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INFORMS 2008 c
2008 INFORMS |isbn 978-1-877640-23-0
doi 10.1287/educ.1080.0052
Value-at-Risk vs. Conditional Value-at-Risk in
Risk Management and Optimization
Sergey Sarykalin
Gobal Fraud Risk Management, American Express, Phoenix, Arizona 85021,
sergey.sarykalin@aexp.com
Gaia Serraino and Stan Uryasev
Department of Industrial and Systems Engineering, Risk Management and Financial
Engineering Lab, University of Florida, Gainesville, Florida 32611
{serraino@ufl.edu, uryasev@ufl.edu}
Abstract From the mathematical perspective considered in this tutorial, risk management is a
procedure for shaping a risk distribution. Popular functions managing risk are value-
at-risk (VaR) and conditional value-at-risk (CVaR). The problem of choice between
VaR and CVaR, especially in financial risk management, has been quite popular in
academic literature. Reasons affecting the choice between VaR and CVaR are based on
the differences in mathematical properties, stability of statistical estimation, simplic-
ity of optimization procedures, acceptance by regulators, etc. This tutorial presents
our personal experience working with these key percentile risk measures. We try to
explain strong and weak features of these risk measures and illustrate them with sev-
eral examples. We demonstrate risk management/optimization case studies conducted
with the Portfolio Safeguard package.
Keywords VaR; CVaR; risk measures; deviation measures; risk management; optimization; Port-
folio Safeguard package
1. Introduction
Risk management is a broad concept involving various perspectives. From the mathematical
perspective considered in this tutorial, risk management is a procedure for shaping a loss
distribution (for instance, an investor’s risk profile). Among the vast majority of recent inno-
vations, only a few have been widely accepted by practitioners, despite their active interest
in this area. Conditional value-at-risk (CVaR), introduced by Rockafellar and Uryasev [19],
is a popular tool for managing risk. CVaR approximately (or exactly, under certain con-
ditions) equals the average of some percentage of the worst-case loss scenarios. CVaR risk
measure is similar to the value-at-risk (VaR) risk measure, which is a percentile of a loss dis-
tribution. VaR is heavily used in various engineering applications, including financial ones.
VaR risk constraints are equivalent to the so-called chance constraints on probabilities of
losses. Some risk communities prefer VaR others prefer chance (or probabilistic) functions.
There is a close correspondence between CVaR and VaR: with the same confidence level,
VaR is a lower bound for CVaR. Rockafellar and Uryasev [19, 20] showed that CVaR is
superior to VaR in optimization applications. The problem of the choice between VaR and
CVaR, especially in financial risk management, has been quite popular in academic litera-
ture. Reasons affecting the choice between VaR and CVaR are based on the differences in
mathematical properties, stability of statistical estimation, simplicity of optimization pro-
cedures, acceptance by regulators, etc. Conclusions made from this properties may often be
quite contradictive. Plenty of relevant material on this subject can be found at the website
http://www.gloriamundi.org. This tutorial should not be considered as a review on VaR and
270
Sarykalin et al.: VaR vs.CVaR in Risk Management and Optimization
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CVaR: many important findings are beyond the scope of this tutorial. Here we present only
our personal experience with these key percentile risk measures and try to explain strong
and weak features of these two risk measures and illustrate them with several examples. We
demonstrate risk management/optimization case studies with the Portfolio Safeguard (PSG)
package by American Optimal Decisions (an evaluation copy of PSG can be requested at
http://www.AOrDa.com).
Key observations presented in this tutorial are as follows:
•CVaR has superior mathematical properties versus VaR. CVaR is a so-called “coherent
risk measure”; for instance, the CVaR of a portfolio is a continuous and convex function
with respect to positions in instruments, whereas the VaR may be even a discontinuous
function.
•CVaR deviation (and mixed CVaR deviation) is a strong “competitor” to the standard
deviation. Virtually everywhere the standard deviation can be replaced by a CVaR deviation.
For instance, in finance, a CVaR deviation can be used in the following concepts: the Sharpe
ratio, portfolio beta, one-fund theorem (i.e., optimal portfolio is a mixture of a risk-free
asset and a master fund), market equilibrium with one or multiple deviation measures, and
so on.
•Risk management with CVaR functions can be done quite efficiently. CVaR can be
optimized and constrained with convex and linear programming methods, whereas VaR is
relatively difficult to optimize (although significant progress was made in this direction; for
instance, PSG can optimize VaR for quite large problems involving thousands of variables
and hundreds of thousands of scenarios).
•VaR risk measure does not control scenarios exceeding VaR (for instance you can signif-
icantly increase the largest loss exceeding VaR, but the VaR risk measure will not change).
This property can be both good and bad, depending upon your objectives:
◦The indifference of VaR risk measure to extreme tails may be a good property if poor
models are used for building distributions. VaR disregards some part of the distribution for
which only poor estimates are available. VaR estimates are statistically more stable than
CVaR estimates. This actually may lead to a superior out-of-sample performance of VaR
versus CVaR for some applications. For instance, for a portfolio involving instruments with
strong mean reverting properties, VaR will not penalize instruments with extremely heavy
losses. These instruments may perform very well at the next iteration. In statistics, it is well
understood that estimators based on VaR are “robust” and may automatically disregard
outliers and large losses, which may “confuse” the statistical estimation procedure.
◦The indifference of VaR to extreme tails may be quite an undesirable property, allow-
ing to take high uncontrollable risks. For instance, so-called “naked” option positions involve
a very small chance of extremely high losses; these rare losses may not be picked by VaR.
•CVaR accounts for losses exceeding VaR. This property may be good or bad, depending
upon your objectives:
◦CVaR provides an adequate picture of risks reflected in extreme tails. This is a very
important property if the extreme tail losses are correctly estimated.
◦CVaR may have a relatively poor out-of-sample performance compared with VaR if
tails are not modelled correctly. In this case, mixed CVaR can be a good alternative that
gives different weights for different parts of the distribution (rather than penalizing only
extreme tail losses).
•Deviation and risk are quite different risk management concepts. A risk measure evalu-
ates outcomes versus zero, whereas a deviation measure estimates wideness of a distribution.
For instance, CVaR risk may be positive or negative, whereas CVaR deviation is always
positive. Therefore, the Sharpe-like ratio (expected reward divided by risk measure) should
involve CVaR deviation in the denominator rather than CVaR risk.
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Figure 1. Risk functions: graphical representation of VaR, VaR Deviation, CVaR, CVaR Devia-
tion, Max Loss, and Max Loss Deviation.
VaR, CVaR, deviations
Frequency
VaR
Probability
1–α
Max
Loss
CVaR
CVaR Deviation
Max Loss Deviation
Loss
Mean
VaR Deviation
2. General Picture of VaR and CVaR
2.1. Definitions of VaR and CVaR
This section gives definitions of VaR and CVaR and discusses their use and basic properties.
We refer to Figure 1 for their graphical representation.
Let Xbe a random variable with the cumulative distribution function FX(z)=P{X≤z}.
Xmay have meaning of loss or gain. In this tutorial, Xhas meaning of loss and this impacts
the sign of functions in definition of VaR and CVaR.
Definition 1 (Value-at-Risk). The VaR of Xwith confidence level α∈]0,1[ is
VaRα(X) = min{z|FX(z)≥α}.(1)
By definition, VaRα(X)isalowerα-percentile of the random variable X. Value-at-risk is
commonly used in many engineering areas involving uncertainties, such as military, nuclear,
material, airspace, finance, etc. For instance, finance regulations, like Basel I and Basel II,
use VaR deviation measuring the width of daily loss distribution of a portfolio.
For normally distributed random variables, VaR is proportional to the standard deviation.
If X∼N(µ, σ2) and FX(z) is the cumulative distribution function of X, then (see Rockafellar
and Uryasev [19]),
VaRα(X)=F−1
X(α)=µ+k(α)σ, (2)
where k(α)=√2 erf −1(2α−1) and erf(z)=(2/√π)z
0e−t2dt.
