Behavioral Optimal Insurance : its resolution of a socioeconomic enigma

1. Behavioral Optimal Insurance: its resolution of a socioeconomic enigma CIAS in CUFE, April 2014 Phillip Yam Department of Statistics The Chinese University of Hong Kong IME: Vol. 49, (3), PP 418–428 Joint work with Joseph Sung, S.P. Yung and J.H. Zhou Supported by HKGRF 502909, CUHK Direct Grant 2060444

2. Outline: • Classical Expected Utility Principle: Theory and Limitations • Behavioral finance: Prospect theory • Formulation of the behavioral optimal insurance decision problem • Solution and a sketch of proof • Conclusion

3. Arrow’s theorem on optimal insurance (One-period model) • An individual/entity is risk-averse, i.e. with concave utility function. • Facing at a risk X and having an opportunity to buy an insurance in the market. • Every insurance indemnity I(X) satisfies: (1) 0 < I(X) < X; (2) I(X) is a non-decreasing and continuous function in X. • Suppose all feasible insurances are priced actuarially at the same rate p (for sake of convenience, we assume that there is no risk-loading). • Claim: The optimal insurance which maximizes the expected utility of the terminal wealth w – p – X – I(X) is the deductible insurance. • For more details, see Arrow (1971, and Scand. Act. J., 1974) and Borch (Astin Bull. 1975).

4. Inadequacy of Expected Utility Theory (EUT) • From early 1950s on, both psychologists and economists have uncovered a vast amount of evidence that individuals do not necessarily conform to many of the key assumptions or predictions of the EUT. • Deviations from the EUT are both predictable and systemic. • Allais paradox (Econometrica, 1953). A doubt on an axiom of the expected utility theory, namely: The axiom of independence – if people generally prefer lottery X over lottery Y, then mixing both lotteries with a third lottery, Z, should not alter the preference. • “Utility maximization is neither a necessary nor a sufficient condition for deducing who will buy insurance” See Simon (U. Chicago Press, 1987), Pashigian, Schkade and Menefee (J. Business, 1966). • Edwards (Psy Bulletin, 1954, and Ann. Rev. Psy., 1961). People can make “smarter” decisions under uncertainty after using tools and having trainings.

5. Negative response from the “Reality” Laboratory Studies • Slovic, Fischhoff, Lictenstein (J. Risk&Ins, 1977), Schoemaker and Kunreuther (J. Risk&Ins, 1979): (1) Insureds preferred low-deductible policies despite their disproportionately high premiums; (2) The failure of individuals to buy insurance even when the premiums have been highly subsidized; (3) Individuals will not use insurance to protect themselves against rare, large losses. They pay more attention to probable small losses. • Huberman, Mayers and Smith (Bell J. Econ., 1983) : “… ‘troubling’ fact that upper limits on coverage are pervasive for just those risks with virtually unbounded losses. … explanation simply recognizes that bankruptcy statutes provide individuals with a form of limited liability. …” • Fuchs (1974, J. Law&Econ., 1976, and 1991): “… (C)onsumers should prefer major medical (catastrophe) insurance, that is, plans with substantial deductibles or co-payment provisions for moderate expenses but ample coverage for very large expenses. Instead, we observe a strong preference for ‘first dollar’ or shallow coverage. …”

6. Extension of Arrow’s result • Non-Expected Utility theory: (1) Choquet Expected Utility (CEU) maximizer (see, e.g. Dana and Shahidi (2000), Machina (Encylopedia of Actuarial Science, 2004) (2) Rank-dependent expected utility model (RDEU): expectation of utility under a new probability measure which is a distortion of the original cumulative probability function. • Others: Cummins and Mahul (J. Risk&Ins., 2004) and Carlier and Dana (J. Math. Econ., 2007). Using RDEU, DARA (Decreasing Absolute Risk-aversion), and their modifications to provide an explanation clients’ preference for limited coverage of insurance. • Limitations of previous works: (1) Lack of immediate connection to mainstream economic theory; (2) Some of the maximizers are even discontinuous.

7. Behavioral Science (Economics/Finance) Ultimatum Game • Player 1: right to choose two choices of divisions of money, e.g. (80%, 20%) or (50%, 50%) • Player 2: Determine whether to accept the chosen division or not. • Humans normally rejects the unfair (behavioral, not too rational) • Chimpanzees normally accept whatever offers (risk-averse and highly “rational”) • Quoted from “Chimpanzees Are Rational Maximizers in an Ultimatum Game” by Keith Jensen et. al., Science 5, Oct 2007.

8. Prospect Theory (Cumulative) Prospect Theory developed by Kahneman and Tversky (Econometrica, 1979, and Amer. Psycho. 1984; Cumulative, J. Risk&Uncertainty, 1992). Also see Kahneman, Slovic and Tversky (CUP, 1982)). A unified theory that can resolve the limitations of EUT on one hand, and “truly” reflect the human behavior on the other hand.

