On coherency and other properties of arXiv:1703.10981v1 [q-fin.MF] 31 Mar 2017
MAXVAR
∗
Jie Sun† and Qiang Yao‡ April 3, 2017
Abstract Consider the MAXVAR risk measure on L 2 space. We present a simple and direct proof of its coherency and aversity. Based on the observation that the MAXVAR measure is a continuous convex combination of the CVAR measure, we provide an explicit formula for the risk envelope of MAXVAR.
2010 MR subject classification: 49N15, 91G40 Key words: Coherent risk measure, risk averse, risk envelope.
1
Introduction
In Cherny & Madan [2] and Cherny & Orlov [3], a new kind of risk measure –“MAXVAR” – is proposed, which is useful in the analysis of large portfolios. Given a probability space (Ω, Σ, P0 ) and a random variable X : L 2 (Ω, Σ, P0 ) → R, where L 2 (Ω, Σ, P0 ) is the square integrable Lebesgue space (L 2 for short), the MAXVAR is defined as MAXVARn (X) := E(max{X1 , · · · , Xn }), ∗ †
This paper is dedicated to Michel Th´era in celebration of his 70th birthday. School of Mathematical Sciences, Chongqing Normal University, PRC and School of Science and CIC,
Curtin University, Australia. Email:
[email protected]. Research partially supported by Australian Research Council under Grant DP160102819. ‡ School of Statistics, East China Normal University, Shanghai, China; E-mail:
[email protected]. Research partially supported by NSFC grants (No.11201150 and No.11126236), the Fundamental Research Funds for the Central Universities (No.239201278210075), and the 111 Project (B14019).
1
where X1 , · · · , Xn are i.i.d. copies of X. We call MAXVAR(·) the “MAXVAR risk measure”. Note that MAXVAR(·) is always finite on L 2 since |MAXVARn (X)| ≤ nE(|X|) < +∞ for any X ∈ L 2 . In [2, 3], the name of “MINVAR risk measure” was used. Since we treat risk measures as a nondecreasing function, we use “MAXVAR risk measure” instead. Obviously, we have MAXVARn (X) = −MINVARn (−X). Different from papers [2, 3], which considered coherency of MINVAR in L ∞ space, this paper deals with the L 2 space. Our proof of the coherency of MAXVAR risk measure is direct and independent of [2, 3]. Moreover, we show the risk averseness and give an explicit formula for the risk envelope of MAXVAR. In Section 2, we present a simple proof for the coherency of MAXVAR. We show its aversity in Section 3. Section 4 is devoted to the discussion of a continuous representation of MAXVAR and Section 5 provides an explicit formula for its risk envelope.
2
Coherency of MAXVAR
In this section, we show that MAXVAR is a coherent risk measure in basic sense of Rockafellar. Definition 2.1 (Rockafellar 2007) A functional R ∈ L 2 is a coherent risk measure in basic sense if it satisfies (A1) R(C) = C for all constant C; (A2) (“convexity”) R(λX + (1 − λ)Y ) ≤ λ · R(X) + (1 − λ) · R(Y ) for any X, Y ∈ L 2 and 0 ≤ λ ≤ 1; (A3) (“monotonicity”) R(X) ≤ R(Y ) for any X, Y ∈ L 2 satisfying X ≤ Y ; (A4) (“closedness”) If kX k − Xk2 → 0 and R(X k ) ≤ 0 for all k ∈ N, then R(X) ≤ 0; (A5) (“positive homogeneity”) R(λX) = λ · R(X) for any λ > 0 and X ∈ L 2 .
