Entropic risk measures: coherence vs. convexity, model ambiguity, and robust large deviations

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1 Entropic risk measures: coherence vs. convexity, model ambiguity, robust large deviations Hans Föllmer Thomas Knispel 2 Abstract We study a coherent version of the entropic risk measure, both in the lawinvariant case in a situation of model ambiguity. In particular, we discuss its behavior under the pooling of independent risks its connection with a classical a robust large deviations bound. Key words: risk measures; premium principles; model ambiguity; large deviations. AMS Subject Classification (200): 60F0, 62P05, 9B6, 9B30 Introduction A monetary risk measure specifies the capital which should be added to a given financial position to make that position acceptable. If the monetary outcome of a financial position is described by a bounded rom variable on some probability space (Ω, F, P ), then a monetary risk measure is given by a monotone translation invariant functional ρ on L (Ω, F, P ). In the law-invariant case the value ρ(x) only depends on the distribution of X under P. Typical examples are Value at Risk (VaR), Average Value at Risk (AVaR), also called Conditional Value at Risk (CVaR) or Tail Value at Risk (TVaR), the entropic risk measure defined by e γ (X) := γ log E P [e γx ] = sup{e Q [ X] γ H(Q P )} Q for parameters γ [0, ), where e 0 (X) := E P [ X] H(Q P ) denotes the relative entropy of Q with respect to P. VaR is the one which is used most widely, but it has various deficiencies; in particular it is not convex may thus penalize a desirable diversification. AVaR is a coherent risk measure, i. e., convex also positively homogeneous. As shown by Kusuoka [3] in the coherent by Kunze [2] Frittelli & Rosazza Gianin [8] in the general convex case, AVaR is a basic building block for any law-invariant convex risk measure. The entropic risk measures e γ are convex, they are additive for independent positions. From an actuarial point of view, however, this property may not be desirable. Indeed, if e γ (X X n ) is viewed as the total premium for a homogeneous portfolio of i. i. d. rom variables X,..., X n, then the premium per contract would simply be e γ (X ), no matter how large n is. Thus the pooling of independent risks Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 0099 Berlin, Germany, foellmer@math.hu-berlin.de 2 Leibniz Universität Hannover, Institut für Mathematische Stochastik, Welfengarten, 3067 Hannover, Germany, knispel@stochastik.uni-hannover.de

2 does not have the effect that the premium per contract decreases to the fair premium, i. e., to the expected loss from a single contract, as the number of contracts increases. In this note we focus on a fourth example, namely on a coherent version of the entropic risk measure defined by ρ c (X) := sup E Q [ X]. Q:H(Q P ) c In Section 3 we clarify the connection between the coherent entropic risk measures ρ c the convex entropic risk measures e γ. In Section 4 we show that the capital requirements computed in terms of ρ c have the desired behavior under the pooling of independent risks X,..., X n. In fact it turns out that the asymptotic analysis of ρ c (X +...+X n ) simply amounts to a reformulation, in the language of risk measures, of Cramér s classical proof of the upper bound for large deviations of the average loss. In Section 5 we extend the discussion beyond the law-invariant case by taking model ambiguity into account. Instead of fixing a probability measure P we consider a whole class P of probabilistic models. We define corresponding robust versions e P,γ ρ P,c of the entropic risk measures derive some of their basic properties. In particular, we show that the pooling of risks has the desired effect if premia are computed in terms of ρ P,c, that this corresponds to a robust version of Cramér s theorem for large deviations. 2 Notation definitions Let X be the linear space of bounded measurable functions on some measurable space (Ω, F). Consider a set A X such that A R R Then the functional ρ : X R defined by is X A, Y X, Y X Y A. i) monotone, i. e., ρ(x) ρ(y ) if X Y, ρ(x) := inf{m R X + m A} () ii) cash-invariant, i. e., ρ(x + m) = ρ(x) m for X X m R. Definition 2.. A functional ρ : X R with properties i) ii) is called a monetary risk measure. Any monetary risk measure is of the form () with A ρ := {X X ρ(x) 0}. If X X is interpreted as the uncertain monetary outcome of a financial position A as a class of acceptable positions, then ρ(x) can be regarded as a capital requirement, i. e., as the minimal capital which should be added to the position to make it acceptable. Definition 2.2. A monetary risk measure is called a convex risk measure if it is quasi-convex, i. e., ρ(λx + ( λ)y ) max{ρ(x), ρ(y )} 2

