Generalized Performance Ratios and Risk Optimization

In this paper, we generalize the notion of the performance measure by using a variety of coherent risk measures. We prove that these classes of coherent risk measures assure the properties of the acceptability indexes. In separate sections classes associated to Expected Shortfall and Shortfall Risk are examined both with their sensitivity, and also the general static optimization problem of these ratios is studied. Mathematics Subject Classifications: 46B40, 91G40


Introduction
In the present paper, we extend the notion of the performance ratios, by using a variety of coherent risk measures and pricing functionals of a portfolio under a static framework.Specifically, we suppose that the space E in which the "tomorrow" value variables of a portfolio lie in, is a subspace of some L 0 (Ω, F, P), where (Ω, F, P) is a non-atomic probability space which describes the uncertainty "today".E is some partially ordered normed space.It is well-known that a seminal paper on the topic of the acceptability indexes is [4].In the present paper, the properties of the performance ratios considered, are compared to the properties of the acceptability indexes considered in [4].Specifically, the main properties are: monotonicity, scale -invariance, arbitrageconsistency, concavity and Fatou continuity.About the risk measures being used in the paper for the performance ratio formulation, are actually general coherent risk measures defined on ordered dual systems consisted by reflexive and specifically non -reflexive spaces and also the expected shortfall and risk measures related to shortfall risk.The presence of non-reflexive spaces is important due to the fact that ES a is defined on L 1 .The sensitivity of performance ratios being built on coherent isk measures relying on Expected Shortfall, such as spectral risk measures is also studied in a separate section.The general problem of the maximization of a performance ratio over a set of not necessarily convex set of constraints is also solved under interior point methods and for L 2 -spaces.This is also the way that the main content of the paper is organized.Finally, the Appendix is separated into the first part, in which the basics about strictly positive functionals in spaces with the Riesz Decomposition Property are mentioned.In the second part, the weak compactness of the representation set of the Expected Shortfall functionals' is deduced, while in the third part of the Appendix the basics about the Henig Dilating Cones are mentioned -which are used in the section regarding the optimization problem.In financial practice, RAROC is a performance ratio, which is widely used.The results of the present paper generalize the RAROC, especially in the form which expresses the ratio of the expected value of some financial position divided by the economic capital of it.In this exact form, may by equal to , where ρ corresponds to some coherent risk measure, or in a better way to the Expected Shortfall ES a .

General Performance Ratios
We consider the ordered Banach space E, a cone with interior points P in E and x 0 ∈ intP .We also consider the coherent risk measure ρ with respect to P and x 0 : and also a subspace G of E. We also suppose that 0 belongs to the σ(E * , G)closure of the base B x 0 of P 0 , then by [15,Cor.13],G has the property that ρ(G) ≥ 0. Hence we may define the following performance measure: where f is some strictly positive functional of E.
Notice that the division c 0 = ∞, if c > 0 and 0 is a result of a calculation on some positive variable.
Theorem 3. a ρ,f scale invariant, monotone and arbitrage-consistent with respect to G.
Proof: For abbreviation we denote a ρ,f by a.We also take some t ∈ R + .
Theorem 4. We assume some t ∈ R + .If f, ρ take positive values, then a ρ,f is a concave function, if ρ is a convex risk measure.
3 Performance Ratios on L p Spaces

Expected -Shortfall Based Perfomance Ratios
First of all, we assume a probability space (Ω, F, P).Looking back at the seminal article on Performance Measures - [4] and specifically going to [4, Th.1] of Section 2.2, we notice that a possible family of D x , x ∈ R + can be The family D x , x ∈ R + may define the acceptability index , due to the dual representation of the Expected Shortfall, see in [11,Th.4.1].Specifically, The existence of the extreme measure Q * x (x) is implied in this case by the weak-star compactness of the order-interval [0, x1].The relevant cones A x may be defined as either via, or without Expected Shortfall.As a consequence For a detailed proof of the above, see Appendix 6.1.
Proof: Except the law-invariance, which is a consequence of the Expected Shortfall, the proof of the other arguments arises from Theorem 3. Proof: If some sequence x n converges to x in P, then there exists some subsequence x kn → x, P -a.e.This implies E(x kn ) → E(x) and Theorem 10.ESR t is arbitrage-consistent with respect to any solid subspace of L 1 (Ω, F, P).
3.2 Performance ratios related to coherent risk measures in L p spaces In [10, Th.1.1],the dual representation of any continuous coherent risk measure ρ : where G ⊆ L q + , such that sup g∈G g q < ∞, and E(g) = 1 for any g ∈ G.This Theorem denotes both by Krein -Ŝmulian Theorem, that the sup remains the same, if we take the set K = co(G).Then the set G may be assumed to be convex, closed and norm-bounded, hence weakly (or weak -star, in the case of L ∞ ) compact.The performance ratio a ρ,f , where ρ(x) = sup g∈K E((−x)g) and f is a strictly positive, continuous functional of L p + , has the following properties: Theorem 11. a ρ,f is concave, scale-invariant and monotone.Also, any a ρ,f satisfies the Fatou property.
Proof: The concavity is implied by Theorem 4. For the scale invariance For the case where ρ(x)ρ(y) < 0 the proof is analogous.Suppose that a ρ,f (x n ) ≥ t, t > 0. If the sequence x n converges to x in P, then there exists some subsequence x kn → x, P -a.e.This implies x kn L p → x and ρ(x kn ) → ρ(x), since ρ is Lipschitz continuous with respect to .p -norm.Hence a ρ,f (x) ≥ t, if ρ(x) > 0.