The ease and intuitiveness of VaR are counterbalanced by its mathematical properties.
As a function of the confidence level, for discrete distributions, VaRα(X) is a nonconvex,
discontinuous function. For a discussion of numerical difficulties of VaR optimization, see,
for example, Rockafellar [17] and Rockafellar and Uryasev [19].
Sarykalin et al.: VaR vs.CVaR in Risk Management and Optimization
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Definition 2 (Conditional Value-at-Risk). An alternative percentile measure of risk
is conditional value-at-risk (CVaR). For random variables with continuous distribution func-
tions, CVaRα(X) equals the conditional expectation of Xsubject to X≥Va Rα(X). This
definition is the basis for the name of conditional value-at-risk. The term conditional value-
at-risk was introduced by Rockafellar and Uryasev [19]. The general definition of conditional
value-at-risk (CVaR) for random variables with a possibly discontinuous distribution func-
tion is as follows (see Rockafellar and Uryasev [20]).
The CVaR of Xwith confidence level α∈]0,1[ is the mean of the generalized α-tail
distribution:
CVaRα(X)=∞
−∞
zdFα
X(z),(3)
where
Fα
X(z)=
0,when z<VaRα(X),
FX(z)−α
1−α,when z≥VaRα(X).
Contrary to popular belief, in the general case, CVaRα(X) is not equal to an average of
outcomes greater than VaRα(X). For general distributions, one may need to split a prob-
ability atom. For example, when the distribution is modelled by scenarios, CVaR may be
obtained by averaging a fractional number of scenarios. To explain this idea in more detail,
we further introduce alternative definitions of CVaR. Let CVaR+
α(X), called “upper CVaR,”
be the conditional expectation of Xsubject to X>VaRα(X):
CVaR+
α(X)=E[X|X>Va R α(X)].
CVaRα(X) can be defined alternatively as the weighted average of VaRα(X) and
CVaR+
α(X), as follows. If FX(VaRα(X)) <1, so there is a chance of a loss greater than
VaRα(X), then
CVaRα(X)=λα(X)VaRα(X)+(1−λα(X))CVaR+
α(X),(4)
where
λα(X)=FX(VaRα(X)) −α
1−α,(5)
whereas if FX(VaRα(X)) = 1, so that VaRα(X) is the highest loss that can occur, then
CVaRα(x)=VaR
α(x).(6)
The definition of CVaR as in Equation (4) demonstrates that CVaR is not defined as a
conditional expectation. The function CVaR−
α(X)=E[X|X≥VaRα(X)], called “lower
CVaR,” coincides with CVaRα(X) for continuous distributions; however, for general distri-
butions it is discontinuous with respect to αand is not convex. The construction of CVaRα
as a weighted average of VaRαand CVaR+
α(X) is a major innovation. Neither VaR nor
CVaR+
α(X) behaves well as a measure of risk for general loss distributions (both are discon-
tinuous functions), but CVaR is a very attractive function. It is continuous with respect to
αand jointly convex in (X, α). The unusual feature in the definition of CVaR is that VaR
atom can be split. If FX(x) has a vertical discontinuity gap, then there is an interval of
confidence level αhaving the same VaR. The lower and upper endpoints of that interval are
α−=FX(VaR−
α(X)) and α+=FX(VaRα(X)), where FX(VaR−
α(X)) = P{X<Va R α(X)}.
When FX(VaR−
α(X)) <α<F
X(VaRα(X)) <1, the atom VaRα(X) having total probability
α+−α−is split by the confidence level αin two pieces with probabilities α+−αand α−α−.
Equation (4) highlights this splitting.
The definition of CVaR is illustrated further with the following examples. Suppose we have
six equally likely scenarios with losses f1···f6. Let α=2
3(see Figure 2). In this case, αdoes
Sarykalin et al.: VaR vs.CVaR in Risk Management and Optimization
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Figure 2. CVaR Example 1: computation of CVaR when αdoes not spilt the atom.
CVaR
6
1
6
1
VaRLoss CVaR+
6
1
6
1
6
1
6
1
f1f2f3
f4f5f6
CVaR–
not split any probability atom. Then VaRα(X)<CVaR−
α(X)<CVaRα(X)=CVaR
+
α(X),
λα(X)=(FX(VaRα(X)) −α)/(1 −α) = 0 and CVaRα(X)=CVaR
+
α(X)=1
2f5+1
2f6, where
f5and f6are losses number five and six, respectively. Now, let α=7
12 (see Figure 3).
In this case, αdoes split the VaRα(X) atom, λα(X)=(FX(VaRα(X)) −α)/(1 −α)>0, and
CVaRα(X) is given by
CVaRα(X)= 1
5VaRα(X)+ 4
5CVaR+
α(X)=1
5f4+2
5f5+2
5f6.
In the last case, we consider four equally likely scenarios and α=7
8splits the last atom
(see Figure 4). Now VaRα(X)=CVaR
−
α(X)=CVaR
α(X), upper CVaR, CVaR+
α(X)isnot
defined, λα(X)=(FX(VaRα(X)) −α)/(1 −α)>0, and CVaRα(X)=VaR(X)=f4. The
PSG package defines the CVaR function for discrete distributions equivalently to (4) through
the lower CVaR and upper CVaR. Suppose that VaRα(X) atom having total probability
α+−α−is split by the confidence level αin two pieces with probabilities α+−αand α−α−.
Then,
CVaRα(X)= α+−α
α+−α−
1−α−
1−αCVaR−
α(X)+ α−α−
α+−α−
1−α+
1−αCVaR+
α(X),(7)
where
CVaR−
α(X)=E[X|X≥VaRα(X)],CVaR+
α(X)=E[X|X>Va R α(X)].(8)
Pflug [15] followed a different approach and suggested to define CVaR via an optimization
problem, which he borrowed from Rockafellar and Uryasev [19]:
CVaRα(X) = min
CC+1
1−αE[X−C]+,where [t]+= max{0,t}.(9)
One more equivalent representation of CVaR was given by Acerbi [1], who showed that
CVaR is equal to “expected shortfall” defined by
CVaRα(X)= 1
αα
0
VaRβ(X)dβ.
Figure 3. CVaR Example 2: computation of CVaR when αsplits the atom.
CVaR
CVaR–
VaRLoss ½f5+½f6 = CVaR+
6
1
6
1
6
1
6
1
6
1
6
112
1
f1f2f3
f4f5f6
Sarykalin et al.: VaR vs.CVaR in Risk Management and Optimization
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Figure 4. CVaR Example 3: computation of CVaR when αsplits the last atom.
CVaR
VaRLoss
4
1
4
1
8
1
4
1
4
1
f1f2f3f4
For normally distributed random variables, a CVaR deviation is proportional to the standard
deviation. If X∼N(µ, σ2), then (see Rockafellar and Uryasev [19])
CVaRα(X)=E[X|X≥Va R α(X)] = µ+k1(α)σ, (10)
where
k1(α)=√2πexp(erf−1(2α−1))2(1 −α)−1
and erf(z)=(2/√π)z
0e−t2dt.
2.2. Risk Measures
Axiomatic investigation of risk measures was suggested by Artzner et al. [3]. Rockafellar [17]
defined a functional R:L2→]−∞,∞]asacoherent risk measure in the extended sense if
R1: R(C)=Cfor all constant C;
R2: R((1 −λ)X+λX)≤(1 −λ)R(X)+λR(X) for λ∈]0,1[ (convexity);
R3: R(X)≤R(X) when X≤X(monotonicity);
R4: R(X)≤0 when Xk−X2→0 with R(Xk)≤0 (closedness).
A functional R:L2→]−∞,∞] is called a coherent risk measure in the basic sense if it
satisfies axioms R1, R2, R3, R4, and additionally the axiom
R5: R(λX)=λR(X) for λ>0 (positive homogeneity).
A functional R:L2→]−∞,∞] is called an averse risk measure in the extended sense if it
satisfies axioms R1, R2, R4, and
R6: R(X)>EX for all nonconstant X(aversity).
