Theory of coherent measures
Miloš Kopa
Faculty of Mathematics and Physics
Charles University in Prague
Czech Republic
e-mail:[email protected]
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Previous lecture
Measure of risk assigns a real number to any random variable L
(loss).
Favorite risk mesures:
variance: var (L) = E(L − EL)2
1
standard deviation sd(L) = (E(L − EL)2 ) 2
h
i
semivariance: rs (L) = E max (0, L − EL)2
mean absolute deviation: ra (L) = E|L − EL|
mean absolute semideviation: ras (L) = E [max (0, L − EL)]
Value at Risk (VaR):
VaRα (L) = inf {l ∈ R, P (L > l) ≤ 1 − α}
Conditional Value
n at Risk (CVaR):
o
1
CVaRα (L) = inf a ∈ R, a + 1−α
E [max (0, L − a)] ,
alternatively:
CVaRα (L) = βE (L|L > VaRα (L)) + (1 − β)VaRα (L) with
some β ∈ [0, 1]
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Questions for today
Risk measures:
What are the “reasonable” properties that should have all
“good” risk measures?
Which of the considered measures has the properties?
Is it possible to generalize a very well-known and popular
standard deviation (variance)?
What is the dual expression of measures with these properties?
Multiobjective optimization:
How to formulate an optimization problem when multipple
objective are considered?
What are the best solutions of such problems?
How to find all these best solutions?
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Coherent risk measures
CRM: R : L2 (Ω) → (−∞, ∞] that satisfies
(R1) Translation equivariance: R(L + C ) = R(L) + C for
all X and constants C ,
(R2) Positive homogeneity: R(0) = 0, and
R(λL) = λR(L) for all L and all λ > 0,
(R3) Subaditivity: R(L + M) ≤ R(L) + R(M) for all L
and M,
(R4) Monotonicity: R(L) ≥ R(M) when L ≥ M.
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Coherent risk measures
CRM: R : L2 (Ω) → (−∞, ∞] that satisfies
(R1) Translation equivariance: R(L + C ) = R(L) + C for
all X and constants C ,
(R2) Positive homogeneity: R(0) = 0, and
R(λL) = λR(L) for all L and all λ > 0,
(R3) Subaditivity: R(L + M) ≤ R(L) + R(M) for all L
and M,
(R4) Monotonicity: R(L) ≥ R(M) when L ≥ M.
Strictly expectation bounded risk measures satisfy (R1), (R2),
(R3), and
(R5) R(L) > E[L] for all nonconstant L, whereas
R(L) = E[L] for constant L.
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Coherent risk measures
CRM: R : L2 (Ω) → (−∞, ∞] that satisfies
(R1) Translation equivariance: R(L + C ) = R(L) + C for
all X and constants C ,
(R2) Positive homogeneity: R(0) = 0, and
R(λL) = λR(L) for all L and all λ > 0,
(R3) Subaditivity: R(L + M) ≤ R(L) + R(M) for all L
and M,
(R4) Monotonicity: R(L) ≥ R(M) when L ≥ M.
Strictly expectation bounded risk measures satisfy (R1), (R2),
(R3), and
(R5) R(L) > E[L] for all nonconstant L, whereas
R(L) = E[L] for constant L.
Note: (R2)&(R3) implies convexity of R: for each a ∈ [0, 1] we
have:
R(aL + (1 − a)M) ≤ R(aL) + R((1 − a)M) = aR(L) + (1 − a)R(M)
Other classes of risk measures and functionals: Follmer and Schied
(2002), Pflug and Romisch (2007).
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Coherent properties for the popular risk mesures
variance: none of (R1)-(R5)
standard deviation: (R2)
semivariance: none of (R1)-(R5)
mean absolute deviation: (R2), (R3)
mean absolute semideviation: (R2), (R3)
Value at Risk (VaR): (R1),(R2), (R4)
Conditional Value at Risk (CVaR): (R1)-(R5)
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Dual representation of coherent risk measures
Consider a measurable space (Ω, A) and the set P of all probability
measures on the space.
Definition
A set Q ⊂ P is called a risk envelope if for each Q ∈ Q one has: Q ≥ 0
and EQ = 1.
