Conditional Value-at-Risk (CVaR) Norm: Stochastic Case

Transcription

1 Conditional Value-at-Risk (CVaR) Norm: Stochastic Case Alexander Mafusalov, Stan Uryasev RESEARCH REPORT 03-5 Risk Management and Financial Engineering Lab Department of Industrial and Systems Engineering 303 Weil Hall, University of Florida, Gainesville, FL 36. s: First draft: April 03, This draft: August 04 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- 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- 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 <. Linear regression problems were solved by minimizing CVaR norm of regression residuals. Keywords: CVaR norm, L p norm, Conditional Value-at-Risk, CVaR

2 . 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 R n was introduced and developed in [7], [5]. This section provides a short introduction in the CVaR norm in R n 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 R n and in the space of random variables, which is an extension of CVaR norm (but it is not actually a norm). Section. 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. proves properties of CVaR norm as a function of α. Section.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.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 R n and in the space of random variables. We use symbols x and x i for a vector and an i-th vector component in R n, i.e. x = (x,..., x n ). We use symbol X for a random variable. l p norms are broadly used in R n, and L p norms are considered in the space of random variables. For p [, ] the norms l p and L p are dened as follows : ( ) /p n l p (x) = x i p, L p (X) = (E X p ) /p, n i= where E is the expectation sign. The most popular cases are p =,,, i.e., l (x) = n n i= x i, L (X) = E X ; l (x) = max i=,...,n x i, L (X) = sup X ; l (x) = ( n n i= x i ) /, L (X) = (EX ) / ; It is known that l (x) l (x) l (x) and L (X) L (X) L (X), see []. Also, the following inequalities holds: l p (x) l q (x) and L p (X) L q (X) for p < q, see []. Note that the classic denition for l p norm is l p (x) = ( n i= x i p ) /p, it does not satisfy inequality l p (x) l q (x) for p < q. This paper uses an equivalent scaled version of this norm l p (x) = ( n n i= x i p) /p, which satises that inequality. Lp norm is commonly dened as L p (f) = f p ( S f p dµ ) /p, where S is a considered space. It is known (see, e.g., []) that for p and q norms inequality f p µ(s) p q f q holds for 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) = and inequality L p (X) L q (X) holds for p q, where L p (X) = (E X p ) /p and E is an expectation sign.

3 Further we will illustrate the general concept of CVaR norm in R n, as well as in spaces of discrete and continuous random variables. CVaR norm in R n is considered in [7] and [5]. According to [7], the CVaR norm in x R n is dened as follows. Let x (i) be ordered absolute values of components of x R n, i.e. x,..., x n } = x (),..., x (n) }, and x (i) x (i+), for i =,..., n. Then, for j = 0,..., n and α j = j/n, scaled CVaR norm (or just CVaR norm in this paper) is dened by x S α j = ( x (j+) x (n) )/(n j). For j = n we have α j = j/n = and the norm is x S α n = x (i) = max i x i. For α j < α < α j+, the norm x S α equals to the weighted sum, where x S α = µ x S α j + ( µ) x S α j+, µ = (α j+ α)( α j ) (α j+ α j )( α). A similar norm, called D-norm, was introduced in [3] in a dierent way: for p [, n], M =,..., n}, S is cardinality of a set S, p = maxl Z l p} (i.e., p is a maximal integer number which is not greater than p) } x p = max S t}:s M, S p,t M\S} j S x j + (p p ) x t For α [ ] 0, n n, the CVaR norm coinsides with the D-norm x p with parameter p dened by p = n( α), see [7]. The second variant of the norm, called non-scaled CVaR norm, is dened in [7] as follows : ( α) x S x α = α, for 0 α < ; 0, for α =. For example, x j/n = ( x (j+) x (n) )/n. This paper shows that the α norm in R n 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. Suppose that a random variable X takes values x i } N i= with probabilities p i } N i=, and N i= p i =, where x i R and N N or N = (we use the notation N for the set of natural numbers). Let us denote by the sequence x (i) } N i= an ordered sequence x i } N i=, i.e. x (i) x (i+). Let us denote for discretely distributed X by p (i) } N i= a corresponding to the x (i) } N i= sequence of values from the p i } N i=, i.e. if x (i) corresponds to x j, then p (i) = p j. Let. In [7] it is dened as x α = n( α) x S α, but in this paper we will stick to slightly dierent denition in the sake of consistency with stochastic case, where X α = ( α) X S α. 3