Aversity has the interpretation that the risk of loss in a nonconstant random variable X
cannot be acceptable; i.e., R(X)<0, unless EX < 0.
A functional R:L2→]−∞,∞] is called an averse risk measure in the basic sense if it
satisfies R1, R2, R4, R6, and also R5.
Examples of coherent measures of risk are R(X)=µX =E[X]orR(X) = sup X. However,
R(X)=µ(X)+λσ(X) for some λ>0 is not a coherent measure of risk because it does not
satisfy the monotonicity axiom R3.
R(X)=VaR
α(X) is not a coherent nor an averse risk measure. The problem lies in the
convexity axiom R2, which is equivalent to the combination of positive homogeneity and
subadditivity, this last defined as R(X+X)≤R(X)+R(X). Although positive homo-
geneity is obeyed, the subadditivity is violated. It has been proved, for example, in Acerbi
and Tasche [2], Pflug [15], and Rockafellar and Uryasev [20], that for any probability level
α∈]0,1[, R(X)=CVaR
α(X) is a coherent measure of risk in the basic sense. CVaRα(X)is
also an averse measure of risk for α∈]0,1]. An averse measure of risk might not be coherent;
a coherent measure might not be averse.
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2.3. Deviation Measures
In this section, we refer to Rockafellar [17] and Rockafellar et al. [24]. A functional D:L2→
[0,∞] is called a a deviation measure in the extended sense if it satisfies
D1: D(C) = 0 for constant C, but D(X)>0 for nonconstant X;
D2: D((1 −λ)X+λX)≤(1 −λ)D(X)+λD(X) for λ∈]0,1[ (convexity);
D3: D(X)≤dwhen Xk−X2→0 with D(Xk)≤d(closedness).
A functional is called a deviation measure in the basic sense when it satisfies axioms
D1, D2, D3, and, furthermore,
D4: D(λX)=λD(X) for λ>0 (positive homogeneity).
A deviation measure in the extended or basic sense is called a coherent measure in the
extended or basic sense if it additionally satisfies
D5: D(X)≤sup X−E[X] for all X(upper range boundedness).
An immediate example of a deviation measure in the basic sense is the standard deviation
σ(X)=(E[X−EX]2)1/2,
which satisfies axioms D1, D2, D3, D4, but not D5. In other words, standard deviation is not
a coherent deviation measure. Here are more examples of deviation measures in the basic
sense:
Standard semideviations:
σ+(X)=(E[max{X−EX,0}]2)1/2,
σ−(X)=(E[max{EX −X, 0}]2)1/2;
Mean Absolute Deviation:
MAD(X)=E[|X−EX|].
Moreover, we define the α-value-at-risk deviation measure and the α-conditional value-at-
risk deviation measure as
α−VaR∆
α(X)=VaR
α(X−EX) (11)
and
α−CVaR∆
α(X)=CVaR
α(X−EX),(12)
respectively. The VaR deviation measure VaR∆
α(X) is not a deviation measure in the general
or basic sense because the convexity axiom D2 is not satisfied. The CVaR deviation measure
CVaR∆
α(X) is a coherent deviation measure in the basic sense.
2.4. Risk Measures vs. Deviation Measures
Rockafellar et al. (originally in [24], then in [17]), obtained the following result:
Theorem 1. A one-to-one correspondence between deviation measures Din the extended
sense and averse risk measures Rin the extended sense is expressed by the relations
R(X)=D(X)+EX,
D(X)=R(X−EX);
additionally,
Ris coherent ↔D is coherent.
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Moreover, the positive homogeneity is preserved:
Ris positively homogeneous ↔Dis positively homogeneous.
In other words, for an averse risk measures Rin the basic sense and a deviation measures
Din the basic sense the one-to-one correspondence is valid, and, additionally, coherent R
↔coherent D.
With this theorem we obtain that for the standard deviation, σ(X), which is a deviation
measure in the basic sense, the counterpart is the standard risk EX +σ(X), which is a risk
averse measure in the basic sense. For CVaR deviation, CVaR∆
α(X), which is a coherent
deviation measure in the basic sense, the counterpart is CVaR risk, CVaRα(X), which is a
risk-averse coherent measure in the basic sense.
Another coherent deviation measure in the basic sense is the so-called Mixed Deviation
CVaR, which we think is the most promising for risk management purposes. Mixed Deviation
CVaR is defined as
Mixed-CVaR∆
α(X)=
K
k=1
λkCVaR∆
αk(X)
for λk≥0, K
k=1 λk= 1, and αkin ]0,1[. The counterpart to the Mixed Deviation CVaR is
the Mixed CVaR, which is the coherent averse risk measure in the basic sense, defined by
Mixed-CVaRα(X)=
K
k=1
λkCVaRαk(X).
3. VaR and CVaR in Optimization and Statistics
3.1. Equivalence of Chance and VaR Constraints
For this section, we refer to Rockafellar [18]. Several engineering applications deal with
probabilistic constraints such as the reliability of a system or the system’s ability to meet
demand; in portfolio management, often it is required that portfolio loss at a certain future
time is, with high reliability, at most equal to a certain value. In these cases an optimization
model can be set up so that constraints are required to be satisfied with some probability
level rather than almost surely. Let x∈nand let ω∈Ω be a random event ranging over the
set Ω of all random events. For a given x, we may require that most of the time some random
functions fi(x, ω), i=1,...,m, satisfy the inequalities fi(x, ω)≤0, i=1,...,m; that is, we
may want that
Prob{fi(x, ω)≤0}≥pifor i=1,...,m, (13)
where 0 ≤pi≤1. Requiring this probability to be equal to 1 is the same as requiring that
fi(x, ω)≤0 almost surely. In most applications, this approach can lead to modelling and
technical problems. In modelling, there is little guidance on what level of pito set; moreover,
one has to deal with the issue of constraint interactions and decide whether, for instance,
it is better to require Prob{f1(x, ω)≤0}≥p1=0.99 and Prob{f2(x, ω)≤0}≥p2=0.95 or
to work with a joint condition like Prob{fi(x, ω )≤0}≥p. Dealing numerically with the
functions Fi(x) = Prob{fi(x, ω)≤0}leads to the task of finding the relevant properties
of Fi; a common difficulty is that the convexity of fi(x, ω ) with respect to xmay not carry
over to the convexity of Fi(x) with respect to x.
Chance constraints and percentiles of a distribution are closely related. Let VaRα(x)be
the VaRαof a loss function f(x, ω ); that is,
VaRα(x) = min{: Prob{f(x, ω )≤}≥α}.(14)
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Then, the following constraints are equivalent:
Prob{f(x, ω)≤}≥α↔Prob{f(x, ω)>}≤1−α↔VaRα(x)≤. (15)
Generally, VaRα(x) is nonconvex with respect to x; therefore, VaRα(x)≤and
Prob{f(x, ω)≤}≥αmay be nonconvex constraints.
