Combined functionals as risk measures Arcady Novosyolov Institute of computational modeling SB RAS, Krasnoyarsk, Russia,

A. NovosyolovCombined functionals2 Structure of the presentation Risk Risk measure Relations among risk measures RM: Expectation RM: Expected utility RM: Distorted probability RM: Combined functional Anticipated questions Illustrations

A. NovosyolovCombined functionals3 Risk Risk is an almost surely bounded random variable Another interpretation: risk is a real distribution function with bounded support Correspondence: Why bounded? Back to structure NextPrevious

A. NovosyolovCombined functionals4 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 Back to structure 1 NextPrevious

A. NovosyolovCombined functionals5 Risk treatment Here risk is treated as gain (the more, the better). Examples: Return on a financial asset x0%20% p x-$1,000,000$0 p Insurable risk Profit/loss distribution (in thousand dollars) x p Back to structure NextPrevious

A. NovosyolovCombined functionals6 Risk measure Risk measure is a real-valued functional or Risk measures allowing both representations with Back to structure are called law invariant. NextPrevious

A. NovosyolovCombined functionals7 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 Back to structure NextPrevious

A. NovosyolovCombined functionals8 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 utilityExpected utility risk measure may be treated as a combination of expectation and dollar transform. Distorted probabilityDistorted probability risk measure may be treated as a combination of expectation and probability transform. Back to structure NextPrevious Combined functionalCombined functional is essentially the application of both transforms to the expectation.

A. NovosyolovCombined functionals9 RM: Expected utility Back to structure 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.paradoxes NextPrevious Is EU a certain equivalent?

A. NovosyolovCombined functionals10 EU as a dollar transform Back to structure Valuex1x2…xn Probp1p2…pn ValueU(x1)U(x2)…U(xn) Probp1p2…pn NextPrevious

A. NovosyolovCombined functionals11 EU is linear in probability Indifference "curves" on a set of probability distributions: parallel straight lines Back to structure Expected utility functional is linear with respect to mixture of distributions. NextPrevious

A. NovosyolovCombined functionals12 EU: Rabin's paradox Back to structure Consider equiprobable gambles implying loss L or gain G with probability 0.5 each, with initial wealth x. Here 0 L and any gain G0, no matter how large. Valuex-Lx+G prob0.5 Example: let L = $100, G = $125. Then expected utility maximizer would reject any equiprobable gamble with loss L0= $600. NextPrevious

A. NovosyolovCombined functionals13 RM: Distorted probability Back to structure 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]. Distortion function Distorted probability is positive homogeneous, that may lead to improper insurance premium calculation. NextPrevious

A. NovosyolovCombined functionals14 DP as a probability transform Back to structure Valuex1x2…xn Probp1p2…pn Valuex1x2…xn Probq1q2…qn NextPrevious

A. NovosyolovCombined functionals15 DP is positively homogeneous Back to structure 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. NextPrevious Distorted probability is a positively homogeneous functional, which is an undesired property in insurance premium calculation.

A. NovosyolovCombined functionals16 RM: Combined functional Back to structure Combined functional involves both dollar and probability transforms:dollar probability Discrete case: Recall expected utility and distorted probability functionals:expected utilitydistorted probability NextPrevious

A. NovosyolovCombined functionals17 CF, risk aversion Back to structure Combined functional exhibits risk aversion in a flexible manner: if its distortion function g satisfies risk aversion condition, then its utility function U need not be concave. The latter may be even convex, thus resolving Rabin's paradox. Next slides display an illustration.Rabin's paradox NextPrevious 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.risk aversion condition

A. NovosyolovCombined functionals18 CF, example parameters Back to structure NextPrevious

A. NovosyolovCombined functionals19 CF: Rabin's paradox resolved Back to structure 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: L0G0 $600$2500 $1000$4100 $2000$8100 NextPrevious

A. NovosyolovCombined functionals20 Relations among risk measures Generalization Partial generalization Back to structure NextPrevious Legend

A. NovosyolovCombined functionals21 Legend for relations Back to structure NextPrevious - expectation - expected utility - distorted probability - combined functional RDEU – rank-dependent expected utility, Quiggin, 1993 Coherent risk measure – Artzner et al, 1999

A. NovosyolovCombined functionals22 Illustrations Back to structure NextPrevious Expected utility indifference curves Distorted probability indifference curves Combined functional indifference curves

A. NovosyolovCombined functionals23 EU: indifference curves Over risks in R2Over distributions in R3 Back to structure NextPrevious

A. NovosyolovCombined functionals24 DP: indifference curves Back to structure Over risks in R2Over distributions in R3 NextPrevious

A. NovosyolovCombined functionals25 CF: indifference curves Back to structure Over risks in R2Over distributions in R3 NextPrevious

A. NovosyolovCombined functionals26 A few anticipated questions Why are risks assumed bounded? NextPrevious Back to structure Is EU a certain equivalent?

A. NovosyolovCombined functionals27 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. Back to Risk Back to structure NextPrevious

A. NovosyolovCombined functionals28 Is EU a certain equivalent? Back to EU Back to structure NextPrevious 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: