Aversion to Risk and Downside Risk in the Large and in the Small under Non-Expected Utility: A Quantile Approach

Affiliation(s)

^{1}
University of Wisconsin-Madison, Madison, USA.

^{2}
Seoul National University, Seoul, South Korea.

ABSTRACT

This paper proposes a decomposition of the cost of risk (as measured by a risk premium) across intervals/quantiles of the payoff distribution. The analysis is based on general smooth risk preferences. While this includes the expected utility model as a special case, the investigation is done under a broad class of non-expected utility models. We decompose the risk premium into additive components across quantiles. Defining downside risk as the risk associated with a lower quantile, this provides a basis to evaluate the cost of exposure to downside risk. We derive a local measure of the cost of risk associated with each quantile. It establishes linkages between the cost of risk, risk preferences and the distribution of risky prospects across quantiles (as measured by quantile variance and skewness). The analysis gives new and useful information on how risk aversion, exposure to downside risk and departures from the expected utility model interact as they affect the risk premium.

This paper proposes a decomposition of the cost of risk (as measured by a risk premium) across intervals/quantiles of the payoff distribution. The analysis is based on general smooth risk preferences. While this includes the expected utility model as a special case, the investigation is done under a broad class of non-expected utility models. We decompose the risk premium into additive components across quantiles. Defining downside risk as the risk associated with a lower quantile, this provides a basis to evaluate the cost of exposure to downside risk. We derive a local measure of the cost of risk associated with each quantile. It establishes linkages between the cost of risk, risk preferences and the distribution of risky prospects across quantiles (as measured by quantile variance and skewness). The analysis gives new and useful information on how risk aversion, exposure to downside risk and departures from the expected utility model interact as they affect the risk premium.

Cite this paper

Chavas, J. and Kim, K. (2015) Aversion to Risk and Downside Risk in the Large and in the Small under Non-Expected Utility: A Quantile Approach.*Theoretical Economics Letters*, **5**, 784-804. doi: 10.4236/tel.2015.56090.

Chavas, J. and Kim, K. (2015) Aversion to Risk and Downside Risk in the Large and in the Small under Non-Expected Utility: A Quantile Approach.

References

[1] Roy, A.D. (1952) Safety First and the Holding of Assets. Econometrica, 20, 431-449. http://dx.doi.org/10.2307/1907413

[2] Kahneman, D. and Tversky, A. (1979) Prospect Theory: An Analysis of Decision under Risk. Econometrica, 47, 263-291. http://dx.doi.org/10.2307/1914185

[3] Gul, F. (1991) A Theory of Disappointment Aversion. Econometrica, 59, 667-686.

http://dx.doi.org/10.2307/2938223

[4] Routledge, B.R. and Zin, S.E. (2010) Generalized Disappointment Aversion and Asset Prices. Journal of Finance, 65, 1303-1332. http://dx.doi.org/10.1111/j.1540-6261.2010.01571.x

[5] Fishburn, P.C. (1977) Mean-Risk Analysis with Risk Associated with Below-Target Returns. American Economic Review, 67, 116-126.

[6] Weitzman, M.L. (2009) On Modeling and Interpreting the Economics of Catastrophic Climate Change. Review of Economics and Statistics, 91, 1-19. http://dx.doi.org/10.1162/rest.91.1.1

[7] Bawa, V.S. (1975) Optimal Rules for Ordering Uncertain Prospects. Journal of Financial Economics, 2, 95-121. http://dx.doi.org/10.1016/0304-405X(75)90025-2

[8] Menezes, C., Geiss, C. and Tressler, J. (1980) Increasing Downside Risk. American Economic Review, 70, 921-932.