3 for X, Y X λ [0, ]. In that case A ρ is convex, this implies that ρ is a convex functional on X; cf. Föllmer & Schied [7], Proposition 4.6. A convex risk measure is called coherent if it is positively homogeneous, i. e., for X X λ 0. ρ(λx) = λρ(x) Remark 2.. Convex risk measures are closely related to actuarial premium principles; cf., e. g., Kaas et al. []. For example, it is shown in Deprez & Gerber [2], that a convex premium principle H is of the form H(X) = ρ( X) for some convex risk measure ρ if it satisfies the no rip-off condition H(X) sup X. Typically, a convex risk measure admits a robust representation of the form ρ(x) = sup {E Q [ X] α(q)}, (2) Q M where M denotes the class of all probability measures on X, where the penalty function α : M (, ] is defined by α(q) := sup X A ρ E Q [ X]; cf., e. g., Artzner et al. [], Delbaen [4], Frittelli & Rosazza Gianin [8], Föllmer & Schied [7], Chapter 4 for criteria examples. In the coherent case we have α(q) {0, }, (2) reduces to ρ(x) = sup E Q [ X], Q Q where Q := {Q M α(q) = 0}. Now suppose that P is a probability measure on (Ω, F) that ρ(x) = ρ(y ) if X = Y P -a. s.. Then ρ can be regarded as a convex risk measure on L := L (Ω, F, P ). In this case the representation (2) holds if ρ is continuous from above, i. e., ρ(x n ) ρ(x) whenever X n decreases to X in X, M can be replaced by the class M (P ) := {Q M Q P }. A monetary risk measure ρ is called law-invariant if ρ(x) only depends on the distribution of X under the given probability measure P. For a convex risk measure which is continuous from above, this is the case if only if the penalty α(q) of Q M (P ) only depends on the law of dq dp under P ; cf., e. g., [7], Theorem A large class of examples arises if acceptability is defined in terms of expected utility, i. e., if A = {X L E P [u(x)] u(0)} for some concave increasing function u. In this case the resulting risk measure is convex law-invariant, its penalty function can be computed in terms of the conjugate function of u; cf. [7], Theorem Let us now take an exponential utility of the form u(x) = e γx for some γ > 0. In that case the corresponding risk measure is given by its robust representation takes the form e γ (X) = γ log E P [e γx ], X L, (3) e γ (X) = sup {E Q [ X] γ H(Q P )}, Q M 3

4 where H(Q P ) = { EQ [log dq dp ] if Q P + otherwise denotes the relative entropy of Q with respect to P. Definition 2.3. The convex risk measure e γ entropic risk measure with parameter γ. defined by (3) is called the (convex) It is easy to see that e γ (X) is increasing in γ strictly increasing as soon as X is not constant P -a. s.. Moreover, lim e γ (X) = E P [ X] lim e γ (X) = ess sup( X); (4) γ 0 γ cf., e. g., [], Theorem.3.2. As noted already by de Finetti [3], the entropic risk measures can be characterized as the only monetary risk measures ρ which are, up to a change of sign, also a certainty equivalent, i. e., u( ρ(x)) = E P [u(x)] for some strictly increasing concave utility function u. In this case the utility function is exponential, ρ = e γ for some γ [0, ). The actuarial premium principle H(X) = e γ ( X) corresponding to the entropic risk measure is usually called the exponential principle; cf. Deprez & Gerber [2] Gerber [9]. 3 Coherent entropic risk measures In this section we focus on the following coherent version of an entropic risk measure. Definition 3.. For each c > 0, the risk measure ρ c defined by ρ c (X) := sup E Q [ X], X L, (5) Q M :H(Q P ) c will be called the coherent entropic risk measure at level c. Clearly, ρ c is a coherent risk measure. It is also law-invariant; this follows from Theorem 4.54 in Föllmer & Schied [7], also from the representation (8) in Proposition 3. below; cf. Corollary 3.. For X L we denote by the exponential family induced by P X, i. e., Q P,X = {Q γ γ R} (6) dq γ dp = e γx E P [e γx ]. If p(x) := P [X = ess inf X] > 0, then we include as limiting case the measure Q := lim γ Q γ = P [ X = ess inf X]. The following proposition shows that the supremum in (5) is attained by some probability measure in the exponential family Q P,X. 4