3.2. CVaR Optimization
For this section, we refer to Rockafellar and Uryasev [19] and Uryasev [29]. Nowadays, VaR
has achieved the high status of being written into industry regulations (for instance, in
regulations for finance companies). It is difficult to optimize VaR numerically when losses
are not normally distributed. Only recently VaR optimization was included in commercial
packages such as PSG. As a tool in optimization modeling, CVaR has superior properties
in many respects. CVaR optimization is consistent with VaR optimization and yields the
same results for normal or elliptical distributions (see definition of elliptical distribution
in Embrechts et al. [6]); for models with such distributions, working with VaR, CVaR, or
minimum variance (Markowitz [11]) is equivalent (see Rockafellar and Uryasev [19]). Most
importantly, CVaR can be expressed by a minimization formula suggested by Rockafellar
and Uryasev [19]. This formula can be incorporated into the optimization problem with
respect to decision variables x∈X∈
n, which are designed to minimize risk or shape
it within bounds. Significant shortcuts are thereby achieved while preserving the crucial
problem features like convexity. Let us consider that a random loss function f(x, y) depends
upon the decision vector xand a random vector yof risk factors. For instance, f(x, y)=
−(y1x1+y2x2) is the negative return of a portfolio involving two instruments. Here, x1,x
2
are positions and y1,y
2are rates of returns of two instruments in the portfolio. The main
idea in Rockafellar and Uryasev [19] is to define a function that can be used instead of
CVaR:
Fα(x, ζ)=ζ+1
1−αE{[f(x, y)−ζ]+}.(16)
The authors proved that
1. Fα(x, ζ) is convex with respect to (w.r.t.) α;
2. VaRα(x) is a minimum point of function Fα(x, ζ ) w.r.t. ζ;
3. minimizing Fα(x, ζ ) w.r.t. ζgives CVaRα(x):
CVaRα(x) = min
αFα(x, ζ).(17)
In optimization problems, CVaR can enter into the objective or constraints or both. A big
advantage of CVaR over VaR in that context is the preservation of convexity; i.e., if f(x, y)
is convex in x, then CVaRα(x)isconvexinx. Moreover, if f(x, y)isconvexinx, then
the function Fα(x, ζ)isconvexinbothxand ζ. This convexity is very valuable because
minimizing Fα(x, ζ) over (x, ζ )∈X×results in minimizing CVaRα(x):
min
x∈XCVaRα(x) = min
(x, ζ)∈X× Fα(x, ζ ).(18)
In addition, if (x∗,ζ∗) minimizes Fαover X×, then not only does x∗minimize CVaRα(x)
over Xbut also
CVaRα(x∗)=Fα(x∗,ζ∗).
In risk management, CVaR can be utilized to “shape” the risk in an optimization model.
For that purpose, several confidence levels can be specified. Rockafellar and Uryasev [19]
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showed that for any selection of confidence levels αiand loss tolerances ωi,i=1,...,l, the
problem
min
x∈Xg(x)
s.t. CVaRαi(x)≤ωi,i=1,...,l
(19)
is equivalent to the problem
min
x, ζ1,...,ζl,∈X××··· g(x)
s.t. Fαi(x, ζi)≤ωi,i=1,...,l.
(20)
When Xand gare convex and f(x, y)isconvexinx, the optimization problems (18)
and (19) are ones of convex programming and, thus, especially favorable for computation.
When Yis a discrete probability space with elements yk,k=1,...,N having probabilities
pk,k=1,...,N,wehave
Fαi(x, ζi)=ζi+1
1−αi
N
k=1
pk[f(x, yk)−ζi]+.(21)
The constraint Fα(x, ζ)≤ωcan be replaced by a system of inequalities by introducing
additional variables ηk:
ηk≥0,f(x, yk)−ζ−ηk≤0,k=1,...,N,
ζ+1
1−α
N
k=1
pkηk≤ω. (22)
The minimization problem in (19) can be converted into the minimization of g(x) with
the constraints Fαi(x, ζi)≤ωibeing replaced as presented in (22). When fis linear in x,
constraints (22) are linear.
3.3. Generalized Regression Problem
In linear regression, a random variable Yis approximated in terms of random variables
X1,X
2,...,X
nby an expression c0+c1X1+···+cnXn. The coefficients are chosen by
minimizing the mean square error:
min
c0,c1,...,cn
E(Y−[c0+c1X1+···+cnXn])2.(23)
Mean square error minimization is equivalent to minimizing the standard deviation with the
unbiasedness constraint (see, Rockafellar et al. [21, 26]):
min σ(Y−[c0+c1X1+···+cnXn])
s.t. E[c0+c1X1+···+cnXn]=EY. (24)
Rockafellar et al. [21, 26] considered a general axiomatic setting for error measures and
corresponding deviation measures. They defined an error measure as a functional E:L2(Ω) →
[0,∞] satisfying the following axioms:
E1: E(0) = 0, E(X)>0 for X=0, E(C)<∞for constant C;
E2: E(λX)=λE(X) for λ>0 (positive homogeneity);
E3: E(X+X)≤E(X)+E(X) for all Xand X(subadditivity);
E4: {X∈L2(Ω) |E(X)≤c}is closed for all c<∞(lower semicontinuity).
For an error measure E, the pro jected deviation measure Dis defined by the equation
D(X) = minCE(X−C), and the statistic S(X) is defined by S(X) = argminCE(X−C).
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Their main finding is that the general regression problem
min
c0,c1,...,cnE(Y−[c0+c1X1+···+cnXn]) (25)
is equivalent to
min
c1,...,cnD(Y−[c1X1+···+cnXn])
s.t.c
0∈S(Y−[c1X1+···+cnXn]).
The equivalence of optimization problems (23) and (24) is a special case of this theorem.
This leads to the identification of a link between statistical work on percentile regression
(see Koenker and Basset [7]) and CVaR deviation measure: minimization of the Koenker
and Bassett [7] error measure is equivalent to minimization of CVaR deviation. Rockafellar
et al. [26] show that when the error measure is the Koenker and Basset [7] function,
Eα
KB(X)=E[max{0,X}+(α−1−1) max{0,−X}], the projected measure of deviation is
D(X)=CVaR
∆
α(X) = CVaRα(X−EX), with the corresponding averse measure of risk and
associated statistic given by
R(X) = CVaRα(X),
S(X)=VaR
α(X).
Then,
min
C∈(E[X−C]++(α−1−1)E[X−C]−)=CVaR
∆
α(X),
arg min
C∈
(E[X−C]++(α−1−1)E[X−C]−)=VaR
α(X).
A similar result is available for the “mixed Koenker and Bassett error measure” and the
corresponding mixed deviation CVaR; see Rockafellar et al. [26].
3.4. Stability of Estimation
The need to estimate VaR and CVaR arises typically when we are interested in estimating
tails of distributions. It is of interest, in this respect, to compare the stability of estimates of
VaR and CVaR based on a finite number of observations. The common flaw in such compar-
isons is that some confidence level is assumed and estimations of VaRα(X) and CVaRα(X)
are compared with the common value of confidence level α, usually, 90%, 95%, and 99%. The
problem with such comparisons is that VaR and CVaR with the same confidence level mea-
sure “different parts” of the distribution. In reality, for a specific distribution, the confidence
levels α1and α2for comparison of VaR and CVaR should be found from the equation
VaRα1(X) = CVaRα2(X).(26)
For instance, in the credit risk example in Serraino et al. [27], we find that CVaR with
confidence level α=0.95 is equal to VaR with confidence level α=0.99. The paper by
Yamai and Yoshiba [30] can be considered a “good” example of a “flawed” comparison of
VaR and CVaR estimates. Yamai and Yoshiba [30] examine VaR and CVaR estimations for
the parametrical family of stable distributions. The authors ran 10,000 simulations of size
1,000 and compared standard deviations of VaR and CVaR estimates normalized by their
mean values. Their main findings are as follows. VaR estimators are generally more stable
than CVaR estimators with the same confidence level. The difference is most prominent for
fat-tailed distributions and is negligible when the distributions are close to normal. A larger
sample size increases the accuracy of CVaR estimation. We provide here two illustrations of
Yamai and Yoshiba’s [30] results of these estimators’ performances.
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In the first case, the distribution of an equity option portfolio is modelled. The portfolio
consists of call options based on three stocks with joint log-normal distribution. VaR and
CVaR are estimated at the 95% confidence level on 10,000 sets of Monte Carlo simulations
with a sample size of 1,000. The resulting loss distribution for the portfolio of at-the-money
options is quite close to normal; estimation errors of VaR and CVaR are similar. The result-
ing loss distribution for the portfolio of deep out-of-the-money options is fat tailed; in this
case, the CVaR estimator performed significantly worse than the VaR estimator.
In the second case, estimators are compared on the distribution of a loan portfolio, con-
sisting of 1,000 loans with homogeneous default rates of 1% through 0.1%. Individual loan
amounts obey the exponential distribution with an average of $100 million. Correlation
coefficients between default events are homogeneous at levels 0.00, 0.03, and 0.05. Results
show that estimation errors of CVaR and VaR estimators are similar when the default rate
is higher; for lower default rates, the CVaR estimator gives higher errors. Also, the higher
the correlation between default events, the more the loan portfolio distribution becomes fat
tailed, and the higher is the error of CVaR estimator relative to VaR estimator.