[9] Antle, J.M. (1987) Econometric Estimation of Producers’ Risk Attitudes. American Journal of Agricultural Economics, 69, 509-522. http://dx.doi.org/10.2307/1241687

[10] Modica, S. and Scarsini, M. (2005) A Note on Comparative Downside Risk Aversion. Journal of Economic Theory, 122, 267-271. http://dx.doi.org/10.1016/j.jet.2004.06.008

[11] Ang, A., Chen, J. and Xing, Y. (2006) Downside Risk. Review of Financial Studies, 19, 1191-1239. http://dx.doi.org/10.1093/rfs/hhj035

[12] Crainich, D. and Eeckhoudt, L. (2007) On the Intensity of Downside Risk Aversion. Journal of Risk and Uncertainty, 36, 267-276. http://dx.doi.org/10.1007/s11166-008-9037-x

[13] Jindapon, P. and Nielson, W. (2007) Higher Order Generalizations of Arrow-Pratt and Ross Risk Aversion: A Comparative Statics Approach. Journal of Economic Theory, 136, 719-728.

http://dx.doi.org/10.1016/j.jet.2006.03.010

[14] Keenan, D. and Snow, A. (2009) Greater Downside Risk Aversion in the Large. Journal of Economic Theory, 144, 1092-1101. http://dx.doi.org/10.1016/j.jet.2008.08.007

[15] Watt, R. and Vasquez, F.J. (2013) Allocative Downside Risk Aversion. International Journal of Economic Theory, 9, 267-277. http://dx.doi.org/10.1111/j.1742-7363.2013.12019.x

[16] Pratt, J.W. (1964) Risk Aversion in the Small and in the Large. Econometrica, 32, 122-136. http://dx.doi.org/10.2307/1913738

[17] Arrow, K.J. (1965) Aspects of the Theory of Risk Bearing. Yrjo Jahnssonin Saatio, Helsinki.

[18] Allais, M. (1953) Le Comportement de l’Homme Rationnel Devant le Risque: Critique des Postulats et Axiomes de l’école Américaine. Econometrica, 21, 503-546. http://dx.doi.org/10.2307/1907921

[19] Machina, M.J. (1982) Expected Utility Analysis without the Independence Axiom. Econometrica, 50, 277-323. http://dx.doi.org/10.2307/1912631

[20] Quiggin, J. (1993) Generalized Expected Utility Theory: The Rank-Dependent Expected Utility Model. Kluwer-Nijhoff, Amsterdam. http://dx.doi.org/10.1007/978-94-011-2182-8

[21] Jia, J., Dyer, J.S. and Butler, J.C. (2001) Generalized Disappointment Models. Journal of Risk and Uncertainty, 22, 59-78. http://dx.doi.org/10.1023/A:1011153523672

[22] Delquié, P. and Cillo, A. (2006) Disappointment without Prior Expectation: A Unified Perspective on Decision under Risk. Journal of Risk and Uncertainty, 33, 197-215. http://dx.doi.org/10.1007/s11166-006-0499-4

[23] Wang, T. (1993) Lp-Fréchet Differentiable Preference and “Local Utility” Analysis. Journal of Economic Theory, 61, 139-159. http://dx.doi.org/10.1006/jeth.1993.1062

[24] Quiggin, J. (1982) A Theory of Anticipated Utility. Journal of Economic Behavior and Organization, 3, 323-343. http://dx.doi.org/10.1016/0167-2681(82)90008-7

[25] Tversky, A. and Kahneman, D. (1992) Advances in Prospect Theory: Cumulative Representation of Uncertainty. Journal of Risk and Uncertainty, 5, 297-323. http://dx.doi.org/10.1007/BF00122574

[26] Chew, S.H. (1983) A Generalization of the Quasi-Linear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox. Econometrica, 51, 1065-1092. http://dx.doi.org/10.2307/1912052

[27] Bassett, G.W., Koenker, R. and Kordas, G. (2004) Pessimistic Portfolio Allocation and Choquet Expected Utility. Journal of Financial Econometrics, 2, 477-492. http://dx.doi.org/10.1093/jjfinec/nbh023

[28] Kim,K., Chavas, J.P., Barham, B. and Foltz, J. (2014) Rice, Irrigation and Downside Risk: A Quantile Analysis of Risk Exposure and Mitigation on Korean Farms. European Review of Agricultural Economics, 41, 775-815. http://dx.doi.org/10.1093/erae/jbt041

[29] Markowitz, H.M. (1959) Portfolio Selection. Yale University Press, New Haven.

[30] Jorion, P. (1996) Value at Risk: A New Benchmark for Measuring Derivatives Risk. Irwin Professional Publishing, Chicago.

[31] Rockafellar, R.T. and Uryasev, S. (2000) Optimization of Conditional Value at Risk. Journal of Risk, 2, 21-41.