Theorem
R is a coherent risk measure if and only if there exists a risk envelope Q
such that:
R(L) = max E (QL)
Q∈Q
and Q can be chosen as a convex set.
Interpretation: A coherent risk measure can be understood as a
worst-case expectation with respect to some class of probability
distributions on (Ω, A), It means for some distribution P 0 . If the
0
probability distribution of L is P then Q = dP
dP .
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Example - Risk envelope for CVaR
To simplify the situation consider a measurable space with M
atoms (discrete distributions). Moreover let L has a discrete
uniform distribution on the space - atoms are equiprobable, i.e.
discrete distribution with M equiprobable scenarios lj ,
j = 1, 2, ..., M. Assume that M(1 − α) is an integer number. Then:
CVaRα (L) = min a +
a,zj
M
X
1
zj
(1 − α) M
j=1
s. t. zj ≥ lj − a, j = 1, .., M
zj ≥ 0, j = 1, .., M
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Example - Risk envelope for CVaR
And dual problem:
CVaRα (L) = max
yj
s. t.
M
X
yj lj
j=1
M
X
yj = 1
j=1
yj ≤
1
(1 − α) M
yj ≥ 0, j = 1, .., M
Note that optimal solution: yj∗ = 0 for j = 1, 2, ..., Mα
1
yj∗ = (1−α)M
for j = Mα + 1, ..., M. Hence the risk envelope for
CVaR is:
1
Q = {Q : EQ = 1, 0 ≤ Q ≤
}
(1 − α)
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Coherent risk and return measures
A return measure is defined as a functional E(L) = −R(L) for a
coherent risk measure R. It is obvious that the expectation
belongs to this class.
Miloš Kopa
Theory of coherent measures and multiobjective optimization
General deviation measures
Rockafellar, Uryasev and Zabarankin (2006A, 2006B): GDM are
introduced as an extension of standard deviation but they need not
to be symmetric with respect to upside X − EX and downside
EX − X of a random variable X .
Miloš Kopa
Theory of coherent measures and multiobjective optimization
General deviation measures
Rockafellar, Uryasev and Zabarankin (2006A, 2006B): GDM are
introduced as an extension of standard deviation but they need not
to be symmetric with respect to upside X − EX and downside
EX − X of a random variable X .
Any functional D : L2 (Ω) → [0, ∞] is called a general deviation
measure if it satisfies
(D1) D(X + C ) = D(X ) for all X and constants C ,
(D2) D(0) = 0, and D(λX ) = λD(X ) for all X and all
λ > 0,
(D3) D(X + Y ) ≤ D(X ) + D(Y ) for all X and Y ,
(D4) D(X ) ≥ 0 for all X , with D(X ) > 0 for nonconstant
X.
(D2) & (D3) ⇒ convexity
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Deviation measures
Standard deviation
D(X ) = σ(X ) =
q
E kX − EX k2
Mean absolute deviation
D(X ) = E |X − EX | .
Mean absolute lower and upper semideviation
D− (X ) = E min(0, X − EX ) , D+ (X ) = E max(0, X − EX ) .
Worst-case deviation
D(X ) = sup |X (ω) − EX |.
ω∈Ω
See Rockafellar et al (2006 A, 2006 B) for another examples.
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Mean absolute deviation from (1 − α)-th quantile
CVaR deviation
For any α ∈ (0, 1) a finite, continuous, lower range dominated
deviation measure
Dα (X ) = CVaRα (X − EX ).
(1)
The deviation is also called weighted mean absolute deviation
from the (1 − α)-th quantile, see Ogryczak, Ruszczynski (2002),
because it can be expressed as
Dα (X ) = min
ξ∈R
1
E[max{(1 − α)(X − ξ), α(ξ − X )}]
1−α
(2)
with the minimum attained at any (1 − α)-th quantile. In relation
with CVaR minimization formula, see Pflug (2000), Rockafellar
and Uryasev (2000, 2002).
Miloš Kopa
Theory of coherent measures and multiobjective optimization
General deviation measures
According to Proposition 4 in Rockafellar et al (2006 A):
if D = λD0 for λ > 0 and a deviation measure D0 , then D is
a deviation measure.
if D1 , . . . , DK are deviation measures, then
D = max{D1 , . . . , DK } is also deviation measure.