As we pointed out, these numerical experiments compare VaR and CVaR with the same
confidence level, and some other research needs to be done to compare stability of estimators
for the same part of the distribution.
3.5. Decomposition According to Contributions of Risk Factors
This subsection discusses decomposition of VaR and CVaR risk according to risk factor
contributions; see, for instance, Tasche [28] and Yamai and Yoshiba [30]. Consider a portfolio
loss X, which can be decomposed as
X=
n
i=1
Xizi,
where Xiare losses of individual risk factors and ziare sensitivities to the risk factors, i=
1,...,n. The following decompositions of VaR and CVaR hold for continuous ditributions:
VaRα(X)=
n
i=1
∂VaRα(X)
∂zi
zi=E[Xi|X=VaR
α(X)]zi,(27)
CVaRα(X)=
n
i=1
∂CVaRα(X)
∂zi
zi=E[Xi|X≥VaRα(X)]zi.(28)
When a distribution is modelled by scenarios, it is much easier to estimate quantities
E[Xi|X≥VaRα(X)] in the CVaR decomposition than quantities E[Xi|X=VaR
α(X)]
in the VaR decomposition. Estimators of ∂VaRα(X)/∂ziare less stable than estimators of
∂CVaRα(X)/∂ zi.
3.6. Generalized Master-Fund Theorem and CAPM
The one-fund theorem is a fundamental result of the classical portfolio theory. It establishes
that any mean-variance-efficient portfolio can be constructed as a combination of a single
master-fund portfolio and a risk-free instrument. Rockafellar et al. [22] investigated in detail
the consequences of substituting the standard deviation in the setting of classical theory with
general deviation measures, in cases where rates of return may have discrete distributions,
mixed discrete-continuous distributions (which can arise from derivatives, such as options)
or continuous distributions. Their main result is that the one-fund theorem holds regardless
of the particular choice of the deviation measure. The optimal risky portfolio needs not
always be unique and it might not always be expressible by a master fund as traditionally
conceived, even when only the standard deviation is involved. The authors introduce the
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concepts of a basic fund as the portfolio providing the minimum portfolio deviation δfor
a gain of exactly 1 over the risk-free rate, with the corresponding basic deviation, defined
as the minimum deviation amount. Then they establish the generalized one-fund theorem,
according to which for any level of risk premium ∆ over the risk-free rate, the solution of
the portfolio optimization problem is given by investing an amount ∆ in the basic fund
and an amount 1 −∆ in the risk-free rate. In particular, the price of the basic fund can be
positive, negative, or equal to zero, leading respectively to a long position, short position, or
no investment in the basic fund. Rockafellar et al., first in Rockafellar et al. [23] and further
in Rockafellar et al. [25], developed these concepts to obtain the generalized capital asset
pricing (CAPM) relations. They find that the generalized CAPM equilibrium holds under
the following assumptions: there are several groups of investors, each with utility function
Uj(ERj,Dj(Rj)) based on deviation measure Dj; the utility functions depend on mean
and deviation and they are concave w.r.t. mean and deviation, increasing w.r.t. mean, and
decreasing w.r.t. deviation; investors maximize their utility functions sub ject to the budget
constraint. The main finding is that an equilibrium exists w.r.t. Diin which each group of
investors has its own master fund and investors invest in the risk-free asset and their own
master funds. A generalized CAPM holds and takes the form
¯rij −r0=βij (¯rjM −r0),
βij =cov(Gj,r
ij )
D(−rjM),
where
¯rij is expected return of asset iin group j,
r0is risk-free rate,
¯rjM is expected return of market fund for investor group j,
Gjis the risk identifier for the market fund j.
From this general statement we can obtain that in classical framework when all investors
are interested in standard deviation, βiis defined as
βi=cov(ri,r
M)
σ2(rM).
Similarly, when all investors are interested in semideviation D(X)=σ−(X), then
βi=cov(ri,r
M)
σ2
−(−rM),
and when D(X) = CVaR∆
α(X) with continuously distributed random values, then
βi=E[ri−¯ri|−rM≥Va Rα(−rM)]
CVaR∆
α(−rM).
It is interesting to observe in the last case that “beta” picks up only events when market is
in α·100% of its highest losses; i.e., −rM≥VaRα(−rM).
4. Comparative Analysis of VaR and CVaR
4.1. VaR Pros and Cons
4.1.1. Pros. VaR is a relatively simple risk management notion. Intuition behind
α-percentile of a distributions is easily understood and VaR has a clear interpretation: how
much you may lose with certain confidence level. VaR is a single number measuring risk,
defined by some specified confidence level, e.g., α=0.95. Two distributions can be ranked
by comparing their VaRs for the same confidence level. Specifying VaR for all confidence
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levels completely defines the distribution. In this sense, VaR is superior to the standard
deviation. Unlike the standard deviation, VaR focuses on a specific part of the distribution
specified by the confidence level. This is what is often needed, which made VaR popular in
risk management, including finance, nuclear, airspace, material science, and various military
applications.
One of important properties of VaR is stability of estimation procedures. Because VaR
disregards the tail, it is not affected by very high tail losses, which are usually difficult to
measure. VaR is estimated with parametric models; for instance, covariance VaR based on
the normal distribution assumption is very well known in finance, with simulation models
such as historical or Monte Carlo or by using approximations based on second-order Taylor
expansion.
4.1.2. Cons. VaR does not account for properties of the distribution beyond the confi-
dence level. This implies that VaRα(X) may increase dramatically with a small increase
in α. To adequately estimate risk in the tail, one may need to calculate several VaRs with
different confidence levels. The fact that VaR disregards the tail of the distribution may
lead to unintentional bearing of high risks. In a financial setting, for instance, let us consider
the strategy of “naked” shorting deep out-of-the-money options. Most of the time, this will
result in receiving an option premium without any loss at expiration. However, there is a
chance of a big adverse market movement leading to an extremely high loss. VaR cannot
capture this risk.
Risk control using VaR may lead to undesirable results for skewed distributions. Later on
we will demonstrate this phenomenon with the case study comparing risk profiles of VaR
and CVaR optimization. In this case, the VaR optimal portfolio has about 20% longer tail
than the CVaR optimal portfolio, as measured by the max loss of those portfolios.
VaR is a nonconvex and discontinuous function for discrete distributions. For instance,
in financial setting, VaR is a nonconvex and discontinuous function w.r.t. portfolio posi-
tions when returns have discrete distributions. This makes VaR optimization a challenging
computational problem. Nowadays there are codes, such as PSG, that can work with VaR
functions very efficiently. PSG can optimize portfolios with a VaR performance function and
also shape distributions of the portfolio with multiple VaR constraints. For instance, in port-
folio optimization it is possible to maximize expected return with several VaR constraints
at different confidence levels.
4.2. CVaR Pros and Cons
4.2.1. Pros. CVaR has a clear engineering interpretation. It measures outcomes that hurt
the most. For example, if Lis a loss then the constraint CVaRα(L)≤¯
Lensures that the
average of (1−α)% highest losses does not exceed ¯
L.
Defining CVaRα(X) for all confidence levels αin (0,1) completely specifies the distribution
of X. In this sense, it is superior to standard deviation.
Conditional value at risk has several attractive mathematical properties. CVaR is a coher-
ent risk measure. CVaRα(X) is continuous with respect to α. CVaR of a convex combina-
tion of random variables CVaRα(w1X1+···+wnXn) is a convex function with respect to
(w1,...,w
n). In financial setting, CVaR of a portfolio is a convex function of portfolio posi-
tions. CVaR optimization can be reduced to convex programming, in some cases to linear
programming (i.e., for discrete distributions).
