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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
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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 )
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Why bounded?
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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
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x2

xn
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t
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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)
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x
0%
20%
p
0.1
0.9
$0
0.98
x
-20
p
0.01
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-5
10
30
0.13 0.65 0.21
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Risk measure
Risk measure is a real-valued functional
:XR
or
 : F  R.
Risk measures allowing both representations with
 ( X )   ( FX )
are called law invariant.
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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
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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.
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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?
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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
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p1
p2
n
U ( X )   U ( xk ) p k
k 1
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EU is linear in probability
U (aF  (1  a)G)  aU ( 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
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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
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loss L0= $600.
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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.
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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 
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Q  (q1 , q2 ,..., qn )
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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.
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RM: Combined functional
Recall expected utility and distorted probability functionals:
1
1
U ( X )   U ( F (v))dv,  g ( X )   FX1 (v)dg (1  v)
0
1
X
0
Combined functional involves both dollar and
probability transforms:
1
U , g ( X )   U ( FX1 (v))dg (1  v).
Discrete case:
0
n
U , g ( X )   U ( xk )qk  EQU ( X )
k 1
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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.
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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
t0
 0.5 exp( t ),
U (t )  
t  0.5 exp( t ), t  0
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g (v)  v 2.33
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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
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G0
$2500
$4100
$8100
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Relations among risk measures
Generalization
Partial generalization
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Legend
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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
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Illustrations
Expected utility indifference curves
Distorted probability indifference curves
Combined functional indifference curves
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EU: indifference curves
Over risks in R2
Over distributions in R3
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DP: indifference curves
Over risks in R2
Over distributions in R3
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CF: indifference curves
Over risks in R2
Over distributions in R3
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A few anticipated questions
Why are risks assumed bounded?
Is EU a certain equivalent?
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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.
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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
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