4 us dene α j = j i= p (i). The non-scaled CVaR norm with condence level α j for the discretely distributed random variable X is dened by the following expression: N i=j+ X αj = ( α j )CVaR αj ( X ) = x (i) p (i), for j = 0,..., N ; 0, for j = N, here CV ar α ( X ) denotes conditional value at risk for random variable X, see [8]. If N =, then X αj = x (i) p (i), for j N. i=j+ Similarly to the denition of non-scaled CVaR norm in R n, for α j < α < α j+ the X α equals to the weighted sum X α = ( λ) X αj + λ X αj+, where λ = (α α j )/(α j+ α j ). If N = n, p i = /n and x = (x,..., x n ) R n, then the CVaR norm of X coincides with the deterministic CVaR norm of x: X α = x α. Let us illustrate the denition of non-scaled CVaR norm in stochastic case with the following example. Example. Consider the random variable X taking the values x i = i with the probabilities p i = i for i N. Since x i > 0 and x i < x i+, then x i = x (i). For α = α j, X αj = ( i ) i = i=j+ i=j+ For j, there is convergence α j and i 4 i = (j+) 4 (j+) 4 = j 4 j /3. X αj 0 = ( )CV ar ( X ). For α = 0., which is between α 0 = 0 and α = 0.5, the value X 0. is a weighted sum of X 0 and X 0.5 with coecient λ = (α α 0 )/(α α 0 ) = 0./0.5 = 0.4. We have therefore, X 0 = /3 = /3, X 0.5 = /3 =.5/3, X 0. = ( λ) X 0 + λ 0.5 X 0.5 = 0.6 / /3 =.7/ Figure shows a plot of X α depending upon α. Notice that X α is a non-increasing, concave and piecewise-linear function w.r.t. α. Paper [7] showed these properties for CVaR norm in R n. Section. proves these properties of X α in stochastic case. 4

5 0.7 non scaled CVaR norm X α α Figure : Non-scaled CVaR norm X α of random variable X with atoms x i = i and probabilities p i = i as a function of α. For discrete random variable X, the scaled CVaR norm is dened as follows: X S α α = X α, for α < ; sup X, for α =, where sup X denotes the essential supremum 3 of the random variable. If N = n, p i = /n and x = (x,..., x n ) R n, then X S α = x S α. Example. Consider the random variable X taking the values x i = i with the probabilities p i = i for i N. For α = α j, values of the X S α norm are: X S α j = j 4 j /3 j i= i = j 4 j /3 j = j /3. For j, there is convergence value α j and CVaR norm X S α j = X S = sup X. For α = 0., X S 0. = (./ /3)/ Figure shows a plot of X S α 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 R n. Section. proves these properties of X S α for stochastic case. 3 By denition, ess sup X = infa R P (X > a) = 0}. 5

6 scaled CVaR norm X α S α Figure : Scaled CVaR norm X α of random variable X with atoms x i = i and probabilities p i = 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 F X (x) be a continuous cumulative distribution function of a random variable X, and F X (α) be an inverse function to F X(x) (i.e., F X (F X(x)) = x), and q α (X) be an α-quantile of the random variable X (i.e., P (X q α (X)) = α, q α (X) = F X (α)). CVaR(X), in this case, is a conditional expectation: CV ar α (X) = E(X X > F X (α)), or, equivalently, CVaR α(x) = q α α p(x)dp, see [8]. In continuous case, the CVaR norm is dened as follows: X S α = CVaR α ( X ). We prove in Section. that CVaR norm is indeed a norm. Also, we show that X S 0 = L (X) and X S = L (X). We use the sign as a notation for words ¾distributed by. For example, X N (0, ) means that the random variable X is normally distributed with mean µ = 0 and variance σ =. 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 and with cumulative distribution function f X (x) = λe λx, x 0; 0, x < 0 }, F X (x) = e λx, x 0; 0, x < 0 }. () 6

7 Since X is a non negative random variable, then X = X. The expression () for F X (x) implies the following equation for quantile q α (X): e λqα(x) = α. Consequently, λq α (X) = ln( α), q α (X) = ln( α). λ For α =, the quantile q (X) =. Then, for α [0, ] X S α = [ ] xλe λx dx α q [ α(x) = q α (X)( α) + α = [ ln( α)]. λ = α ( λ ) e λx [ ( xe ) λx + q α(x) ] q α(x) = α q α(x) ] λe λx dx = [q α (X)( α) + λ ] ( α) = For α = 0, X S 0 = λ = EX = E X = L (X). For α =, X S = = sup X = sup X = L (X). Figure 3 shows a plot of the probability density function f X (x) of X and Figure 4 shows a plot of the function X S α as a function of α. Similar to the discrete distribution case, we consider the non-scaled CVaR norm: X α = ( α) X S α. CVaR α ( X )). It is easy to see that X α = E(I( X > F X (α)) X ), where I(A) is an indicator function:, if A is true; I(A) = 0, if A is false. Section. shows that the non-scaled CVaR norm is a decreasing concave function of α and changes from L (X) to 0 when α changes from 0 to. Let us illustrate the denition of non-scaled CVaR norm X α 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, ), the non-scaled CVaR norm equals X α = α [ ln( α)]. λ 7

