# Portfolio allocation problems between risky and ambiguous assets

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## Abstract

This paper considers a portfolio allocation problem between a risky asset and an *ambiguous asset*, and investigates how greater ambiguity aversion influences the optimal proportion invested in the two assets. We derive several sufficient conditions under which greater ambiguity aversion decreases the optimal proportion invested in the ambiguous asset. Furthermore, we consider an international diversification problem as an application and show that ambiguity aversion partially resolves the home bias puzzle.

## Keywords

Uncertainty modelling Home bias puzzle Portfolio allocation problem Smooth ambiguity model Greater ambiguity aversion## JEL Classification

G11 D81## Notes

### Acknowledgements

We acknowledge an anonymous reviewer whose comments improved this paper substantially. We are grateful to Masamitsu Ohnishi and participants at Paris Financial Management Conference 2017 and the 2017 Annual Meeting of the Nippon Finance Association. Needless to say, we are responsible for any remaining errors. This research is financially supported by the JSPS KAKENHI Grant Nos. 26380240, 26380411, 26705004, 16H02026, 16H03619, 16K03558, 17K03806, and the Joint Research Program of KIER.

## References

- Anantanasuwong, K., Kouwenberg, R., Mitchell, O. S., & Peijnenburg, K. (2019).
*Ambiguity attitudes about investments: Evidence from the field*. Wharton Pension Research Counsil Working Papers, University of Pennsylvania.Google Scholar - Arrow, K. J. (1965).
*Aspects of the theory of risk-bearing*. Helsinki: Yrjo Jahnsonin Saatio.Google Scholar - Bianchi, M., & Tallon, J.-M. (2018). Ambiguity preferences and portfolio choices: Evidence from the field.
*Management Science*. https://doi.org/10.1287/mnsc.2017.3006. - Billingsley, P. (1995).
*Probability and measure*(3rd ed.). Hoboken: Wiley.Google Scholar - Borgonovo, E., Cappelli, V., Maccheroni, F., & Marinacci, M. (2018). Risk analysis and decision theory: A bridge.
*European Journal of Operational Research*,*264*, 280–293.CrossRefGoogle Scholar - Boyle, P., Garlappi, L., Uppal, R., & Wang, T. (2012). Keynes meets Markowitz: The trade-off between familiarity and diversification.
*Management Science*,*58*, 253–272.CrossRefGoogle Scholar - Chateauneuf, A. (1994). Modeling attitudes towards uncertainty and risk through the use of choquet integral.
*Annals of Operations Research*,*52*, 3–20.CrossRefGoogle Scholar - Chiu, W. H., Eeckhoudt, L., & Rey, B. (2012). On relative and partial risk attitudes: Theory and implications.
*Economic Theory*,*50*, 151–167.CrossRefGoogle Scholar - Clark, E., & Jokung, O. (1999). A note on asset proportions, stochastic dominance, and the 50% rule.
*Management Science*,*45*, 1724–1727.CrossRefGoogle Scholar - Driouchi, T., Trigeorgis, L., & So, R. H. Y. (2018). Option implied ambiguity and its information content: Evidence from the subprime crisis.
*Annals of Operations Research*,*262*, 463–491.CrossRefGoogle Scholar - Eeckhoudt, L., Fiori, A. M., & Gianin, E. R. (2016). Loss-averse preferences and portfolio choices: An extension.
*European Journal of Operational Research*,*249*, 224–230.CrossRefGoogle Scholar - Eeckhoudt, L., & Gollier, C. (1995). Demand for risky assets and the monotone probability ratio order.
*Journal of Risk and Uncertainty*,*11*, 113–122.CrossRefGoogle Scholar - Eeckhoudt, L., & Schlesinger, H. (2006). Putting risk in its proper place.
*American Economic Review*,*96*, 280–289.CrossRefGoogle Scholar - Ekern, S. (1980). Increasing \(N\)th degree risk.
*Economics Letters*,*6*, 329–333.CrossRefGoogle Scholar - Ellsberg, D. (1961). Risk, ambiguity, and the savage axioms.
*Quarterly Journal of Economics*,*75*, 643–669.CrossRefGoogle Scholar - Epstein, L. G., & Miao, J. (2003). A two-person dynamic equilibrium under ambiguity.
*Journal of Economic Dynamics and Control*,*27*, 1253–1288.CrossRefGoogle Scholar - Epstein, L. G., & Schneider, M. (2008). Ambiguity, information quality, and asset pricing.
*Journal of Finance*,*63*, 197–228.CrossRefGoogle Scholar - Fang, Y., & Post, T. (2017). Higher-degree stochastic dominance optimality and efficiency.
*European Journal of Operational Research*,*261*, 984–993.CrossRefGoogle Scholar - Fishburn, P. C., & Porter, R. B. (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 - French, K. R., & Poterba, J. M. (1991). Investor diversification and international equity markets.
*American Economic Review*,*81*, 221–226.Google Scholar - Ghirardato, P., & Marinacci, M. (2001). Risk, ambiguity, and the separation of utility and beliefs.
*Mathematics of Operations Research*,*26*, 864–890.CrossRefGoogle Scholar - Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with non-unique priors.
*Journal of Mathematical Economics*,*18*, 141–153.CrossRefGoogle Scholar - Gilboa, I., & Schmeidler, D. (1994). Additive representations of non-additive measures and the choquet integral.