Performance ratios related to Shortfall Risk
According to the recent literature on elicitable risk measures, see [3], and their relation to shortfall risk, mentioned widely in [8,Sect.3].Consider some increasing, convex loss function : R → R and the expected loss E( (−x)) if x lies in the space L ∞ (Ω, F, P).Definition 12.The Expected Loss Ratio, associated to the loss function is defined as follows: Theorem 13.ELR is monotone and law invariance if is continuous and strictly increasing, either on (−∞, ) or on ( , +∞) for some > 0.
Proof: If x ≤ y, P-a.e., then E(x) ≤ E(y).Since is increasing and −y ≤ −x, then E( (−y)) ≤ E( (−x)).Then . From [3, Def.4.2], the shortfall measure being defined by as is law invariance since by [8] is continuous, the value ρ (F ) on the distribution F is the solution of the equation Theorem 14. ELR is monotone and law invariance if is continuous and strictly increasing, either on (−∞, ) or on ( , +∞) for some > 0.
Proof: If x ≤ y, P-a.e., then E(x) ≤ E(y).Since is increasing and −y ≤ −x, then E( (−y)) ≤ E( (−x)).Then . From [3, Def.4.2], the shortfall measure being defined as is law invariance since by [8] is continuous, the value ρ (F ) on the distribution F is the solution of the equation Theorem 15.ELR satisfies the Fatou property, probably passing to subsequences, if is continuous and strictly increasing, either on (−∞, ) or on ( , +∞) for some > 0.
Proof: If some sequence x n converges to x in P, then there exists some subsequence x kn → x, P -a.e.This implies E(x kn ) → E(x) and ELR (x kn ) → ELR (x), since 1 (x kn ) → 1 (x) .If E( (x)) = 0, and

Strong Sensitivity of Performance Ratios
We recall the following Definition of Strict Sensitivity obtained by [9]: We also need another notion of sensitivity on L 1 : Definition 17.A monetary risk measure ρ on L 1 is strongly sensitive, if and only if x = y, µ-a.e.⇒ ρ(x) = ρ(y).
Also, the sensitivity of a performance ratio a ρ,f may be defined in an equivalent way: Definition 18.A performance ratio a ρ,f on L 1 is strongly sensitive, if and only if x = y, µ-a.e.⇒ a ρ,f (x) = a ρ,f (y).
Theorem 19.For any a ∈ (0, 1], the Expected Shortfall ES a is strongly sensitive.
Proof: By the dual representation theorem of Expected Shortfall [?, Th.4.1], we have such that π is a Radon -Nikodym derivative of some probability measure which implies that ES a is strongly sensitive.
Theorem 20.For any a ∈ (0, 1], b > 1 such that 1 b < 1 a , the Adjusted Expected Shortfall is strongly sensitive.Proof: By [16, Lem.6] regarding the dual representation of Adjusted Expected Shortfall, we get that hence the specific risk measure is strongly sensitive on L 1 . Theorem 21.Any spectral risk measure of the form M m (x) = 1 0 aES a (x)dm(a) defined on L 1 is strongly sensitive.
Proof: By [1, Th.2.5], any spectral risk measure M m is a continuous, coherent risk measure on L 1 , since 1 0 adm(a) = 1 and ES a is a continuous, coherent risk measure on L 1 .The coherence of M m is implied by [1].The continuity of M m is implied by relation ( 6) in [1].More specifically, Theorem 22.The pointwise limit of spectral risk measures on L 1 under the same measure of risk spectrum m is a strongly sensitive coherent risk measure. Proof: , for any n ∈ N, hence there is some c > 1 ≥ a n > 0 for any n, such that for any n ∈ N. The last inequality implies which completes the proof.
Theorem 23.Any Kusuoka Representable risk measure on L 1 is strongly sensitive.
Proof: By [18, Pr.1], any such risk measure ρ is a coherent risk measure on L 1 , which admits the representation where Z ∈ L 1 and M denotes the closure under weak topology of a set of probability measures M on [0, 1].More specifically, for some b > 1 such that 0 < a ≤ 1 < b, which implies strong sensitivity.
Theorem 24.The pointwise limit of a sequence of Kusuoka Representable risk measures on L 1 is a strongly sensitive coherent risk measure.
Proof: By [18, Pr.1], any such risk measure ρ n of the sequence (ρ n ) n∈N is a coherent risk measure on L 1 , which admits the representation where Z ∈ L 1 and M n denotes the closure under weak topology of a set of probability measures M n on [0, 1].More specifically, We may quote on the coherence of pointwise limits of a sequence of coherent risk measures -being defined for example on L 1 -at this point.Proposition 25.The pointwise limit ρ of a sequence of coherent risk measures Proof: It suffices to prove that ρ satisfies the four properties of coherence.
From the uniqueness of the limit of the sequence of real numbers (ρ n (Z + a1)) n∈N , which is ρ(Z + a1) is equal to ρ(Z) − a.By the same way we deduce the Positive Homogeneity of ρ.About Subadditivity of ρ, we notice that for any x, y ∈ L 1 and n ∈ N, the inequality holds in the set on real numbers.This implies that for the limit of this sequence the same inequality is true.Finally, if x ≥ y, P − a.e. for any n ∈ N, the inequality ρ n (x) − ρ n (y) ≤ 0, holds in the set of real numbers.This implies that for the limit ρ(x) − ρ(y), the same inequality holds.About the notion of strong sensitivity of risk measures, see [14].
Proposition 26.If ρ : L 1 → R is strongly sensitive, then the associated performance ratio a ρ,f : L 1 → R, where a ρ,f (x) = f (x) ρ(x) and f ∈ L ∞ + being a strictly positive functional of L 1 , is strongly sensitive.

