Advertisement

Risk Management

, Volume 20, Issue 1, pp 77–94 | Cite as

The two-moment decision model with additive risks

  • Xu Guo
  • Andreas Wagener
  • Wing-Keung Wong
  • Lixing Zhu
Original Article

Abstract

With multiple additive risks, the mean–variance approach and the expected utility approach of risk preferences are compatible if all attainable distributions belong to the same location–scale family. Under this proviso, we survey existing results on the parallels of the two approaches with respect to risk attitudes, the changes thereof, and the comparative statics for simple, linear choice problems under risks. In mean–variance approach all effects can be couched in terms of the marginal rate of substitution between mean and variance. We provide some simple proofs of some previous results. We apply the theory we stated or developed in our paper to study the behavior of banking firm and study risk-taking behavior with background risk in the mean–variance model.

Keywords

Mean–variance model Location–scale family Background risk Multiple additive risks Expected utility approach 

JEL Classification

C0 D81 G11 

Notes

Acknowledgements

The authors are grateful to Professor Ira Horowitz for his valuable comments that have significantly improved this manuscript. The third author would like to thank Professors Robert B. Miller and Howard E. Thompson for their continuous guidance and encouragement. The authors are grateful to the Editor and two anonymous referees for constructive comments and suggestions that led to a significant improvement of an early manuscript. This research has been partially supported by the Fundamental Research Funds for the Central Universities, grants from the National Natural Science Foundation of China (11626130, 11601227, 11671042), Natural Science Foundation of Jiangsu Province, China (BK20150732), Asia University, Hang Seng Management College, Lingnan University, Shanghai University of International Business and Economics, Hong Kong Baptist University, the Research Grants Council (RGC) of Hong Kong (project numbers 12502814 and 12500915), and Ministry of Science and Technology (MOST), R.O.C.

