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Combined functionals as risk measures Arcady Novosyolov Institute of computational modeling SB RAS, Krasnoyarsk, Russia, 660036 [email protected] http://www.geocities.com/novosyolov/ 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 A. Novosyolov Combined functionals 2 Risk Risk is an almost surely bounded random variable X X L (, B, P) Another interpretation: risk is a real distribution function with bounded support F F Correspondence: X FX , FX (t ) P( X t ) Previous Next Back to structure Why bounded? A. Novosyolov Combined functionals 3 Example: Finite sample space Let the sample space be finite: | | n. Then: Probability distribution is a vector P ( p1 , p2 ,..., pn ) n Random variable is a vector X ( x1 , x2 ,..., xn ) R Distribution function is a step function 1 FX (t ) pn p2 p1 x1 A. Novosyolov x2 xn Combined functionals t Previous Next Back to structure 4 Risk treatment Here risk is treated as gain (the more, the better). Examples: • Return on a financial asset • Insurable risk x -$1,000,000 p 0.02 • Profit/loss distribution (in thousand dollars) A. Novosyolov x 0% 20% p 0.1 0.9 $0 0.98 x -20 p 0.01 Combined functionals -5 10 30 0.13 0.65 0.21 Previous Next Back to structure 5 Risk measure Risk measure is a real-valued functional :XR or : F R. Risk measures allowing both representations with ( X ) ( FX ) are called law invariant. Previous Next Back to structure A. Novosyolov Combined functionals 6 Using risk measures Certain equivalent of a risk X ( FX ) Price of a financial asset, portfolio X ( FX ) Insurance premium for a risk X ( FX ) Goal function in decision-making problems Previous Next Back to structure A. Novosyolov Combined functionals 7 RM: Expectation ( X ) EX ( F ) tdF (t ) 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 A. Novosyolov Combined functionals 8 RM: Expected utility U ( F ) U (t )dF (t ) U ( X ) EU ( X ) 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. Is EU a certain equivalent? A. Novosyolov Combined functionals Previous Next Back to structure 9 EU as a dollar transform X: U (X ) : n Value x1 x2 … xn Prob p1 p2 … pn Value U(x1) U(x2) … U(xn) Prob … pn EX x k p k k 1 A. Novosyolov p1 p2 n U ( X ) U ( x k ) p k k 1 Combined functionals Previous Next Back to structure 10 EU is linear in probability U (aF (1 a)G) aU ( F ) (1 a) U (G), p2 a [0,1]; Expected utility functional is linear with respect to mixture of distributions. p1 p3 F, G F Indifference "curves" on a set of probability distributions: parallel straight lines A. Novosyolov Combined functionals Previous Next Back to structure 11 EU: Rabin's paradox Rx ( L, G) : Value x-L x+G prob 0.5 0.5 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 Previous Next loss L0= $600. Back to structure A. Novosyolov Combined functionals 12 RM: Distorted probability 0 0 g ( X ) [ g (1 FX (t )) 1]dt g (1 FX (t ))dt Distortion function g : [0,1] [0,1], g (0) 0, g (1) 1 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 A. Novosyolov Combined functionals 13 DP as a probability transform X: X: n EX x k p k k 1 Value x1 x2 … xn Prob p1 p2 … pn Value x1 x2 … xn Prob q1 q2 … qn n g ( X ) xk qk EQ X k 1 n n qk g pi g pi , k 1,..., n i k i k 1 Previous Next Q (q1 , q2 ,..., qn ) Back to structure A. Novosyolov Combined functionals 14 DP is positively homogeneous g (aX ) a g ( X ), a 0, X X 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 A. Novosyolov Combined functionals 15 RM: Combined functional Recall expected utility and distorted probability functionals: 1 1 U ( X ) U ( F (v))dv, g ( X ) FX1 (v)dg (1 v) 0 1 X 0 Combined functional involves both dollar and probability transforms: 1 U , g ( X ) U ( FX1 (v))dg (1 v). Discrete case: 0 n U , g ( X ) U ( xk )qk EQU ( X ) k 1 A. Novosyolov Previous Next Back to structure Combined functionals 16 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 U need 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 A. Novosyolov Combined functionals 17 CF, example parameters U(t) g(v) 2 1.0 1 0.5 0 0.0 -2 0 2 0.0 0.5 1.0 v t0 0.5 exp( t ), U (t ) t 0.5 exp( t ), t 0 A. Novosyolov Combined functionals g (v) v 2.33 Previous Next Back to structure 18 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: L0 $600 $1000 $2000 A. Novosyolov G0 $2500 $4100 $8100 Combined functionals Previous Next Back to structure 19 Relations among risk measures Generalization Partial generalization A. Novosyolov Combined functionals Legend Previous Next Back to structure 20 Legend for relations EX U (X ) g (X ) U , g ( X ) - 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 A. Novosyolov Combined functionals 21 Illustrations Expected utility indifference curves Distorted probability indifference curves Combined functional indifference curves Previous Next Back to structure A. Novosyolov Combined functionals 22 EU: indifference curves Over risks in R2 Over distributions in R3 Previous Next Back to structure A. Novosyolov Combined functionals 23 DP: indifference curves Over risks in R2 Over distributions in R3 Previous Next Back to structure A. Novosyolov Combined functionals 24 CF: indifference curves Over risks in R2 Over distributions in R3 Previous Next Back to structure A. Novosyolov Combined functionals 25 A few anticipated questions Why are risks assumed bounded? Is EU a certain equivalent? Previous Next Back to structure A. Novosyolov Combined functionals 26 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 A. Novosyolov Combined functionals 27 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: U ( X ) U ( U ( X )), X X 1 Previous Next Back to structure Back to EU A. Novosyolov Combined functionals 28