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Sufficient conditions under which SSD- and MR-efficient sets are identical. (English) Zbl 1339.91053
Summary: Three approaches are commonly used for analyzing decisions under uncertainty: expected utility (EU), second-degree stochastic dominance (SSD), and mean-risk (MR) models, with the mean-standard deviation (MS) being the best-known MR model. Because MR models generally lead to different efficient sets and thus are a continuing source of controversy, the specific concern of this article is not to suggest another MR model. Instead, we show that the SSD- and MR-efficient sets are identical, as long as (a) the risk measure satisfies both positive homogeneity and consistency with respect to the M. Rothschild and J. E. Stiglitz [“Increasing risk. I: A definition”, J. Econ. Theory 2, No. 3, 225–243 (1970; doi:10.1016/0022-0531(70)90038-4)] definition(s) of increasing risk and (b) the choice set includes the riskless asset and satisfies a generalized location and scale property, which can be interpreted as a market model. Under these conditions, there is no controversy among MR models and they all have a decision-theoretic foundation. They also offer a convenient way to compare the estimation error related to the empirical implementation of different MR models.

##### MSC:
 91B16 Utility theory 91B06 Decision theory 91B26 Auctions, bargaining, bidding and selling, and other market models 91B30 Risk theory, insurance (MSC2010)
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##### References:
 [1] Adrian, T., Etula, E., & Muir, T. (2013). Financial intermediaries and the cross-section of asset returns. Staff Report No. 464, Federal Reserve Bank of New York. [2] Artzner, P.; Delbaen, F.; Eber, J.-M.; Heath, D., Coherent measures of risk, Mathematical Finance, 9, 203-228, (1999) · Zbl 0980.91042 [3] Auer, B. R.; Schuhmacher, F., Robust evidence on the similarity of sharpe ratio and drawdown-based hedge fund performance rankings, Journal of International Financial Markets, Institutions and Money, 24, 153-165, (2013) [4] Aumann, R.; Serrano, R., An economic index of riskiness, Journal of Political Economy, 116, 810-836, (2008) · Zbl 1341.91040 [5] Banz, R. W., The relationship between return and market value of common stocks, Journal of Financial Economics, 9, 3-18, (1981) [6] Basu, S., Investment performance of common stocks in relation to their price-earnings ratios: A test of the efficient market hypothesis, Journal of Finance, 32, 663-682, (1977) [7] Bawa, V. S., Optimal rules for ordering uncertain prospects, Journal of Financial Economics, 2, 95-121, (1975) [8] Bawa, V. S., Safety-first, stochastic dominance, and optimal portfolio choice, Journal of Financial and Quantitative Analysis, 13, 255-271, (1978) [9] Bodie, Z.; Kane, A.; Marcus, A. J., Investments, (2011), McGraw Hill New York [10] Campbell, J. Y., Asset pricing at the millennium, Journal of Finance, 55, 1515-1567, (2000) [11] Capocci, D.; Hübner, G., Analysis of hedge fund performance, Journal of Empirical Finance, 11, 55-89, (2004) [12] Chamberlain, G., A characterization of the distributions that imply mean-variance utility functions, Journal of Economic Theory, 29, 185-201, (1983) · Zbl 0495.90009 [13] Chen, L.; He, S.; Zhang, S., When all risk-adjusted performance measures are the same: in praise of the sharpe ratio, Quantitative Finance, 11, 1439-1447, (2011) · Zbl 1258.91097 [14] Ding, B.; Shawky, H. A., The performance of hedge fund strategies and the asymmetry of return distributions, European Financial Management, 13, 309-331, (2007) [15] Eling, M., Does the measure matter in the mutual fund industry?, Financial Analysts Journal, 64, 54-66, (2008) [16] Eling, M.; Schuhmacher, F., Does the choice of performance measure influence the evaluation of hedge funds?, Journal of Banking and Finance, 31, 2632-2647, (2007) [17] Fama, E. F.; French, K. R., The cross-section of expected returns, Journal of Finance, 47, 427-465, (1992) [18] Fama, E. F.; French, K. R., Common risk factors in the returns of stocks and bonds, Journal of Financial Economics, 33, 3-56, (1993) · Zbl 1131.91335 [19] Farinelli, S.; Tibiletti, L., Sharpe thinking in asset ranking with one-sided measures, European Journal of Operational Research, 185, 1542-1547, (2008) · Zbl 1161.91415 [20] Fishburn, P. C., Mean-risk analysis with risk associated with below-target returns, American Economic Review, 67, 116-126, (1977) [21] Gaivoronski, A. A.; Pflug, G., Value-at-risk in portfolio optimization: properties and computational approach, Journal of Risk, 7, 1-31, (2005) [22] Grootveld, H.; Hallerbach, W., Variance vs downside risk: Is there really that much difference?, European Journal of Operational Research, 114, 304-319, (1999) · Zbl 0935.91021 [23] Hadar, J.; Russel, W. R., Stochastic dominance and diversification, Journal of Economic Theory, 3, 288-305, (1971) [24] Homm, U.; Pigorsch, C., Beyond the sharpe ratio: an application of the Aumann-serrano index to performance measurement, Journal of Banking and