8 probability density function f X (x) x Figure 3: Probability density function f X (x) for X Exp(). non scaled CVaR norm X α α Figure 4: Non-scaled CVaR norm X α for X Exp(), as a function of α. 8

9 8 scaled CVaR norm X α S α Figure 5: Scaled CVaR norm X S α for X Exp(), as a function of α. For α = 0, X α = λ = EX = E X = L (X). For α =, X α = 0. Figure 5 shows a plot of the function X α. 9

10 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 X α 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) = X α (see Section.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 ( 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 l and α norms: From the denition of x α follows: r α (x) = l (x) x α. r α j (x) = ( x () x (j) )/n, for α = α j = j/n, r 0 (x) = 0, for α = 0, r (x) = l (x), for α =. We also dene scaled version of negative CVaR function as follows From the denition of x S α follows: rα,s (x) = α (l (x) ( α) x S α). r α j (x) = ( x () x (j) )/j, for α = α j = j/n, r 0 (x) = min i x i, for α = 0, r (x) = l (x), for α =. 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 x S α and rα,s (x) in R for dierent values of α. The function rα,s is a natural extension of S α. When α variates from 0 to, the function rα,s (x) changes from min i x i to l (x) = n n i= x i, and the function x S α changes from l (x) to max i x i. 0

11 3.5 x S 0.5 = max( x, x )= x S 0.5 = x S 0 = l (x) = r,s (x)= r,s 0.75 (x)= r,s 0.5 (x) = min( x, x )= x x Figure 6: Level-sets of scaled CVaR norm x S α for α = 0, 0.5, 0.5 and level-sets of scaled negative CVaR function rα,s (x) for α = 0.5, 0.75, in R space. For α [0.5, ] norm x S α = max i x i. For α [0, 0.5] function rα,s (x) = min i x i. Equality x S 0 = l (x) = r,s (x) holds.

12 . CVaR Norm in Stochastic Case This section gives a formal denition of CVaR norm in stochastic case and proves various properties of the norm... CVaR Norm Denition and Properties Let us denote [x] + = max0, x}, [x] = max0, x}. Consider cumulative distribution function F X (x) = P (X x). If, for a probability level α (0, ), there is a unique x such that F X (x) = α, then this x is called the α- quantile q α (X). In general, however, the value x is not unique, or may not even exist any. There are two values to consider as extremes: q + α (X) = infx F X (x) > α}, q α (X) = supx F X (x) < α}. We will call by the quantile the entire interval between the two extreme values, q α (X) = [q α (X), q + α (X)]. () We will use notation q p (X)dp q p (X)dp, which is a reasonable since q + p (X)dp = q p (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. Let X be a random variable with E X <. Then, CVaR norm of X with parameter α [0, ) is dened as follows: X S α = min c + } E[ X c]+. c α If also X is essentially nite (i.e., exists C R : X < C), then for α = X S = sup X. Denition. Let X be a random variable with E X <. Then CVaR norm of X with parameter α [0, ) is dened as follows: X S α = α α q p ( X )dp. If also X is essentially nite (i.e., exists C R : X < C), then for α = X S = sup X. It immediately follows from the denitions of CVaR (see [8]) that the Denitions and are equivalent. Proposition. Let X be a continuous random variable, i.e., its cumulative distribution function is continuous. Then X S α = E ( X X > q α ( X )).

13 Proof. If cumulative distribution function F X of random variable X is continuous, then F X is continuous, q p ( X ) = F X (p) and X S α = α F X (p)dp α = F xdf X (x) X (α) ( ) ( ) = E X X > F P X > F X (α) X (α). Let X be a discrete random variable, i.e., it takes values x i } N i= with positive probabilities p i } N i= (N also can be ). Let us denote by the sequence x (i) } N i= an ordered sequence x i } N i=, i.e., x (i) x (i+). x i } N i= exists, such that if x (i) x j, then x (i) = x j. We also denote by p (i) } N i= a corresponding to the x (i) } N i= sequence of probabilities from the p i } N i=. Note that ordered sequence x (i) } exists only for special sets of x i }. In particular, it exists in following cases: set x i } is nite; sequence x i } i= has no converging subsequences; all converging subsequences of sequence x i } i= converge to x = sup x i }. If ordered sequence x (i) } exists, then the following proposition holds. Proposition. Let X be a discrete random variable, i.e., it takes values x i } N i=, with positive probabilities p i } N i=, where N i= p i = and N N. exists, such that if x (i) x j, then x (i) = x j. Let us denote by p (i) } N i= a corresponding to the x (i) } N i= sequence of values from the p i} N i=, i.e. if x (i) x j, then p (i) = p j. Then for N <, for N =, for N < and j = 0,..., N X S = x (N), X S = lim i x (i). X S α j = α j where α j = j i= p (i), for N = and j Z + N 0}, for α j < α < α j+, X S α j = α j N i=j+ i=j+ x (i) p (i), x (i) p (i), X S α = ( λ) α j α X S α j + λ α j+ α X S α j+, where λ = (α α j )/(α j+ α j ). Proof. Let us proceed with the proof bullet by bullet. 3