D =P
λ1 D1 + · · · + λK DK is also deviation measure, if λk > 0
K
and k=1 λk = 1.
Rockafellar et al (2006 A, B): Duality representation using risk
envelopes , subdifferentiability and optimality conditions.
Miloš Kopa
Theory of coherent measures and multiobjective optimization
General deviation measures
We say that general deviation measure D is
(LSC) lower semicontinuous (lsc) if all the subsets of
L2 (Ω) having the form {X : D(X ) ≤ c} for c ∈ R
(level sets) are closed;
Miloš Kopa
Theory of coherent measures and multiobjective optimization
General deviation measures
We say that general deviation measure D is
(LSC) lower semicontinuous (lsc) if all the subsets of
L2 (Ω) having the form {X : D(X ) ≤ c} for c ∈ R
(level sets) are closed;
(D5) lower range dominated if
D(X ) ≤ EX − inf ω∈Ω X (ω) for all X .
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Strictly expectation bounded risk measures
Theorem 2 in Rockafellar et al (2006 A):
Theorem
Deviation measures correspond one-to-one with strictly expectation
bounded risk measures under the relations
D(X ) = R(X − EX )
R(X ) = E[−X ] + D(X )
In this correspondence, R is coherent if and only if D is lower
range dominated.
Miloš Kopa
Theory of coherent measures and multiobjective optimization
References
Follmer, H., Schied, A.: Stochastic Finance: An
Introduction In Discrete Time. Walter de Gruyter, Berlin,
2002.
Pflug, G.Ch., Romisch, W.: Modeling, measuring and
managing risk. World Scientific Publishing, Singapore, 2007.
Rockafellar, R.T., Uryasev, S. (2002). Conditional
Value-at-Risk for General Loss Distributions, Journal of
Banking and Finance 26, 1443–1471.
Rockafellar, R.T., Uryasev, S., Zabarankin M. (2006A).
Generalized Deviations in Risk Analysis. Finance and
Stochastics 10 , 51–74.
Rockafellar, R.T., Uryasev, S., Zabarankin M. (2006B).
Optimality Conditions in Portfolio Analysis with General
Deviation Measures. Mathematical Programming 108, No.
2-3, 515–540.
Miloš Kopa
Theory of coherent measures and multiobjective optimization
Convergence of approximate solutions in mean-risk models
Václav Kozmı́k
Faculty of Mathematics and Physics
Charles University in Prague
Supervisor: Miloš Kopa
Introduction
Mean-risk models
aim to find optimal portfolio of assets
analytical solutions for continuous distributions
solutions using generated scenarios
comparison of the approaches mentioned above
predetermined, for instance by historical data
generated with the assumption of continuous distribution
generated according to few moment estimators
convergence and its properties
different continuous distributions
different risk measures
computational part
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data processing and generating the scenarios
optimization tasks in GAMS
Efficient portfolios
we consider portfolio based on N assets
P
weights of the assets w, N
i=1 wi = 1
expected returns uw (always using expectation)
different risk measures rw
minimal required returns ue
Definition
Portfolio of given N assets with weights w is (mean-risk)
efficient,
PN
∗
∗
∗
if there are no other weights w1 , .., wN such that i=1 wi = 1 and
uw∗ ≥ uw a rw∗ ≤ rw .
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Classical optimization task
Efficient portfolios can be obtained while solving following task:
min rw
w
s. t. uw ≥ ue
N
X
wi = 1
i=1
wi ∈ R, i = 1, .., N.
Non-negativity condition:
w1 , .., wN ≥ 0.
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Risk measures
variance
VaR
cVaR
absolute deviation
semivariance
Definition
Let α ∈ (0, 1) be the threshold and L random variable which
represents the loss from holding the portfolio.Then we define VaRα as:
VaRα (L) = inf {l ∈ R, P (L > l) ≤ 1 − α}
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Risk measures
Definition
cVaRα is defined as:
1
cVaRα (L) = inf a ∈ R, a +
E [max (0, L − a)] .
1−α
Absolute deviation can be calculated as:
ra (L) = E|L − EL|.
