Conditional Value-at-Risk (CVaR) Norm: Stochastic Case
Alexander Mafusalov, Stan Uryasev
RESEARCH REPORT 2013-5
Risk Management and Financial Engineering Lab
Department of Industrial and Systems Engineering
303 Weil Hall, University of Florida, Gainesville, FL 32611.
E-mails:
[email protected], [email protected].
First draft: April 2013, This draft: August 2014
Correspondence should be addressed to: Stan Uryasev
Abstract
The concept of Conditional Value-at-Risk (CVaR) is used in various applications in uncertain environment. This paper introduces CVaR norm for a random variable, which is
by denition CVaR of absolute value of this random variable. It is proved that CVaR
norm is indeed a norm in the space of random variables. CVaR norm is dened in two
variations: scaled and non-scaled. L-1 and L-innity norms are limiting cases of the CVaR
norm. In continuous case, scaled CVaR norm is a conditional expectation of the random
variable. A similar representation of CVaR norm is valid for discrete random variables.
Several properties for scaled and non-scaled CVaR norm, as a function of condence level,
were proved. Dual norm for CVaR norm is proved to be the maximum of L-1 and scaled
L-innity norms. CVaR norm, as a Measure of Error, generates a Regular Risk Quadrangle. Negative CVaR function, which is a non convex extension for CVaR norm, is
introduced analogously to function L-p for p < 1. Linear regression problems were solved
by minimizing CVaR norm of regression residuals.
Keywords: CVaR norm, Lp norm, Conditional Value-at-Risk, CVaR
1. Introduction
The concept of Conditional Value-at-Risk (CVaR) is widely used in risk management
and various applications in uncertain environment. This paper introduces a concept of
CVaR norm in the space of random variables. CVaR norm in Rn was introduced and
developed in [7], [5]. This section provides a short introduction in the CVaR norm in Rn
and shows the relation with the CVaR norm in the space of random variables. Also, we
consider special cases of CVaR norm for random variables with discrete and continuous
distributions and provide some examples. The negative CVaR function is dened, both
in Rn and in the space of random variables, which is an extension of CVaR norm (but it
is not actually a norm).
Section 2.1 gives a formal denition of CVaR norm in stochastic case and proves
that CVaR norm is indeed a norm. This section also shows an equivalence of denitions
for special cases given in the introductory section and the general denition. CVaR norm
is a parametric family of norms with respect to the condence parameter α. Section 2.2
proves properties of CVaR norm as a function of α. Section 2.3 denes the dual norm to
the CVaR norm and proves several basic statements about normed space generated by the
CVaR norm (Banach and reexive space). Section 2.4 gives a short introduction to the
concept of Risk Quadrangle (see [9]). We dene the quadrangle generated by the CVaR
norm as a measure of error and we prove that this quadrangle is regular. Section 3 denes
the negative CVaR function and proves several basic properties. Section 4 illustrates
properties of CVaR norm with a case study.
The concept of this paper is motivated by applications of norms in optimization. We
consider norms in Rn and in the space of random variables. We use symbols x and xi for
a vector and an i-th vector component in Rn , i.e. x = (x1 , . . . , xn ). We use symbol X for
a random variable. lp norms are broadly used in Rn , and Lp norms are considered in the
space of random variables. For p ∈ [1, ∞] the norms lp and Lp are dened as follows1 :
!1/p
n
1X
|xi |p
, Lp (X) = (E|X|p )1/p ,
lp (x) =
n i=1
where E is the expectation sign. The most popular cases are p = 1, 2, ∞, i.e.,
P
• l1 (x) = n1 ni=1 |xi |, L1 (X) = E|X|;
• l∞ (x) = maxi=1,...,n |xi |,
1/2
P
• l2 (x) = n1 ni=1 x2i
,
L∞ (X) = sup |X|;
L2 (X) = (EX 2 )1/2 ;
It is known that l1 (x) ≤ l2 (x) ≤ l∞ (x) and L1 (X) ≤ L2 (X) ≤ L∞ (X), see [2]. Also, the
following inequalities holds: lp (x) ≤ lq (x) and Lp (X) ≤ Lq (X) for p < q , see [2].
1
Note that the classic denition for lp norm is lp (x) = ( ni=1 |xi |p )1/p , it does not satisfy inequality lp (x) ≤ lq (x) for p < q . This paper uses an equivalent scaled version of this norm
1/p
Pn
lp (x) = n1 i=1 |xi |p
, which satises that inequality. Lp norm is commonly dened as Lp (f ) =
1/p
R
p
kf kp ≡ S |f | dµ
, where S is a considered space. It is known (see, e.g., [2]) that for k · kp and k · kq
1
1
norms inequality kf kp ≤ µ(S) p − q kf kq holds for 1 ≤ p ≤ q ≤ ∞, where S is a considered space and µ(S)
is the measure of the space S . When S is a probability space, µ(S) = 1 and inequality Lp (X) ≤ Lq (X)
holds for 1 ≤ p ≤ q ≤ ∞, where Lp (X) = (E|X|p )1/p and E is an expectation sign.
P
2
Further we will illustrate the general concept of CVaR norm in Rn , as well as in
spaces of discrete and continuous random variables.
CVaR norm in Rn is considered in [7] and [5]. According to [7], the CVaR norm
in x ∈ Rn is dened as follows. Let |x|(i) be ordered absolute values of components of
x ∈ Rn , i.e.
{|x1 |, . . . , |xn |} = {|x|(1) , . . . , |x|(n) },
and
|x|(i) ≤ |x|(i+1) , for i = 1, . . . , n − 1.
Then, for j = 0, . . . , n − 1 and αj = j/n, scaled CVaR norm (or just CVaR norm in this
paper) is dened by
hhxiiSαj = (|x|(j+1) + . . . + |x|(n) )/(n − j).
For j = n we have αj = j/n = 1 and the norm is hhxiiSαn = |x|(i) = maxi |xi |. For
αj < α < αj+1 , the norm hhxiiSα equals to the weighted sum,
hhxiiSα = µhhxiiSαj + (1 − µ)hhxiiSαj+1 ,
where
µ=
(αj+1 − α)(1 − αj )
.
(αj+1 − αj )(1 − α)
A similar norm, called D-norm, was introduced in [3] in a dierent way: for p ∈ [1, n],
M = {1, . . . , n}, |S| is cardinality of a set S , bpc = max{l ∈ Z|l ≤ p} (i.e., bpc is a maximal
integer number which is not greater than p)
)
(
X
|||x|||p =
max
|xj | + (p − bpc)|xt | .
{S∪{t}:S⊂M,|S|≤bpc,t∈M \S}
j∈S
, the CVaR norm coinsides with the D-norm |||x|||p with parameter p
For α ∈ 0, n−1
n
dened by p = n(1 − α), see [7].
The second variant of the norm, called non-scaled CVaR norm, is dened in [7] as
follows2 :
(
(1 − α)hhxiiSα , for 0 ≤ α < 1;
hhxiiα =
0,
for α = 1.
For example, hhxiij/n = (|x|(j+1) + . . . + |x|(n) )/n. This paper shows that the hh·iiα norm
in Rn can be considered as a special case of CVaR norm in the space of discrete random
variables, which is dened in the following paragraph.
Now, we consider CVaR norm for discretely distributed random variables.
PN Suppose
N
N
that a random variable X takes values {xi }i=1 with probabilities {pi }i=1 , and i=1 pi = 1,
where xi ∈ R and N ∈ N or N = ∞ (we use the notation N for the set of natural numbers).
N
Let us denote by the sequence {|x|(i) }N
i=1 an ordered sequence {|xi |}i=1 , i.e. |x|(i) ≤ |x|(i+1) .