Theorem 2.1 MAXVAR(·) is a coherent risk measure in basic sense. Proof. (A1) is obvious by definition. (A5) is also easy to check since if X1 , · · · , Xn are i.i.d. copies of X and λ > 0, then λX1 , · · · , λXn are i.i.d. copies of λX. 2
Proof of (A2): For any X ∈ L 2 and 0 ≤ λ ≤ 1. Take (X1 , Y1 ), · · · , (Xn , Yn ) as i.i.d. copies of (X, Y ). Then for any 0 ≤ λ ≤ 1, λX1 + (1 − λ)Y1 , · · · , λXn + (1 − λ)Yn are i.i.d. copies of λX + (1 − λ)Y . In particular, X1 , · · · , Xn are i.i.d. copies of X, Y1 , · · · , Yn are i.i.d. copies of Y . Since max{λX1 +(1−λ)Y1, · · · , λXn +(1−λ)Yn} ≤ λ·max{X1 , · · · , Xn }+(1−λ)·max{Y1 , · · · , Yn }, by definition of MAXVAR we get MAXVARn (λX + (1 − λ)Y ) = E(max{λX1 + (1 − λ)Y1, · · · , λXn + (1 − λ)Yn }) ≤ λ · E(max{X1 , · · · , Xn }) + (1 − λ) · E(max{Y1 , · · · , Yn }) = λ · MAXVARn (X) + (1 − λ) · MAXVARn (Y ). Proof of (A3): For any X ≤ Y satisfying X, Y ∈ L 2 , suppose X1 , · · · , Xn are i.i.d. copies of X and Y1 , · · · , Yn are i.i.d. copies of Y . We can see that P0 (X ≤ t) ≥ P0 (Y ≤ t) for any t ∈ R since X ≤ Y . Then we have Z 0 Z +∞ MAXVARn (X) = [P0 (max{X1 , · · · , Xn } > t) − 1]dt + P0 (max{X1 , · · · , Xn } > t)dt −∞ 0 Z 0 Z +∞ n =− (P0 (X ≤ t)) dt + [1 − (P0 (X ≤ t))n ]dt (2.1) −∞ 0 Z 0 Z +∞ n ≤− (P0 (Y ≤ t)) dt + [1 − (P0 (Y ≤ t))n ]dt −∞ 0 Z 0 Z +∞ = [P0 (max{Y1, · · · , Yn } > t) − 1]dt + P0 (max{Y1 , · · · , Yn } > t)dt 0
−∞
= MAXVARn (Y ). Proof of (A4): Suppose X k (k = 1, 2, · · · ), X ∈ L 2 and kX k − Xk2 → 0 as k tends to infinity. Then X k → X in distribution. Denote by Fk (t) the distribution function of X k (k = 1, 2, · · · ) and by F (t) the distribution of X. Then lim Fk (t) = F (t) for all k→∞
continuous points of F (·). It implies that lim [Fk (t)]n = [F (t)]n for all continuous points of k→∞
[F (·)]n . Note that [Fk (t)]n is the distribution function of max{X1k , · · · , Xnk } and [F (t)]n is the distribution function of max{X1 , · · · , Xn }, where X1k , · · · , Xnk are i.i.d. copies of X k (k = 1, 2, · · · ) and X1 , · · · , Xn are i.i.d. copies of X. Therefore, we have max{X1k , · · · , Xnk } → max{X1 , · · · , Xn } in distribution, and MAXVARn (X k ) = E(max{X1k , · · · , Xnk }) → E(max{X1 , · · · , Xn }) = MAXVARn (X) as k tends to infinity. Thus, if MAXVARn (X k ) ≤ 0 for all k = 1, 2, · · · then MAXVARn (X) ≤ 0. The proof of the theorem is completed.
✷ 3
3
Risk-averseness of MAXVAR
Suppose R is a functional from L 2 to (−∞, +∞]. Recall that an averse risk measure is defined by axioms (A1), (A2), (A4), (A5) and (A6) R(X) > E(X) for all non-constant X. We then have the next theorem. Theorem 3.1 If n ≥ 2, then MAXVARn (·) is averse.
F¨ollmer and Schied [4] proved that if R is a coherent, law-invariant risk measure in L ∞ (not L 2 ) other than E(·), then R is averse, where “law-invariant” stands for that R(X) = R(Y ) whenever X and Y have the same distribution under P0 . Since we are now considering the L 2 case, we cannot use the result in F¨ollmer and Schied [4] directly. We next give a separate proof. Proof of Theorem 3.1 On one hand, for any X ∈ L 2 , let X1 , · · · , Xn be i.i.d. copies of X. Then we have MAXVARn (X) = E(max{X1 , · · · , Xn }) ≥ E(X1 ) = E(X). On the other hand, if MAXVARn (X) = E(X) (n ≥ 2), then E(max{X1 , · · · , Xn }) = E(X1 ). So max{X1 , · · · , Xn } = X1 almost surely. Similarly, max{X1 , · · · , Xn } = X2 almost surely. Therefore, X1 = X2 almost surely. But X1 and X2 are independent, we must have X1 equals to a constant almost surely, which is equivalent to say X equals to a constant almost surely. Therefore, MAXVARn (X) > E(X) for nonconstant X, which implies that MAXVARn (·) is averse when n ≥ 2.
