Optimization of Expected Shortfall on Convex Sets

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Optimization of Expected Shortfall on Convex Sets Christos E. Kountzakis Department of Statistics and Actuarial - Financial Mathematics University of the Aegean

7th Conference in Actuarial Science and Finance on Samos May 31-June 3, 2012

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Expected Shortfall -ESa

• Expected Shortfall ESa, where a ∈ (0, 1) denotes a level of significance, is identical to Conditional Value at Risk CV aRa and also ESa is a coherent risk measure on L1(Ω, F, µ) (see C. Acerbi, D. Tasche (2002)), where (Ω, F, µ) is a probability space. •

1 ESa(X) = (E(X1{X≤qa(X)}) + qa(X)(a − µ(X ≤ qa(X))), a where qa(X) denotes the a-lower quantile of X. R 1 a − a 0 qu(X)du

• The expression ESa(X) = indicates that ESa is the building block for law invariant, coherent risk measures, according to the results containing in (S. Kusuoka (2001)).

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• ESa admits the dual representation ESa(X) = max EQ(−X), Q∈Za

1 ≤ uschendorf where Za = {Q ∈ M1| dQ dµ a , µ-a.e. }, see (M. Kaina, L. R¨ (2009)).

• M1 denotes the set of µ-continuous probability measures on the measurable space (Ω, F).

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•

dQ dµ ∈ ∞

[0, a1 1] holds with respect to the usual (pointwise) partial ordering on ∞ L (Ω, F, µ), hence dQ ∈ L (Ω, F, µ) for any Q ∈ Za. dµ

• Da = {π ∈ L1(Ω, F, µ)|π =

dQ dµ , Q

∈ Za}.

• The max instead of sup in the dual representation is due to the following Lemma 1. The set Da is a weak-star compact set of L∞(Ω, F, µ).

∞

1



• The Lemma’s proof is a consequence of the fact that L , L is a Riesz pair for a probability measure µ, hence any order-interval in L∞(Ω, F, µ) is σ(L∞, L1)-compact.

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Minimization of ESa over a set of financial positions

• Consider a (convex) set of financial positions, namely a subset X ⊆ L1(Ω, F, µ). • Also, we consider the following minimization problem M inimize ESa(X) subject to X ∈ X

(1)

• According to the dual representation of the ESa, the expression of the (possibly equal to +∞) optimal value of the problem is inf ( sup π(−X)).

X∈X π∈Da

• This expression corresponds to a saddle-point solution of this problem.

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• We use the following saddle-point existence Theorem, mentioned in (F.Delbaen (2002)): Let K be a compact, convex subset of a locally convex space Y . Let L be a convex subset of an arbitrary vector space E. Suppose that u is a bilinear function u : E × Y → R. For each l ∈ L, we suppose that the partial (linear) function u(l, .) is continuous on Y . Then we have that inf sup u(l, k) = sup inf u(l, k).

l∈L k∈K

k∈K l∈L

• By the above Theorem we obtain the Theorem 1. If X is a convex set of L1(Ω, F, µ), then the risk minimization problem 1 has a solution. • u : L1 × L∞ → R, where u(X, π) = π(−X), K = Da, L = X .

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• Also, it is well-known that: A function satisfying the min-max equality if and only if it has a saddle-point. • The saddle -point (X∗, πQ∗ ) ∈ X × Da is such that ESa(X∗) = inf ESa(X) = X∈X

= inf

sup u(X, πQ) = sup

X∈X π ∈Da Q

inf u(X, πQ) = u(X∗, πQ∗ ).

πQ ∈Da X∈X

• The important point is that no topological properties are assumed for X in the above proof of solution for 1 except the geometric assumption of convexity.