4.2.2. Cons. CVaR is more sensitive than VaR to estimation errors. If there is no good
model for the tail of the distribution, CVaR value may be quite misleading; accuracy of
CVaR estimation is heavily affected by accuracy of tail modelling. For instance, historical
scenarios often do not provide enough information about tails; hence, we should assume
a certain model for the tail to be calibrated on historical data. In the absence of a good
tail model, one should not count on CVaR. In financial setting, equally weighted portfolios
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may outperform CVaR-optimal portfolios out of sample when historical data have mean
reverting characteristics.
4.3. What Should You Use, VaR or CVaR?
VaR and CVaR measure different parts of the distribution. Depending on what is needed,
one may be preferred over the other.
Let us illustrate this topic with financial applications of VaR and CVaR and examine
the question of which measure is better for portfolio optimization. A trader may prefer
VaR to CVaR, because he may like high uncontrolled risks; VaR is not as restrictive as
CVaR with the same confidence level. Nothing dramatic happens to a trader in case of
high losses. He will not pay losses from his pocket; if fired, he may move to some other
company. A company owner will probably prefer CVaR; he has to cover large losses if they
occur; hence, he “really” needs to control tail events. A board of directors of a company
may prefer to provide VaR-based reports to shareholders and regulators because it is less
than CVaR with the same confidence level. However, CVaR may be used internally, thus
creating asymmetry of information between different parties.
VaR may be better for optimizing portfolios when good models for tails are not available.
VaR disregards the hardest to measure events. CVaR may not perform well out of sample
when portfolio optimization is run with poorly constructed set of scenarios. Historical data
may not give right predictions of future tail events because of mean-reverting characteristics
of assets. High returns typically are followed by low returns; hence, CVaR based on history
may be quite misleading in risk estimation.
If a good model of tail is available, then CVaR can be accurately estimated and CVaR
should be used. CVaR has superior mathematical properties and can be easily handled in
optimization and statistics.
When comparing stability of estimation of VaR and CVaR, appropriate confidence levels
for VaR and CVaR must be chosen, avoiding comparison of VaR and CVaR for the same
level of αbecause they refer to different parts of the distribution.
5. Numerical Results
This section reports results of several case studies exemplifying how to use VaR and CVaR
in optimization settings.
5.1. Portfolio Safeguard
We used PSG to do the case studies. We posted MATLAB files to run these case studies
in a MATLAB environment on the MathWorks website (http://www.mathworks.com), in
the file exchange-optimization area. PSG is designed to solve a wide range of risk manage-
ment and stochastic optimization problems. PSG has built-in algorithms for evaluating and
optimizing various statistical and deterministic functions. For instance, PSG includes the
statistical functions Mean, Variance, Standard Deviation, VaR, CVaR, CDaR, MAD, Max-
imum Loss, Partial Moment, Probability of Exceeding a Threshold, and the deterministic
functions Cardinality, Fixed Charge, and Buyin. For a complete list of functions, see Table
A.1 in Appendix I. Required data inputs are matrices of Scenarios or Covariance Matrices
on which statistical functions are evaluated. PSG uses a new design for defining optimiza-
tion problems. A function is defined by just providing an underlying Matrix to a Function.
Figure 5 illustrates PSG procedure for function definition (Mean and Max Loss Functions are
defined: Matrix of Scenarios ⇒Loss Vector ⇒Functions). With PSG design all needed data,
including names of variables, are taken from a Matrix of Scenarios or Covariance Matrix.
PSG can calculate values and sensitivities of defined functions on decision vectors. Also,
you can place functions to objective and constraint and define an optimization problem.
Linear combinations of functions can be placed in the constraints. PSG includes many case
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Figure 5. Example of evaluation of two PSG functions: Mean and Max Loss.
INPUT DATA: Matrix of scenarios
xy
Scenario benchmark Scenario probability
Loss vector
Loss function Scenario probability
Function evaluation
Function value
Mean
Max Loss
θ11 θ01
θ21
θ12
θ22 θ02
θ01 –(θ11 *X+θ21 *Y)
p1*[θ01 –(θ11 *X+θ21 *Y)] + p2*[θ02 –(θ12 *X+θ22 *Y)]
max[θ01 –(θ11 *X+θ21*Y)], [θ02 –(θ12 *X+θ22 *Y)]
θ02 –(θ12 *X+θ22 *Y)
p1
p2
p1
p2
Notes. These functions are completely defined by the matrix of scenarios. Both names of variables and
function values are calculated using matrix of scenarios.
studies, most of them motivated by financial applications (such as collateralized debt obli-
gation structuring, portfolio management, portfolio replication, and selection of insurance
contracts) that are helpful in learning the software. To build a new optimization problems,
you can simply make a copy of the case study and change input matrices and parameters.
PSG can solve problems with simultaneous constraints on various risks at different time
intervals (e.g., multiple constraints on standard deviation obtained by resampling, combined
with VaR and drawdown constraints), thus allowing robust decision making. It has built-in
efficient algorithms for solving large-scale optimization problems (up to 1,000,000 scenarios
and up to 10,000 decision variables in callable MATLAB and C++ modules). PSG offers
several tools for analyzing solutions, generated by optimization or through other procedures.
Among these tools are sensitivities of risk measures to changes in decision variables, and
the incremental impact of decision variables on risk measures and various functions of risk
measures. Analysis can reveal decision variables having the biggest impact on risk and other
functions, such as expected portfolio return. For visualization of these and other characteris-
tics, PSG provides tools for building and plotting various characteristics combining different
functions, points, and variables. It is user friendly and callable from MATLAB and C/C++
environment.
5.2. Risk Control Using VaR (PSG MATLAB Environment)
Risk control using VaR may lead to paradoxical results for skewed distributions. In this case,
minimization of VaR may lead to a stretch of the tail of the distribution exceeding VaR. The
purpose of VaR minimization is to reduce extreme losses. However, VaR minimization may
lead to an increase in the extreme losses that we try to control. This is an undesirable feature
of VaR optimization. Larsen et al. [9] showed this phenomenon on a credit portfolio with the
loss distribution created with a Monte Carlo simulation of 20,000 scenarios of joint credit
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Figure 6. VaR minimization.
VaR
CVaR
320
290
260
230
200
170
0 5 10 15 20 25 30 35 40 45 50 55
MUSD
Iteration
Notes. 99%-VaR and 99%-CVaR for algorithm A2 in Larsen et al. [9]. The algorithm lowered VaR at the
expense of an increase in CVaR.
states. The distribution is skewed with a long fat right tail. A more detailed description of
this portfolio can be found in Bucay and Rosen [4] and Mausser and Rosen [13, 14]. Larsen
et al. [9] suggested two heuristic algorithms for optimization of VaR. These algorithms are
based on the minimization of CVaR. The minimization of VaR leads to about 16% increase of
the average loss for the worst 1% scenarios, compared with the worst 1% scenarios in CVaR
minimum solution. These numerical results are consistent with the theoretical results: CVaR
is a coherent, whereas VaR is not a coherent, measure of risk. Figure 6 reproduces results
from Larsen et al. [9] showing how iteratively VaR is decreasing and CVaR is increasing.
We observed a similar performance of VaR on a small portfolio consisting of 10 bonds
and modelled with 1,000 scenarios. We solved two optimization problems with PSG. In the
first one, we minimized 99%-CVaR deviation of losses subject to constraints on budget and
required return. In the second one, we minimized 99%-VaR deviation of losses subject to
the same constraints.
Problem 1: min CVaR∆
α(x)
s.t.
n
i=1
rixi≥¯r,
n
i=1
xi=1.
(29)
Problem 2: min VaR∆
α(x)
s.t.
n
i=1
rixi≥¯r,
n
i=1
xi=1.
