On Distortion Functionals

On Distortion Functionals Georg Ch. Pflug1

Abstract Distorted measures have been used in pricing of insurance contracts for a long time. This paper reviews properties of related acceptability functionals in risk management, called distortion functionals. These functionals may be characterized by being mixtures of average values-at-risk. We give a dual representation of these functionals and show how they functionals may be used in portfolio optimization. An iterative numerical procedure for the solution of these portfolio problems is given which is based on duality.

Keywords: Risk measures, insurance premium, portfolio optimization

1

Introduction: Distortion functionals as insurance premia

Let L be a random variable describing the (nonnegative) loss distribution of an insurance contract. Let GL be the pertaining distribution function GL (u) = P{L ≤ u}. How much premium should the insurance company ask for coverage of L? Obviously, the premium should be greater than E[L] otherwise the insurance company will go bankrupt for sure. Based on the well known formula Z 1 E[L] = (1 − GL (u)) du 0

a safe insurance premium can be defined by Z 1 πψ [L] = ψ(1 − GL (u)) d(u),

(1)

0 1

Department of Statistics and Decision Support Systems, Universitaetsstrasse 5, University of Vienna, 1090 Wien-Vienna, Austria; e-mail: [email protected]

1

where ψ is function mapping [0,1] to [0,1], such that ψ(u) ≥ u

for u ∈ [0, 1].

(2)

The condition (2) guarantees that the premium is not smaller than the expectation. However one usually considers more specific functions ψ. Definition. Distortion functions. A function ψ is called a distortion function, if it is is monotonic, left continuous and satisfies ψ(u) ≥ u, ψ(0) = 0, ψ(1) = 1. If ψ is a distortion function, πψ called a distortion insurance premium. Distortion functions as premium principles were introduced by Deneberg ([8]) and further developed by S.S. Wang ([19]) among others. For instance, the following distortion functions have been proposed • The power distortion ψ(u) = ur ,

0 < r < 1.

• The Wang distortion ψ(u) = Φ(Φ−1 (u) + λ);

λ>0

where Φ is the distribution function of the standard normal. 1

0.9

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0.5

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0.1

0

0

0.1

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Fig. 1. The power distortion function (r = 0.3, dotted) and the Wang distortion function (λ = 1, dashed) Notice that the function ψ(u) = κu for κ > 1 gives a valid insurance premium by (2), called the proportional loading, it is however not a distortion function in the sense of Definition 1. The formula (1) may be interpreted in two ways. Firstly, one may argue that instead of the loss distribution function GL one considers the ψ-distorted version GL,ψ (u) = 1 − ψ(1 − GL (u)). and takes the expectation of the latter. Notice that if ψ is a distortion function, then GL,ψ is a probability distribution function and (2) implies that GL,ψ dominates GL in the first order sense. 2

R1 Secondly, as will be argued below, πψ [L] = − 0 G−1 L (p) dψ(1 − p). For a distortion function ψ, −dψ(1 − p) is a probability distribution on [0,1]. While the expectation of L equals the expectation of G−1 L (U ) with a uniform [0,1] U , the insurance premium πψ [L] is calculated by changing the uniform distribution dp on [0,1] to a new distribution −dψ(1 − u), which puts more weight to larger loss values. This justifies the alternative name change-ofmeasure premium principle. If V has distribution P{V ≤ u} = 1 − ψ(1 − u) on [0,1], then the random variable Z = G−1 L (V ) has the distorted distribution GL,ψ .

2

Distortion acceptability functionals

In the previous section we considered nonnegative loss variables L. In this section we extend the concept to arbitrary profit variables Y , for which the analogon to the insurance premium is the acceptability value: Let H be a monotonic, right continuous bounded function on [0,1] satisfying H(0) = 0. We do not necessarily require that H generates to a probability measure, i.e. H(1) = 1, but sometimes we will. Assume that the profit variable Y has distribution function G and that G−1 is its left continuous inverse, called the quantile function, i.e. G−1 (p) = inf{u : G(u) ≥ p}. Definition. Distortion acceptability functionals. Let GH be the set of distributions G for which G−1 is dH integrable. For G in GH we define the distortion acceptability functional Z 1 AH {G} = G−1 (p) dH(p). 0