References

  1. Battermann, H., U. Broll, and J.E. Wahl. 2002. Insurance demand and the elasticity of risk aversion. OR Spectrum 24: 145–150.CrossRefGoogle Scholar
  2. Broll, U., M. Egozcue, W.K. Wong, and R. Zitikis. 2010. Prospect theory, indifference curves, and hedging risks. Applied Mathematics Research Express 2: 142–153.Google Scholar
  3. Broll, U., X. Guo, P. Welzel, and W.K. Wong. 2015. The banking firm and risk taking in a two-moment decision model. Economic Modelling 50: 275–280.CrossRefGoogle Scholar
  4. Broll, U., and S. Mukherjee. 2017. International trade and firms’ attitude towards risk. Economic Modelling 64: 69–73.CrossRefGoogle Scholar
  5. Broll, U., J.E. Wahl, and W.K. Wong. 2006. Elasticity of risk aversion and international trade. Economics Letters 91: 126–130.CrossRefGoogle Scholar
  6. Caballé, J., and A. Pomansky. 1997. Complete monotonicity, background risk, and risk aversion. Mathematical Social Sciences 34: 205–222.CrossRefGoogle Scholar
  7. Chamberlain, G. 1983. A characterization of the distributions that imply mean–variance utility functions. Journal of Economic Theory 29: 185–201.CrossRefGoogle Scholar
  8. Chan, R.H., S.C. Chow, and W.K. Wong. 2016. Central moments, stochastic dominance and expected utility. Social Science Research Network Working Paper Series 2849715.Google Scholar
  9. Chiu, W.H. 2010. Skewness preference, risk taking and expected utility maximization. The Geneva Risk and Insurance Review 35: 108–129.CrossRefGoogle Scholar
  10. Choi, G., I. Kim, and A. Snow. 2001. Comparative statics predictions for changes in uncertainty in the portfolio and savings problems. Bulletin of Economic Research 53: 61–72.CrossRefGoogle Scholar
  11. Dionne, G., and Ch. Gollier. 1992. Comparative statics under multiple sources of risk with applications to insurance demand. The Geneva Papers on Risk and Insurance Theory 17: 21–33.CrossRefGoogle Scholar
  12. Eeckhoudt, L., C. Gollier, and H. Schlesinger. 1996. Changes in background risk and risk taking behavior. Econometrica 64: 683–689.CrossRefGoogle Scholar
  13. Egozcue, M., L. Fuentes García, W.K. Wong, and R. Zitikis. 2011. Do investors like to diversify? A study of Markowitz preferences. European Journal of Operational Research 215: 188–193.CrossRefGoogle Scholar
  14. Eichner, T. 2008. Mean variance vulnerability. Management Science 54: 586–593.CrossRefGoogle Scholar
  15. Eichner, T., and A. Wagener. 2003a. More on parametric characterizations of risk aversion and prudence. Economic Theory 21: 895–900.CrossRefGoogle Scholar
  16. Eichner, T., and A. Wagener. 2003b. Variance vulnerability, background risks, and mean–variance preferences. The Geneva Papers on Risk and Insurance—Theory 28: 173–184.CrossRefGoogle Scholar
  17. Eichner, T., and A. Wagener. 2004. Relative risk aversion, relative prudence and comparative statics under uncertainty: The case of (mu, sigma)-preferences. Bulletin of Economic Research 562: 159–170.CrossRefGoogle Scholar
  18. Eichner, T., and A. Wagener. 2005. Measures of risk attitude: Correspondences between mean–variance and expected-utility approaches. Decisions in Economics and Finance 28: 53–65.CrossRefGoogle Scholar
  19. Eichner, T., and A. Wagener. 2009. Multiple risks and mean–variance preferences. Operations Research 57: 1142–1154.CrossRefGoogle Scholar
  20. Eichner, T., and A. Wagener. 2011a. Portfolio allocation and asset demand with mean–variance preferences. Theory and Decision 70: 179–193.CrossRefGoogle Scholar
  21. Eichner, T., and A. Wagener. 2011b. Increases in skewness and three-moment preferences. Mathematical Social Sciences 61 (2): 109–113.CrossRefGoogle Scholar
  22. Eichner, T., and A. Wagener. 2012. Tempering effects of (dependent) background risks: A mean–variance analysis of portfolio selection. Journal of Mathematical Economics 48: 422–430.CrossRefGoogle Scholar
  23. Eichner, T., and A. Wagener. 2014. Insurance demand and first-order risk increases under \((\mu, \sigma )\)-preferences revisited. Finance Research Letters 11: 326–331.CrossRefGoogle Scholar
  24. Fang, K.-T., S. Kotz, and K.-W. Ng. 1990. Symmetric multivariate and related distributions. London/New York: Chapman and Hall.CrossRefGoogle Scholar
  25. Fishburn, P.C., and R.B. Porter. 1976. Optimal portfolios with one safe and one risky asset: Effects of changes in rate of return and risk. Management Science 22: 1064–1073.CrossRefGoogle Scholar
