(x), FX , FY are Cumulative Distribution Functions (CDFS). Note: X-axis shows the threshold. Y-axis shows the value of CDFs. FX,FYintersect O(x) at SX,SY, and L≤SX,SY≤b. Define UN L=a,SN X,UN M=SN X,SN Y,UN H=SN Y,b. UN 1=nx∈[a,b]F(N) X(ϕ)≤F(N) Y(ϕ)o,UN 2=nx∈[a,b]F(N) X(ϕ)>F(N) Y(ϕ)o. CN=RUN 1F(N) Y(ϕ)−F(N) X(ϕ)dϕ, CN 1=RUN 1∩UN LF(N) Y(ϕ)−F(N) X(ϕ)dϕ+RUN 1∩UN MF(N) Y(ϕ)−O(N)(ϕ)dϕ, CN 2=RUN 1∩UN HF(N) Y(ϕ)−F(N) X(ϕ)dϕ+RUN 1∩UN MO(N)(ϕ)−F(N) Y(ϕ)dϕ, CN 1+CN 2=CN; VN=RUN 2F(N) X(ϕ)−F(N) Y(ϕ)dϕ, VN 1=RUN 2∩UN LF(N) X(ϕ)−F(N) Y(ϕ)dϕ,VN 2=RUN 2∩UN HF(N) X(ϕ)−F(N) Y(ϕ)dϕ, And +VN 2=VN. If X≥ANSD Y: VN CN+VN=RUN 2F(N) X(ϕ)−F(N) Y(ϕ)dϕ RF(N) Y(ϕ)−F(N) X(ϕ)∨dϕ ≤εN, Therefore, there exists a constant making VN≤δN∗CN, and δN=εN 1−εN. J. Risk Financial Manag. 2022,15, 438 19 of 20 Denote ωN 1=Rb SN XO(N)(ϕ)−F(N) X(ϕ)dϕand ωN 2=RSN X aF(N) X(ϕ)−O(N)(ϕ)dϕ, ΩN X(L) = RF(N) X≤O(N)O(N)(ϕ)−F(N) X(ϕ)dϕ RF(N) X>O(N)F(N) X(ϕ)−O(N)(ϕ)dϕ=ωN 1 ωN 2 ΩN Y(L) = RF(N) Y≤O(N)O(N)(ϕ)−F(N) Y(ϕ)dϕ RF(N) Y>O(N)F(N) Y(ϕ)−O(N)(ϕ)dϕ=ωN 1+VN 2−CN 2 ωN 2+CN 1−VN 1 ΩN X(L)−ΩN Y(L) = ωN 1 ωN 2 −ωN 1+VN 2−CN 2 ωN 2+CN 1−VN 1 =ωN 1(ωN 2+CN 1−VN 1)−ωN 2(ωN 1+VN 2−CN 2) ωN 2(ωN 2+CN 1−VN 1) =ωN 1(CN 1−VN 1)−ωN 2(VN 2−CN 2) ωN 2(ωN 2+CN 1−VN 1) The proof is similar to N= 1, and we conclude that, When LN∈LN 1,LN 2 , ΩN X(L)−ΩN Y(L)≥ 0 holds, where. ΩN XLN 1=δN,ΩN XLN 2=1 δN. In conclusion, If X≥ANSD Y,ΩN X(L)≥ΩN Y(L), with LN∈LN 1,LN 2. References Bai, Zhidong, Li Huai, McAleer Michael, and Wong Wing-Keung Wong. 2015. Stochastic Dominance Statistics for Risk Averters and Risk Seekers: An Analysis of Stock Preferences for USA and China. Quantitative Finance 15: 889–900. [CrossRef] Balder, Sven, and Nikolaus Schweizer. 2017. 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