Corollary 27. The performance ratio ESR
ESt(x) , is strongly sensitive.
Proof: It arises from the Proposition 26.
Corollary 28.The performance ratio SR φ : and φ is a spectral risk measure, is strongly sensitive.
Proof: It arises from the Proposition 26.

Optimization of Performance Ratios
The general static optimization problem for some performance ratio a ρ,f is the following: where S is a proper subset of some normed linear space E.
Theorem 29.If E has a well-based cone K, and for the constraints' set that where K n , n ≥ 2 is some of the Henig Dilating Cones, (see Appendix 6.2), then the problem takes the form where z S is continuous, convex.
Proof: We apply the Separation Theorem [6,Th.2.3.6].Namely, we pose A = S, D = K n , n ≥ 2. Also, we take some k 0 ∈ K n+2 , k 0 = 0, and according to the same Lemma C = K n+2 , being a cone, satisfying the condition C + int(D) ⊆ C. Hence, the conclusion of [6, Th.2.3.6],(since the recession conditions 2.22, 2.28 also hold for K n , k 0 ), implies the existence of a continuous, convex functional z Kn,k 0 , such that the scalarization of the constraint of the above problem, as follows: M aximize a ρ,f (x) s.t.z Kn (x) ≥ 0, where z S : E → R is the continuous, convex functional separating S and −int(K n ).If we would like to transfer the scalar constraint itself inside the objective function, the problem becomes namely an unconstrained maximization problem of some continuous, concave functional.By following [6], z Kn,k 0 (x) = inf{t ∈ R|x ∈ tk 0 − K n }.Proof: and the order-interval [0, From the fact that dQ λ dP , λ ∈ Λ, we obtain that dQ λ dP ∈ L 1 (Ω, F, P).We have to prove that f is a Radon-Nikodym derivative of some measure Q 1 ∈ Z a with respect to P. Let us consider the map Q 1 : F → [0, 1] where and I A is the characteristic random variable of A. In order to show that Q 1 is a probability measure, which is the limit lim λ∈Λ Ω dQ λ and every of the terms of the net of real numbers Ω dQ λ λ∈Λ , is equal to 1.By the same argument, we may deduce that Q where N denotes the set of natural numbers.For n → ∞ and the definition of Q 1 , the fact that any characteristic function I A , A ∈ F belongs to L 1 (Ω, F, P).We may also refer to the Monotone Convergence Theorem [2,Th.11.18],where the restriction of the Q 1 on the set ∪ ∞ n=1 A n is the integrable function which is mentioned in the Theorem, while f n is the restriction of Q 1 on a set of the form ∪ n k=1 A k .For the P-continuity of Q 1 , we have that if for a set A ∈ F P(A) = 0 holds, then since Q λ , λ ∈ Λ is P-continuous, for any A ∈ F. This implies that A f dP ∈ [0, 1 a ] for any A ∈ F. This implies 0 ≤ f ≤ 1 a 1 P-a.e., since if we suppose that this does not hold, then there exists some B ∈ F with P(B) > 0 such that either f (ω) > 1 a , or f (ω) < 0 for any ω ∈ B. Then, we would have either B f dP > 1 a , or B f dP < 0, a contradiction.Finally, the set Z a is a weak-star closed subset of a weak-star compact set which is the set D a .
(iv) K n is a cone n ≥ 2. Proof:

Theorem 8 .
ESR t is concave.Proof: It arises from the Theorem 4. Theorem 9. ESR t satisfies the Fatou property, probably passing to subsequences.