# Risk measures based on behavioural economics theory

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## Abstract

Coherent risk measures (Artzner et al. in Math. Finance 9:203–228, 1999) and convex risk measures (Föllmer and Schied in Finance Stoch. 6:429–447, 2002) are characterized by desired axioms for risk measures. However, concrete or practical risk measures could be proposed from different perspectives. In this paper, we propose new risk measures based on behavioural economics theory. We use rank-dependent expected utility (RDEU) theory to formulate an objective function and propose the smallest solution that minimizes the objective function as a risk measure. We also employ cumulative prospect theory (CPT) to introduce a set of acceptable regulatory capitals and define the infimum of the set as a risk measure. We show that the classes of risk measures derived from RDEU theory and CPT are equivalent, and they are all monetary risk measures. We present the properties of the proposed risk measures and give sufficient and necessary conditions for them to be coherent and convex, respectively. The risk measures based on these behavioural economics theories not only cover important risk measures such as distortion risk measures, expectiles and shortfall risk measures, but also produce new interesting coherent risk measures and convex, but not coherent risk measures.

## Keywords

Distortion risk measure Expectile Coherent risk measure Convex risk measure Monetary risk measure Stop-loss order preserving Rank-dependent expected utility theory Cumulative prospect theory## Mathematics Subject Classification (2010)

91B16 91B30 91G99## JEL Classification

C60 G10 D81## Notes

### Acknowledgements

The authors thank the two anonymous referees, the Associate Editor, and the Editor for insightful suggestions that improved the presentation of the paper. Tiantian Mao is grateful for the support from National Science Foundation of China (grant Nos. 71671176, 11371340). Jun Cai is grateful to the support from the Natural Sciences and Engineering Research Council (NSERC) of Canada (grant No. RGPIN-2016-03975).

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