  26. Franke, G., H. Schlesinger, and R.C. Stapleton. 2006. Multiplicative background risk. Management Science 52: 146–153.CrossRefGoogle Scholar
  27. Hadar, J., and T. Seo. 1990. The effects of shifts in a return distribution on optimal portfolios. International Economic Review 31: 721–736.CrossRefGoogle Scholar
  28. Honda, Y. 1985. Downside risk and the competitive firm. Metroeconomica 37: 231–240.CrossRefGoogle Scholar
  29. Kimball, M. 1990. Precautionary saving in the small and in the large. Econometrica 58: 53–73.CrossRefGoogle Scholar
  30. Lajeri, F., and L.T. Nielsen. 2000. Parametric characterizations of risk aversion and prudence. Economic Theory 15: 469–476.CrossRefGoogle Scholar
  31. Lajeri-Chaherli, F. 2002. More on properness: The case of mean–variance preferences. The Geneva Papers on Risk and Insurance Theory 27: 49–60.CrossRefGoogle Scholar
  32. Lajeri-Chaherli, F. 2004. Proper prudence, standard prudence and precautionary vulnerability. Economics Letters 82: 29–34.CrossRefGoogle Scholar
  33. Lajeri-Chaherli, F. 2005. Proper and standard risk aversion in two-moment decision models. Theory and Decision 57: 213–225.CrossRefGoogle Scholar
  34. Landsman, Z., and A. Tsanakas. 2006. Stochastic ordering of bivariate elliptical distributions. Statistics and Probability Letters 76: 488–494.CrossRefGoogle Scholar
  35. Levy, H., and M. Levy. 2004. Prospect theory and mean–variance analysis. Review of Financial Studies 17: 1015–1041.CrossRefGoogle Scholar
  36. Leung, P.L., and W.K. Wong. 2008. On testing the equality of the multiple Sharpe ratios, with application on the evaluation of iShares. Journal of Risk 10: 1–16.CrossRefGoogle Scholar
  37. Menezes, C.F., and D.L. Hanson. 1970. On the theory of risk aversion. International Economic Review 11: 481–487.CrossRefGoogle Scholar
  38. Meyer, J. 1987. Two-moment decision models and expected utility maximization. American Economic Review 77: 421–430.Google Scholar
  39. Meyer, J. 1992. Beneficial changes in random variables under multiple sources of risk and their comparative statics. The Geneva Papers on Risk and Insurance Theory 17: 7–19.CrossRefGoogle Scholar
  40. Mueller, A., and M. Scarsini. 2001. Stochastic comparison of random vectors with a common copula. Mathematics of Operations Research 26: 723–740.CrossRefGoogle Scholar
  41. Niu, C.Z., W.K. Wong, and Q.F. Xu. 2017. Kappa ratios and (higher-order) stochastic dominance. Risk Management, forthcoming.Google Scholar
  42. Ormiston, M.B., and E.E. Schlee. 2001. Mean-variance preferences and investor behaviour. Economic Journal 111: 849–861.CrossRefGoogle Scholar
  43. Pratt, J.W., and R. Zeckhauser. 1987. Proper Risk Aversion. Econometrica 55 (1): 143–54.CrossRefGoogle Scholar
  44. Sandmo, A. 1971. On the theory of the competitive firm under price uncertainty. American Economic Review 61: 65–73.Google Scholar
  45. Tobin, J. 1958. Liquidity preference as behavior towards risk. Review of Economic Studies 25: 65–86.CrossRefGoogle Scholar
  46. Tsetlin, I., and R.L. Winkler. 2005. Risky choices and correlated background risk. Management Science 51: 1336–1345.CrossRefGoogle Scholar
  47. Wagener, A. 2002. Prudence and risk vulnerability in two-moment decision models. Economics Letters 74: 229–235.CrossRefGoogle Scholar
  48. Wagener, A. 2003. Comparative Statics under Uncertainty: The Case of Mean-Variance Preferences. European Journal of Operational Research 151: 224–232.CrossRefGoogle Scholar
  49. Wong, W.K., and R. Chan. 2008. Markowitz and prospect stochastic dominances. Annals of Finance 4: 105–129.CrossRefGoogle Scholar
  50. Wong, W.K., and C.H. Ma. 2008. Preferences over location-scale family. Economic Theory 37: 119–146.CrossRefGoogle Scholar

Copyright information

© Macmillan Publishers Ltd 2017

Authors and Affiliations

  • Xu Guo
    • 1
  • Andreas Wagener
    • 2
  • Wing-Keung Wong
    • 3
    • 4
    • 5
  • Lixing Zhu
    • 6
    • 7
  1. 1.School of StatisticsBeijing Normal UniversityBeijingChina
  2. 2.Institute of Economic PolicyUniversity of HannoverHannoverGermany
  3. 3.Department of Finance and Big Data Research CenterAsia UniversityTaichung CityTaiwan, ROC
  4. 4.Department of Economics and FinanceHang Seng Management CollegeHong KongChina
  5. 5.Department of EconomicsLingnan UniversityHong KongChina
  6. 6.School of Statistics and InformationShanghai University of International Business and EconomicsShanghaiChina
  7. 7.Department of MathematicsHong Kong Baptist UniversityHong KongChina

Personalised recommendations