If the random variable Y has distribution function G, we also use the notation AH [Y ], if no confusion may occur. We allow the distribution G to have jumps to cover finite sample situations. The following formula for partial integration is well known Z ∞ Z 0 Z ∞ E[Y ] = u dG(U ) = − G(u) du + (1 − G(u)) du. −∞

−∞

0

The next Lemma generalizes this formula. Notice that we allow H to have jumps at values G(u), for which G(u) and G(u−) = limv↑u G(v) are different. A similar formula is found in [13]. Lemma 1. Let G ∈ GH . Then Z ∞ Z 0 Z ∞ −1 ¯ − G(u)) du (3) G (p) dH(p) = − H(G(u)) du + H(1 −∞

−∞

0

3

with

¯ H(u) = H(1) − H(1 − u).

Proof. We start with stating that Z 1 Z −1 G (p) dH(p) = 0

Notice first that Z 1

∞

u dH(G(u)). −∞

Z −1

∞

G (p) dH(p) = 0

G−1 (G(u)) dH(G(u)).

−∞ −1

Now notice that G (G(u)) ≤ u and G−1 (G(u)) < u only if there is a v < u such that G(v) = G(u) and consequently H(G(v)) = H(G(u)). Thus setting A = {u : G−1 (G(u)) = u}, we see that Z Z ∞ −1 G−1 (G(u)) dH(G(u)) G (G(u)) dH(G(u)) = −∞ ZA Z ∞ = u dH(G(u)) = u dH(G(u)) A

−∞

The partial integration formula for Stieltjes integrals is Z Z K1 (u+) dK2 (u)+ K2 (u−) dK1 (u) = K1 (b+)K2 (b+)−K1 (a−)K2 (a−). [a,b]

[a,b]

An application of this formula gives Z Z Z ∞ u dH(G(u)) = u dH(G(u)) + u dH(G(u)) −∞ (−∞,0] (0,∞) Z Z = u dH(G(u)) − u d[H(1) − H(G(u))] (−∞,0] (0,∞) Z = − H(G(u)) du + 0 · H(G(0)) − (−∞) · H(G(−∞)) (−∞,0] Z + [H(1) − H(G(u))] du − (0,∞)

∞ · [H(1) − H(G(∞))] + 0 · [H(1) − H(G(0))] Z 0 Z ∞ ¯ − G(u)) du. = − H(G(u)) du + H(1 −∞

0

2 Remark. If Y is nonnegative and H is a probability distribution, then AH [Y ] = πH¯ [Y ] as is an easy consequence of the Lemma. 4

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1

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distorted df. H(G)

H 0.6

original df. G

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0 −3

1

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G(0)

Fig. 2. The (nonconcave) distortion function H(u) = u + (1 − u)2 + (1 − u)3 − (1 − u)4 − (1 − u)5 (left), an original distribution G(·) (the standard normal) and its distortion H(G(·)) (right). Distortion acceptability functionals appear also under the name of spectral measures (Acerbi [1]). Their role in determining the needed risk capital has been emphasized by Artzner et al. [5] and H¨ urlimann [12]. Examples for distortion acceptability functionals. • Setting H(p) = p, one gets the expectation Z Z 0 Z 1 −1 G(u) du + G (p) dp = − E[Y ] =

(1 − G(u)) du. 0

−∞

0

∞

• Setting H(p) = 1l[α,1] (p) one gets the value-at-risk V@Rα [Y ] = G−1 (α). • Setting H(p) = min(p/α, 1) one gets the average value-at-risk Z 1 α −1 AV@Rα [Y ] = G (p) dp α 0 Z 0 Z ∞ = − min(G(u)/α, 1) du + max(1 − G(u)/α, 0) du. −∞

0

Several other names have been proposed for this functional, such as conditional value-at-risk (Rockefellar and Uryasev [18]), expected shortfall (Acerbi and Tasche [2]) and tail value-at-risk (Artzner et al. [4]). The name average value-at-risk is due to F¨ollmer and Schied ([10]). Rp • If H has a density, i.e. H(p) = 0 h(q) dq the pertaining class coincides with Yaari’s dual functionals, see [20]. 5