Finance, 36, 2274-2284, (2012) [25] Kijima, M.; Ohnishi, M., Mean-risk analysis of risk aversion and wealth effects on optimal portfolios with multiple investment opportunities, Annals of Operations Research, 45, 147-163, (1993) · Zbl 0785.90012 [26] Konno, H.; Yamazaki, H., Mean-absolute deviation portfolio optimization model and its application to Tokyo stock market, Management Science, 37, 519-531, (1991) [27] Krokhmal, P.; Palmquist, J.; Uryasev, S., Portfolio optimization with conditional value-at-risk objective and constraints, Journal of Risk, 4, 43-68, (2002) [28] Krokhmal, P.; Zabarankin, M.; Uryasev, S., Modeling and optimization of risk, Surveys in Operations Research and Management Science, 16, 49-66, (2011) [29] Kroll, Y.; Levy, H.; Markowitz, H. M., Mean-variance versus direct utility maximization, Journal of Finance, 39, 47-61, (1984) [30] Lettau, M.; Ludvigson, S., Consumption, aggregate wealth and expected stock returns, Journal of Finance, 56, 815-849, (2001) [31] Lettau, M.; Ludvigson, S., Resurrecting the C(CAPM): A cross-sectional test when risk premia are time-varying, Journal of Political Economy, 109, 1238-1287, (2001) [32] Levy, H., Two-moment decision models and expected utility maximization: comment, American Economic Review, 79, 597-600, (1989) [33] Levy, H., Stochastic dominance and expected utility: survey and analysis, Management Science, 38, 555-593, (1992) · Zbl 0764.90004 [34] Lintner, J., Security prices, risk, and maximal gains from diversification, Journal of Finance, 20, 587-615, (1965) [35] Ludvigson, S. C.; Ng, S., The empirical risk-return relation: A factor analysis approach, Journal of Financial Economics, 83, 171-222, (2007) [36] Ludvigson, S. C.; Ng, S., Macro factors in bond risk premia, Review of Financial Studies, 22, 5027-5067, (2009) [37] Markowitz, H., Portfolio selection, Journal of Finance, 7, 77-91, (1952) [38] Meyer, J., Two moment decision models and expected utility maximization, American Economic Review, 77, 421-430, (1987) [39] Meyer, J.; Rasche, R. H., Sufficient conditions for expected utility to imply mean-standard deviation rankings: empirical evidence concerning the location and scale condition, Economic Journal, 102, 91-106, (1992) [40] Ogryczak, W.; Ruszczyński, A., From stochastic dominance to mean-risk models: semideviations as risk measures, European Journal of Operational Research, 116, 33-50, (1999) · Zbl 1007.91513 [41] Ortobelli, S.; Rachev, S. T.; Stoyanov, S.; Fabozzi, F. J.; Biglova, A., The proper use of risk measures in portfolio theory, International Journal of Theoretical and Applied Finance, 8, 1107-1133, (2005) · Zbl 1117.91035 [42] Owen, J.; Rabinovitch, R., On the class of elliptical distributions and their applications to the theory of portfolio choice, Journal of Finance, 38, 745-752, (1983) [43] Pedersen, C. S.; Satchell, S. E., An extended family of financial-risk measures, Geneva Papers on Risk and Insurance Theory, 23, 89-117, (1998) [44] Piazzesi, M.; Schneider, M.; Tuzel, S., Housing, consumption and asset pricing, Journal of Financial Economics, 83, 531-569, (2007) [45] Porter, R. B., Semivariance and stochastic dominance: A comparison, American Economic Review, 64, 200-204, (1974) [46] Rockafellar, R. T.; Uryasev, S., Optimization of conditional value-at-risk, Journal of Risk, 2, 21-41, (2000) [47] Rockafellar, R.; Uryasev, S.; Zabarankin, M., Generalized deviations in risk analysis, Finance and Stochastics, 10, 51-74, (2006) · Zbl 1150.90006 [48] Rothschild, M.; Stiglitz, J. E., Increasing risk: I. A definition, Journal of Economic Theory, 2, 225-243, (1970) [49] Roy, A. D., Safety first and the holding of assets, Econometrica, 20, 431-449, (1952) · Zbl 0047.38805 [50] Santos, T.; Veronesi, P., Labor income and predictable stock returns, Review of Financial Studies, 19, 1-44, (2006) [51] Schuhmacher, F., Der anwendungsbereich der sharpe ratio als performancemaß ist größer, als viele vermuten, Journal of Business Economics, 82, 685-705, (2012) [52] Schuhmacher, F.; Eling, M., Sufficient conditions for expected utility to imply drawdown-based performance rankings, Journal of Banking and Finance, 35, 2311-2318, (2011) [53] Schuhmacher, F.; Eling, M., A decision-theoretic foundation for reward-to-risk performance measures, Journal of Banking and Finance, 36, 2077-2082, (2012) [54] Shanken, J., Multivariate tests of the zero-beta CAPM, Journal of Financial Economics, 14, 327-348, (1985) [55] Sharpe, W. F., Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19, 425-442, (1964) [56] Sinn, H. W., Economic decisions under uncertainty, (1983), North-Holland Amsterdam, New York, and Oxford · Zbl 0519.90001 [57] Szegö, G., Measures of risk, European Journal of Operational Research, 163, 5-19, (2005) · Zbl 1066.91061 [58] Wong, W.-K.; Ma, C., Preferences over location-scale family, Economic Theory, 37, 119-146, (2008) · Zbl 1147.91042 [59] Yitzhaki, S., Stochastic dominance, mean variance, and gini’s mean difference, American Economic Review, 72, 178-185, (1982)
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