Jenaer Schriften zur Wirtschaftswissenschaft
A Decision Rule Based on
the Conditional Value at Risk
Werner Jammernegg und Peter Kischka
09/2005
Arbeits- und Diskussionspapiere
der Wirtschaftswissenschaftlichen Fakultät
der Friedrich-Schiller-Universität Jena
ISSN 1611-1311
Herausgeber: Wirtschaftswissenschaftliche Fakultät Friedrich-Schiller-Universität Jena Carl-Zeiß-Str. 3, 07743 Jena www.wiwi.uni-jena.de Schriftleitung:Prof. Dr. Hans-Walter Lorenz
Prof. Dr. Armin Scholl
A Decision Rule Based on the Conditional Value at Risk
Werner Jammernegg
Vienna University of Economics and Business Administration, Department of Information Systems and Operations, A-1090 Wien, Austria
Tel.: (0043) 1-3 13 36 56 33, Fax: (0043) 1-3 13 36 56 10 ([email protected])
Peter Kischka
Friedrich Schiller University Jena, Department of Business Statistics, D-07743 Jena, Germany Tel.: +49 3641 94 33 00, Fax: +49 3641 94 33 02
Abstract:
We introduce a decision rule where the risk dimension is measured by the conditional value of risk. We characterize the risk attitudes implied by the decision rule in a way similar to the well known mean variance framework. We show that the rule is consistent with Yaari’s dual theory for all risk attitudes. Finally a reformulation of the decision rule is presented which is based on two conditional expected values.
1 Definition of the Decision Rule
Let I be a random variable, e. g. an income stream. Let F be the distribution function of I: F(x) = P(I ≤ x). Within the well known mean variance framework the utility of I is given by (1.1) E(I)−δ Var(I)
with some constant δ, expected value E(I) and variance Var(I). For two income streams I and I ' we have: I is preferred to I ' if and only if
E(I)−δ Var(I)>E(I ')−δ Var(I ').
In (1.1) the expected value represents the value dimension of I and the variance represents the risk dimension of I.
It is well known that the variance is not a coherent measure of risk as defined in (Artzner et al., 1999), indeed it does not fulfil any of the four axioms. In the following, we will replace the variance in (1.1) by a coherent risk measure, the negative conditional value at risk (CVaR). For 0 < α ≤ 1 the CVaRα with level α is defined for a random variable I as the conditional expected value of I given I is below the α-quantile. For invertible distribution function F we have
(1.2) CVaR (I)α =E(I | I≤F ( ))−1 α
In general, with F ( )−1 α =inf x | F(x)
{
≥α}
we have(1.2' ) 1 0 1 CVaR (I) F (t)dt α α = α
∫
−For invertible F (1.2) and (1.2' ) coincide (Acerbi/Tasche, 2002).
It is reasonable to consider an approach where the utility of I is measured by the value dimension E (I) and the risk dimension -CVaR; instead of (1.1) we will consider the decision rule.
(1.3) E(I)−δ (-CVaR (I)α ) or
(1.3' ) E(I)+δ CVaR (I)α .
Such an approach with positive δ is presented in (Hanisch, 2004); moreover, it is the result from the maximization of the expected net present value (Pflug/Ruczczynski, 2004):
Let z denote consumption which is derived from I. If z exceeds I there is a shortfall (z –I), if I exceeds z the surplus I –z has to be discounted.
3 f (z) : z= + ⋅a max (0, I− −z) q max (0, z−I)
where 0 < a < 1 is a discount factor and q > 1 determines the insurance premium q E(z−I)+. There is an equivalent definition of the CVaRα as a solution from a maximization problem (Rockafeller/Uryasev, 2000) which implies
z
max f (z)=aE(I)+ −(1 a) CVaR (I)α
with 1 a
q a α= −
− (see Pflug/Ruczczynski, 2004): Dividing by a > 0 we get the decision rule (1.3' ) with 1 a
a
δ= − . Note that δ >0 since a < 1 in this approach. In the following we will consider the case δ ≤0 (a≥1), too.