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Motivation for this problem

• The motivation for the problem 1 is the calculation of the solvency capital with respect to ESa over a set of financial or actuarial positions. The capital which eliminates the risk for a single position U with respect to ESa is ESa(U ). • If the initial surplus of an insurance company is u ∈ R+ and X denotes the (convex) set of possible surplus variables at the date T , we may suppose that the actual set of constraints of the problem 1 in this case is the convex set X˜ = {U ∈ X |ESa(U ) ≤ u}. • The solvency capital for the company in this case is the optimal value of the problem 1 over X˜ . For any position U ∗ ∈ X˜ which minimizes ESa over this set, ESa(U ∗) is the surplus which has to be subtracted from u so that the insurance company to eliminate the risk which arises from its obligations because this surplus is going to be spent for this aim in any case.

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Stochastic models for the surplus and motivation

• We will present some models for the surplus of an insurance company related to the motivation of 1. • We may refer to a diffusion model for the capital of an insurance company, (Ct)t∈[0,T ] which evolves according to the following stochastic differential equation (1)

dCt = (m(t, Ct) + φtrtBt)dt − θts(t, Ct)dWt

,

µ − a.e. ,

where C0 = c is some constant which denotes the initial capital of it and W (1) is some one-dimensional Brownian motion with respect to an appropriate filtration F of (Ω, F, µ) (see also C. Hipp (2004)).

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Rt

• The process pt = 0 m(s, Cs)ds may be interpreted as the total premium process, or else it indicates the total premium payments Rthat the insurance t (1) company receives till time-period t. The pure diffusion term 0 θus(u, Cu)dWu corresponds to the payoff of the investment of these received premiums to some risky asset whose volatility-term is s(t, Ct), t ∈ [0, T ] with respect to an one-dimensional Brownian motion W (1). This part of the premiums is invested to this asset as long as they are received by the company, for this reason θt = θ(t, Ct) = θtm(t, Ct), t ∈ [0, T ]. Another part of these premiums equal to φt = φ(t, Ct) = φtm(t, Ct), t ∈ [0, T ] is invested to the risk -free asset, whose interest-rate is rt, t ∈ [0, T ].

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• We may suppose that the interest -rate process (rt)t∈[0,T ] is uniformly bounded, or else that rt(ω) ≤ M for any (t, ω) ∈ [0, T ] × Ω. Φ is a set of admissible trading strategies for the risk-free asset, while Θ and Φ have the same properties. The integrability properties that the investment strategies φ ∈ Φ, θ ∈ Θ have to satisfy are the following: Z

T

Z |θtrt|dt < ∞,

0

T

Z |φtrt|dt < ∞,

0

T

θt2s2(t, Ct)dt < ∞, µ − a.e.

0

These integrability conditions are implied from (I.Karatzas, S. Shreve (1998)). • We may suppose that θ ∈ Θ, φ ∈ Φ if (φt)θt(ω) = at, at ∈ R, t ∈ [0, T ]. • The above conditions hold since for θ, φ |θt(ω)| < 1, |φt(ω)| < 1, (t, ω) ∈ [0, T ] × Ω.

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• For the claim-process St, t ∈ [0, T ] we consider a compound Poisson process PNt St = i=1 Yi, t ∈ [0, T ], where Nt, t ∈ [0, T ] is a Poisson process with rate λ > 0, and claim size random variables Yi, i = 1, 2, ... are positive, i.i.d. with cumulative distribution function G and generic element denoted by Y . More specifically, the inter-occurrence random time θ is exponentially distributed with P n mean λ1 and the random times Tn = k=1 θk , , n = 1, 2, ... of the claims’ occurrence form a counting process Nt = N um{Tn ≤ t, n = 1, 2, ...}, t ∈ [0, T ].

