# The two-moment decision model with additive risks

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## 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

- Battermann, H., U. Broll, and J.E. Wahl. 2002. Insurance demand and the elasticity of risk aversion.
*OR Spectrum*24: 145–150.CrossRefGoogle Scholar - 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 - 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 - Broll, U., and S. Mukherjee. 2017. International trade and firms’ attitude towards risk.
*Economic Modelling*64: 69–73.CrossRefGoogle Scholar - Broll, U., J.E. Wahl, and W.K. Wong. 2006. Elasticity of risk aversion and international trade.
*Economics Letters*91: 126–130.CrossRefGoogle Scholar - Caballé, J., and A. Pomansky. 1997. Complete monotonicity, background risk, and risk aversion.
*Mathematical Social Sciences*34: 205–222.CrossRefGoogle Scholar - Chamberlain, G. 1983. A characterization of the distributions that imply mean–variance utility functions.
*Journal of Economic Theory*29: 185–201.CrossRefGoogle Scholar - 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
- Chiu, W.H. 2010. Skewness preference, risk taking and expected utility maximization.
*The Geneva Risk and Insurance Review*35: 108–129.CrossRefGoogle Scholar - 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 - 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 - Eeckhoudt, L., C. Gollier, and H. Schlesinger. 1996. Changes in background risk and risk taking behavior.
*Econometrica*64: 683–689.CrossRefGoogle Scholar - 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 - Eichner, T. 2008. Mean variance vulnerability.
*Management Science*54: 586–593.CrossRefGoogle Scholar - Eichner, T., and A. Wagener. 2003a. More on parametric characterizations of risk aversion and prudence.
*Economic Theory*21: 895–900.CrossRefGoogle Scholar - 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 - 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 - 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 - Eichner, T., and A. Wagener. 2009. Multiple risks and mean–variance preferences.
*Operations Research*57: 1142–1154.CrossRefGoogle Scholar - Eichner, T., and A. Wagener. 2011a. Portfolio allocation and asset demand with mean–variance preferences.
*Theory and Decision*70: 179–193.CrossRefGoogle Scholar - Eichner, T., and A. Wagener. 2011b. Increases in skewness and three-moment preferences.
*Mathematical Social Sciences*61 (2): 109–113.CrossRefGoogle Scholar - 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 - 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 - Fang, K.-T., S. Kotz, and K.-W. Ng. 1990.
*Symmetric multivariate and related distributions*. London/New York: Chapman and Hall.CrossRefGoogle Scholar - 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 - Franke, G., H. Schlesinger, and R.C. Stapleton. 2006. Multiplicative background risk.
*Management Science*52: 146–153.CrossRefGoogle Scholar - 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 - Honda, Y. 1985. Downside risk and the competitive firm.
*Metroeconomica*37: 231–240.CrossRefGoogle Scholar - Kimball, M. 1990. Precautionary saving in the small and in the large.
*Econometrica*58: 53–73.CrossRefGoogle Scholar - Lajeri, F., and L.T. Nielsen. 2000. Parametric characterizations of risk aversion and prudence.
*Economic Theory*15: 469–476.CrossRefGoogle Scholar - 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 - Lajeri-Chaherli, F. 2004. Proper prudence, standard prudence and precautionary vulnerability.
*Economics Letters*82: 29–34.CrossRefGoogle Scholar - Lajeri-Chaherli, F. 2005. Proper and standard risk aversion in two-moment decision models.
*Theory and Decision*57: 213–225.CrossRefGoogle Scholar - Landsman, Z., and A. Tsanakas. 2006. Stochastic ordering of bivariate elliptical distributions.
*Statistics and Probability Letters*76: 488–494.CrossRefGoogle Scholar - Levy, H., and M. Levy. 2004. Prospect theory and mean–variance analysis.
*Review of Financial Studies*17: 1015–1041.CrossRefGoogle Scholar - 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 - Menezes, C.F., and D.L. Hanson. 1970. On the theory of risk aversion.
*International Economic Review*11: 481–487.CrossRefGoogle Scholar - Meyer, J. 1987. Two-moment decision models and expected utility maximization.
*American Economic Review*77: 421–430.Google Scholar - 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 - Mueller, A., and M. Scarsini. 2001. Stochastic comparison of random vectors with a common copula.
*Mathematics of Operations Research*26: 723–740.CrossRefGoogle Scholar - Niu, C.Z., W.K. Wong, and Q.F. Xu. 2017. Kappa ratios and (higher-order) stochastic dominance.
*Risk Management*, forthcoming.Google Scholar - Ormiston, M.B., and E.E. Schlee. 2001. Mean-variance preferences and investor behaviour.
*Economic Journal*111: 849–861.CrossRefGoogle Scholar - Pratt, J.W., and R. Zeckhauser. 1987. Proper Risk Aversion.
*Econometrica*55 (1): 143–54.CrossRefGoogle Scholar - Sandmo, A. 1971. On the theory of the competitive firm under price uncertainty.
*American Economic Review*61: 65–73.Google Scholar - Tobin, J. 1958. Liquidity preference as behavior towards risk.
*Review of Economic Studies*25: 65–86.CrossRefGoogle Scholar - Tsetlin, I., and R.L. Winkler. 2005. Risky choices and correlated background risk.
*Management Science*51: 1336–1345.CrossRefGoogle Scholar - Wagener, A. 2002. Prudence and risk vulnerability in two-moment decision models.
*Economics Letters*74: 229–235.CrossRefGoogle Scholar - Wagener, A. 2003. Comparative Statics under Uncertainty: The Case of Mean-Variance Preferences.
*European Journal of Operational Research*151: 224–232.CrossRefGoogle Scholar - Wong, W.K., and R. Chan. 2008. Markowitz and prospect stochastic dominances.
*Annals of Finance*4: 105–129.CrossRefGoogle Scholar - Wong, W.K., and C.H. Ma. 2008. Preferences over location-scale family.
*Economic Theory*37: 119–146.CrossRefGoogle Scholar