Higher-Order Risk Measure and (Higher-Order) Stochastic Dominance

Niu, Cuizhen and Wong, Wing-Keung and Xu, Qunfang (2017): Higher-Order Risk Measure and (Higher-Order) Stochastic Dominance.

Abstract

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 relationship 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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Stochastic Dominance, Kappa ratio, Omega Ratio, Sortino ratio, mean-risk analysis, risk aversion, risk seeking
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty; G - Financial Economics > G1 - General Financial Markets > G10 - General

References

Bai, Z.D., Hui, Y.C., Wong, W.K., Zitikis, R., 2012, Evaluating Prospect Performance: Making a Case for a Non-Asymptotic UMPU Test, Journal of Financial Econometrics 10(4), 703-732.

Bai, Z.D., Lui, H.X., Wong, W.K., (2009), Enhancement of the Applicability of Markowitz's Portfolio Optimization by Utilizing Random Matrix Theory, Mathematical Finance, 19(4), 639-667.

Bai, Z.D., Li, H., McAleer, M., Wong, W.K. (2015). Stochastic dominance statistics for risk averters and risk seekers: an analysis of stock preferences for USA and China. Quantitative Finance 15(5), 889-900.

Bai, Z.D., Hui, Y.C., Wong, W.K., Zitikis, R., 2012, Evaluating Prospect Performance: Making a Case for a Non-Asymptotic UMPU Test, Journal of Financial Econometrics 10(4), 703-732.

Bai, Z.D., Phoon, K.F., Wang, K.Y., Wong, W.K., 2013, The Performance of Commodity Trading Advisors: A Mean-Variance-Ratio Test Approach, North American Journal of Economics and Finance, 25, 188-201.

Beedles, W.L. 1979. Return, Dispersion, and Skewness: Synthesis and Investment Strategy, Journal of Financial Research 2, 71-80.

Brockett, P.L., Garven, J.R., 1998, A Reexamination of the Relationship Between Preferences and Moment Orderings by Rational Risk-Averse Investors, GENEVA Papers on Risk and Insurance Theory, 23(2), 127-137.

Brockett, P.L., Kahane, Y. 1992, Risk, Return, Skewness and Preference, Management Science, 38(6), 851-866.

Chan, C.Y., de Peretti, C., Qiao, Z., Wong, W.K., 2012, Empirical test of the efficiency of the UK covered warrants market: Stochastic dominance and likelihood ratio test approach, Journal of Empirical Finance 19(1), 162-174.

Clark, E., Qiao, Z., Wong, W.K. 2016, Theories of Risk: Testing Investor Behaviour on the Taiwan Stock and Stock Index Futures Markets, Economic Inquiry 54(2), 907-924.

Darsinos, T., Satchell, S., 2004. Generalizing universal performance measures. Risk (June), 80-84.

Fong, W.M., Wong, W.K., Lean, H.H., (2005), International momentum strategies: a stochastic dominance, Journal of Financial Markets, 8, 89-109.

Fong, W.M., Wong, W.K., Lean, H.H., (2005), International momentum strategies: a stochastic dominance, Journal of Financial Markets, 8, 89-109.

Gasbarro, D., Wong, W.K., Zumwalt, J.K. (2007), Stochastic Dominance Analysis of iShares, European Journal of Finance 13, 89-101.

Guo, B. and Xiao, Y. G. 2016. A note on why doesn't the choice of performance measure matter? Finance Research Letters 16, 248-254.

Guo, X., Jiang, X.J., Wong, W.K. 2016. A note on Stochastic Dominance and the Omega Ratio. Social Science Research Network Working Paper Series 2827058

Guo, X., Wong, W.K. (2016). Multivariate Stochastic Dominance for Risk Averters and Risk Seekers. RAIRO - Operations Research 50(3), 575-586.

Hammond, J. S. (1974). Simplifying the choice between uncertain prospects where preference is nonlinear, Management Science 20(7), 1047-1072.

Hanoch G., Levy, H. (1969). The Efficiency Analysis of Choices Involving Risk. Review of Economic studies 36, 335-346.

Hoang, T.H.V., Wong, W.K., Zhu, Z.Z., 2015, Is gold different for risk-averse and risk-seeking investors? An empirical analysis of the Shanghai Gold Exchange, Economic Modelling 50, 200-211.

Kaplan, P.D. , Knowles, J.A. , 2004. Kappa: A generalized downside risk-adjusted performance measure. J. Perform. Meas. 8(3), 42-54.

Leitner, J. (2005). A short note on second-order stochastic dominance preserving coherent risk measures. Mathematical Finance 15(4), 649-651.

Leung, P.L., Wong, W.K., 2008, On testing the equality of the multiple Sharpe Ratios, with application on the evaluation of iShares, Journal of Risk, 10(3), 1-16.

Levy, H., 1969, A Utility Function Depending on the First Three Moments, Journal of Finance 24(4), 715-719.

Levy, H. (2015). Stochastic Dominance: Investment Decision Making Under Uncertainty. Third Edition, Springer, New York.

Li, C.K., Wong, W.K., 1999, Extension of stochastic dominance theory to random variables, RAIRO - Operations Research 33(4), 509-524.

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. (1952a). Portfolio Selection, Journal of Finance 7, 77-91.

Markowitz, H.M. (1952b). The utility of wealth, Journal of Political Economy 60, 151-156.

Niu, C.Z., Wong, W.K., Zhu, L.X. 2016. First Stochastic Dominance and Risk Measurement, MPRA Paper 75027, University Library of Munich, Germany.

Ogryczak, W., Ruszczynski, R. (1999). From stochastic dominance to mean-risk models: semideviations as risk measures. European Journal of Operational Research 116, 33-50.

Ogryczak, W., Ruszczynski, A. (2002). Dual stochastic dominance and related mean-risk models. SIAM Journal of Optimization 13, 60-78.

Schuhmacher, F., Eling, M. (2012). A decision-theoretic foundation for reward-to-risk performance measures. Journal of Banking Finance 36(7), 2077-2082.

Shadwick, W.F., Keating, C. , 2002. A universal performance measure. J. Perform. Meas. 6(3), 59-84.

Sharpe, W.F., 1966, Mutual Funds Performance. Journal of Business 39, 119-138.

Sortino, F.A., van der Meer, R., 1991. Downside risk. Journal of Portfolio Management 17, 27-31.

Sriboonchitta, S., Wong, W.K., Dhompongsa, S., Nguyen, H.T. 2009, Stochastic Dominance and Applications to Finance, Risk and Economics, Chapman and Hall/CRC, Taylor and Francis Group, Boca Raton, Florida, USA.

Stoyan, D., Daley, D.J. (1983). Comparison methods for queues and other stochastic models, John Wiley & Sons, New York, USA.

Vinod, H.D., 2004. Ranking mutual funds using unconventional utility theory and stochastic dominance, Journal of Empirical Finance, 11(3), 353-377.

Wong, W.K., 2007. Stochastic dominance and mean-variance measures of profit and loss for business planning and investment, European Journal of Operational Research 182, 829-843.

Wong, W.K., Li, C.K., 1999. A Note on Convex Stochastic Dominance Theory, Economics Letters 62, 293-300.

Wong, W.K., Ma, C. (2008). Preferences over Location-scale Family, Economic Theory 37, 119-146.