3

Dual representations

If U is uniform [0,1], then G−1 (U ) has the same distribution as Y . The converse is not true: If G has jumps, then G(Y ) is not uniformly distributed, it is in fact stochastically larger than a uniform [0,1] distribution, P{G(Y ) ≤ p} ≤ P{U ≤ p}. To correct the distribution of G(Y ) towards a uniform distribution, let U be the countable set of all jump points of G. For u ∈ U , let (Vu ) be a collection of independent random variables, which are Uniform [0, G(u) − G(u−)] distributed and which are independent of Y . For u ∈ / U , set Vu = 0. By composition, one may form the random variable VY . Lemma 2. G(Y ) − VY is uniformly [0,1] distributed and G−1 (G(Y ) − VY ) = Y a.s. Proof. Let 0 < p < 1. We have to show that P{G(Y ) − VY ≤ p} = p. One may find an u such that G(u−) ≤ p ≤ G(u). If G(u−) = G(u), then Vu = 0 and P{G(Y ) ≤ p} = p. Otherwise G(Y ) − VY ≤ p iff Y < u or Y = u and Vu ≥ G(u) − p, i.e. P{G(Y ) − VY ≤ p} = G(u−) + [G(u) − G(u−)][p − G(u−)]/[G(u) − G(u−)] = p. To prove the second assertion, notice that conditional on Y = u, G(u)−Vu lies in the interval [G(u−), G(u)] and with probability 1 in (G(u−), G(u)]. However in the latter interval, G−1 equals 2 R p u. Lemma 3. Suppose that H(p) = 0 h(α) dα. Then AH [Y ] = E[Y h(G(Y ) − VY )]. Proof. Let U = G(Y ) − VY . Then by Lemma 2, U is uniform[0,1] and G−1 (U ) = Y a.s. and therefore AH (Y ) = E[G−1 (U )h(U )] = E[Y h(G(Y ) − VY )]. 2 The next result shows the dual representation of AH . Proposition 1. (see also Pflug [16]) AH [Y ] = inf{E[Y Z] : Z = h(U ), where U is uniformly [0,1] distributed}. which is the same as AH [Y ] = inf{E[Y Z] : Z ≺CXD Z ∗ , where Z ∗ = h(U ), with U uniformly [0, 1] distributed }. The infimum in (4) is attained for Z = h(Y − VY ). Proof. By virtue of Lemma 3, AH [Y ] ≥ inf{E(Y Z) : Z = h(U ),U Uniform[0,1]}. To prove the other inequality, notice Hoeffdings Lemma first ([11], see also [15]): Z Z ∞

∞

−∞

−∞

Cov(Y, Z) =

GY,Z (u, v) − GY (u) · GZ (v) du dv 6

where GY,Z is the joint distribution and GY , GZ are the marginal distributions of the vector (Y, Z). If we fix the marginals, the covariance Cov(Y, Z) (and equivalently E(Y Z)) is minimized for the joint distribution, which is the lower Fr´echet bound GY,Z (u, v) = max(GY (u) + GZ (v) − 1, 0), i.e. if Y and Z are antimonotone coupled. Since h is nonincreasing and G is nondecreasing, it is easy to see that the variables Y and h(G(Y ) − VY ) are indeed antimonotone coupled and therefore (4) is the lower bound of E(Y Z) : Z = h(U ). For the second representation, we use a result by Dentcheva and Ruszczynski ([9]): L conv {Z : Z = V } = {Z : Z ≺CXD V }. Here Z ≺CXD V iff E[f (Z)] ≤ E[f (V )] for all convex functions f . 2 Since the functions H(p) = min(p/α, 1) are the extremal elements in the set of all concave, monotonic probability distribution functions on [0,1], there is a Choquet representation for distortion functionals (This particular representation is not due to Choquet [6], but in his spirit). Proposition 2. Choquet representation. (AcerbiR[1], [13]). Suppose p that H concave, i.e. has a representation as H(p) = 0 h(α) dα with a nonincreasing h(p). Then Z 1 AH {G} = AV@Rα {G} dK(α) (4) 0