14 It follows directly from the denition that CVaR ( X ) = sup x i } N i=. Since x (i) < x (i+), then in case N <, X S = sup x i } N i= = max i x (i) = x (N). Consider the case N =. Since x (i) } i= is a non decreasing sequence, then it has a nite or an innite limit lim i x (i). Since also all probabilities p (i) > 0, then X S = sup x (i) } i= = lim i x (i). If X is a discrete random variable, then F X (x) is a step function. Then, q p ( X ) is a step function of p. Length of a step number i is α i α i = p (i). Height of the step number i is q αi ( X ) = x (i). It implies that the area under the step equals to the following integral: αi α i q p ( X )dp = p (i) x (i). Then, = α j X S α j = q p ( X )dp = α j α j N i=j+ αi α i q p ( X )dp = α j N i=j+ p (i) x (i). Since q p ( X ) is a step function of p, then it is a constant function of α on the interval (α j, α j+ ). Then, the integral q α p( X )dp is a linear function of α for α j < α < α j+. It implies that this integral is the linear combination α q p ( X )dp = ( λ) q p ( X )dp + λ α j q p ( X )dp, α j+ where λ = (α α j )/(α j+ α j ). Then, X S α = ( λ) α j α X S α j + λ α j+ α X S α j+. Denition immediately implies that X S 0 = L (X), X S = L (X). Furthermore, we prove that X S α is a norm for α (0, ). Proposition 3. Let X be a random variable. X S α variables. is a norm in the space of random Proof. By denition, function d(x) is a norm if. d(x) = 0 X 0,. d(λx) = λ d(x), 3. d(x + Y ) d(x) + d(y ). 4

15 Since CVaR α (X) is a regular Measure of Risk (see [9]), then for λ > 0 and CVaR α (λx) = λcvar α (X), CVaR α (X + Y ) CVaR α (X) + CVaR α (Y ). (3) By denition, f(x) is monotonic if X Y implies f(x) f(y ). Since CVaR α (X) is monotonic (see [9]), then for X Y CVaR α (X) CVaR α (Y ). (4) Let us prove the axioms of a norm for X S α = CVaR α ( X ).. CVaR α ( X ) = 0 X 0. The statement X 0 CVaR α ( X ) = 0 is obvious. If X 0, then q p ( X ) > 0 for p (0, ), and CVaR α ( X ) = α α q p ( X )dp > 0. If α = 0 and X 0, then CVaR 0 ( X ) = E X > 0. If α = and X 0, then CVaR ( X ) = sup( X ) > 0.. CVaR α ( λx ) = λ CVaR α ( X ). Since λ > 0, then CVaR α ( λx ) = CVaR α ( λ X ) = λ CVaR α ( X ). 3. CVaR α ( X + Y ) CVaR α ( X ) + CVaR α ( Y ). Since X + Y X + Y, using (4) we have Finally, which follows from (3). CVaR α ( X + Y ) CVaR α ( X + Y ). CVaR α ( X + Y ) CVaR α ( X ) + CVaR α ( Y ), The next proposition provides an alternative way to calculate X S α. Proposition 4. Let X be a random variable. Let Y be a random variable dened as follows X, with probability Y =, X, with probability. Then, X S α = CVaR (+α)/ (Y ). 5

16 Proof. Minimization form denition of CVaR, see [8] or Denition, implies that c + CVaR (+α)/ (Y ) = min c since /( ( + α)/) = /(( α)/). Dene c + c Y = arg min c E[Y c]+ ( α)/ E[Y c]+ ( α)/ }. }, (5) Notice that Y is symmetric, therefore 0 q Y (/). Notice also and (+α)/ /, since α 0. Since optimal solution in CVaR denition (5) is the quantile q (+α)/ (Y ), then c Y 0. Then, } CVaR (+α)/ (Y ) = min c + E[Y c]+ = (6) c ( α)/ = c Y + ( α)/ E[Y c Y ] + = (7) } = min c 0 = min c 0 c + E[Y c]+ ( α)/ ( α)/ E[Y + c] + c + = (8) }, (9) where equality between (6) and (7) follows from denition of c Y ; equality between (7) and (8) follows from c Y 0; equality between (8) and (9) follows from [Y c] + = [Y + c] + for c 0. Note that Y + X, with probability =, 0, with probability. Therefore, CVaR (+α)/ (Y ) = min c 0 = min c 0 } c + E[ X c]+ = ( α)/ c + } E[ X c]+ = X S α α. (0) Last equation in (0) follows from CVaR norm Denition and from arg min c + } E[ X c]+ 0, c α which holds since arg min is a quantile q α ( X ) and X 0, therefore q α ( X ) 0... 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, q α p(x)dp is a continuous increasing function of α, α 6