[32] Antle, J.M. (2010) Asymmetry, Partial Moments and Production Risk. American Journal of Agricultural Economics, 92, 1294-1309. http://dx.doi.org/10.1093/ajae/aaq077

[33] Schmeidler, D. (1989) Subjective Probability and Expected Utility without Additivity. Econometrica, 57, 571-587. http://dx.doi.org/10.2307/1911053

[34] Wakker, P. (1990) Under Stochastic Dominance Choquet-Expected Utility and Anticipated Utility Are Identical. Theory and Decision, 29, 119-132. http://dx.doi.org/10.1007/BF00126589

[35] Winkler, R.L., Roodman, G.M. and Britney, R.R. (1972) The Determination of Partial Moments. Management Science, 19, 290-296. http://dx.doi.org/10.1287/mnsc.19.3.290

[36] Gonzalez, R. and Wu, G. (1999) On the Shape of the Probability Weighting Function. Cognitive Psychology, 38, 129-166. http://dx.doi.org/10.1006/cogp.1998.0710

[1] Roy, A.D. (1952) Safety First and the Holding of Assets. Econometrica, 20, 431-449. http://dx.doi.org/10.2307/1907413

[2] Kahneman, D. and Tversky, A. (1979) Prospect Theory: An Analysis of Decision under Risk. Econometrica, 47, 263-291. http://dx.doi.org/10.2307/1914185

[3] Gul, F. (1991) A Theory of Disappointment Aversion. Econometrica, 59, 667-686.

http://dx.doi.org/10.2307/2938223

[4] Routledge, B.R. and Zin, S.E. (2010) Generalized Disappointment Aversion and Asset Prices. Journal of Finance, 65, 1303-1332. http://dx.doi.org/10.1111/j.1540-6261.2010.01571.x

[5] Fishburn, P.C. (1977) Mean-Risk Analysis with Risk Associated with Below-Target Returns. American Economic Review, 67, 116-126.

[6] Weitzman, M.L. (2009) On Modeling and Interpreting the Economics of Catastrophic Climate Change. Review of Economics and Statistics, 91, 1-19. http://dx.doi.org/10.1162/rest.91.1.1

[7] Bawa, V.S. (1975) Optimal Rules for Ordering Uncertain Prospects. Journal of Financial Economics, 2, 95-121. http://dx.doi.org/10.1016/0304-405X(75)90025-2

[8] Menezes, C., Geiss, C. and Tressler, J. (1980) Increasing Downside Risk. American Economic Review, 70, 921-932.

[9] Antle, J.M. (1987) Econometric Estimation of Producers’ Risk Attitudes. American Journal of Agricultural Economics, 69, 509-522. http://dx.doi.org/10.2307/1241687

[10] Modica, S. and Scarsini, M. (2005) A Note on Comparative Downside Risk Aversion. Journal of Economic Theory, 122, 267-271. http://dx.doi.org/10.1016/j.jet.2004.06.008

[11] Ang, A., Chen, J. and Xing, Y. (2006) Downside Risk. Review of Financial Studies, 19, 1191-1239. http://dx.doi.org/10.1093/rfs/hhj035

[12] Crainich, D. and Eeckhoudt, L. (2007) On the Intensity of Downside Risk Aversion. Journal of Risk and Uncertainty, 36, 267-276. http://dx.doi.org/10.1007/s11166-008-9037-x

[13] Jindapon, P. and Nielson, W. (2007) Higher Order Generalizations of Arrow-Pratt and Ross Risk Aversion: A Comparative Statics Approach. Journal of Economic Theory, 136, 719-728.

http://dx.doi.org/10.1016/j.jet.2006.03.010

[14] Keenan, D. and Snow, A. (2009) Greater Downside Risk Aversion in the Large. Journal of Economic Theory, 144, 1092-1101. http://dx.doi.org/10.1016/j.jet.2008.08.007

[15] Watt, R. and Vasquez, F.J. (2013) Allocative Downside Risk Aversion. International Journal of Economic Theory, 9, 267-277. http://dx.doi.org/10.1111/j.1742-7363.2013.12019.x

[16] Pratt, J.W. (1964) Risk Aversion in the Small and in the Large. Econometrica, 32, 122-136. http://dx.doi.org/10.2307/1913738

[17] Arrow, K.J. (1965) Aspects of the Theory of Risk Bearing. Yrjo Jahnssonin Saatio, Helsinki.