for a monotonically increasing K, which satisfies K(0) = 0, K(1) = H(1). Thus for concave H, the distortion functionals are mixtures of AV@R’s. Proof. We may assume that the nonincreasing function h defined on [0, 1] is continuous from the left and has the representation Z 1 h(p) = 1l[0,α] (p) dK(α). (5) (0,1] α To show (5), let hn be the largest nonincreasing left continuous stepfunction, which is dominated by h and which Rjumps only at dyadic rational points k/2n . Clearly, one may write hn (p) = (0,1] α1 1l[0,α] (p) dKn (α) with Kn (0) = 0, R1 Kn (1) = 0 hn (p) dp ≤ H(1). The sequence dKn is a sequence of bounded measures on [0,1], which has a weak limit dK with K(0) = 0, K(1) =

7

limn

R1 0

hn (p) dp = H(1). Now Z

1

Z

1

G (p) dH(p) = G−1 (p)h(p) dp 0 Z0 1 Z 1 = G−1 (p) 1l[0,α] (p) dK(α) dp 0 (0,1] α Z Z 1 1 −1 = G (p)1l[0,α] (p) dp dK(α) (0,1] α 0 · Z α ¸ Z Z 1 −1 = G (p) dp dK(α) = AV@Rα (G) dK(α). (0,1] α 0 (0,1]

AH {G} =

−1

2 Proposition 3. Properties of distortion functionals. (see also De Giorgi [7]) R1 Distortion functionals AH [Y ] = 0 G−1 (u) dH(u) are (i) version independent, i.e. they depend on the random variable Y only through its distribution function G (ii) linear in the quantile function G−1 and hence comonotone additive, (iii) positively homogeneous, i.e. AH [cY ] = cAH [Y ] for c > 0 (iv) isotonic w.r.t. first order stochastic dominance (≺F SD ). If dH is a probability measure (i.e. H(1) = 1) then AH is (iv) translation equivariant, i.e. AH [Y + a] = AH [Y ] + a. Rp If H is concave (i.e. H(p) = 0 h(p) dp) for a nonincreasing h, then (vi) concave, i.e. A[λY1 +(1−λ)Y2 ] ≥ λAH [Y1 ]+(1−λ)A[Y2 ] for 0 ≤ λ ≤ 1, (vii) isotonic w.r.t. second order stochastic dominance (≺SSD ). (viii) If H is concave and h ∈ Lq [0, 1], then Y 7→ AH (Y ) is continuous in the Lp sense, where 1/p + 1/q = 1. Proof. (i) is obvious. (ii). Recall that a functional F is called comonotone additive if for two comonotone random variables Y1 and Y2 , F[Y1 + Y2 ] = F[Y1 ] + F[Y2 ]. Two random variables are comonotone, if the joint distribution G1,2 (u) is related to the marginal distributions G1 , G2 by G1,2 (u, v) = min(G1 (u), G2 (v)). For comonotone variables, the quantile function of Y1 +Y2 8

−1 is the sum of the marginal quantile functions: G−1 1 (u) + G2 (u). (iii). Notice that cY has quantile function cG−1 (u) for positive c. (iv) Y1 is dominated in the first order sense if G1 (u) ≥ G2 (u), which is equivalent to −1 G−1 1 (u) ≤ G2 (u) for all u. (iv) The variable a + Y has quantile function a + G−1 (u). (vi). If H is concave then AH [Y ] has a dual representation as given in Proposition 1. Thus AH is the infimum of linear (in Y ) functions and hence concave. Finally, for concave H, AH (Y ) has by Proposition 2 a representation as a mixture of AV@R’s. The assertion (vii) is proved, if we show it holds true for all AV@Rα . It is well known (cite here Uryasev and Rockafellar [17]) that 1 AV@Rα [Y ] = max{a − E([Y − a]− ) : a ∈ R}. α 1 For fixed a, the function Y 7→ a − α E([Y − a]− ) is monotonic w.r.t. second order stochastic dominance, since the function y 7→ a− α1 [y−a]− is monotonic and concave. Thus also the maximum is monotonic w.r.t. second order stochastic dominance and proves (vii). Finally, notice that because of (4) AH is upper semicontinuous. If AH is not continuous, one may find a sequence Yn converging to Y in Lp sense, such that