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• Then the surplus process in this model is Kt = Ct − St, t ∈ [0, T ] and the set of the attainable surplus variables is Z X = {U ∈ L1(Ω, F, µ)|U = u+

T

Z (m(t, Ct)+φtrtBt)dt−

0

T

(1)

θts(t, Ct)dWt −ST ,

0

θ ∈ Θ, φ ∈ Φ}. • If the distribution G (and mainly the tail of it) is such that the moment Eµ(Y ) does not exist, then X is not a subset of L1, since Eµ(U ), U ∈ X does not exist, since Eµ(ST ) does not exist. On the other hand, if Eµ(Y ) exist, then Eµ(ST ) exist and X is a subset of L1. Also, if both Eµ(Y ) and Eµ(Y 2) exist, then X ⊆ L1. • The set X is convex, because the sets Φ, Θ are convex sets. This allows for the consideration of 1 over X .

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• A more general form of this surplus model is to suppose that the claim payments’ process is a sum of a diffusion process and a compound Poisson process St, t ∈ [0, T ], described as above. This is the case where the claim payments’ process is At = Zt + St, t ∈ [0, T ], where (2)

dZt = q(t, Zt)dt + σ(t, Zt)dWt

,

µ − a.e. ,

where W (2) is an one dimensional Brownian motion with respect to an appropriate filtration F of (Ω, F, µ) If we pose Z to be the zero process, we take the above model case and the process equation A = S holds. • The constraint ESa(U ) ≤ u due to the coherence of ESa (Translation Invariance) is equivalent to Z ESa( 0

T

Z (m(t, Ct) + φtrtBt)dt −

T

(1)

θts(t, Ct)dWt

− ST ) ≤ 0,

0

or else to the requirement that the initial surplus to be adequate for the elimination of risk in any case.

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Possible Extensions

• Specification of the optimal strategy-variables at any time-period t ∈ [0, T ]. • Uniqueness of the saddle-point. • Model Uncertainty-other risk measures.

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Appendix

• If F : A × B → R and A, B non-empty sets, the min-max equality is verified by F if maxx∈A miny∈B F (x, y) = miny∈B maxx∈A F (x, y) and we always have supx∈A inf y∈B F (x, y) ≤ inf y∈B supx∈A F (x, y). The min-max condition is satisfied if and only if the following conditions are satisfied (i) supx∈A inf y∈B F (x, y) = inf y∈B supx∈A F (x, y). (ii) there is x ∈ A such that inf y∈B F (x, y) = supx∈A inf y∈B F (x, y). (iii) there is y ∈ Y such that supx∈A F (x, y) = inf y∈B supx∈A F (x, y). • F (x, y) is the saddle-value of F . inf y∈B F (x, y) is attained at y, while supx∈A F (x, y) is attained at x. Hence F (x, y) ≤ inf F (x, y) = sup inf F (x, y) = y∈B

x∈A y∈B

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= sup F (x, y) ≤ F (x, y). x∈A

By the above relation we have that F (x, y) = inf F (x, y) = sup inf F (x, y). y∈B

x∈A y∈B

The above equality indicates sup F (x, y) = inf F (x, y) = F (x, y), x∈A

y∈B

namely F (x, y) ≤ inf F (x, y) ≤ F (x, y), y∈B

for any (x, y) ∈ A × B. The last pair of inequalities is the definition of the saddle-point.

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• A Riesz pair is a dual pair hE, E ∗i of Riesz spaces (vector lattices), where E ∗ is an ideal of the order dual E 0. • An ideal A of a Riesz space E is a subspace such that if |y| ≤ |x|, x ∈ A, then y ∈ A. • The order-dual E 0 of E contains all the order-continuous linear functionals f : E → R. • Order-convergence of a net xa, a ∈ A in a Riesz space E to some x ∈ E we have when there is some net ya, a ∈ A, such that ya ↓ 0 and |xa − x| ≤ ya for o any a. The order-convergence is denoted by xa → x. Order -continuity of a o o linear functional f means that if xa → x, then f (xa) → f (x). • The positive cone of Lp, 1 ≤ p ≤ ∞, is Lp+ = {f ∈ Lp|f (ω) ≥ 0, µ − a.s.} • An order-interval in Lp is [x, y] = (x + Lp+) ∩ (y − Lp+), x, y ∈ Lp.

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References

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