N
Let us denote for discretely distributed X by {|p|(i) }i=1 a corresponding to the {|x|(i) }N
i=1
sequence of values from the {pi }N
,
i.e.
if
|x|
corresponds
to
|x
|
,
then
|p|
=
p
.
Let
j
j
(i)
(i)
i=1
2 In
[7] it is dened as hhxiiα = n(1 − α)hhxiiSα , but in this paper we will stick to slightly dierent
denition in the sake of consistency with stochastic case, where hhXiiα = (1 − α)hhXiiSα .
3
P
us dene αj = ji=1 |p|(i) . The non-scaled CVaR norm with condence level αj for the
discretely distributed random variable X is dened by the following expression:
(P
N
i=j+1 |x|(i) |p|(i) , for j = 0, . . . , N − 1;
hhXiiαj = (1 − αj )CVaRαj (|X|) =
0,
for j = N,
here CV aRα (|X|) denotes conditional value at risk for random variable |X|, see [8]. If
N = ∞, then
∞
X
|x|(i) |p|(i) , for j ∈ N.
hhXiiαj =
i=j+1
Similarly to the denition of non-scaled CVaR norm in Rn , for αj < α < αj+1 the
hhXiiα equals to the weighted sum
hhXiiα = (1 − λ)hhXiiαj + λhhXiiαj+1 ,
where λ = (α − αj )/(αj+1 − αj ).
If N = n, pi = 1/n and x = (x1 , . . . , xn ) ∈ Rn , then the CVaR norm of X coincides
with the deterministic CVaR norm of x: hhXiiα = hhxiiα .
Let us illustrate the denition of non-scaled CVaR norm in stochastic case with the
following example.
Example 1.
Consider the random variable X taking the values xi = 1 − 2−i with the probabilities
pi = 2−i for i ∈ N. Since xi > 0 and xi < xi+1 , then xi = |x|(i) . For α = αj ,
hhXiiαj =
∞
X
−i
(1 − 2 )2
i=j+1
−i
=
∞
X
2−i − 4−i =
i=j+1
2−(j+1)
4−(j+1)
−
= 2−j − 4−j /3.
1 − 2−1 1 − 4−1
For j → ∞, there is convergence αj → 1 and
hhXiiαj → 0 = (1 − 1)CV aR1 (|X|).
For α = 0.2, which is between α0 = 0 and α1 = 0.5, the value hhXii0.2 is a weighted sum
of hhXii0 and hhXii0.5 with coecient λ = (α − α0 )/(α1 − α0 ) = 0.2/0.5 = 0.4. We have
hhXii0 = 1 − 1/3 = 2/3, hhXii0.5 = 0.5 − 0.25/3 = 1.25/3,
therefore,
hhXii0.2 = (1 − λ)hhXii0 + λ · 0.5 · hhXii0.5 = 0.6 · 2/3 + 0.4 · 1.25/3 = 1.7/3 ≈ 0.567.
Figure 1 shows a plot of hhXiiα depending upon α. Notice that hhXiiα is a non-increasing,
concave and piecewise-linear function w.r.t. α. Paper [7] showed these properties for
CVaR norm in Rn . Section 2.2 proves these properties of hhXiiα in stochastic case.
4
non−scaled CVaR norm 〈〈X〉〉α
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
α
0.6
0.7
0.8
0.9
1
Figure 1: Non-scaled CVaR norm hhXiiα of random variable X with atoms xi = 1 − 2−i and probabilities
pi = 2−i as a function of α.
For discrete random variable X , the scaled CVaR norm is dened as follows:
(
1
hhXiiα , for α < 1;
hhXiiSα = 1−α
sup |X|,
for α = 1,
where sup |X| denotes the essential supremum3 of the random variable. If N = n, pi = 1/n
and x = (x1 , . . . , xn ) ∈ Rn , then hhXiiSα = hhxiiSα .
Example 2.
Consider the random variable X taking the values xi = 1 − 2−i with the probabilities
pi = 2−i for i ∈ N. For α = αj , values of the hhXiiSα norm are:
hhXiiSαj =
2−j − 4−j /3
2−j − 4−j /3
=
= 1 − 2−j /3.
Pj
−j
−i
2
1 − i=1 2
For j → ∞, there is convergence value αj → 1 and CVaR norm hhXiiSαj → 1 = hhXiiS1 =
sup |X|.
For α = 0.2,
hhXiiS0.2 = (1.2/3 + 0.5/3)/0.8 ≈ 0.708.
Figure 2 shows a plot of hhXiiSα depending upon α. Notice that the norm is an increasing
continuous function w.r.t. α. Paper [7] showed these properties for CVaR norm in Rn .
Section 2.2 proves these properties of hhXiiSα for stochastic case.
3 By
denition, ess sup X = inf{a ∈ R|P (X > a) = 0}.
5
α
scaled CVaR norm 〈〈X〉〉S
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0
0.1
0.2
0.3
0.4
0.5
α
0.6
0.7
0.8
0.9
1
Figure 2: Scaled CVaR norm hhXiiα of random variable X with atoms xi = 1 − 2−i and probabilities
pi = 2−i as a function of α.
Furthermore, we consider the space of random variables with continuous distribution
functions. This is an important special case.
Let FX (x) be a continuous cumulative distribution function of a random variable X ,
and FX−1 (α) be an inverse function to FX (x) (i.e., FX−1 (FX (x)) = x), and qα (X) be an
α-quantile of the random variable X (i.e., P (X ≤ qα (X)) = α, qα (X) = FX−1 (α)).
CVaR(X), in this case, is a conditional
R 1 expectation: CV aRα (X) = E(X|X >
−1
1
FX (α)), or, equivalently, CVaRα (X) = 1−α α qp (X)dp, see [8].
In continuous case, the CVaR norm is dened as follows: hhXiiSα = CVaRα (|X|). We
prove in Section 2.1 that CVaR norm is indeed a norm. Also, we show that hhXiiS0 = L1 (X)
and hhXiiS1 = L∞ (X).
We use the sign ∝ as a notation for words ¾distributed by¿. For example, X ∝
N (0, 1) means that the random variable X is normally distributed with mean µ = 0 and
variance σ 2 = 1.
Let us illustrate the denition of CVaR norm in the space of continuous random
variables with the following example.
Example 3.
Consider an exponentially distributed random variable X ∝ Exp(λ) with the probability density function
fX (x) = λe−λx , x ≥ 0; 0, x < 0 ,
and with cumulative distribution function
FX (x) = 1 − e−λx , x ≥ 0; 0, x < 0 .
6
(1)
Since X is a non negative random variable, then X = |X|. The expression (1) for FX (x)
implies the following equation for quantile qα (X):
1 − e−λqα (X) = α.
Consequently,
−λqα (X) = ln(1 − α),
1
qα (X) = − ln(1 − α).
λ
For α = 1, the quantile q1 (X) = ∞. Then, for α ∈ [0, 1]
Z ∞
Z ∞
1
1
−λx
S
−λx
−λx ∞
λe dx =
hhXiiα =
xλe dx =
−xe
+
qα (X)
1 − α qα (X)
1−α
qα (X)
"
∞ #
1
1
1
1 −λx =
=
qα (X)(1 − α) + (1 − α) =
qα (X)(1 − α) + − e
1−α
λ
1−α
λ
qα (X)
=
1
[1 − ln(1 − α)].
λ
For α = 0,
hhXiiS0 =
1
= EX = E|X| = L1 (X).
λ
For α = 1,
hhXiiS1 = ∞ = sup X = sup |X| = L∞ (X).
