Combined functionals as risk measures

1. Combined functionals as risk measures Arcady Novosyolov Institute of computational modeling SB RAS, Krasnoyarsk, Russia, 660036 anov@icm.krasn.ruhttp://www.geocities.com/novosyolov/

2. Structure of the presentation Risk Risk measure RM: Expectation RM: Expected utility RM: Distorted probability RM: Combined functional Relations among risk measures Illustrations Anticipated questions Combined functionals

3. Risk Risk is an almost surely bounded random variable Another interpretation: risk is a real distribution function with bounded support Correspondence: Previous Next Why bounded? Back to structure Combined functionals

4. Example: Finite sample space Let the sample space be finite: Then: Probability distribution is a vector Random variable is a vector Distribution function is a step function 1 Previous Next Back to structure Combined functionals

5. Risk treatment Here risk is treated as gain (the more, the better). Examples: •Return on a financial asset • Insurable risk • Profit/loss distribution (in thousand dollars) Previous Next Back to structure Combined functionals

6. Risk measure Risk measure is a real-valued functional or Risk measures allowing both representations with are called law invariant. Previous Next Back to structure Combined functionals

7. Using risk measures • Certain equivalent of a risk • Price of a financial asset, portfolio • Insurance premium for a risk • Goal function in decision-making problems Previous Next Back to structure Combined functionals

8. RM: Expectation Expectation is a very simple law invariant risk measure, describing a risk-neutral behavior. Being almost useless itself, it is important as a basic functional for generalizations. Expected utility risk measure may be treated as a combination of expectation and dollar transform. Distorted probability risk measure may be treated as a combination of expectation and probability transform. Combined functional is essentially the application of both transforms to the expectation. Previous Next Back to structure Combined functionals

9. RM: Expected utility Expected utility is a law invariant risk measure, exhibiting risk averse behavior, when its utility function U is concave (U''(t)<0). Expected utility is linear with respect to mixture of distributions, a disadvantageous feature, that leads to effects, perceived as paradoxes. Previous Next Is EU a certain equivalent? Back to structure Combined functionals

10. EU as a dollar transform Previous Next Back to structure Combined functionals

11. EU is linear in probability Expected utility functional is linear with respect to mixture of distributions. Indifference "curves" on a set of probability distributions: parallel straight lines Previous Next Back to structure Combined functionals

12. EU: Rabin's paradox Consider equiprobable gambles implying loss L or gain G with probability 0.5 each, with initial wealth x. Here 0<L<G. Rabin had discovered the paradox: if an expected utility maximizer rejects such gamble for any initial wealth x, then she would reject similar gambles with some loss L0> L and any gain G0, no matter how large. Example: let L = $100, G = $125. Then expected utility maximizer would reject any equiprobable gamble with loss L0= $600. Previous Next Back to structure Combined functionals

13. RM: Distorted probability Distortion function Distorted probability is a law invariant risk measure, exhibiting risk averse behavior, when its distortion function g satisfies g(v)<v, all v in [0,1]. Distorted probability is positive homogeneous, that may lead to improper insurance premium calculation. Previous Next Back to structure Combined functionals

14. DP as a probability transform Previous Next Back to structure Combined functionals

15. DP is positively homogeneous Distorted probability is a positively homogeneous functional, which is an undesired property in insurance premium calculation. Consider a portfolio containing a number of "small" risks with loss $1,000 and a few "large" risks with loss $1,000,000 and identical probability of loss. Then DP functional assigns 1000 times larger premium to large risks, which seems intuitively insufficient. Previous Next Back to structure Combined functionals

16. RM: Combined functional Recall expected utility and distorted probability functionals: Combined functional involves both dollar and probability transforms: Discrete case: Previous Next Back to structure Combined functionals

17. CF, risk aversion Combined functional exhibits risk aversion in a flexible manner: if its distortion function g satisfies risk aversion condition, then its utility function Uneed not be concave. The latter may be even convex, thus resolving Rabin's paradox. Next slides display an illustration. Note that if distortion function g of a combined functional does not satisfy risk aversion condition, then the combined functional fails to exhibit risk aversion. Concave utility function alone cannot provide "enough" risk aversion. Previous Next Back to structure Combined functionals

18. CF, example parameters Previous Next Back to structure Combined functionals

19. CF: Rabin's paradox resolved Given the combined functional with parameters from the previous slide (with t measured in hundred dollars), the equiprobable gamble with L = $100, G = $125 is rejected at any initial wealth, and the following equiprobable gambles are acceptable at any wealth level: Previous Next Back to structure Combined functionals

20. Relations among risk measures Legend Generalization Previous Next Partial generalization Back to structure Combined functionals

21. Legend for relations - expectation - expected utility - distorted probability - combined functional RDEU – rank-dependent expected utility, Quiggin, 1993 Coherent risk measure – Artzner et al, 1999 Previous Next Back to structure Combined functionals

22. Illustrations Expected utility indifference curves Distorted probability indifference curves Combined functional indifference curves Previous Next Back to structure Combined functionals

23. EU: indifference curves Over risks in R2 Over distributions in R3 Previous Next Back to structure Combined functionals

24. DP: indifference curves Over risks in R2 Over distributions in R3 Previous Next Back to structure Combined functionals

25. CF: indifference curves Over risks in R2 Over distributions in R3 Previous Next Back to structure Combined functionals

26. A few anticipated questions Why are risks assumed bounded? Is EU a certain equivalent? Previous Next Back to structure Combined functionals

27. Why are risks assumed bounded? Boundedness assumption is a matter of convenience. Unbounded random variables and distributions with unbounded support may be considered as well, with some additional efforts to overcome technical difficulties. Previous Next Back to structure Back to Risk Combined functionals

28. Is EU a certain equivalent? Strictly speaking, the value of expected utility functional itself is not a certain equivalent. However, the certain equivalent can be easily obtained by applying the inverse utility function: Previous Next Back to structure Back to EU Combined functionals