AUSTRIAN JOURNAL OF STATISTICS
Volume 31 (2002), Number 1, 73-75
Coherent Risk Measures and Convex Combinations of the Conditional Value at Risk (
C V@R)
Georg Ch. Pflug
Department of Statistics and Decision Support Systems University of Vienna
Dedicated to Gerhart Bruckmann on the occasion of his 70th birthday Summary: The conditional-value-at-risk (CV@R) has been widely used as a risk measure. It is well known, that CV@R is coherent in the sense of Artzner, Delbaen, Eber, Heath (1999). The class of coherent risk measures is convex. It was conjectured, that all coherent risk measures can be represented as convex combinations ofCV@R’s. In this note we show that this conjecture is wrong.
Let the random variableY represent the future value of a portfolio. To measure the risk contained inY is an important task in stochastic finance. Among the enormous group of statistical parameters, which can be associated toY, like expectation, median, variance, mean absolute deviation, coefficient of variation, Gini-measure etc., only some qualify as acceptable risk measures.
Artzner, Delbaen, Eber, Heath (1999) call a statistical parameterF(Y)coherent, if it has the following properties:
(i) First order stochastic monotonicity.
PfY
1
ugPfY
2
ugfor alluimpliesF(Y1
)F(Y
2 ):
(ii) Positive homogeneity.
F(Y )=F(Y); for>0:
(iii) Subadditivity.
IfY1andY2are two random variables, independent or dependent, then
F(Y
1 +Y
2
)F(Y
1
)+F(Y
2 ):
LetF be the set of all coherent risk measures. Evidently,F is a convex set of functio- nals. There is practically only one known risk functional, which satisfies coherence: The conditional value-at-riskCV@R.
The conditional value-at-risk CV@R is defined as follows. LetGbe the probability distribution function ofY, i.e.G(u)=PfY ug. Then
CV@R (Y)=
1 Z
G 1
(u)du:
74 Austrian Journal of Statistics, Vol. 31 (2002), No. 1, 73-75
It is known thatCV@Ris coherent in the above sense (see Pflug, 2000). IfH is any monotonic, right continuous function on [0,1], then
CV@RH(Y)=
Z
CV@R(Y)dH( ) is also coherent.
CV@RH may be represented as
CV@RH (Y) =
Z
1
0 Z
1
0 1
1l[0;℄
(u)G 1
(u)dH( )du
= Z
1
0 Z
1
u 1
dH( )G 1
(u)du
= Z
1
0
h(u)G 1
(u)du: (1)
Here
h(u)= Z
1
u 1
dH( ): (2)
his monotonically decreasing inuand conversely, every monotonically decreasing func- tionh has a representation (2). Thus the convex hull of all CV@R is given by all risk measures of the form (1).
It was proved by Uryasev and Rockafellar (2000) that
CV@R
(G)=maxfa 1
Z
[u a℄ dG(u):a2Rg:
It was conjectured that the class CV@RH coincides with the class of coherent risk measures. This is however not true: Consider the new measure
F
(G)=maxf(a
1 +a
2 )=2
1
2
1 Z
[u a
1
℄ dG(u) 1
2
2 Z
[u a
2
℄ dG(u):a
1
a
2 g
(3) Proposition.The risk measure (3) is coherent, but is not of the form (1).
Proof.LetH(a1
;a
2
;Y)=(a
1 +a
2 )=2
1
2
1 R
[u a
1
℄ dG
Y (u)
1
2
2 R
[u a
2
℄ dG
Y (u)=
(a
1 +a
2 )=2
1
2
1
E([Y a
1
℄ ) 1
2
2
E([Y a
2
℄ ).
F
(G
Y
) = inffH(a
1
;a
2
;Y) : a
1
a
2
g. Since H(a1;a2;Y) is stochastically monotone in Y, the same is true for F, i.e. (1) is fulfilled. Since for all > 0,
H(a
1
;a
2
;Y) = H(a
1
;a
2
;Y) and since also the condition a1
a
2 is homoge- neos, F is positively homogeneous. FinallyH(a1;a2;X+Y) H(a1=2;a2=2;X)+
H(a
1
=2;a
2
=2;Y). and this implies thatF is subadditive.
If the constrainta1
a
2 was not present, the infimum would be
1
2
CV@R1 (Y)+
1
2
CV@R2 (Y):
However, the constraint does not change the property of coherence, but introduces a new type of measure.
G. Pflug 75
Let us finally remark, that if one weakens the assumption of monotonicity w.r.t. first order stochastic dominance by the assumption of second order stochastic dominance (see Fishburn, 1980)
(i’)
R
v
1 PfY
1
ugdv R
v
1 PfY
2
ugdvfor allv impliesF(Y1
)F(Y
2 );
then more functionals satisfy (i’),(ii),(iii). Examples are
F(Y)=E(Y) q
E([Y EY℄ ) 2
or
F(Y)=E(Y) 1
2
E(jY EYj):
Literatur
P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk.Mathema- tical Finance, 9:203–228, 1999.
P.C. Fishburn. Stochastic dominance and moments of distributions. Mathematics of Operations Research, 5:94–100, 1980.
G. Pflug. Some remarks on the Value-at-Risk and the conditional Value-at-Risk. In S. Uryasev, editor,Probabilistic Constrained Optimization – Methodology and Ap- plications, pages 272–281, Kluwer Academic Publishers, 2000.
S. Uryasev and R.T. Rockafellar. Optimization of conditional Value-at-Risk. The Jour- nal of Risk, 2(3):21–41, 2000.
S. Uryasev. Conditional Value-at-Risk: Optimization algorithms and applications. Fi- nancial Engineering News14, February 2000.
Author’s address:
Prof. Dr. Georg Ch. Pflug
Department of Statistics and Decision Support Systems University of Vienna
Universit¨atsstraße 5 A-1010 Wien, Austria
E-mail: georg.pflug@univie.ac.at