(30)
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Table 1. Value of risk functions.
min CVaR∆
0.99 min VaR∆
0.99 Ratio
CVaR0.99 0.0073 0.0083 1.130
CVaR∆
0.99 0.0363 0.0373 1.026
VaR0.99 0.0023 0.0005 0.231
VaR∆
0.99 0.0313 0.0295 0.944
Max loss = CVaR10.0133 0.0148 1.116
Max loss deviation = CVaR∆
10.0423 0.0438 1.036
Notes. Column “min CVaR0.99
α” reports the value of risk functions for the portfolio obtained by
minimizing 99%-CVaR deviation, column “minVaR∆
α” reports the value of risk functions for the
portfolio obtained by minimizing 99%-VaR deviation, and column “Ratio” contains the ratio of
every cell of column “minVaR∆
α” to every cell of column “minCVaR∆
α.”
where xis the vector of portfolio weights, riis the rate of return of asset i,¯ris the lower
bound on estimated portfolio return. We then evaluated different risk functions at the opti-
mal points for Problems 1 and 2. Results are shown in Table 1.
Suppose that we start with the portfolio having minimal 99%-CVaR deviation. Minimiza-
tion of 99%-VaR deviation leads to 13% increase in 99%-CVaR, compared with 99%-CVaR
in the optimal 99%-CVaR deviation portfolio. We found that even in a problem with a rela-
tively small number of scenarios, if the distribution is skewed, minimization of VaR deviation
may lead to a stretch of the tail compared with the CVaR optimal portfolio. This result is
quite important when we look at financial risk management regulations like Basel II that
are based on minimization of VaR deviation.
5.3. Linear Regression-Hedging: VaR, CVaR, Mean Absolute, and
Standard Deviations (PSG MATLAB Environment)
This case study investigates performance of optimal hedging strategies based on different
deviation measures measuring quality of hedging. The objective is to build a portfolio of
financial instruments that mimics the benchmark portfolio. Weights in the replicating port-
folio are chosen such that the deviation between the value of this replicating portfolio and the
value of the benchmark is minimized. Benchmark value and replicating financial instruments
values are random valves. Determining the optimal hedging strategy is a linear regression
problem where the response is the benchmark portfolio valve, the predictors are the repli-
cating financial instrument valves, and the coefficients of the predictors to be determined
are the portfolio weights. Let ˆ
θbe the replicating portfolio value, θ0be the benchmark
portfolio value, θ1,...,θ
Ibe replicating instrument values, and x1,...,x
Ibe their weights.
The replicating portfolio value can be expressed as follows:
ˆ
θ=x1θ1+···+xIθI.(31)
The coefficients x1,...,x
Ishould be chosen to minimize a replication error function depend-
ing upon the residual θ0−ˆ
θ. The intercept in this case is absent. According to the equivalence
of (25) and (26), an error minimization problem is equivalent to the minimization of the
appropriate deviation; see Rockafellar et al. [21] and [26]. This case study considers hedging
pipeline risk in the mortgage underwriting process. Hedging instruments are 5% Mortgage-
Backed Securities (MBS) forward, 5.5% MBS, and call options on 10-year treasury note
futures. Changes of values of the benchmark and the hedging instruments are driven by
changes in the mortgage rate. We minimize five different deviation measures: Standard Devi-
ation, Mean Absolute Deviation, CVaR Deviation, Two-Tailed 75%-VaR, and Two-Tailed
90%-VaR. We tested in-sample and out-of-sample performance of the hedging strategies.
On our set of scenarios, we found that Two-Tailed 90%-VaR has the best out-of-sample
performance, whereas the standard deviation has the worst out-of-sample performance.
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We think that the out-of-sample performance of hedging strategies based on different
deviation measures significantly depends on the skewness of the distribution. In this case,
the distribution of residuals is quite skewed.
We use here PSG definitions of Loss and Deviation functions. The Loss Function is defined
as follows:
Loss Function = L(x, θ)=L(x1,...,x
I,θ
0,...,θ
I)=θ0−
I
i=1
θixi.(32)
The loss function has jscenarios, L(x, θ1),...,L(x, θJ), each with probability pj,j=1,...,J.
Here are deviation measures considered in this case study:
Mean Absolute Deviation = MAD(L(x, θ)),(33)
Standard Deviation = σ(L(x, θ )),(34)
α%-VaR Deviation = VaR∆
α(L(x, θ)),(35)
α%-CVaR Deviation = CVaR∆
α(L(x, θ)),(36)
Two-Tail α%-VaR Deviation = TwoTailVaR∆
α(L(x, θ))
=VaR
α(L(x, θ))+VaR
α(−L(x, θ)).(37)
We solved the following minimization problems:
Minimize 90%-CVaR Deviation
min
xCVaR∆
0.9(L(x, θ)) (38)
Minimize Mean Absolute Deviation
min
xMAD(L(X,θ)) (39)
Minimize Standard Deviation
min
xσ(L(x, θ)) (40)
Minimize Two-Tail 75%-VaR Deviation
min
xTwoTailVaR∆
0.75(L(X, θ)) (41)
Minimize Two-Tail 90%-VaR Deviation
min
xTwoTailVaR∆
0.9(L(X,θ)).(42)
The data set for the case study includes 1,000 scenarios of value changes for each hedging
instrument and for the benchmark. For the out-of-sample testing, we partitioned the 1,000
scenarios into 10 groups with 100 scenarios in each group. Each time, we selected one group
for the out-of-sample test and we calculated optimal hedging positions based on the remain-
ing nine groups containing 900 scenarios. For each group of 100 scenarios, we calculated the
ex ante losses (i.e., underperformances of hedging portfolio versus target) with the optimal
hedging positions obtained from 900 scenarios. We repeated the procedure 10 times, once
for every out-of-sample group with 100 scenarios. To estimate the out-of-sample perfor-
mance, we aggregated the out-of-sample losses from the 10 runs and obtained a combined
set including 1,000 out-of-sample losses. Then, we calculated five deviation measures on the
out-of-sample 1,000 losses: Standard Deviation, Mean Absolute Deviation, CVaR Deviation,
Two-Tail 75%-VaR Deviation, and Two-Tail 90%-VaR Deviation. In addition, we calculated
three downside risk measures: 90%-CVaR, 90%-VaR, and Max Loss = 100%-CVaR on the
out-of-sample losses. Tables 2 and 3 show the results.
By minimizing the Two-Tail 90%-VaR Deviation, we obtained the best values for all three
considered downside risk measures (negative loss indicates gain). Minimization of CVaR
deviation lead to good results, whereas minimization of standard deviation gave the worst
level for three downside risk measures.
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Tutorials in Operations Research, c
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Table 2. Out-of-sample performance of various deviations on optimal hedging portfolios.
Optimal points CVaR∆
0.9MAD σTwoTailVaR∆
0.75 TwoTailVaR∆
0.9
CVaR∆
0.90.690 0.815 1.961 0.275 1.122
MAD 1.137 0.714 1.641 0.379 1.880
σ1.405 0.644 1.110 0.979 1.829
TwoTailVaR∆
0.75 1.316 0.956 1.955 0.999 1.557
TwoTailVaR∆
0.90.922 0.743 1.821 0.643 1.256
Notes. Each row reports values of five different deviation measures evaluated at optimal hedging points
obtained with five hedging strategies.
5.4. Example of Equivalence of Chance and VaR Constraints
(PSG MATLAB Environment)
This case study illustrates the equivalence between chance constraints and VaR constraints,
as explained in §3.1. We will illustrate numerically the equivalence
Prob{L(x, θ)>}≤1−α↔Va R α(L(x, θ)) ≤, (43)
where
Loss Function L(x, θ )=L(x1,...,x
I,θ
1,...,θ
I)=−
I
i=1
θixi.(44)
The case study is based on a data set including 1,000 return scenarios for 10 clusters of loans.
Here, I= 10 is the number of instruments, θ1,...,θ
Iare rates of returns of instruments,
x1,...,x
Iare instrument weights, and L(x, θ ) is a portfolio loss. We solved two portfolio
optimization problems. In both cases we maximized the estimated return of the portfolio. In
the first problem, we imposed a constraint on probability; in the second problem, we imposed
an equivalent constraint on VaR. In particular, in the first problem we require the 95%-VaR
of the optimal portfolio to be at most equal to the constant , whereas in the second problem
we require the probability of losses greater than to be lower than 1 −α=1−0.95=0.05.