Semivariance can be calculated as::
h
i
rs (L) = E max (0, L − EL)2
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Elliptical distributions
generalization of normal distribution
include normal distribution, Student distribution, logistic elliptical
distribution and others
symmetrical around the mean
simple analysis of linear combinations
Theorem
Let X ∼ E (µ, Σ, ψ), A ∈ Rm×n , b ∈ Rm . Then it holds:
AX + b ∼ E Aµ + b, AΣAT , ψ .
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Variance
We get classical optimization task which can be used for all elliptical
distributions:
min wT Vw
w
s. t. wT µ ≥ ue
N
X
wi = 1
i=1
wi ∈ R, i = 1, .., N
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Normal distribution
VaR
cVaR
√
VaRα (L) = −wT µ + qα wT Vw
n 2o
exp − q2α √
√
cVaRα (L) = −wT µ +
wT Vw
(1 − α) 2π
absolute deviation
r
ra (L) =
semivariance
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2√ T
w Vw
π
i
1 h
rs (L) = E (L − EL)2
2
Student distribution
VaR
cVaR
√
VaRα (L) = VaRα (L) = −wT µ + tα,ν wT Σw
cVaRα (L) = −wT µ +
Γ
Γ
− ν−1
2
2
tα,ν
ν−1 √
ν
1
+
√
2
ν
wT Σw
√
ν−2
(1
−
α)
(ν
−
2)
π
2
absolute deviation
√
√
2 νΓ ν+1
2
ra (L) =
wT Σw
√
(ν − 1) πΓ ν2
semivariance
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i
1 h
rs (L) = E (L − EL)2
2
Variance - scenarios
suppose we have M scenarios of possible stock prices
we can use mean and variance-covariance estimators l̂ and V̂ to
minimize variance
allows us to process estimators before running the optimization
task and therefore is quick
min wT V̂w
w
s. t. wT l̂ ≥ ue
N
X
wi = 1
i=1
wi ∈ R, i = 1, .., N.
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VaR - scenarios
general case could be nonconvex
we reformulate the task using integer programming
still hardly computable - M binary variables
min ν
ν,w,δ j
s. t. − wT lj ≤ ν + K δ j , j = 1, .., M
M
X
δ j = b(1 − α) Mc
j=1
δ j ∈ {0, 1} , j = 1, .., M
M
1 X Tj
w l ≥ ue ...
M
j=1
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cVaR - scenarios
linear programming task, can be solved quickly
min a +
a,w,z j
M
X
1
zj
(1 − α) M
j=1
j
T j
s. t. z ≥ −w l − a, j = 1, .., M
z j ≥ 0, j = 1, .., M
M
1 X Tj
w l ≥ ue
M
j=1
N
X
wi = 1
i=1
wi ∈ R, i = 1, .., N.
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Absolute deviation - scenarios
linear programming task
min
w,z j
M
1 X j
z
M
j=1
s. t. wT lj −
M
1 X Ti
w l ≤ z j , j = 1, .., M
M
i=1
M
1 X Ti
−w l +
w l ≤ z j , j = 1, .., M
M
T j
i=1
1
M
...
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M
X
j=1
wT lj ≥ ue
Semivariance - scenarios
quadratic programming task
min
w,z j
M
1 X j 2
z
M
j=1
s. t. z j ≥ −wT lj +
M
1 X Ti
w l , j = 1, .., M
M
i=1
z j ≥ 0, j = 1, .., M
M
1 X Tj
w l ≥ ue
M
j=1
...