17 lim α q α α p(x)dp = sup X, α q α 0 p(x)dp is a continuous non-decreasing function of α, α 0 q p(x)dp is a convex function of α, α 0 q p(x)dp is a piecewise-linear function of α for discretely distributed X. Proof. We provide references for these statements bullet by bullet. CVaR α is a continuous increasing function of α, see [8]. Then, integral form denition of CVaR α (X) = q α α p(x)dp implies that the integral q α α p(x)dp is a continuous increasing function of α. lim α CVaR α (X) = sup X, see [8]. Therefore, lim α q α α p(x)dp = sup X. Notice that q p ( X) = q p (X), then α α q p ( X)dp = α α q p (X)dp = α α 0 q p (X)dp. The rst bullet of this proposition implies that q α p( X)dp is a decreasing α continuous function of α, therefore, integral q α 0 p(x)dp is a continuous nondecreasing function of α. Integral α 0 q p(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 p. If α = F X (x) and α = α + p, then q α (X) = x for α (α, α ). Therefore, α q 0 p(x)dp is linear on α (α, α ). Since α q 0 p(x)dp is also continuous, and X is dicretely distributed, then α q 0 p(x)dp is a piecewise-linear function of α. Let X be a random variable. Take constant C, such that L (X) C L (X). The following corollary assures that there exists a single α such that CVaR α ( X ) = C. Therefore, for any p there is a single α such that CVaR α ( X ) = L p (X). Corollary. Let X be a random variable. The norm X S α function of α. is a continuous increasing 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, ] is dened as follows: ( α) X S X α = α, for α [0, ), 0, for α =. 7

18 Corollary. Let X be a random variable. The norm X α is a concave and decreasing function of α. Furthermore, if X is discretely distributed, X α is a piecewise-linear function of α. Proof. By denition, If α < α, then X α = ( α)cvar α ( X ) = X α = α q p ( X )dp α q p ( X )dp. α q p ( X )dp = X α. Therefore, X α is a decreasing function of α. To prove that X α is a concave function of α, consider Y = X and apply Proposition 5 to Y. Since α q 0 p( X )dp is convex, then X α = E X α q 0 p( X )dp is a concave function of α. For piecewise-linearity, take Y = X, note that X α = EY α q 0 p(y )dp, and apply Proposition 5. Corollary 3. Let X be a random variable. ( α)cvar α (X) is a concave function of α. Furthermore, it is a piecewise-linear function of α if X is discretely distributed. Proof. ( α)cvar α (X) = q α p(x)dp = EX α q 0 p(x)dp, therefore, it is concave w.r.t. α, and it is piecewise-linear for discretely distributed X, see Proposition Dual Norm to CVaR Norm and CVaR Normed Space Denition 4. Let X be a normed space over R with norm (i.e., X 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 of f is dened by } f(x) f = sup f(x) : x X, x } = sup x : x X, x 0. The asterisk denotes the dual norm to a norm. Therefore, Y S α denotes the norm dual to the CVaR norm X S α. Proposition 6. Let X be a random variable. The norm Y S α = maxe Y, ( α) sup Y } is dual to the norm X S α for α (0, ). Proof. Paper [9] proved that CVaR α (X) = sup Q Q EXQ, where Q = Q 0 Q } α, EQ =. Then, X S α = CVaR α ( X ) = sup Q Q E X Q. Let us prove that sup E X Q = sup EXY, Q Q Y Y 8

19 where Y = Y Y, E Y }. α First, (I(X > 0) I(X < 0))Q Y sup E X Q sup EXY, Q Q Y Y since X(I(X > 0) I(X < 0))Q = X Q. Second, sup EXY sup E X Y sup E X Q. Y Y Y Y Q Q Finally, sup EXY = sup E X Q = CV ar α ( X ) = X S α. Y Y Q Q Then, Y = Y EXY X S α} must be a convex hull of Y. The set Y is closed: and convex: Y k Y, Y k α Y α, Y k Y, E Y k E Y, Y, Y Y λy + ( λ)y λ Y + ( λ) Y (λ + ( λ)), α E λy + ( λ)y λe Y + ( λ)e Y (λ + ( λ)). Then, Y = Y and Y is a unit ball in the dual norm to the CVaR norm Y S EXY α = sup, X 0 X S α since Y S α EXY X S α for all X. Then, the unit sphere in the dual norm is the set } } Y sup Y = α, E Y Y sup Y α, E Y =. Then, the dual norm equals Y S α = maxe Y, ( α) sup Y }. Denition 5. A Banach space is a vector space X over R, which is equipped with a norm and which is complete with respect to that norm. By denition, completeness means that for every Cauchy sequence x n } n= in X (i.e., for every ε > 0 exists N such that x m x n < ε for all m, n > N), there exists an element x in X such that lim x n = x, i.e., lim x n x = 0. n n The next statement follows directly from Denition 5 of a Banach space. Proposition 7. Let L be a norm, X L = X L(X) < } and space (X L, L) is a Banach space. Let L be a norm such that exist such C, C + > 0 that C L(X) L(X) C + L(X)for all X. Then, X L = X L and (X L, L) is a Banach space. 9