[18] Allais, M. (1953) Le Comportement de l’Homme Rationnel Devant le Risque: Critique des Postulats et Axiomes de l’école Américaine. Econometrica, 21, 503-546. http://dx.doi.org/10.2307/1907921

[19] Machina, M.J. (1982) Expected Utility Analysis without the Independence Axiom. Econometrica, 50, 277-323. http://dx.doi.org/10.2307/1912631

[20] Quiggin, J. (1993) Generalized Expected Utility Theory: The Rank-Dependent Expected Utility Model. Kluwer-Nijhoff, Amsterdam. http://dx.doi.org/10.1007/978-94-011-2182-8

[21] Jia, J., Dyer, J.S. and Butler, J.C. (2001) Generalized Disappointment Models. Journal of Risk and Uncertainty, 22, 59-78. http://dx.doi.org/10.1023/A:1011153523672

[22] Delquié, P. and Cillo, A. (2006) Disappointment without Prior Expectation: A Unified Perspective on Decision under Risk. Journal of Risk and Uncertainty, 33, 197-215. http://dx.doi.org/10.1007/s11166-006-0499-4

[23] Wang, T. (1993) Lp-Fréchet Differentiable Preference and “Local Utility” Analysis. Journal of Economic Theory, 61, 139-159. http://dx.doi.org/10.1006/jeth.1993.1062

[24] Quiggin, J. (1982) A Theory of Anticipated Utility. Journal of Economic Behavior and Organization, 3, 323-343. http://dx.doi.org/10.1016/0167-2681(82)90008-7

[25] Tversky, A. and Kahneman, D. (1992) Advances in Prospect Theory: Cumulative Representation of Uncertainty. Journal of Risk and Uncertainty, 5, 297-323. http://dx.doi.org/10.1007/BF00122574

[26] Chew, S.H. (1983) A Generalization of the Quasi-Linear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox. Econometrica, 51, 1065-1092. http://dx.doi.org/10.2307/1912052

[27] Bassett, G.W., Koenker, R. and Kordas, G. (2004) Pessimistic Portfolio Allocation and Choquet Expected Utility. Journal of Financial Econometrics, 2, 477-492. http://dx.doi.org/10.1093/jjfinec/nbh023

[28] Kim,K., Chavas, J.P., Barham, B. and Foltz, J. (2014) Rice, Irrigation and Downside Risk: A Quantile Analysis of Risk Exposure and Mitigation on Korean Farms. European Review of Agricultural Economics, 41, 775-815. http://dx.doi.org/10.1093/erae/jbt041

[29] Markowitz, H.M. (1959) Portfolio Selection. Yale University Press, New Haven.

[30] Jorion, P. (1996) Value at Risk: A New Benchmark for Measuring Derivatives Risk. Irwin Professional Publishing, Chicago.

[31] Rockafellar, R.T. and Uryasev, S. (2000) Optimization of Conditional Value at Risk. Journal of Risk, 2, 21-41.

[32] Antle, J.M. (2010) Asymmetry, Partial Moments and Production Risk. American Journal of Agricultural Economics, 92, 1294-1309. http://dx.doi.org/10.1093/ajae/aaq077

[33] Schmeidler, D. (1989) Subjective Probability and Expected Utility without Additivity. Econometrica, 57, 571-587. http://dx.doi.org/10.2307/1911053

[34] Wakker, P. (1990) Under Stochastic Dominance Choquet-Expected Utility and Anticipated Utility Are Identical. Theory and Decision, 29, 119-132. http://dx.doi.org/10.1007/BF00126589

[35] Winkler, R.L., Roodman, G.M. and Britney, R.R. (1972) The Determination of Partial Moments. Management Science, 19, 290-296. http://dx.doi.org/10.1287/mnsc.19.3.290

[36] Gonzalez, R. and Wu, G. (1999) On the Shape of the Probability Weighting Function. Cognitive Psychology, 38, 129-166. http://dx.doi.org/10.1006/cogp.1998.0710