Figure 3 shows a plot of the probability density function fX (x) of X and Figure 4 shows
a plot of the function hhXiiSα as a function of α.
Similar to the discrete distribution case, we consider the non-scaled CVaR norm :
hhXiiα = (1 − α)hhXiiSα .
−1
CVaRα (|X|)). It is easy to see that hhXiiα = E(I(|X| > F|X|
(α))|X|), where I(A) is an
indicator function:
(
1, if A is true;
I(A) =
0, if A is f alse.
Section 2.2 shows that the non-scaled CVaR norm is a decreasing concave function of α
and changes from L1 (X) to 0 when α changes from 0 to 1.
Let us illustrate the denition of non-scaled CVaR norm hhXiiα in the space of continuous random variables with the following example.
Example 4.
Consider an exponentially distributed random variable X ∝ Exp(λ). For α ∈ [0, 1),
the non-scaled CVaR norm equals
hhXiiα =
1−α
[1 − ln(1 − α)].
λ
7
probability density function fX(x)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
x
Figure 3: Probability density function fX (x) for X ∝ Exp(1).
non−scaled CVaR norm 〈〈X〉〉α
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
α
0.6
0.7
0.8
0.9
Figure 4: Non-scaled CVaR norm hhXiiα for X ∝ Exp(1), as a function of α.
8
1
scaled CVaR norm 〈〈X〉〉S
α
8
7
6
5
4
3
2
1
0
0.1
0.2
0.3
0.4
0.5
α
0.6
0.7
0.8
0.9
Figure 5: Scaled CVaR norm hhXiiSα for X ∝ Exp(1), as a function of α.
For α = 0,
hhXiiα =
1
= EX = E|X| = L1 (X).
λ
For α = 1,
hhXiiα = 0.
Figure 5 shows a plot of the function hhXiiα .
9
1
Risk Quadrangle denes risk R(X), deviation D(X), regret V(X), error E(X) and
statistic S(X), satisfying some axioms, see [9]. Considered functionals can be regular or
non-regular. further work that if R(X) is a regular Measure of Risk, than R(|X|) is a
norm and a regular Measure of Error. This paper proves that hhXiiα is a regular Measure
of Error and nds the corresponding functions R(X), D(X), V(X) and S(X) in quadrangle generated by the Measure of Error E(X) = hhXiiα (see Section 2.4).
This paper denes also non convex functions closely related to CVaR norm. In
deterministic case, by denition, CVaR norm is the average of the biggest by absolute
value (1 − α)n components of a vector. The negative CVaR function is dened as an
average of the smallest by absolute value αn components of a vector. We dene nonscaled version of negative CVaR function as the dierence of l1 and hh·iiα norms:
rα− (x) = l1 (x) − hhxiiα .
From the denition of hhxiiα follows:
rα−j (x) = (|x|(1) + . . . + |x|(j) )/n, for α = αj = j/n,
r0− (x) = 0, for α = 0,
r1− (x) = l1 (x), for α = 1.
We also dene scaled version of negative CVaR function as follows
rα−,S (x) =
1
(l1 (x) − (1 − α)hhxiiSα ).
α
From the denition of hhxiiSα follows:
rα−j (x) = (|x|(1) + . . . + |x|(j) )/j, for α = αj = j/n,
r0− (x) = min |xi |, for α = 0,
i
r1− (x)
= l1 (x), for α = 1.
A general denition of the negative CVaR function, both in deterministic and stochastic cases, is considered in Section 3.
Figure 6 shows level-sets of hhxiiSα and rα−,S (x) in R2 for dierent values of α. The
S
function rα−,S is a natural extension of hh·iiP
α . When α variates from 0 to 1, the function
1
−,S
rα (x) changes from mini |xi | to l1 (x) = n ni=1 |xi |, and the function hhxiiSα changes from
l1 (x) to maxi |xi |.
10
3
〈〈x〉〉S
= max(|x1|, |x2|)=1
0.5
〈〈x〉〉S
=1
0.25
2.5
〈〈x〉〉S
= l1(x) = r−,S
(x)=1
0
1
r−,S
(x)=1
0.75
2
r−,S
(x) = min(|x1|, |x2|)=1
0.5
1.5
1
x2
0.5
0
−0.5
−1
−1.5
−2
−2.5
−3
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
x1
Figure 6: Level-sets of scaled CVaR norm hhxiiSα for α = 0, 0.25, 0.5 and level-sets of scaled negative CVaR
function rα−,S (x) for α = 0.5, 0.75, 1 in R2 space. For α ∈ [0.5, 1] norm hhxiiSα = maxi |xi |. For α ∈ [0, 0.5]
function rα−,S (x) = mini |xi |. Equality hhxiiS0 = l1 (x) = r1−,S (x) holds.
11
2. CVaR Norm in Stochastic Case
This section gives a formal denition of CVaR norm in stochastic case and proves
various properties of the norm.
2.1. CVaR Norm Denition and Properties
Let us denote
[x]+ = max{0, x}, [x]− = max{0, −x}.
Consider cumulative distribution function FX (x) = P (X ≤ x). If, for a probability
level α ∈ (0, 1), there is a unique x such that FX (x) = α, then this x is called the αquantile qα (X). In general, however, the value x is not unique, or may not even exist any.
There are two values to consider as extremes:
qα− (X) = sup{x|FX (x) < α}.
qα+ (X) = inf{x|FX (x) > α},
We will call by the quantile the entire interval between the two extreme values,
qα (X) = [qα− (X), qα+ (X)].
(2)
R
R −
R +
We
will
use
notation
q
(X)dp
≡
q
(X)dp
,
which
is
a
reasonable
since
qp (X)dp =
p
p
R −
qp (X)dp.
Further we provide two general denitions of CVaR norm, following from the two
equivalent general denitions of CVaR (see [8]). We also show that denitions for the
discrete case and continuous case, made in introduction, are special cases of these general
denitions.
Denition 1. Let X be a random variable with E|X| < ∞. Then, CVaR norm of X
with parameter α ∈ [0, 1) is dened as follows:
hhXiiSα
= min c +
c
1
+
E[|X| − c]
.
1−α
If also X is essentially nite (i.e., exists C ∈ R : |X| < C ), then for α = 1
hhXiiS1 = sup |X|.
Denition 2. Let X be a random variable with E|X| < ∞. Then CVaR norm of X with
parameter α ∈ [0, 1) is dened as follows:
hhXiiSα
1
=
1−α
Z
1
qp (|X|)dp.
α
If also X is essentially nite (i.e., exists C ∈ R : |X| < C ), then for α = 1
hhXiiS1 = sup |X|.
It immediately follows from the denitions of CVaR (see [8]) that the Denitions 1
and 2 are equivalent.
Proposition 1. Let X be a continuous random variable, i.e., its cumulative distribution
function is continuous. Then hhXiiSα = E (|X| ||X| > qα (|X|)) .
12
Proof. If cumulative distribution function FX of random variable X is continuous, then
−1
F|X| is continuous, qp (|X|) = F|X|
(p) and
R1
hhXiiSα =
α
−1
F|X|
(p)dp
1−α
R∞
=
xdF|X| (x)
−1
= E |X| |X| > F|X|
(α) .
−1
P |X| > F|X|
(α)
−1
F|X|
(α)
Let X be a discrete random variable, i.e., it takes values {xi }N
i=1 with positive probN
abilities {pi }i=1 (N also can be ∞).