20 Corollary 4. The norm X S α generates a Banach space for α [0, ]. Proof. E X X S α α E X, for α <. E X = L (X) and it is known that L -norm generates a Banach space. X S = sup X = L (X) and it is known that L -norm generates a Banach space..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) = mine(x C)}, R(X) = minc + V(X C)}, () C S(X) = arg mine(x C)} = arg minc + V(X C)}. (3) Measure of risk R(X) is regular if R(X) (, ], R(X) is closed convex, R(C) = C for any constant C, 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 0, for sequence of random variables X k } k= lim E(X k) = 0 lim EX k = 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 0. 0 C C

21 Measure of regret V(X) is regular if V(X) (, ], V(X) is closed convex, V(0) = 0, V(X) > 0 for any X 0, for sequence of random variables X k } k= lim [V(X k) EX k ] = 0 lim EX k = 0. k k The quadrangle (R, D, E, V, S) is regular if axioms 3 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 3 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 X S α 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) = X S α = CVaR α ( X ) and quadrangle axioms 3 hold, then the risk measure ( is R(X) = αcvar (+α)/(x)+ +αcvar ( α)/(x) and the statistic is S(X) = q( α)/ (X) + q (+α)/ (X) ). Proposition 8. E(X) = X S α 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 X S α is a norm. Let us prove that E(X) is closed and convex. X S α is a norm, therefore, it is a convex function. Closeness is equivalent to the following statement. Let us consider a sequence of random variables X k } k= and a random variable X such that expectations µ(x k X) 0 and variances σ(x k X) 0 for k and E(X k ) C for all k. Then, under these conditions, E(X) C. Here is the proof of this statement. E(X k ) E(X) E(X k X) α E X k X E Xk X α = = σ (X k X) µ α (X k X) 0, (4) for k. It implies that E(X) E(X k ) E(X X k ) = E(X k X) 0, (5) for k. Combining (4) and (5) we have E(X) E(X k ) 0 E(X k ) E(X) E(X) C.

22 E(0) = 0, follows from the fact that X S α is a norm. E(X) > 0 for any X 0, follows from the fact that X S α 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 0. Assume ψ(x) = x. Since CVaR α (X) is a regular measure of risk, it satises the following inequality: CVaR α (X) EX. Therefore, X S α = CVaR α ( X ) E X EX = ψ(ex). Proposition 9. Let X be a random variable. arg min X d S α = ( q( α)/ (X) + q (+α)/ (X) ), d min X d S α = ( + α d α CVaR ( α)/ (X) + α CV ar (+α)/ (X) EX ). Proof. According to Denition of X S α: then min d X S α = min c X d S α = min d c + E[ X c]+ α }, min c + } E[ X d c]+. c α Notice that optimal c 0, because c = q α ( 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. (6) Notice that ( α)c + (d c) = α Combining (6) and (7) we have X d S α = [ α α min min d c + + α ( (d c) + (d + c) + + α (d c). (7) ( (d + c) + E[X (d c)]+ ( + α)/ ) E[X (d + c)]+ + ( α)/ ) ] EX. (8)