*Annals of Operations Research*,*52*, 43–65.CrossRefGoogle Scholar - Gilboa, I., & Schmeidler, D. (1995). Canonical representation of set functions.
*Mathematics of Operations Research*,*20*, 197–212.CrossRefGoogle Scholar - Gollier, C. (2011). Portfolio choice and asset prices: The comparative statics of ambiguity aversion.
*Review of Economic Studies*,*78*, 1329–1344.CrossRefGoogle Scholar - Hadar, J., & Seo, T. K. (1988). Asset propotions in optimal portfolios.
*Review of Economic Studies*,*55*, 459–468.CrossRefGoogle Scholar - Hadar, J., & Seo, T. K. (1990). The effects of shifts in a return distribution on optimal portfolios.
*International Economic Review*,*31*, 721–736.CrossRefGoogle Scholar - Huang, Y.-C., & Tzeng, L. Y. (2018). A mean-preserving increase in ambiguity and portfolio choices.
*Journal of Risk and Insurance*,*85*, 993–1012.CrossRefGoogle Scholar - Jewitt, I., & Mukerji, S. (2017). Ordering ambiguous acts.
*Journal of Economic Theory*,*171*, 213–267.CrossRefGoogle Scholar - Jindapon, P., & Neilson, W. S. (2007). Higher-order generalizations of Arrow–Pratt and ross risk aversion: A comparative statics approach.
*Journal of Economic Theory*,*136*, 719–728.CrossRefGoogle Scholar - Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk.
*Econometrica*,*47*, 263–291.CrossRefGoogle Scholar - Kelsey, D., Kozhan, R., & Pang, W. (2010). Asymmetric momentum effects under uncertainty.
*Review of Finance*,*15*, 603–631.CrossRefGoogle Scholar - Keynes, J. M. (1921).
*A treatise on probability*. London: MacMillan.Google Scholar - Kijima, M., & Ohnishi, M. (1996). Portfolio selection problems via the bivariate characterization of stochastic dominance relations.
*Mathematical Finance*,*6*, 237–277.CrossRefGoogle Scholar - Klibanoff, P., Marinacci, M., & Mukerji, S. (2005). A smooth model of decision making under ambiguity.
*Econometrica*,*73*, 1849–1892.CrossRefGoogle Scholar - Knight, F. H. (1921).
*Risk, uncertainty and profit*. Boston: Houghton Mifflin.Google Scholar - Landsberger, M., & Meilijson, I. (1990). Demand for risky financial assets: A portfolio analysis.
*Journal of Economic Theory*,*50*, 204–213.CrossRefGoogle Scholar - Lehmann, E. L. (2005).
*Testing statistical hypotheses*(3rd ed.). Berlin: Springer.Google Scholar - Levy, H. (1992). Stochastic dominance and expected utility: Survey and analysis.
*Management Science*,*38*, 555–593.CrossRefGoogle Scholar - Lewis, K. K. (1999). Trying to explain home bias in equities and consumption.
*Journal of Economic Literature*,*37*, 571–608.CrossRefGoogle Scholar - Menezes, C., Geiss, C., & Tressler, J. (1980). Increasing downside risk.
*American Economic Review*,*70*, 921–932.Google Scholar - Meyer, D. J., & Meyer, J. (2005). Relative risk aversion: What do we know?
*Journal of Risk and Uncertainty*,*31*, 243–262.CrossRefGoogle Scholar - Neilson, W. (2010). A simplified axiomatic approach to ambiguity aversion.
*Journal of Risk and Uncertainty*,*41*, 113–124.CrossRefGoogle Scholar - Osaki, Y., & Schlesinger, H. (2014).
*Portfolio choice and ambiguous background risk*. Working Paper, University of Alabama. Available at http://hschlesinger.people.ua.edu/uploads/2/6/8/4/26840405/ambiguousbgr.pdf. - Peter, R. (2019). Revisiting precautionary saving under ambiguity.
*Economics Letters*,*174*, 123–127.CrossRefGoogle Scholar - Peter, R., & Ying, J. (2018). Do you trust your insurer? Ambiguity about contract nonperformance and optimal insurance demand.
*Journal of Economic Behavior and Organization*. https://doi.org/10.1016/j.jebo.2019.01.002. - Post, T., & Kopa, M. (2013). General linear formulations of stochastic dominance criteria.
*European Journal of Operational Research*,*230*, 321–332.CrossRefGoogle Scholar - Pratt, J. W. (1964). Risk aversion in the small and the large.
*Econometrica*,*32*, 122–136.CrossRefGoogle Scholar - Roman, D., Mitra, G., & Zverovich, V. (2013). Enhanced indexation based on second-order stochastic dominance.
*European Journal of Operational Research*,*228*, 273–281.CrossRefGoogle Scholar - Rothschild, M., & Stiglitz, J. E. (1970). Increasing risk: I. A definition.
*Journal of Economic Theory*,*2*, 225–243.CrossRefGoogle Scholar - Rothschild, M., & Stiglitz, J. E. (1971). Increasing risk: II. Its economic consequences.
*Journal of Economic Theory*,*3*, 66–84.CrossRefGoogle Scholar - Schmeidler, D. (1989). Subjective probability and expected utility without additivity.
*Econometrica*,*57*, 571–587.CrossRefGoogle Scholar - Segal, U. (1987). The Ellsberg paradox and risk aversion: An anticipated utility approach.
*International Economic Review*,*28*, 175–202.CrossRefGoogle Scholar - Solnik, B., & Zuo, L. (2012). A global equilibrium asset pricing model with home preference.
*Management Science*,*58*, 273–292.CrossRefGoogle Scholar - Solnik, B., & Zuo, L. (2017). Relative optimism and the home bias puzzle.
*Review of Finance*,*21*, 2045–2074.Google Scholar