N
Let us denote by the sequence {|x|(i) }N
i=1 an ordered sequence {|xi |}i=1 , i.e., |x|(i) ≤
|x|(i+1) . {|xi |}N
i=1 exists, such that if |x|(i) ↔ |xj |, then |x|(i) = |xj |. We also denote by
N
{|p|(i) }N
a
corresponding
to the {|x|(i) }N
i=1
i=1 sequence of probabilities from the {pi }i=1 .
Note that ordered sequence {|x|(i) } exists only for special sets of {xi }. In particular,
it exists in following cases: set {xi } is nite; sequence {xi }∞
i=1 has no converging subsequences; all converging subsequences of sequence {|xi |}∞
converge
to x̄ = sup{|xi |}. If
i=1
ordered sequence {|x|(i) } exists, then the following proposition holds.
Proposition 2. Let X be a discreteP
random variable, i.e., it takes values {xi }N
i=1 , with
N
N
positive probabilities {pi }i=1 , where i=1 pi = 1 and N ∈ N ∪ ∞. exists, such that
if |x|(i) ↔ |xj |, then |x|(i) = |xj |. Let us denote by {|p|(i) }N
i=1 a corresponding to the
N
N
{|x|(i) }i=1 sequence of values from the {pi }i=1 , i.e. if |x|(i) ↔ |xj |, then |p|(i) = |pj |. Then
• for N < ∞,
hhXiiS1 = |x|(N ) ,
for N = ∞,
hhXiiS1 = lim |x|(i) .
i→∞
• for N < ∞ and j = 0, . . . , N − 1
hhXiiSαj =
N
X
1
|x|(i) |p|(i) ,
1 − αj i=j+1
where αj = ji=1 |p|(i) ,
for N = ∞ and j ∈ Z+ ≡ N ∪ {0},
P
hhXiiSαj
∞
X
1
=
|x|(i) |p|(i) ,
1 − αj i=j+1
• for αj < α < αj+1 ,
hhXiiSα = (1 − λ)
1 − αj
1 − αj+1
hhXiiSαj + λ
hhXiiSαj+1 ,
1−α
1−α
where λ = (α − αj )/(αj+1 − αj ).
Proof. Let us proceed with the proof bullet by bullet.
13
• It follows directly from the denition that CVaR1 (|X|) = sup{|xi |}N
i=1 .
Since |x|(i) < |x|(i+1) , then in case N < ∞,
hhXiiS1 = sup{|xi |}N
i=1 = max |x|(i) = |x|(N ) .
i
Consider the case N = ∞. Since {|x|(i) }∞
i=1 is a non decreasing sequence, then it
has a nite or an innite limit limi→∞ |x|(i) . Since also all probabilities |p|(i) > 0,
then
hhXiiS1 = sup{|x|(i) }∞
i=1 = lim |x|(i) .
i→∞
• If X is a discrete random variable, then F|X| (x) is a step function. Then, qp (|X|) is
a step function of p. Length of a step number i is αi − αi−1 = |p|(i) . Height of the
step number i is qαi (|X|) = |x|(i) . It implies that the area under the step equals to
the following integral:
Z
αi
qp (|X|)dp = |p|(i) |x|(i) .
αi−1
Then,
hhXiiSαj
1
=
1 − αj
Z
1
qp (|X|)dp =
αj
N Z αi
N
X
X
1
1
qp (|X|)dp =
=
|p|(i) |x|(i) .
1 − αj i=j+1 αi−1
1 − αj i=j+1
• Since qp (|X|) is a step function of p, then it is a constant function of α on the
R1
interval (αj , αj+1 ). Then, the integral α qp (|X|)dp is a linear function of α for
αj < α < αj+1 . It implies that this integral is the linear combination
Z
1
Z
1
qp (|X|)dp = (1 − λ)
α
Z
1
qp (|X|)dp,
qp (|X|)dp + λ
αj
αj+1
where λ = (α − αj )/(αj+1 − αj ). Then,
hhXiiSα = (1 − λ)
1 − αj
1 − αj+1
hhXiiSαj + λ
hhXiiSαj+1 .
1−α
1−α
Denition 2 immediately implies that hhXiiS0 = L1 (X), hhXiiS1 = L∞ (X). Furthermore, we prove that hhXiiSα is a norm for α ∈ (0, 1).
Proposition 3. Let X be a random variable. hhXiiSα is a norm in the space of random
variables.
Proof. By denition, function d(X) is a norm if
1. d(X) = 0 ⇔ X ≡ 0,
2. d(λX) = |λ|d(X),
3. d(X + Y ) ≤ d(X) + d(Y ).
14
Since CVaRα (X) is a regular Measure of Risk (see [9]), then for λ > 0
CVaRα (λX) = λCVaRα (X),
and
CVaRα (X + Y ) ≤ CVaRα (X) + CVaRα (Y ).
(3)
By denition, f (X) is monotonic if X ≤ Y implies f (X) ≤ f (Y ). Since CVaRα (X) is
monotonic (see [9]), then for X ≥ Y
CVaRα (X) ≥ CVaRα (Y ).
(4)
Let us prove the axioms of a norm for hhXiiSα = CVaRα (|X|).
1. CVaRα (|X|) = 0 ⇔ X ≡ 0.
The statement X ≡ 0 ⇒ CVaRα (|X|) = 0 is obvious.
If X 6= 0, then qp (|X|) > 0 for p ∈ (0, 1), and
1
CVaRα (|X|) =
1−α
Z
1
qp (|X|)dp > 0.
α
If α = 0 and X 6= 0, then CVaR0 (|X|) = E|X| > 0.
If α = 1 and X 6= 0, then CVaR1 (|X|) = sup(|X|) > 0.
2. CVaRα (|λX|) = |λ|CVaRα (|X|). Since |λ| > 0, then
CVaRα (|λX|) = CVaRα (|λ||X|) = |λ|CVaRα (|X|).
3. CVaRα (|X + Y |) ≤ CVaRα (|X|) + CVaRα (|Y |).
Since |X + Y | ≤ |X| + |Y |, using (4) we have
CVaRα (|X + Y |) ≤ CVaRα (|X| + |Y |).
Finally,
CVaRα (|X| + |Y |) ≤ CVaRα (|X|) + CVaRα (|Y |),
which follows from (3).
The next proposition provides an alternative way to calculate hhXiiSα .
Proposition 4. Let X be a random variable. Let Y be a random variable dened as
follows
(
X,
with probability 12 ,
Y =
−X, with probability 12 .
Then,
hhXiiSα = CVaR(1+α)/2 (Y ).
15
Proof. Minimization form denition of CVaR, see [8] or Denition 1, implies that
1
+
CVaR(1+α)/2 (Y ) = min c +
,
E[Y − c]
c
(1 − α)/2
(5)
since 1/(1 − (1 + α)/2) = 1/((1 − α)/2). Dene
1
+
.
cY = arg min c +
E[Y − c]
c
(1 − α)/2
Notice that Y is symmetric, therefore 0 ∈ qY (1/2). Notice also and (1 + α)/2 ≥ 1/2, since
α ≥ 0. Since optimal solution in CVaR denition (5) is the quantile q(1+α)/2 (Y ), then
cY ≥ 0. Then,
1
+
CVaR(1+α)/2 (Y ) = min c +
E[Y − c]
=
(6)
c
(1 − α)/2
1
= cY +
E[Y − cY ]+ =
(7)
(1 − α)/2
1
+
= min c +
E[Y − c]
=
(8)
c≥0
(1 − α)/2
1
+
+
= min c +
E[Y − c]
,
(9)
c≥0
(1 − α)/2
where equality between (6) and (7) follows from denition of cY ; equality between (7)
and (8) follows from cY ≥ 0; equality between (8) and (9) follows from [Y −c]+ = [Y + −c]+
for c ≥ 0. Note that
(
|X|, with probability 12 ,
+
Y =
0,
with probability 12 .