23 Notice that if Q(d, c) = G(d + c) + H(d c), then min d min c = min (d+c),(d c) Q(d, c) = min d,c Applying (9) to (8), we have Q(d, c) = min G(d + c) + H(d c) = d,c G(d + c) + H(d c) = min d+c G(d + c) + min H(d c). (9) d c X d S α = [ ( )] α α min (d + c) + E[X (d + c)]+ + (d+c) ( α)/ + [ ( )] + α α min (d c) + E[X (d c)]+ (d c) ( + α)/ α EX = = [ α CVaR (+α)/ (X) + + α ] CVaR ( α)/ (X) EX, α where which implies (d + c) = q (+α)/ (X), (d c) = q ( α)/ (X), d = ( q( α)/ (X) + q (+α)/ (X) ) = arg min d X d S α. Proposition 0. CVaR Norm Quadrangle. Error measure E(X) = X α generates the following regular quadrangle: S(X) = ( q( α)/ (X) + q (+α)/ (X) ), R(X) = α CVaR (+α)/ (X) + + α CVaR ( α)/ (X), D(X) = α CVaR (+α)/ (X EX) + + α CVaR ( α)/ (X EX), V(X) = X α + EX, E(X) = X α. 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 X S α is a regular measure of error and X α = ( α) X S α, then X α is a regular measure of error. From Proposition 9 and equality CVaR α (X) EX = CV ar α (X EX) follows that quadrangle axioms 3 hold. Regularity follows from Proposition 8 and Quadrangle Theorem (see [9]). CVaR Norm Quadrangle from Proposition 0 is similar to Mixed-Quantile-Based quadrangle (see [9]) for α = ( + α)/, α = ( α)/, λ = ( α)/, λ = ( + α)/. (0) 3

24 Dene E αk (X) = E [ ] αk X + + X, V αk (X) = EX +. α k α k With parameters from (0) we obtain following Mixed-Quantile-Based quadrangle S(X) = α R(X) = α D(X) = α q (+α)/ (X) + + α q ( α)/ (X), CVaR (+α)/ (X) + + α CVaR ( α)/ (X), CVaR (+α)/ (X EX) + + α CVaR ( α)/ (X EX), V(X) = min λ V α (X B ) + λ V α (X B ) λ B + λ B = 0}, B,B E(X) = min B,B λ E α (X B ) + λ E α (X B ) λ B + λ B = 0}. 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 E i (X(θ)), () where i = for error from CVaR Norm Quadrangle, and i = for error from Mixed- Quantile-Based quadrangle. Assume that X(θ) = θ 0 + Y ( θ), where θ = (θ 0, θ), and θ 0 is a free parameter. Dene θ i = arg min θ E i (X(θ)). Then θ = θ = arg min θ D(Y ( θ)) = θ. Therefore, Y ( θ ) = Y ( θ ) and two optimal points X(θ ) and X(θ ) for problems () can be obtained from each other by adding constant shift X(θ ) = (θ 0) + Y ( θ ), X(θ ) = (θ 0) + Y ( θ ), X(θ ) X(θ ) = (θ 0) (θ 0). 3. Negative CVaR Function Paper [4] considers a class of functions dened similar to L p norms, but for p [0, ). These functions are not norms and they are concave for some regions of the space they are dened 4. 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 CVaR α (X) for α < 0 or α >. According to CVaR denition in minimization form (see [8] or denition ), CVaR α (X) = min c c + α E[X c]+ }. () Let us prove that CVaR α (X) = for α (, 0) (, ). Consider c < inf X (c may be ), then expression under minimization in formula () equals to c + α E[X c]+ = c α α + EX. (3) α 4 For p [0, ) there is l p (x) in R n and L p (X) in the space of random variables. Concavity holds, for example, for region x 0 in R n and for region X 0 in the space of random variables. 4

25 If α < 0 or α >, then α > 0 and the expression (3) tends to for c. We α see that denition () for α < 0 and α > makes no sense. Further we dene negative CVaR function. Denition 6. Negative CVaR function R α (X) is dened as follows: for α (0, ], R α (X) = α (E X ( α)cvar α( X )), for α = 0, R 0 (X) = inf X. R α (X) can be interpreted as an expectation of X in left α-tail. Note that R α (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, ], for α = 0, R α (X) = α α 0 q p ( X )dp, R 0 (X) = inf X. Two denitions are equivalent since E X = q 0 p( X )dp and X α = q α p( X )dp. The following proposition gives an alternative denition of the negative CVaR function, similar to the denition of CVaR norm. Proposition. Proof. Rα (X) = maxc c α E[ X c] }. R α (X) = α (E X ( α)cvar α( X )) = α (E X min( α)c + E[ X c] + } = c = α max E X c + αc E[ X c] + } = maxc c c α E[ X c] }. For p (0, ) the following inequality holds L p (X) L (X), where L p (X) = (E X p ) /p. Since x p is a concave function for 0 < p <, using Jensen's inequality we have E X p (E X ) p, therefore, (E X p ) /p E X. Similar statement is valid for the negative CVaR function. Proposition. 0 R α (X) L (X) = E X. The following proposition establishes the properties of negative CVaR function similar to the properties of CVaR norm. Proposition 3. The negative CVaR function satises the following properties: R α (λx) = λ R α (X), 5