2 Risk Aversion
A decision maker is risk averse (in the sense of Arrow/Pratt) if the expected value of I is preferred to the stochastic income stream I. In the mean variance context of (1.1) a decision maker therefore is risk averse for δ >0, otherwise a decision maker is risk neutral (δ =0)or risk seeking (δ <0). Whereas risk aversion or risk neutrality are standard assumptions in finance and some other economic fields, these assumption are e. g. no longer obvious in operations management. E. g. for the newsvendor problem it is shown that decision makers show a risk seeking behaviour dealing with low profit products (see Schweitzer/Cachon, 2000). The following proposition shows that with (1.3) all attitudes to risk can be covered.
Proposition 1
A decision maker with decision rule (1.3) is risk averse (risk neutral, risk seeking) if and only if δ >0 (δ =0,δ <0).
Proof:
Since CVaR (I)α ≤E(I) and CVaR (k)α = for any constant k we have for k δ >0
E(I) CVaR (I) E(I) E(I)
E(E(I)) CVaR (E(I))
α α δ δ δ + ≤ + = +
The latter term is the ‘utility’ of E(I) according to (1.3). The cases δ =0,δ <0 can be proved analogously.
Remark:
If I, I' are two income streams with E(I) = E(I' ) a risk seeking decision maker will choose that income stream with the smaller conditional value at risk, i. e. the decision maker over-estimates the impact of the random variable given it is higher than the α-quantile. A detailed analysis of this effect will be given in section 4.
Further insights concerning the risk attitude of a decision maker can be obtained by analyzing the decision rule
(2.1) a E(I)+ −(1 a) CVaR (I)α
for some constant a≥0. For a=0 (2.1) is the decision rule based only on the CVaR, a case not covered by the approach (1.3). This case can be interpreted as extreme risk aversion. For a > 0 (2.1) and (1.3) are equivalent decision rules: Dividing (2.1) by a > 0 (2.1) and (1.3) will yield the same result with 1 a
a
δ= − ; note that δ > −1. Conversely, starting with (1.3) with
1
δ > − the decision rules are equivalent for a 1 1 δ =
+ .
For any a≤1 (2.1) is a convex combination of the value dimension and the risk dimension. For a = 1 we get the expected value criterion. We will give special emphasis to the case a > 1. The following example shows an interesting consequence of this assumption.
Example:
Let I ~ N( ,µ σ . Let ϕ, Φ denote the density and the distribution function of the standard2) normal distribution. We have
1 1 CVaR (I)α µ σ ϕ Φ α( ( )) α − = − ⋅ ⋅
(see Johnson et al., 1994, p. 156 f). Therefore (2.1) is given by 1 1 1 a (1 a) ( ( ( )) 1 (1 a) ( ( )) µ µ σ ϕ Φ α α µ σ ϕ Φ α α − − + − − ⋅ ⋅ = = − − ⋅ ⋅ ⋅
For a > 1 the ‘utility’ of I increases with increasing σ.
3 Relations to Dual Utility Theory
In this section, we show that the criterion (2.1) is consistent with Yaaris’s dual theory. We generalize a result of J. Hanisch (Hanisch, 2004) who has shown this consistency for
0≤ ≤a 1.
Yaaris’s theory (Yaari, 1987) is a special case of the rank dependent utility theory. It is based on the idea, that the probability of a bad result is judged differently from the same probability of a good result. There is some evidence for the dual theory (see e. g. Diedicue/Wakker, 2001) from an empirical point of view. Further evidence is given by a reformulation of (2.1) in section 4 (see proposition 5 and following remarks).
