Post on 28-Dec-2015
Combined functionals as risk measures
Arcady NovosyolovInstitute of computational modeling
SB RAS, Krasnoyarsk, Russia, 660036
anov@icm.krasn.ru 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
Combined functionals 3A. Novosyolov
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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Combined functionals 4A. Novosyolov
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
Back to structure
1
t
)(tFX
1x 2x nx1p
2p np
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Combined functionals 5A. Novosyolov
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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Combined functionals 6A. Novosyolov
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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Combined functionals 7A. Novosyolov
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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Combined functionals 8A. Novosyolov
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.
Combined functionals 9A. Novosyolov
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?
Combined functionals 10A. Novosyolov
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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Combined functionals 11A. Novosyolov
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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Combined functionals 12A. Novosyolov
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
Combined functionals 13A. Novosyolov
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
Combined functionals 14A. Novosyolov
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
Combined functionals 15A. Novosyolov
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.
Combined functionals 16A. Novosyolov
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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Combined functionals 17A. Novosyolov
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.
Combined functionals 18A. Novosyolov
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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Combined functionals 19A. Novosyolov
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
Combined functionals 20A. Novosyolov
Relations among risk measures
GeneralizationPartial generalization Back to structur
e
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Legend
Combined functionals 21A. Novosyolov
Legend for relations
Back to structure
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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
Combined functionals 22A. Novosyolov
Illustrations
Back to structure
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Expected utility indifference curves
Distorted probability indifference curvesCombined functional indifference curves
Combined functionals 23A. Novosyolov
EU: indifference curves
Over risks in R2
Over distributions in R3
Back to structure
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Combined functionals 24A. Novosyolov
DP: indifference curves
Back to structure
Over risks in R2
Over distributions in R3NextPreviou
s
Combined functionals 25A. Novosyolov
CF: indifference curves
Back to structure
Over risks in R2
Over distributions in R3NextPreviou
s
Combined functionals 26A. Novosyolov
A few anticipated questions
Why are risks assumed bounded?
NextPreviousBack to structure
Is EU a certain equivalent?
Combined functionals 27A. Novosyolov
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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Combined functionals 28A. Novosyolov
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