combined functionals as risk measures arcady novosyolov institute of computational modeling sb ras,...

Combined functionals as risk measures

Arcady NovosyolovInstitute of computational modeling

SB RAS, Krasnoyarsk, Russia, 660036

[email protected] http://www.geocities.com/novosyolov/

mailto:[email protected]?subject=via%20Bowles%20Symposium

http://www.geocities.com/novosyolov/

http://www.geocities.com/novosyolov/

Combined functionals 2A. Novosyolov

Structure of the presentation

RiskRisk measure

Relations among risk measures

RM: ExpectationRM: Expected utilityRM: Distorted probabilityRM: Combined functional

Anticipated questions

Illustrations


Risk

Risk is an almost surely bounded random variable

),,( PBX LXAnother interpretation: risk is a real distribution function with bounded support FF

Correspondence:

,XFX )()( tXtFX P

Why bounded? Back to structure

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Example: Finite sample space

Let the sample space be finite:

.|| n Then:

Probability distribution is a vector

),...,,( 21 npppPRandom variable is a vector n

n RxxxX ),...,,( 21

Distribution function is a step function

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1

t

)(tFX

1x 2x nx1p

2p np

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Risk treatment

Here risk is treated as gain (the more, the better). Examples:

• Return on a financial asset

x 0% 20%

p 0.1 0.9

x -$1,000,00

0

$0

p 0.02 0.98

• Insurable risk

• Profit/loss distribution (in thousand dollars)

x -20 -5 10 30

p 0.01 0.13 0.65 0.21

Back to structure

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Risk measure

Risk measure is a real-valued functional

RX:or

.: RFRisk measures allowing both representations with

Back to structure

are called law invariant.

)()( XFX

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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

)( XFX

)( XFX

)( XFX

Back to structure

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RM: ExpectationXX E)(

)()( ttdFF

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.

Back to structure

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Combined functional is essentially the application of both transforms to the expectation.


RM: Expected utility

Back to structure

)()( XUXU E

)()()( tdFtUFU

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.

NextPreviousIs EU a certain equivalent?


EU as a dollar transform

Back to structure

Value

x1 x2 … xn

Prob p1 p2 … pn

n

kkkU pxUX

1

)()(

:X

n

kkk pxX

1

E

Value

U(x1)

U(x2)

… U(xn)

Prob p1 p2 … pn

:)(XU

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EU is linear in probability

Indifference "curves" on a set of probability distributions: parallel straight lines

),()1()())1(( GaFaGaaF UUU ];1,0[a

Back to structure

1p

2p

3p

FGF ,

Expected utility functionalis linear with respect tomixture of distributions.

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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<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.

Value x-L x+G

prob 0.5 0.5

:),( GLRx

Example: let L = $100, G = $125. Then expected utility maximizer would reject any equiprobable gamble with loss L0= $600. NextPreviou

s


RM: Distorted probability

Back to structure

0

0))(1(]1))(1([)( dttFgdttFgX XXg

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 ],1,0[]1,0[: g ,0)0( g 1)1( g

Distorted probability is positive homogeneous, that may lead to improper insurance premium calculation. NextPreviou

s


DP as a probability transform

Back to structure

Value

x1 x2 … xn

Prob p1 p2 … pn

XEqxX Q

n

kkkg

1

)(

:X

n

kkk pxX

1

E

Value

x1 x2 … xn

Prob q1 q2 … qn

:X

nkpgpgqn

kii

n

kiik ,...,1,

1

),...,,( 21 nqqqQ NextPreviou

s


DP is positively homogeneous

Back to structure

),()( XaaX gg ,0a XX

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.

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Distorted probability is a positively homogeneousfunctional, which is an undesired property

in insurance premium calculation.


RM: Combined functional

Back to structure

,))(()(1

0

1 dvvFUX XU 1

0

1 )1()()( vdgvFX XgCombined functional involves both dollar and probability transforms:

.)1())(()(1

0

1, vdgvFUX XgU

Discrete case:)()()(

1, XUEqxUX Q

n

kkkgU

Recall expected utility and distorted probability functionals:

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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.

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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.


CF, example parameters

Back to structure

U(t)

0

1

2

-2 0 2

0),exp(5.0

0),exp(5.0)(

ttt

tttU

33.2)( vvg

g(v)

0.0

0.5

1.0

0.0 0.5 1.0

v

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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:

L0 G0

$600 $2500

$1000 $4100

$2000 $8100 NextPrevious


Relations among risk measures

GeneralizationPartial generalization Back to structur

e

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Legend


Legend for relations

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- expectationXE)(XU - expected utility

)(Xg - distorted probability

)(, XgU - combined functional

RDEU – rank-dependent expected utility, Quiggin, 1993

Coherent risk measure – Artzner et al, 1999


Illustrations

Back to structure

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Expected utility indifference curves

Distorted probability indifference curvesCombined functional indifference curves


EU: indifference curves

Over risks in R2

Over distributions in R3

Back to structure

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DP: indifference curves

Back to structure

Over risks in R2

Over distributions in R3NextPreviou

s


CF: indifference curves

Back to structure

Over risks in R2

Over distributions in R3NextPreviou

s


A few anticipated questions

Why are risks assumed bounded?

NextPreviousBack to structure

Is EU a certain equivalent?


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

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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:

X XXUX UU )),(()( 1

combined functionals as risk measures arcady novosyolov institute of computational modeling sb ras,...

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