26 R α (0) = 0, but also R α (X) = 0 for some X 0, for X, Y such that XY 0 inequality holds R α (λx + ( λ)y ) λr α (X) + ( λ)r α (Y ), function R α (X) is concave in the subspace of positive random variables X 0. Proof. We prove the properties one by one. R α (λx) = λ R α (X) follows from the fact, that E X and ( α)cvar α ( X ) are norms. Assume X = 0 with probability 0.5 and X = with probability 0.5. Then, for α [0, 0.5] function R α (X) = 0. Notice that if XY 0, then X + Y = X + Y, therefore maxc c α E[ X + Y c] } = maxc α E[ X + Y c] } c c X + c Y α E[ X + Y c X c Y ], where c X = arg min c c α E[ X c] }, c Y = arg min c c α E[ Y c] }. Considering that [x] is a concave function, we obtain R α (X + Y ) c X + c Y α (E[ X c X] + E[ Y c Y ] ) = R α (X) + R α (Y ). If X 0 and Y 0, then XY 0, therefore R α (X + Y ) R α (X) + R α (Y ), i.e., R α (X) is concave in subspace of positive random variables X 0. Proposition 3 states that R α (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) 0. Assume that g(x) is concave in the space of random variables. Since g(x) 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 R α (X) is a continuous non-decreasing function w.r.t. α, αr α (X) is a convex non-decreasing function w.r.t. α. Proof. We will prove negative CVaR function properties one by one. 6

27 Consider Y = X and apply Proposition 5 to Y. αr α (X) = L (X) ( α)cvar α ( X ) = L (X) X α. Since X α is a concave, non-increasing function w.r.t. α, then X α is a convex non-decreasing function of α. L (X) does not depend upon α, therefore L (X) X α is also a convex non-decreasing function of α. 4. Case Study We illustrate CVaR Norm Quadrangle, see Proposition 0, with the following case study. The case study results are posted at this link 5. 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 R n be a vector of observations on the dependent variable. Let e R n be a vector of ones. Dene extended matrix X = [e, X], including additional constant term. Let us consider linear regression: ŷ = Xa, where a R d+ is a vector of parameters. We will minimize CVaR norm of vector of residuals y ŷ: min y Xa α. (4) a R d+ 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 000 Value Index (RLV), Russell 000 Growth Index (RUO) and Russell 000 Growth Index (RLG) are taken as independent variables. Data include,64 observations. The CVaR norm is minimized with Portfolio Safeguard [] software package. Con- dence level α in CVaR norm equals α = 0.9. We minimized CVaR instead of CVaR norm, according to Proposition 4. Denote ȳ = [y; y] R n and X = [ X; X] R n d. Proposition 4 implies y Xa S α = CVaR (+α)/ (ȳ Xa). Then, problem (4) is equivalently stated as follows min a R d+ CVaR (+α)/ (ȳ Xa). Optimization results for this problem are in Table. CVaR Norm Quadrangle is a regular quadrangle, see Proposition 0. 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 (+α)/ (X)+q ( α)/ (X))/. 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 cvar-norm-regression/ 6 estimation-of-cvar-through-explanatory-factors-with-mixed-quantile-regression/ 7

28 rlv rlg ruj ruo intercept objective Table : Optimal vector of parameters and objective for linear regression with CVaR norm. Theorem, c S(y Xb ) (we write because, in general, quantile q p (X) is an interval, see (), therefore S(X) is also an interval). At the optimal point, c = 0.00, q0.05(y Xb ) = 0.03, q0.95(y Xb ) = Therefore, S(y Xb ) (q 0.05(y Xb ) + q 0.95(y Xb ))/ = ( )/ = 0.00 = c. Numerical experiment conrm theoretical results for CVaR Norm Quadrangle. References [] Portfolio Safeguard version., [] Banach, S. Theory of linear operations. North-Holland Sole distributors for the U.S.A. and Canada, Elsevier Science Pub. Co, Amsterdam New York, 987. [3] Bertsimas, D., Pachamanova, D., and Sim, M. Robust linear optimization under general norms. Operations Research Letters 3, 6 (November 004), [4] Ge, D., Jiang, X., and Ye, Y. A note on the complexity of Lp minimization. Math. Program. 9, (0), [5] Gotoh, J.-y., and Uryasev, S. Approximation of Euclidean norm by Lp-representable norms and applications. University of Florida, Research Report 03-3 (May 03). ApproxLwithLinearNorms.pdf. [6] Ogryczak, W., and Ruszczynski, A. Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization 3, (00), [7] Pavlikov, K., and Uryasev, S. CVaR Norm and Applications in Optimization. University of Florida, Research Report 03- (September 03). ufl.edu/uryasev/files/03/08/cvar_norm_working_paper.pdf. [8] Rockafellar, R. T., and Uryasev, S. Conditional value-at-risk for general loss distributions. Journal of Banking and Finance (00), [9] Rockafellar, R. T., and Uryasev, S. The Fundamental Risk Quadrangle in Risk Management, Optimization and Statistical Estimation. Surveys in Operations Research and Management Science 8 (03). 8