Therefore,
1
1
+
CVaR(1+α)/2 (Y ) = min c +
E[|X| − c]
=
c≥0
(1 − α)/2 2
1
+
= min c +
E[|X| − c]
= hhXiiSα .
c≥0
1−α
Last equation in (10) follows from CVaR norm Denition 1 and from
1
+
arg min c +
E[|X| − c]
≥ 0,
c
1−α
which holds since arg min is a quantile qα (|X|) and |X| ≥ 0, therefore qα (|X|) ≥ 0.
2.2. CVaR Norm Properties With Respect to α
Let us remind some general properties of integrals of quantile.
Proposition 5. Let X be a random variable. For 0 ≤ α ≤ 1,
•
1
1−α
R1
α
qp (X)dp is a continuous increasing function of α,
16
(10)
R1
1
• limα→1 1−α
q (X)dp = sup X ,
α p
Rα
• α1 0 qp (X)dp is a continuous non-decreasing function of α,
Rα
• 0 qp (X)dp is a convex function of α,
Rα
• 0 qp (X)dp is a piecewise-linear function of α for discretely distributed X .
Proof. We provide references for these statements bullet by bullet.
• CVaRα is a continuous increasing
function of α, see [8]. Then, integral form deniR1
R1
1
1
tion of CVaRα (X) = 1−α α qp (X)dp implies that the integral 1−α
q (X)dp is a
α p
continuous increasing function of α.
R1
1
q (X)dp = sup X .
• limα→1 CVaRα (X) = sup X , see [8]. Therefore, limα→1 1−α
α p
• Notice that qp (−X) = −q1−p (X), then
Z
Z
Z
1 1
1 1
1 α
qp (−X)dp = −
q1−p (X)dp = −
qp (X)dp.
α 1−α
α 1−α
α 0
R1
The rst bullet of this proposition implies that α1 1−α qp (−X)dp is a decreasing
Rα
continuous function of α, therefore, integral α1 0 qp (X)dp is a continuous nondecreasing function of α.
Rα
• Integral 0 qp (X)dp is called the Absolute Lorenz Curve, which is known to be convex
w.r.t. α, e.g. [6].
• Consider X having an atom x with probability
R αp. If α1 = FX (x) and α2 = α1 + p,
then qRα (X) = x for α ∈ (α1 , α2 ). Therefore, 0 qp (X)dp is linear on α R∈ (α1 , α2 ).
α
α
Since 0 qp (X)dp is also continuous, and X is dicretely distributed, then 0 qp (X)dp
is a piecewise-linear function of α.
Let X be a random variable. Take constant C , such that L1 (X) ≤ C ≤ L∞ (X).
The following corollary assures that there exists a single α such that CVaRα (|X|) = C .
Therefore, for any p there is a single α such that CVaRα (|X|) = Lp (X).
Corollary 1. Let X be a random variable. The norm hhXiiSα is a continuous increasing
function of α.
Proof. Consider Y = |X| and apply Proposition 5 to Y .
The following proposition establishes properties of non-scaled CVaR norm with respect to parameter α.
Denition 3. Let X be a random variable with E|X| < ∞. Then, CVaR norm of X
with parameter α ∈ [0, 1] is dened as follows:
(
(1 − α)hhXiiSα , for α ∈ [0, 1),
hhXiiα =
0,
for α = 1.
17
Corollary 2. Let X be a random variable. The norm hhXiiα is a concave and decreasing
function of α. Furthermore, if X is discretely distributed, hhXiiα is a piecewise-linear
function of α.
Proof. By denition,
Z
1
hhXiiα = (1 − α)CVaRα (|X|) =
qp (|X|)dp.
α
If α1 < α2 , then
Z
1
hhXiiα1 =
Z
1
qp (|X|)dp ≥
qp (|X|)dp = hhXiiα2 .
α1
α2
Therefore, hhXiiα is a decreasing function of α.
To prove that RhhXiiα is a concave function of α, consider Y = |X|
R αand apply Proposition 5
α
to Y . Since 0 qp (|X|)dp is convex, then hhXiiα = E|X| − 0 qp (|X|)dp is a concave
function of α.
Rα
For piecewise-linearity, take Y = |X|, note that hhXiiα = EY − 0 qp (Y )dp, and
apply Proposition 5.
Corollary 3. Let X be a random variable. (1 − α)CVaRα (X) is a concave function of
α. Furthermore, it is a piecewise-linear function of α if X is discretely distributed.
R
R
Proof. (1−α)CVaRα (X) = α1 qp (X)dp = EX − 0α qp (X)dp, therefore, it is concave w.r.t.
α, and it is piecewise-linear for discretely distributed X , see Proposition 5.
2.3. Dual Norm to CVaR Norm and CVaR Normed Space
Denition 4. Let X be a normed space over R with norm k·k (i.e., kXk ∈ R for X ∈ X).
Then, the dual (or conjugate) normed space X∗ is dened as the set of all continuous linear
functionals from X into R. For f ∈ X∗ , the dual norm k · k∗ of f is dened by
∗
kf k = sup{|f (x)| : x ∈ X, kxk ≤ 1} = sup
|f (x)|
: x ∈ X, x 6= 0 .
kxk
∗
The asterisk ∗ denotes the dual norm to a norm. Therefore, hhY iiSα denotes the norm
dual to the CVaR norm hhXiiSα .
Proposition 6. Let X be a random variable. The norm hhY iiSα ∗ = max{E|Y |, (1 −
α) sup |Y |} is dual to the norm hhXiiSα for α ∈ (0, 1).
Proof. Paper [9] proved that CVaRα (X) = supQ∈Q EXQ, where
Q = Q 0 ≤ Q ≤
1
, EQ = 1
1−α
.
Then, hhXiiSα = CVaRα (|X|) = supQ∈Q E|X|Q. Let us prove that
sup E|X|Q = sup EXY,
Y ∈Y
Q∈Q
18
1
where Y = Y |Y | ≤ 1−α
, E|Y | ≤ 1 .
First,
(I(X > 0) − I(X < 0))Q ∈ Y ⇒ sup E|X|Q ≤ sup EXY,
Y ∈Y
Q∈Q
since X(I(X > 0) − I(X < 0))Q = |X|Q.
Second,
sup EXY ≤ sup E|X||Y | ≤ sup E|X|Q.
Y ∈Y
Finally,
Y ∈Y
Q∈Q
sup EXY = sup E|X|Q = CV aRα (|X|) = hhXiiSα .
Y ∈Y
Q∈Q
Then, Y ∗ = {Y |EXY ≤ hhXiiSα } must be a convex hull of Y . The set Y is closed:
Yk → Y, |Yk | ≤
1
1
⇒ |Y | ≤
,
1−α
1−α
Yk → Y, E|Yk | ≤ 1 ⇒ E|Y | ≤ 1,
and convex:
Y1 , Y2 ∈ Y ⇒ |λY1 + (1 − λ)Y2 | ≤ λ|Y1 | + (1 − λ)|Y2 | ≤
1
(λ + (1 − λ)),
1−α
E|λY1 + (1 − λ)Y2 | ≤ λE|Y1 | + (1 − λ)E|Y2 | ≤ 1 · (λ + (1 − λ)).
Then, Y ∗ = Y and Y is a unit ball in the dual norm to the CVaR norm
EXY
,
S
X6=0 hhXiiα
∗
hhY iiSα = sup
∗
since hhY iiSα ≤ 1 ⇔ EXY ≤ hhXiiSα for all X . Then, the unit sphere in the dual norm is
the set
1
1
, E|Y | ≤ 1 ∪ Y sup |Y | ≤
, E|Y | = 1 .
