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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.

91B16 Utility theory
91B06 Decision theory
91B26 Auctions, bargaining, bidding and selling, and other market models
91B30 Risk theory, insurance (MSC2010)
Full Text: DOI
[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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