AH (Y ) ≥ lim sup AH (Yn ) + ² n

for some ² ≥ 0. Represent AH [Yn ] = E[Yn h(Un )] with Un a Uniform[0,1] ¯ (say) for all n. If kY − Yn kn < ²/h, ¯ then variable. Notice that kh(Un )kq = h ¯ n − Y kn < AH (Yn ) + ², AH (Y ) ≤ E[Y h(Un )] ≤ E[Yn h(Un )] + hkY a contradiction. Notice that (iii) and (vi) imply that AH is superadditive

2

AH [Y + Y¯ ] ≥ AH [Y ] + AH [Y¯ ]. Remark. Kusuoka [14] has shown that any version independent functional F, which is positively homogeneous, translation equivariant and coincides with its concave bidual can be represented as Z 1 F[Y ] = inf{ AV@Rα (Y ) dM (α) : M ∈ M}, 0

where M is a set of probability measures on [0,1]. If F is comonotone additive, then M contains only one probability measure and F has a Choquet representation of the form (4). In Proposition 2 we have derived this result in 9

a direct manner by relating the mixture measure M to the distortion function H. Proposition 3 has an inverse given by the following result. Proposition 4. Let A{G} be version independent functional, which is finite for bounded distributions. If A is positively homogeneous, monotonic w.r.t. R 1 −1FSD and comonotone additive, then it is of the form A{G} = G (u) dH(u). 0 Proof. W.l.o.g we may assume that A(1l) = 1 and A(0) = 0. By positive homogeneity A(c) = c for c ≥ 0. By A(−c) + A(c) = A(0) = 0, also A(c) = c for c < 0. Let Gp (u) be the distribution which has point masses at 0 (with probability p) and 1 (with probability 1 − p). Let H(p) = 1 − A(Gp ). Then H isRmonotonically nondecreasing, satisfies H(0) = 0, H(1) = 1. Let 1 A0 (G) = 0 G−1 (p) dH(p). Notice that any discrete distribution can be represented as a comonotone sum of discrete distribution with just two point masses. This implies that A coincides with A0 for all discrete distributions. If G is bounded and nondiscrete, one may find, for every ², two discrete distributions G1 and G2 such that G1 ≺F SD G ≺F SD G2 such that A0 (G2 ) − A0 (G1 ) ≤ ². Since also A(G1 ) ≤ A(G) ≤ A(G2 ) one sees that in fact A(G) = A0 (G). 2

4

Portfolio optimization

In this section, we consider a one-period portfolio optimization problem, where the objective is to maximize the expected return under the constraint that a distortion acceptability functional does not fall below a prespecified level. Since the negative distortion risk functional has the interpretation as required risk capital (see [3]), one may equivalently say that the portfolio optimization problem seeks for maximizing the return for a given maximal risk capital. Let ξ = (ξ (1) , . . . , ξ (M ) ) a row vector of random portfolio returns defined on some probability space (Ω, B, P) and x = (x1 , . . . , xM )> the column vector of weights. The total portfolio has the value Yx = ξ · x. The optimization problem reads

10

° ° ° ° ° ° ° ° ° °

Maximize (in x) : E(ξ) · x subject to AH [ξ · x] ≥ q x> 1l = 1 x≥0

(6)