Y sup |Y | =
1−α
1−α
∗
Then, the dual norm equals hhY iiSα = max{E|Y |, (1 − α) sup |Y |}.
Denition 5. A Banach space is a vector space X over R, which is equipped with a norm
k · k and which is complete with respect to that norm. By denition, completeness means
that for every Cauchy sequence {xn }∞
n=1 in X (i.e., for every ε > 0 exists N such that
kxm − xn k < ε for all m, n > N ), there exists an element x in X such that
lim xn = x, i .e.,
n→∞
lim kxn − xk = 0.
n→∞
The next statement follows directly from Denition 5 of a Banach space.
Proposition 7. Let L be a norm, XL = {X|L(X) < ∞} and space (XL , L) is a Banach
space. Let L̄ be a norm such that
exist such C− , C+ > 0 that C− L(X) ≤ L̄(X) ≤ C+ L(X)for all X.
Then, XL̄ = XL and (XL , L̄) is a Banach space.
19
Corollary 4. The norm hhXiiSα generates a Banach space for α ∈ [0, 1].
Proof.
1
E|X|,
1−α
for α < 1. E|X| = L1 (X) and it is known that L1 -norm generates a Banach space.
hhXiiS1 = sup |X| = L∞ (X) and it is known that L∞ -norm generates a Banach space.
E|X| ≤ hhXiiSα ≤
2.4. Risk Quadrangle With CVaR Norm (CVaR Norm Quadrangle)
Risk Quadrangle (see [9]) denes risk R(X), deviation D(X), regret V(X), error
E(X) and statistic S(X) related by the following equations:
V(X) = EX + E(X), R(X) = EX + D(X),
D(X) = min{E(X − C)}, R(X) = min{C + V(X − C)},
C
C
S(X) = arg min{E(X − C)} = arg min{C + V(X − C)}.
C
C
(11)
(12)
(13)
Measure of risk R(X) is regular if
• R(X) ∈ (−∞, ∞],
• R(X) is closed convex,
• R(C) = C for any constant C ,
• R(X) > EX for any nonconstant X .
Measure of deviation D(X) is regular if
• D(X) ∈ [0, ∞],
• D(X) is closed convex,
• D(C) = 0 for any constant C ,
• D(X) > 0 for any nonconstant X .
Measure of error E(X) is regular if
• E(X) ∈ [0, ∞],
• E(X) is closed convex,
• E(0) = 0,
• E(X) > 0 for any X 6= 0,
• for sequence of random variables {Xk }∞
k=1
lim E(Xk ) = 0 ⇒ lim EXk = 0,
k→∞
k→∞
which is equivalent to E(X) ≥ ψ(EX) with a convex function ψ on (−∞, ∞) having
ψ(0) = 0 but ψ(t) > 0 for t 6= 0.
20
Measure of regret V(X) is regular if
• V(X) ∈ (−∞, ∞],
• V(X) is closed convex,
• V(0) = 0,
• V(X) > 0 for any X 6= 0,
• for sequence of random variables {Xk }∞
k=1
lim [V(Xk ) − EXk ] = 0 ⇒ lim EXk = 0.
k→∞
k→∞
The quadrangle (R, D, E, V, S) is regular if axioms 11 13 are hold and if also R(X) is a
regular measure of risk, D(X) is a regular measure of deviation, V(X) is a regular measure
of regret, and E(X) is a regular measure of error.
Quadrangle Theorem (see [9]) implies that if axioms 11 13 are hold for functions R, D,
E , V , S , and if also E(X) is a regular measure of error, then (R, D, E, V, S) is a regular
quadrangle.
We will prove that hhXiiSα is a regular measure of error (it will imply that the quadrangle,
generated by CVaR norm as a measure of error is regular).
We also prove that if E(X) = hhXiiSα = CVaRα (|X|) and quadrangle axioms 11 13
hold, then the risk measure is R(X) = 1−α
CVaR(1+α)/2 (X) + 1+α
CVaR(1−α)/2 (X) and the
2
2
1
statistic is S(X) = 2 q(1−α)/2 (X) + q(1+α)/2 (X) .
Proposition 8. E(X) = hhXiiSα is a regular measure of error.
Proof. We further prove axioms of the regular measure of error.
• E(X) ∈ [0, ∞], follows from the fact that hhXiiSα is a norm.
• Let us prove that E(X) is closed and convex. hhXiiSα is a norm, therefore, it is a
convex function.
Closeness is equivalent to the following statement. Let us consider a sequence of
random variables {Xk }∞
k=1 and a random variable X such that expectations µ(Xk −
X) → 0 and variances σ(Xk − X) → 0 for k → ∞ and E(Xk ) ≤ C for all k . Then,
under these conditions, E(X) ≤ C .
Here is the proof of this statement.
E(Xk ) − E(X) ≤ E(Xk − X) ≤
=
1
1 p
E|Xk − X| ≤
E|Xk − X|2 =
1−α
1−α
1 p 2
σ (Xk − X) − µ2 (Xk − X) → 0,
1−α
(14)
for k → ∞. It implies that
E(X) − E(Xk ) ≤ E(X − Xk ) = E(Xk − X) → 0,
for k → ∞. Combining (14) and (15) we have
|E(X) − E(Xk )| → 0 ⇔ E(Xk ) → E(X) ⇒ E(X) ≤ C.
21
(15)
• E(0) = 0, follows from the fact that hhXiiSα is a norm.
• E(X) > 0 for any X 6= 0, follows from the fact that hhXiiSα is a norm.
• Let us prove that E(X) ≥ ψ(EX) with a convex function ψ on (−∞, ∞) having
ψ(0) = 0 but ψ(t) > 0 for t 6= 0.
Assume ψ(x) = |x|. Since CVaRα (X) is a regular measure of risk, it satises the
following inequality: CVaRα (X) ≥ EX . Therefore,
hhXiiSα = CVaRα (|X|) ≥ E|X| ≥ |EX| = ψ(EX).
Proposition 9. Let X be a random variable.
arg minhhX − diiSα =
d
minhhX −
d
diiSα
1
=
1−α
1
q(1−α)/2 (X) + q(1+α)/2 (X) ,
2
1−α
1+α
CVaR(1−α)/2 (X) +
CV aR(1+α)/2 (X) − EX .
2
2
Proof. According to Denition 2 of hhXiiSα :
hhXiiSα
then
minhhX −
d
diiSα
= min c +
1
+
E[|X| − c]
,
1−α
c
1
+
= min min c +
E[|X − d| − c]
.
c
d
1−α
Notice that optimal c∗ ≥ 0, because c∗ = qα (|X − d|) and inf |X − d| ≥ 0.
The following chain of equalities is valid
E[|X − d| − c]+ = E(|X − d| − c)I(|X − d| − c ≥ 0) =
=E(X − d − c)I(X > d)I(|X − d| − c ≥ 0) + E(d − X − c)I(X ≤ d)I(|X − d| − c ≥ 0) =
=E(X − (d + c))I(X > d)I(X ≥ (d + c)) + E((d − c) − X)I(X ≤ d)I(X ≤ (d − c)) =
=E(X − (d + c))I(X ≥ (d + c)) − E(X − (d − c))I(X ≤ (d − c)) =
=E[X − (d + c)]+ − E(X − (d − c)) + E(X − (d − c))I(X > (d − c)) =
=(d − c) + E[X − (d + c)]+ + E[X − (d − c)]+ − EX.
