Combined functionals as risk measures

Combined functionals as risk measures

Arcady NovosyolovInstitute of computational modeling

SB RAS, Krasnoyarsk, Russia, 660036

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

novosyolov/

A. Novosyolov Combined functionals 2

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.

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


RM: Expected utility

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)()( 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

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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

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1p

2p

3p

FGF ,

Expected utility functionalis linear with respect tomixture of distributions.

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EU: Rabin's paradox

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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

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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

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,))(()(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

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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

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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

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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

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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

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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

Documents

risk measuresrm

expected utility risk

risk averse behavior

riskneutral behavior

probability transformback

structuredistorted probability

identical probability

utility function u