Here 1l is the column vector of length M with entries 1. (6) is a linear optimization problem under a convex constraint. To derive the necessary conditions for optimality, a characterization of the supergradient set ∂AH [Y ] = {Z ∈ Lq : AH [Y¯ ] ≤ AH [Y ] + E[(Y¯ − Y ) · Z] for all Y¯ ∈ Lp } of AH at Y is needed. Assume that Y ∈ Lp (Ω), p ≥ 1 and qR= p/(p − 1) for u p > 1 and q = ∞ for p = 1. Assume further that H(u) = 0 h(v) dv with h ∈ Lq [0, 1]. Lemma 4. The supergradient of AH is ∂AH [Y ] = {h(G(Y ) − VY )}, where VY runs through all random variables on Ω, for which the conditional distribution of VY given Y = u is uniform [0, G(u) − G(u−)]. The supergradient set is a singleton if and only if G is continuous, given that Ω is rich enough to carry a uniform [0,1] distribution. Proof. Let Z = {Z = h(U ), where U is uniformly [0,1] distributed}. By (4) AH [Y ] = E[Y · h(G(Y ) − VY )] = inf{E[Y Z] : Z ∈ Z}. Since AH [Y¯ ] ≤ E[(Y¯ h(G(Y ) − VY )], one sees that h(G(Y ) − VY ) ∈ ∂AH [Y ] for all versions of VY . Conversely, let Z¯ ∈ ∂AH [Y ]. If Z¯ is not in Z, then there is an ² > 0 and ¯ + ² ≤ inf{E[Y¯ Z] : Z ∈ Z} = AH [Y¯ ]. Consequently a Y¯ such that E[Y¯ Z] ¯ ≤ AH [Y ] + AH [Y¯ ] − ² AH [Y + Y¯ ] ≤ AH [Y ] + E[Y¯ Z] ≤ AH [Y + Y¯ ] − ², a contradiction. Thus the first assertion has been proved. If G is continuous, then VY = 0 and thus the supergradient is unique. If however G has jumps, then h(G(Y ) − VY ) may be realized in different ways and is thus not unique. 2 Lemma 5. If H has a nonnegative density h ∈ Lq [0, 1] and G is continuous, then Y¯ 7→ AH [Y¯ ] is strongly differentiable in Lp sense at Y . The derivative is ∇AH [Y ] = h(G(Y )). Consequently also the mapping x 7→ AH [ξ · x] is differentiable at Yx = ξ > x with derivative ∇x A[ξ · x] = E[ξ · h(G(ξ · x)]]. 11

Proof. Let Z = ∂AH (Y ). If AH is not differentiable at Y , then there is a sequence Yn converging to Y in Lp sense such that AH (Yn ) − AH (Y ) − E[(Yn − Y )Z] ≤ −²kYn − Y kp .

(7)

for some ² > 0 and all n. Let Zn ∈ ∂AH (Yn ). Then AH (Y ) − AH (Yn ) − E[(Y − Yn )Zn ] ≤ 0

(8)

(7) and (8) together give kYn − Y kp · kZn − Zkq ≥ E[(Yn − Y )(Z − Zn )] ≥ ²kYn − Y kp .

(9)

By uniform integrability, the sequence (Yn ) must have a weak cluster point, say Z 0 . By (9), Z = 6 Z 0 . On the other hand, looking at AH (Y¯ ) ≤ A(Yn ) + E[(Y¯ − Yn )Zn ], for all Y¯ , using the Lp -continuity of AH (see Proposition 3 (viii)) it follows that letting n → ∞ AH (Y¯ ) ≤ A(Y ) + E[(Y¯ − Y )Z 0 . Thus Z 0 must be a further element of ∂AH (Y ), a contradiction to the assumption. 2 A slight modification of (6) is the problem ° ° Maximize E[ξ] · x ° ° subject to ° ° AH [ξ · x] = q ° > ° x 1l = 1 ° ° x≥0 Under the assumption of differentiability, the necessary Karush-KuhnTucker (KKT-) conditions for this problem are: E(ξ) + λ∇x AH [ξ · x] + µ1l + γ = 0 λ(AH [ξ · x] − q) = 0 µ(x> 1l − 1) = 0 γm xm = 0 for m = 1, . . . , M γ≥0 Suppose that the optimum lies in the interior of X, i.e. asset not present in the portfolio are neglected. Interpreting −AH as the necessary risk capital, the KKT conditions can be formulated as xm em ∇x AH [ξ · x] AH [ξ·x]−AH [ξ·x], E[ξ (m) ]−E[ξ·x] is proportional to PM x e ∇ A [ξ · x] m m x H m=1 12

where em is the m-th unit vector. Introducing the quantity xm em ∇x AH [ξ · x] − PM AH [ξ · x] m=1 xm em ∇x AH [ξ · x] as the local risk capital contribution, the following condition holds at optimality: For each asset in the portfolio, the return contribution minus total return is proportional to the local risk capital contribution minus the total risk capital.