4
✷
MAXVAR as a continuous convex combination of CVaR
An important coherent risk measure in basic sense is the conditional value at risk (CVaR) popularized by Rockafellar and Uryasev [6]. Among several equivalent definitions of CVaR, the most familiar one is probably the following. CVaRα (X) = min β + β∈R
4
1 E(X − β)+ , 1−α
(4.1)
where (t)+ = max(t, 0), the minimum is attained at β ∗ = VaRα (X), and the VaR ( “Valueat-Risk”) is defined as VaRα (X) := inf {ν ∈ R : P0 (X > ν) < 1 − α} .
(4.2)
In this section, we show that MAXVARn (·) is certain “continuous convex combination” of the CVaR measure in the sense that MAXVARn (·) =
Z
1
CVaRα (·)w(α)dα, 0
where w(α) (α ∈ [0, 1]) is the “weight function” which satisfies w(α) ≥ 0 on [0, 1] and R1 w(α)dα = 1. Specifically, we have the next theorem. 0 Theorem 4.1 For any X ∈ L 2 , we have
MAXVARn (X) =
Z
1
CVaRα (X)w(α)dα, 0
where w(α) := n(n − 1)(1 − α)αn−2,
α ∈ [0, 1]
is the weight function. Theorem 4.1 was mentioned in Cherny and Orlov [3] without details. We now give a detailed proof by using the so called “Choquet integral”. First, we need a lemma. For any α ∈ [0, 1], define fα (·) : Σ → [0, 1] in the following way, 1 P (A) if P0 (A) ≤ 1 − α, 1−α 0 fα (A) : = 1 otherwise. = gα [P0 (A)],
where gα (x) :=
1 x 1−α
if x ∈ [0, 1 − α],
1
if x ∈ [1 − α, 1].
(4.3)
We then have the following lemma, which implies that the CVaR measure can be written as the “Choquet integral” with respect to fα (·). Lemma 4.1 For any X ∈ L 2 and α ∈ [0, 1], we have CVaRα (X) =
Z
0
[fα (X > t) − 1]dt +
Z
0
−∞
5
+∞
fα (X > t)dt.
Proof. If VaRα (X) ≤ 0, then Z 0 Z +∞ [fα (X > t) − 1]dt + fα (X > t)dt −∞ 0 Z +∞ Z 0 1 1 P0 (X > t) − 1 dt + P0 (X > t)dt = 1−α 0 VaRα (X) 1 − α Z +∞ 1 · P0 (X > t)dt =VaRα (X) + 1 − α VaRα (X) 1 =VaRα (X) + · E[(X − VaRα (X))+ ] = CVaRα (X). 1−α The last step above is due to (4.1) and (4.2). If VaRα (X) > 0, then Z
=
0
[fα (X > t) − 1]dt +
−∞ Z VaRα (X)
dt +
0
+∞
Z
Z
+∞
fα (X > t)dt 0
1 P0 (X > t)dt 1−α
VaRα (X) Z +∞
1 · P0 (X > t)dt 1 − α VaRα (X) 1 · E[(X − VaRα (X))+ ] = CVaRα (X), =VaRα (X) + 1−α
=VaRα (X) +
which completes the proof.
✷
Proof of Theorem 4.1 Define h(x) := 1 − (1 − x)n ,
x ∈ [0, 1].
It is not difficult to check that h(x) =
Z
1
gα (x)w(α)dα,
x ∈ [0, 1],
(4.4)
0
where gα (x) is as defined in (4.3). By (2.1), for any X ∈ L 2 we have MAXVARn (X) =
Z
0
[h(P0 (X > t)) − 1]dt +
−∞
Z
+∞
h(P0 (X > t))dt.