We expected to obtain at optimality the same objective function value and similar optimal
portfolios for the two problems. Problem formulations are as follows:
Problem 1: max E[−L(x, θ)]
s.t. Prob{L(x, θ)>}≤1−α=0.05,
vi≤xi≤ui,i=1,...,I,
I
i=1
xi=1.
(45)
Table 3. Out-of-sample performance of various downside risks on optimal hedging portfolios.
Optimal points Max Loss CVaR0.9VaR0.9
CVaR∆
0.9−18.01 −18.05 −18.08
MAD −16.49 −17.44 −17.88
σ−13.31 −15.29 −15.60
TwoTailVaR∆
0.75 −15.31 −16.19 −16.71
TwoTailVaR∆
0.9−18.02 −18.51 −18.66
Notes. Each row reports values of three different risk measures evaluated at optimal hedging points obtained
with five hedging strategies (negative loss indicates gain).
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Table 4. Chance vs. VaR constraints.
Optimal weights Prob ≤0.05 VaR ≤
x10.051 0.051
x20.055 0.055
x30.071 0.071
x40.053 0.053
x50.079 0.079
x60.289 0.289
x70.020 0.020
x80.300 0.300
x90.063 0.063
x10 0.020 0.020
Notes. Optimal portfolios obtained on the same data set when we
maximized return with the chance constraint (the first column)
and with the VaR constraint (the second column).
Problem 2: max E[−L(x, θ)]
s.t. VaRα(L(x, θ )) ≤,
vi≤xi≤ui,i=1,...,I,
I
i=1
xi=1.
(46)
In the problems, viis the lower bound on position for asset i, and uiis the upper bound
on position for asset i. We also have the budget constraint: sum of weights is equal to 1.
The two problems at optimality selected the same portfolios and have the same objective
function value of 120.19. Table 4 shows the optimal points.
5.5. Portfolio Rebalancing Strategies: Risk vs. Deviation
(PSG MATLAB Environment)
In this case study we consider a portfolio rebalancing problem. A portfolio manager allocates
his wealth to different funds periodically solving an optimization problem and in each time
period building the portfolio that minimizes a certain risk function, given budget constraints
and bounds on each exposure. We solved the following problem:
min R(x, θ)−k∗E[−L(x, θ)] (47)
s.t.
I
i=1
xi= 1 (48)
vi≤(xi)≤uii=1...,I, (49)
where we denote by R(x, θ) a risk function, E[−L(x, θ)] is the expected portfolio return,
θ1,...,θ
Iare rates of returns of instruments, and x1,...,x
Iare instrument weights. The
scenario data set is composed of 46 monthly return observations for seven managed funds;
we solved the first optimization problem using the first 10 scenarios, then we rebalanced the
portfolio monthly. We used as risk functions VaR, CVaR, VaR Deviation, CVaR Deviation,
and Standard Deviation.
We then evaluated Sharpe ratio and mean value of each sequence of portfolios obtained
by succesively solving the optimization problem with a given ob jective function. Results are
reported in Tables 5 and 6 for different values of the parameter k. Both in terms of Sharpe
ratio and mean portfolio value we found a good performance of VaR and VaR deviation
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Table 5. Out-of-sample Sharpe ratio.
kVaR CVaR VaR Deviation CVaR Deviation Standard Deviation
−1 1.2710 1.2609 1.2588 1.2693 1.2380
−3 1.2711 1.2667 1.2762 1.2652 1.2672
−5 1.2712 1.2666 1.2721 1.2743 1.2628
Notes. Sharpe ratio for the rebalancing strategy when different risk functions are used in the
objective function with several values of parameter k.
Table 6. Out-of-sample portfolio mean return.
kVaR CVaR VaR Deviation CVaR Deviation Standard Deviation
−1 0.2508 0.2556 0.2445 0.2567 0.2425
−3 0.2645 0.2575 0.2631 0.2598 0.2576
−5 0.2663 0.2612 0.2662 0.2617 0.2532
Notes. Portfolio mean return for the rebalancing strategy when different risk functions are used
in the objective function with several values of parameter k.
minimization, whereas standard deviation minimization gives inferior results. Overall, we
rebalanced the portfolios 37 times for each objective function. Results depend on the scenario
data set and on the parameter k; thus, we cannot conclude that minimization of a certain
risk function is always the best choice. However, we observe that in the presence of mean
reversion, the tails of historical distribution are not good predictors of the tails in the
future. In this case, VaR disregarding the tails may lead to a good out-of-sample portfolio
performance. In fact, VaR disregards the unstable part of the distribution.
Appendix I
Table A.1. PSG functions.
Function group Full name PSG version
Deterministic Functions
Linear group Variable Function 1.1
Linear Function 1.1
Linear Multiple 1.1
Nonlinear group Polynomial Absolute 1.1
Relative Entropy 1.1
Maximum Positive 1.2 Beta
Maximum Negative 1.2 Beta
CVaR Positive 1.2 Beta
CVaR Negative 1.2 Beta
VaR Positive 1.2 Beta
VaR Negative 1.2 Beta
Cardinality group Cardinality Positive 1.2 Beta
Cardinality Negative 1.2 Beta
Cardinality 1.2 Beta
Buyin Positive 1.2 Beta
Buyin Negative 1.2 Beta
Buyin 1.2 Beta
Fixed Charge Positive 1.2 Beta
Fixed Charge Negative 1.2 Beta
Fixed Charge 1.2 Beta
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292 Tutorials in Operations Research, c
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Table A.1. Continued.
Function group Full name PSG version
Risk functions
Average group Average Gain 1.1
Average Loss 1.1
VaR group VaR Deviation for Gain 1.1
VaR Deviation for Loss 1.1
VaR Risk for Gain 1.1
VaR Risk for Loss 1.1
CVaR group CVaR Ceviation for Gain 1.1
CVaR Deviation for Loss 1.1
CVaR Risk for Gain 1.1
CVaR Risk for Loss 1.1
CDaR group CDaR Deviation for Gain 1.1
CDaR Deviation for Gain Multiple 1.1
CDaR Deviation 1.1
CDaR Deviation Multiple 1.1
Drawdown Deviation Average for Gain 1.1
Drawdown Deviation Average for Gain Multiple 1.1
Drawdown Deviation Average 1.1
Drawdown Deviation Average Multiple 1.1
Drawdown Deviation Maximum for Gain 1.1
Drawdown Deviation Maximum for Gain Multiple 1.1
Drawdown Deviation Maximum 1.1
Drawdown Deviation maximum Multiple 1.1
Maximum group Maximum Deviation for Gain 1.1
Maximum Deviation for Loss 1.1
Maximum Risk for Gain 1.1
Maximum Risk for Loss 1.1
Mean Abs group Mean Absolute Deviation 1.1
Mean Absolute Penalty 1.1
Mean Absolute Risk for Gain 1.1
Mean Absolute Risk for Loss 1.1
Partial moment group Partial Moment Gain Deviation 1.1
Partial Moment Loss Deviation 1.1
Partial Moment Penalty for Gain 1.1
Partial Moment Penalty for Loss 1.1
Probability group Probability Exceeding Deviation for Gain 1.1
Probability Exceeding Deviation for Gain Multiple 1.1
Probability Exceeding Deviation for Loss 1.1
Probability Exceeding Deviation for Loss Multiple 1.1
Probability Exceeding Penalty for Gain 1.1
Probability Exceeding Penalty for Gain Multiple 1.1
Probability Exceeding Penalty for Loss 1.1
Probability Exceeding Penalty for Loss Multiple 1.1
Standard group Standard Deviation 1.1
Standard Gain 1.1
Standard Penalty 1.1
Standard Risk 1.1
Mean Square Penalty 1.1
Variance 1.1
Utility group Exponential Utility 1.2 Beta
Logarithmic Utility 1.2 Beta
Power Utility 1.2 Beta
Sarykalin et al.: VaR vs.CVaR in Risk Management and Optimization
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