5

Numerical portfolio optimization

A numerical procedure to solve (6) may be based on the dual representation: Using A[Y ] = inf{E[Y Z] : Z ∈ Z} this problem may be solved by the following dual iterative procedure: 1. Set Z˜ = ∅. 2. Outer problem. Solve ° ° Maximize (in x) : E[ξ] · x ° ° subject to ° ° E[Yx Z] ≥ q for all Z ∈ Z˜ ° ° xT 1l = 1 ° ° x≥0 3. Inner problem. With the incumbent solution x, solve v = inf{E[Yx Z] : Z ∈ Z}. If v ≥ q, then stop. Otherwise add the minimizer function Z = argmin {E[Yx Z] : Z ∈ Z} to the set Z˜ and goto 2. Proposition 6. The dual iterative procedure stops only at optimal points. Proof. Suppose that the procedure generates a sequence of solutions x1 , . . . , xn and dual variables Z1 , . . . , Zn and stops then. Notice that at step n the outer problem solves the problem ° ° Maximize (in x) : E(ξ) · x ° ° subject to ° ° An [Yx ] := inf{E[Yx Z] : Z ∈ Zn } ≥ q (10) ° T ° x 1l = 1 ° ° x≥0 13

where Zn is the convex hull of Z1 , . . . , Zn . Notice that An ≥ A, i.e. the constraint set of this outer problem contains the original constraint set. Since the inner problem stopped, we know that AH [Yxn ] ≥ q, i.e. that xn is feasible for the original outer problem. This proves that xn is a solution of the original problem. 2 Let us consider in detail the case of a finite probability space Ω = {ω1 , . . . ωS }. Let us form the probability vector   P (ω1 )   .. p=  . P (ωS ) and the [S × M ] value matrix with entries Ξs,m = ξ (m) (ωs ). The set Z is a set of [1 × S] vectors. The outer problem reads Maximize p> Ξx such that z > diag (p)Ξx ≥ q xT 1l = 1 x≥0

for all z ∈ Z

(11)

Consider the inner problem inf{E[Yx Z] : Z = h(U ), where U is uniformly [0,1] distributed}.

(12)

for a decreasing h. We know from the proof of Proposition 1 that E(Yx Z) is minimized for all Z which have a given distribution, if Yx and Z are coupled in an antimonotone way, i.e. have the Fr´echet lower bound copula C(p, q) = max(p + q − 1, 0). For the discrete model, the portfolio value Yx = Ξx takes the value ys , which is the s-th element of the column vector Ξx with probability ps , s = 1, . . . S. Suppose that the ordered values (ys ) are (y[s:S] ) and ys = y[π(s):S] for some Ps permutation π. Let p˜π(s) = ps be the pertaining probabilities and p¯s = i=1 p˜i and Z p¯s z˜s = h(u) du s = 1, . . . S. p¯s−1

Let finally zs = z˜π(s) . Then the minimal value of (12) is S X

y[s:S] · z˜s =

s=1

S X s=1

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ys · zs

and the minimizer takes the value zs for the scenario s. Form the vector z = (z1 , . . . , zS )> . then the constraint to be added for the outer problem is u> x ≥ q where u> = z > diag (p) Ξ. Proposition 7. For a finite probability space, the dual iterative procedure stops after finitely many steps at a solution. Proof. The set Z is a polyhedral set of vectors in Rs . This set has only finitely many extremal points. The inner problem generates at each step a new extremal point. Assume that the procedure does not stop at step n. ˜ < inf{E[Yxn z] : z ∈ {z1 , . . . zn }} and therefore Then inf{E[Yxn z] : z ∈ Z} ˜ cannot be contained in the zn+1 , the minimizer of inf{E[Yxn z] : z ∈ Z}, convex hull of z1 , . . . , zn−1 . Thus the procedure must stop in finitely many steps. 2

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