5
Lemma
For 0< <α 1 and for 0 a 1 1 α ≤ ≤ − the function (3.1) : [0,1] [0,1] x x ax (1 a) min ,1 ϑ α → ⎛ ⎞ → + − ⎜ ⎟ ⎝ ⎠
is a continuous, monotonically increasing function with (0)ϑ =0, (1)ϑ =1.
Proof: x : (x) a x (1 a) with '(x) a 1 1 x : (x) a x (1 a)x with '(x) a (1 a) α ϑ ϑ α ϑ ϑ α α ≥ = + − = ≤ = + − = + − So, for 0 a 1 1 α ≤ ≤
− ϑ has the required properties.
In Yaari’s dual theory I is preferred to I' if for some function ϑ with the properties stated in
the above lemma and for the (generalized) inverse distribution functions F, F' we have (3.2) 1 1 1 1 0 0 F (x) d (x)− ϑ ≥ F ' (x) d (x)− ϑ
∫
∫
(see e. g. Guriev, 2001).Note that the above integrals exist for continuous functions ϑ since any distribution function is of bounded variation (see e. g. Apostol, 1974, p 128, 139).
Proposition 2
For ϑ defined in (3.1) and for 0 a 1 1 α ≤ ≤ − we have 1 1 0
aE(I)+ −(1 a) CVaR (I)α =
∫
F (x) d (x)− ϑProof:
From the linearity properties of the Riemann-Stieltjes integral (see e. g. Apostol, 1974), p 142) we have
1 1 1 1 0 0 1 1 0 1 0 1 1 1 1 1 0 0 x F (x) d (x) F (x) d(ax (1 a) min( ,1)) a F (x) dx x (1 a) F (x) d(min( ,1)) x (1 a) F (x) d(min( ,1)) 1 a F (x) dx (1 a) F (x)dx a E(I) (1 a) CVaR (I)
α α α α ϑ α α α α − − − − − − − = + − = + + − + − = + − = + −
∫
∫
∫
∫
∫
∫
∫
So from proposition 2 and (3.2) we can conclude that a decision maker with a decision rule given by (1.3) or (2.1) satisfies the axioms of the dual utility theory.
4 A characterization of risk behaviour
From proposition 1 we know that a decision maker using (1.3) as a decision rule is risk averse if and only if δ >0. From (2.1) we conclude that a decision maker is risk averse if and only if a < 1, since (2.1) can be transformed to (1.3) by dividing by a. For convenience we assume that F is invertible in the following.
Further characterizations of the risk behaviour can be derived from the following decision rule (4.1) λE(I | I≤F ( ))−1 α + −(1 λ)E(I | I>F ( ))−1 α
with some constant λ ∈[0,1].
Proposition 3
We have
1 1
E(I | I F ( )) (1 )E(I | I F ( )) a E(I) (1 a) CVaR (I)α
λ ≤ − α + −λ > − α = + − with a 1 1 λ α − = − .
7 Proof: 1 1 1 F ( ) E(I | I F ( )) (1 )E(I | I F ( )) 1 CVaR (I) (1 ) x dF(x) 1 1
CVaR (I) (1 ) (E(I) CVaR (I))
1 1
E(I) CVaR (I).
1 1 α α α α α λ α λ α λ λ α λ λ α α λ λ α α α − − − ∞ ≤ + − > = ⋅ + − − = ⋅ + − − − − − = + − −
∫
So (2.1) and (4.1) define the same decision rule. In (4.1)E(I | I>F ( ))−1 α may be interpreted as a value dimension. The smaller λ the smaller is the impact of the risk dimension
1
E(I | I≤F ( ))− α .The reformulation (4.1) of (2.1) is an additional hint that the criterion (1.3) or (2.1) respectively can be interpreted as a special case of the dual theory, since the decision is based on a ranking of the possible realizations.
The relations between α and λ in (4.1) determine the risk attitude.
Proposition 4
A decision maker using (4.1) is risk averse (risk neutral, risk seeking) if and only if
( , )
λ α λ α λ α> = < .
Proof:
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