(16)
Notice that
(1 − α)c + (d − c) =
1+α
1−α
(d + c) +
(d − c).
2
2
(17)
Combining (16) and (17) we have
1
1−α
1
+
hhX −
=
min min
(d + c) +
E[X − (d + c)]
+
1−α d c
2
(1 − α)/2
1+α
1
+
(d − c) +
E[X − (d − c)]
− EX .
(18)
+
2
(1 + α)/2
diiSα
22
Notice that if Q(d, c) = G(d + c) + H(d − c), then
min min Q(d, c) = min Q(d, c) = min G(d + c) + H(d − c) =
c
d
=
d,c
d,c
G(d + c) + H(d − c) = min G(d + c) + min H(d − c).
min
d+c
(d+c),(d−c)
d−c
(19)
Applying (19) to (18), we have
1
1−α
1
S
+
hhX − diiα =
min
(d + c) +
E[X − (d + c)]
+
1 − α (d+c)
2
(1 − α)/2
1
1
1+α
1
+
+
−
min
(d − c) +
E[X − (d − c)]
EX =
1 − α (d−c)
2
(1 + α)/2
1−α
1−α
1+α
1
CVaR(1+α)/2 (X) +
CVaR(1−α)/2 (X) − EX ,
=
1−α
2
2
where
(d + c)∗ = q(1+α)/2 (X), (d − c)∗ = q(1−α)/2 (X),
which implies
d∗ =
1
q(1−α)/2 (X) + q(1+α)/2 (X) = arg minhhX − diiSα .
d
2
Proposition 10. CVaR Norm Quadrangle. Error measure E(X) = hhXiiα generates
the following regular quadrangle:
1
q(1−α)/2 (X) + q(1+α)/2 (X) ,
2
1−α
1+α
R(X) =
CVaR(1+α)/2 (X) +
CVaR(1−α)/2 (X),
2
2
1−α
1+α
D(X) =
CVaR(1+α)/2 (X − EX) +
CVaR(1−α)/2 (X − EX),
2
2
V(X) = hhXiiα + EX,
E(X) = hhXiiα .
S(X) =
Proof. It was proved in [9] that if E(X) is a regular measure of error, then λE(X) is a
regular measure of error for any positive λ. Since hhXiiSα is a regular measure of error and
hhXiiα = (1 − α)hhXiiSα , then hhXiiα is a regular measure of error. From Proposition 9 and
equality CVaRα (X) − EX = CV aRα (X − EX) follows that quadrangle axioms 11 13
hold. Regularity follows from Proposition 8 and Quadrangle Theorem (see [9]).
CVaR Norm Quadrangle from Proposition 10 is similar to Mixed-Quantile-Based
quadrangle (see [9]) for
α1 = (1 + α)/2, α2 = (1 − α)/2, λ1 = (1 − α)/2, λ2 = (1 + α)/2.
23
(20)
Dene
αk
1
+
−
Eαk (X) = E
X + X , Vαk (X) =
EX + .
1 − αk
1 − αk
With parameters from (20) we obtain following Mixed-Quantile-Based quadrangle
1−α
1+α
q(1+α)/2 (X) +
q(1−α)/2 (X),
2
2
1+α
1−α
CVaR(1+α)/2 (X) +
CVaR(1−α)/2 (X),
R(X) =
2
2
1−α
1+α
D(X) =
CVaR(1+α)/2 (X − EX) +
CVaR(1−α)/2 (X − EX),
2
2
V(X) = min {λ1 Vα1 (X − B1 ) + λ2 Vα2 (X − B2 )|λ1 B1 + λ2 B2 = 0} ,
S(X) =
B1 ,B2
E(X) = min {λ1 Eα1 (X − B1 ) + λ2 Eα2 (X − B2 )|λ1 B1 + λ2 B2 = 0} .
B1 ,B2
Note that CVaR Norm Quadrangle and Mixed-Quantile-Based quadrangle have the
same deviation and risk measures. Therefore, suppose one is optimizing measure of error
over some parametric family X(θ):
min
θ
Ei (X(θ)),
(21)
where i = 1 for error from CVaR Norm Quadrangle, and i = 2 for error from MixedQuantile-Based quadrangle. Assume that X(θ) = θ0 + Y (θ̃), where θ = (θ0 , θ̃), and θ0 is a
free parameter. Dene θi∗ = arg minθ Ei (X(θ)). Then θ̃1∗ = θ̃2∗ = arg minθ̃ D(Y (θ̃)) = θ̃∗ .
Therefore, Y (θ̃1∗ ) = Y (θ̃2∗ ) and two optimal points X(θ1∗ ) and X(θ2∗ ) for problems (21) can
be obtained from each other by adding constant shift
X(θ1∗ ) = (θ0∗ )1 + Y (θ̃∗ ), X(θ2∗ ) = (θ0∗ )2 + Y (θ̃∗ ), X(θ1∗ ) − X(θ2∗ ) = (θ0∗ )1 − (θ0∗ )2 .
3. Negative CVaR Function
Paper [4] considers a class of functions dened similar to Lp norms, but for p ∈ [0, 1).
These functions are not norms and they are concave for some regions of the space they
are dened4 . Such norms are used in optimization problems to achieve a sparsity of a
solution vector. We will dene similar functions in terms of CVaR concept.
First, let us consider a classic CVaRα (X) for α < 0 or α > 1. According to CVaR
denition in minimization form (see [8] or denition 2),
CVaRα (X) = min{c +
c
1
E[X − c]+ }.
1−α
(22)
Let us prove that CVaRα (X) = −∞ for α ∈ (−∞, 0) ∪ (1, ∞).
Consider c < inf X (c may be −∞), then expression under minimization in formula (22)
equals to
c+
1
−α
1
E[X − c]+ = c
+
EX.
1−α
1−α 1−α
4 For
(23)
p ∈ [0, 1) there is lp (x) in Rn and Lp (X) in the space of random variables. Concavity holds, for
example, for region x ≥ 0 in Rn and for region X ≥ 0 in the space of random variables.
24
−α
If α < 0 or α > 1, then 1−α
> 0 and the expression (23) tends to −∞ for c → −∞. We
see that denition (22) for α < 0 and α > 1 makes no sense.
Further we dene negative CVaR function.
Denition 6. Negative CVaR function Rα− (X) is dened as follows:
for α ∈ (0, 1],
Rα− (X) =
for α = 0,
1
(E|X| − (1 − α)CVaRα (|X|)),
α
R0− (X) = inf |X|.
Rα− (X) can be interpreted as an expectation of |X| in left α-tail. Note that Rα− (X)
is then the average quantile of the random variable |X|.
Denition 7. Negative CVaR function Rα− (X) is dened as follows:
for α ∈ (0, 1],
Rα− (X)
1
=
α
Z
α
qp (|X|)dp,
0
for α = 0,
R0− (X) = inf |X|.
R1
R1
Two denitions are equivalent since E|X| = 0 qp (|X|)dp and hhXiiα = α qp (|X|)dp.
The following proposition gives an alternative denition of the negative CVaR function, similar to the denition of CVaR norm.
Proposition 11.
Rα− (X) = max{c −
c
1
E[|X| − c]− }.
α
Proof.
1
1
(E|X| − (1 − α)CVaRα (|X|)) = (E|X| − min{(1 − α)c + E[|X| − c]+ } =
c
α
α
1
1
= max{E|X| − c + αc − E[|X| − c]+ } = max{c − E[|X| − c]− }.
c
α c
α
Rα− (X) =
For p ∈ (0, 1) the following inequality holds Lp (X) ≤ L1 (X), where Lp (X) =
(E|X|p )1/p . Since xp is a concave function for 0 < p < 1, using Jensen's inequality
we have E|X|p ≤ (E|X|)p , therefore, (E|X|p )1/p ≤ E|X|. Similar statement is valid for
the negative CVaR function.