9. Prospect Theory (Cumulative) Facing a lottery • Game A: 25% Win $8000 or 75% Nothing • Game B: 25% Win $3000, 25% Win $5000 or 50% Nothing According to expected utility theory, 0.25*u(8000)<0.25*u(3000) +0.25*u(5000) which implies (u(8000) – u(5000))/3000 < (u(3000) – u(0))/3000. • Most of you are risk-averse towards GAIN.

10. Prospect Theory (Cumulative) Facing a risk • Game C: 20% Lose $10,000 • Game D: 20% Lose $ 4,000 or 20% Lose $6,000 0.2*u(10,000)>0.2*u(4,000) +0.2*u(6,000) which implies (u(10,000) – u(6,000))/4,000 > (u(4,000) – u(0))/4,000. Contrary to the concavity assumption of utility function. • Most of you are ACTUALLY risk-seeking towards LOSS.

11. S-shaped value function (utility)

12. Prospect Theory (Cumulative) Distortion • Game E: 0.001% Win $2,000,000 or otherwise Nothing • Game F: 0.002% Win $1,000,000 or otherwise Nothing “0.001%” * u(2M) > “0.002%” *u(1M) which implies “0.001%”/ “0.002%” > u(1M)/u(2M) > 1/2 (using the concavity of utility . • Most of you may perceive probability of occurrence of a contingent event in a “distorted” way.

13. g-Distortion • Slovic, Fischhoff, Lictenstein (J. Risk&Ins, 1977), Schoemaker and Kunreuther (J. Risk&Ins, 1979): Individuals will not use insurance to protect themselves against rare, large losses. They pay more attention to probable small losses. • g(0) = 0 and g(1) = 1, and g’ is increasing. Also see Yaari’s (1987).

14. Feasible insurances in the market • Voice from clients: an additional amount of compensation could be ensured if there would be an additional amount of damage, i.e. I(X) is increasing. • Thoughts of insurers and regulators: (1) There cannot be a “bonus” whenever a reported loss exceeds any particular finite value, i.e. I(X) should be continuous in X. (2) Insurance is a protection against contingent events and it is not a lottery, an additional unit of loss claimed by the insureds cannot result in more than a unit of redemption; the insureds also have to take the initiative and responsibility to reduce as much damage as possible, i.e. I(X) <X. • Violation of (1) and/or (2) may result in moral issue, e.g. insurance swindle over their losses.

15. Mathematical formulation of the optimization problem (1) • Maximize the value function: where subject to: • For illustration, we assume that any insurance indemnity I(X) is priced actuarially, i.e. • I(X) is continuous, non-decreasing, and I(X) <X, in particular, we have:

16. Solution to Problem (1) • Splitting into three sub-problems: Given a feasible indemnity I(X), define: Note that both I1 and I2 are continuous and

17. Theorem: Given an indemnity I*(X), define: is optimal to Problem (1) if, and only if, I1*(X) and I2*(X) are optimal to the following two optimization Problems (2) and (3) respectively with the ordered pair (A*, P1*) which is also optimal to Problem (4).

18. Problems (2), (3) and (4) Given a positive number P1 and a measurable set A, • Problem (2) Subject to • Problem (3) Subject to • Problem (4) where a is a real number such that

19. Further reduction • Under the condition on our indemnities I(X), it can easily be shown that A* can be chosen in the form as an interval [0, b]. • Problem (2’) • Problem (3’)

20. Solution to Problem (2’)

21. For any non-negative random variable Z and h is a continuous non-increasing function where Yg1 ~ g1 o Pr(X < y) • Define

22. Note that • Therefore,

23. Solution to Problem (3’)

24. Combing solutions in Problems (2’) and (3’)

25. For every A* = [0, b], there is a better A** = [0, d3], in the sense that

26. Form of an optimal insurance • Insurance layer

27. Corollaries • Either the initial wealth w goes to infinity or the reference point goes to negative infinity, the optimal insurance converges in Lp to the standard deductible insurance as deduced by Arrow. • Treating the premium rate P as a variable, and optimizing with respect to P, we deduce that the optimal insurance should have zero deductible with limited coverage. • If g-distortion is getting “serious”, the optimal premium rate will approaches to zero; in other words, the client prefers not to buy the insurance.

28. Corollaries • There is a critical point separating the region of low risk loading, which favors the insurance buying, from the region of high risk loading in which no insurance buying is more preferable

29. Conclusion and future works • Implications on policy-making: boosting the distortion g through education; • Pareto optimal nature of the “natural” features of all feasible insurances in the market; • Extension to dynamic settings; • Problem subject to more financial constraints.

30. Thank you!