So by (4.4), (4.5) and Lemma 4.1, together with Fubini’s theorem and the fact that 1, we get MAXVARn (X) =
Z
0 −∞
Z
1
[fα (X > t) − 1]w(α)dαdt +
0
Z
0
6
(4.5)
0
+∞
Z
0
R1 0
w(α)dα =
1
fα (X > t)w(α)dαdt
= =
Z
Z
1 0
Z
0
[fα (X > t) − 1]dt +
0
−∞
1
Z
+∞
fα (X > t)dt w(α)dα
CVaRα (X)w(α)dα 0
for any X ∈ L 2 , as desired.
✷
Remark. Theorem 4.1 says that MAXVARn (·) is a continuous convex combination of the CVaR measure, its coherency in basic sense follows from Proposition 2.1 of Ang et al [1]. Therefore, Theorem (4.1) provides an alternative proof of the coherency of MAXVARn (·).
5
The risk envelope of MAXVAR
Since MAXVARn (·) =
Z
1
CVaRα (·)w(α)dα
0
is a coherent risk measure on L 2 , by the dual representation theorem (Rockafellar [5]), there exists a unique, nonempty, convex and closed set Qn ⊆ L 2 , called “the risk envelope of MAXVARn (·)” such that MAXVARn (X) = sup E(XQ) Q∈Qn
for any X ∈ L 2 . In this section we aim at characterizing the risk envelope of MAXVARn (·)”. First recall the following well-known result for the discrete convex combination of the CVaR measure, which can be found in Rockafellar [5] and whose proof can be found in Ang et al [1]. Proposition 5.1 Let R(·) =
n P
λi CVaRαi (·) with positive weights λi adding up to 1. Then
i=1
R is a coherent risk measure in the basic sense and its risk envelope is ( n ) X 1 λi Qi : 0 ≤ Qi ≤ , E(Qi ) = 1, ∀ı = 1, 2, · · · , n . 1 − α i i=1
A continuous version of Proposition 5.1 gives the risk envelope of MAXVAR as follows. Theorem 5.1 The risk envelope of MAXVAR is Z 1 1 , E(Qα ) = 1, ∀α ∈ [0, 1] , Qn := cl Qα w(α)dα, 0 ≤ Qα ≤ 1−α 0 7
(5.1)
where w(α) := n(n − 1)(1 − α)αn−2
α ∈ [0, 1]
is the weight function (00 is defined to be 1), and “cl” stands for the closure in L 2 . R1 Proof. Note that the integration “ 0 Qα w(α)dα” in (5.1) is defined pointwise. That is, R1 R1 1 Y = 0 Qα w(α)dα means Y (ω) = 0 Qα (ω)w(α)dα for any ω ∈ Ω. Since 0 ≤ Qα ≤ 1−α for any α ∈ [0, 1], we have Z 1 Z 1 0≤ Qα (ω)w(α)dα ≤ n(n − 1)αn−2dα = n 0
0
∞
2
for any ω ∈ Ω. Therefore, Qn ⊆ L ⊆ L . In addition, we can check that Z 1 MAXVARn (X) = CVaRα (X)w(α)dα 0 Z 1 Z 1 = sup E X · Qα w(α)dα : Qα w(α)dα ∈ Qn 0
(5.2)
0
for any X ∈ L 2 . Furthermore, it is easy to check the convexity of Qn . Since Qn is closed in L 2 , it follows from the dual representation theorem that Formula (5.2) implies that (5.1) is the risk envelope of MAXVARn (·).
✷
References [1] M. Ang, J. Sun and Q. Yao (2017) On the dual representation of coherent risk measures, Ann. Oper. Res. to appear. DOI: 10.1007/s10479-017-2441-3. [2] A. Cherny and D. Madan (2009) New measures for performance evaluation, Rev. Finan. Stud. 22 2571-2606. [3] A. Cherny and D. Orlov (2011) On two approaches to coherent risk contribution, Math. Finance 23 557-571. [4] H. F¨ollmer and A. Schied (2002).Stochastic Finance. Walter de Gruyter, Berlin, Germany. [5] R. T. Rockafellar (2007) Coherent approaches to risk in optimization under uncertainty, Tutorials in Operations Research, INFORMS 38-61. [6] R.T. Rockafellar and S. Uryasev (2000). Optimization of conditional value-at-risk. Journal of Risk, 2(3)3, 21-42.
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