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Computational part
own software in C++ & GAMS
software configuration
maximal number of scenarios
50 iterations of generating scenarios, average of the solutions found
scenarios are always generated from scratch
non-negativity condition
ML estimators of distribution parameters
threshold 95%
different minimal expected returns - small, medium, high
for VaR approx. 1 000 scenarios
other risk measures up to 50 000 scenarios
data used
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stock market indices
Japan, USA, Great Britain, Czech Rep. and Germany
from 15.9.2008 to 18.9.2009
Convergence issues
we can experience difficulties with large number of scenarios
Distance from the optimal solution
Convergence -normal distribution - semivariance
0,3
0,25
0,2
0,15
0,1
0,05
0
100
200
500
1000
2000
5000
10000
Number of scenarios
solutions from repeated iterations form clusters
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20000
50000
Convergence issues - normal distribution
Weight of the index in the portfolio
Semivariance - cluster analysis
1,0
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0,0
cluster 1
cluster 2
Japan
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USA
Britain
Czech R. Germany
Convergence issues - Student distribution
Weight of the index in the portfolio
Semivariance - cluster analysis
1,0
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0,0
cluster 1
cluster 2
cluster 3
cluster 4
Japan
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USA
Britain
Czech R. Germany
Cluster analysis
we use the k-means procedure
given the number of clusters we choose the mean of largest one as
the optimal solution
how to choose the optimal number of clusters
we use Bayes Information Criterion (BIC) modified for
multidimensional case
if the information criterion decreases while adding more clusters we
use the last largest cluster which had more than half of the
observations included
needed mostly for elliptical distributions and semivariance
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Variance
Variance
0,35
Distance from the optimal solution
0,3
0,25
normal (small returns)
normal (medium returns)
0,2
normal (high returns)
Student 7 (small returns)
Student 7 (medium returns)
0,15
Student 7 (high returns)
Student 5 (small returns)
0,1
Student 5 (medium returns)
Student 5 (high returns)
0,05
0
100
200
500
1000
2000
Number of scenarios
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5000
10000
20000
50000
VaR
VaR
0,3
Distance from the optimal solution
0,25
0,2
normal (small returns)
normal (medium returns)
normal (high returns)
0,15
Student 7 (small returns)
Student 7 (medium returns)
Student 7 (high returns)
0,1
Student 5 (small returns)
Student 5 (medium returns)
Student 5 (high returns)
0,05
0
100
200
300
500
Number of scenarios
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700
1000
cVaR
cVaR
0,35
Distance from the optimal solution
0,3
0,25
normal (small returns)
normal (medium returns)
0,2
normal (high returns)
Student 7 (small returns)
Student 7 (medium returns)
0,15
Student 7 (high returns)
Student 5 (small returns)
0,1
Student 5 (medium returns)
Student 5 (high returns)
0,05
0
100
200
500
1000
2000
Number of scenarios
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5000
10000
20000
50000
Absolute deviation
Absolute deviation
0,35
Distance from the optimal solution
0,3
0,25
normal (small returns)
normal (medium returns)
0,2
normal (high returns)
Student 7 (small returns)
Student 7 (medium returns)
0,15
Student 7 (high returns)
Student 5 (small returns)
0,1
Student 5 (medium returns)
Student 5 (high returns)
0,05
0
100
200
500
1000
2000
Number of scenarios
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5000
10000
20000
50000
Semivariance
Semivariance
0,35
Distance from the optimal solution
0,3
0,25
normal (small returns)
normal (medium returns)
0,2
normal (high returns)
Student 7 (small returns)
Student 7 (medium returns)
0,15
Student 7 (high returns)
Student 5 (small returns)
0,1
Student 5 (medium returns)
Student 5 (high returns)
0,05
0
100
200
500
1000
2000
Number of scenarios
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5000
10000
20000
50000
Normal distribution
Normal distribution (medium returns)
0,14
Distance from the optimal solution
0,12
0,1
0,08
variance
cVaR
0,06
abs. deviation
semivariance
0,04
0,02
0
100
200
500
1000
2000
Number of scenarios
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5000
10000
20000
50000
Moment generated scenarios
Moment generated scenarios
0,25
Distance from the optimal solution
0,2
0,15
variance
cVaR
abs. deviation
0,1
semivariance
0,05
0
100
200
500
1000
2000
Number of scenarios
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5000
10000
20000
50000
Conclusion
the convergence could not be achieved without proper analysis of
repeated experiments
using cluster analysis we achieved convergence for all risk measures
the convergence properties depend on the minimal expected
returns
even though we have smaller set of feasible solutions the convergence
properties could be worse
moment generated scenarios should be used only for a small
sample of initial scenarios
best convergence properties
28 z 29
small differences between distributions
variance or cVaR - fast computation
Conclusion
Thank you for your attention!
Miloš Kopa
e-mail:
[email protected]
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