Proposition 12.
0 ≤ Rα− (X) ≤ L1 (X) = E|X|.
The following proposition establishes the properties of negative CVaR function similar to the properties of CVaR norm.
Proposition 13. The negative CVaR function satises the following properties:
• Rα− (λX) = |λ|Rα− (X),
25
• Rα− (0) = 0, but also Rα− (X) = 0 for some X 6= 0,
• for X, Y such that XY ≥ 0 inequality holds
Rα− (λX + (1 − λ)Y ) ≥ λRα− (X) + (1 − λ)Rα− (Y ),
• function Rα− (X) is concave in the subspace of positive random variables X ≥ 0.
Proof. We prove the properties one by one.
• Rα− (λX) = |λ|Rα− (X) follows from the fact, that E|X| and (1 − α)CVaRα (|X|) are
norms.
• Assume X = 0 with probability 0.5 and X = 1 with probability 0.5. Then, for
α ∈ [0, 0.5] function Rα− (X) = 0.
• Notice that if XY ≥ 0, then |X + Y | = |X| + |Y |, therefore
max{c −
c
1
1
E[|X + Y | − c]− } = max{c − E[|X| + |Y | − c]− } ≥
c
α
α
1
≥ cX + cY − E[|X| + |Y | − cX − cY ]− ,
α
where cX = arg minc {c − α1 E[|X| − c]− }, cY = arg minc {c − α1 E[|Y | − c]− }. Considering that [x]− is a concave function, we obtain
Rα− (X + Y ) ≥ cX + cY −
1
(E[|X| − cX ]− + E[|Y | − cY ]− ) = Rα− (X) + Rα− (Y ).
α
• If X ≥ 0 and Y ≥ 0, then XY ≥ 0, therefore Rα− (X + Y ) ≥ Rα− (X) + Rα− (Y ), i.e.,
Rα− (X) is concave in subspace of positive random variables X ≥ 0.
Proposition 13 states that Rα− (X) is a concave function for X ≥ 0. Notice that this
property cannot be strengthened to concavity in the whole space of random variables.
Consider a function g(X) such that g(X) ≥ 0, g(0) = 0 and g(X) 6≡ 0. Assume that
g(X) is concave in the space of random variables. Since g(X) 6≡ 0, then exists X such
that g(X) > 0. Then
g(X) + g(−X) > 0 = g(0) = g(X − X),
which implies that g(X) is not a concave function.
Let us prove some properties of CVaR negative function with respect to parameter
α.
Corollary 5. Let X be a random variable. Then
• Rα− (X) is a continuous non-decreasing function w.r.t. α,
• αRα− (X) is a convex non-decreasing function w.r.t. α.
Proof. We will prove negative CVaR function properties one by one.
26
• Consider Y = |X| and apply Proposition 5 to Y .
• αRα− (X) = L1 (X)−(1−α)CVaRα (|X|) = L1 (X)−hhXiiα . Since hhXiiα is a concave,
non-increasing function w.r.t. α, then −hhXiiα is a convex non-decreasing function
of α. L1 (X) does not depend upon α, therefore L1 (X) − hhXiiα is also a convex
non-decreasing function of α.
4. Case Study
We illustrate CVaR Norm Quadrangle, see Proposition 10, with the following case
study. The case study results are posted at this link5 .
Let us consider a linear regression problem with CVaR norm error. Let X be a n × d
design matrix, where n is a number of observations, d is a number of explanatory variables.
Let y ∈ Rn be a vector of observations on the dependent variable. Let e ∈ Rn be a vector
of ones. Dene extended matrix X̃ = [e, X], including additional constant term. Let us
consider linear regression: ŷ = X̃a, where a ∈ Rd+1 is a vector of parameters. We will
minimize CVaR norm of vector of residuals y − ŷ:
min
a∈Rd+1
hhy − X̃aiiα .
(24)
We consider the dataset from the case study ¾Estimation of CVaR through Explanatory Factors with Mixed Quantile Regression¿6 . The data contains returns of the Fidelity
Magellan Fund as a dependent variable. Russell Value Index (RUJ), Russell 1000 Value
Index (RLV), Russell 2000 Growth Index (RUO) and Russell 1000 Growth Index (RLG)
are taken as independent variables. Data include 1,264 observations.
The CVaR norm is minimized with Portfolio Safeguard [1] software package. Condence level α in CVaR norm equals α = 0.9. We minimized CVaR instead of CVaR
norm, according to Proposition 4. Denote ȳ = [y; −y] ∈ R2n and X̄ = [X̃; −X̃] ∈ R2n×d .
Proposition 4 implies
hhy − X̃aiiSα = CVaR(1+α)/2 (ȳ − X̄a).
Then, problem (24) is equivalently stated as follows
min
a∈Rd+1
CVaR(1+α)/2 (ȳ − X̄a).
Optimization results for this problem are in Table 1.
CVaR Norm Quadrangle is a regular quadrangle, see Proposition 10. According
to the Regression Theorem, see [9], the interceipt, obtained in regression, equals to the
Statistic of a modied residuals. In CVaR Norm Quadrangle, Statistic equals S(X) =
(q(1+α)/2 (X)+q(1−α)/2 (X))/2. Denote the optimal vector of parameters obtained in regression by a∗ = [c∗ , b∗ ], where c∗ ∈ R is an optimal interceipt. According to the Regression
5 http://www.ise.ufl.edu/uryasev/research/testproblems/advanced-statistics/
cvar-norm-regression/
6 http://www.ise.ufl.edu/uryasev/research/testproblems/financial_engineering/
estimation-of-cvar-through-explanatory-factors-with-mixed-quantile-regression/
27
rlv
0.578
rlg
0.484
ruj
-0.07
ruo
-0.008
intercept
-0.002
objective
0.015
Table 1: Optimal vector of parameters and objective for linear regression with CVaR norm.
Theorem, c∗ ∈ S(y − Xb∗ ) (we write ∈ because, in general, quantile qp (X) is an interval, see (2), therefore S(X) is also an interval). At the optimal point, c∗ = −0.002,
−
−
q0.05
(y − Xb∗ ) = −0.013, q0.95
(y − Xb∗ ) = 0.009. Therefore,
−
−
S(y − Xb∗ ) ≈ (q0.05
(y − Xb∗ ) + q0.95
(y − Xb∗ ))/2 = (−0.013 + 0.009)/2 = −0.002 = c∗ .
Numerical experiment conrm theoretical results for CVaR Norm Quadrangle.
References
[1] Portfolio Safeguard version 2.1, 2009. http://www.aorda.com/aod/welcome.action.
[2]
Banach, S.
Theory of linear operations. North-Holland Sole distributors for the
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[3]
Bertsimas, D., Pachamanova, D., and Sim, M.
Robust linear optimization under
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[4]
Ge, D., Jiang, X., and Ye, Y.
[5]
Gotoh,
Program. 129, 2 (2011), 285299.
J.-y.,
and
A note on the complexity of Lp minimization. Math.
Uryasev,
Approximation of Euclidean norm by
S.
Lp-representable norms and applications. University of Florida, Research Report 2013-3 (May 2013). http://www.ise.ufl.edu/uryasev/files/2013/06/
ApproxL2withLinearNorms.pdf.
[6]
Dual stochastic dominance and related
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ufl.edu/uryasev/files/2013/08/cvar_norm_working_paper.pdf.
[8]
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Ogryczak, W., and Ruszczynski, A.
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