5 Proposition 3.. For c (0, log p(x)) we have ρ c (X) = max E Q[ X] = E Qγc [ X], (7) Q M :H(Q P ) c where Q γc Q P,X γ c > 0 is such that H(Q γc P ) = c, If p(x) > 0 c log p(x), then ρ c (X) = min γ>0 { c γ + e γ(x)} = c γ c + e γc (X). (8) ρ c (X) = E Q [ X] = ess sup( X). (9) Proof. We exclude the trivial case, where X is P -a. s. constant. ) Assume that 0 < c < log p(x) take Q such that H(Q P ) c. For any γ > 0 for Q γ Q P,X, H(Q P ) = H(Q Q γ ) + γe Q [ X] log E P [e γx ] (0) with H(Q Q γ ) 0 H(Q Q γ ) = 0 iff Q = Q γ. Thus ρ c (X) = E Q [ X] c γ + e γ(x), () sup E Q [ X] Q M :H(Q P ) c inf γ>0 { c γ + e γ(x)}. Both the supremum the infimum are attained, they coincide. Indeed, we can choose γ c > 0 such that H(Q γc P ) = c, then we get equality in () for Q = Q γ γ = γ c. Such a γ c > 0 exists is unique. Indeed, H(Q γ P ) = γe Qγ [ X] log E p [e γx ] is continuous strictly increasing in γ, lim γ H(Q γ P ) = log p(x) since lim γ Q γ [A a ] = for A a := { X > a} any a < ess sup( X). 2) If p(x) > 0, then Q satisfies H(Q P ) = log p(x) E Q [ X] = ess sup( X). For c log p(x) we thus obtain ρ c (X) ess sup( X), hence (9), since the converse inequality in (9) is clear. Remark 3.. Conversely, the convex entropic risk measure e γ can be expressed in terms of the coherent entropic risk measures ρ c as follows: e γ (X) = min c>0 {ρ c(x) c γ } = ρ c γ (X) cγ γ, where c γ := H(Q γ P ); this follows immediately from (8). Remark 3.2. Note that the parameter γ c in (7) depends both on c on X. For a fixed value γ > 0, the resulting functional ρ(x) := E P [( X)e γx ](E P [e γx ]) is neither coherent nor convex, but the corresponding actuarial premium principle H(X) = ρ( X) is well-known as the Esscher principle; cf., e. g., Deprez & Gerber [2]. 5

6 Corollary 3.. The coherent risk measure ρ c is law-invariant, continuous from above, even continuous from below. Moreover, ρ c is increasing in c satisfies lim ρ c (X) = E P [ X] lim ρ c (X) = ess sup( X). (2) c 0 c Proof. Law-invariance follows from (8) since each e γ is law-invariant. Continuity from below follows from (7), i. e., from the representation ρ c (X) = max Q Q E Q[ X] with Q = {Q M H(Q P ) c}; cf. [7], Corollary In particular, ρ c is continuous from above; cf. [7], Corollary 4.35 together with Theorem 4.3. The convergence in (2) follows easily from Proposition 3.. Indeed, (8) implies lim c 0 ρ c (X) e γ (X) for each γ > 0, hence the first equality in (2), due to (4). As to the second equality, it is enough to consider the measures Q = P [ A a ] for the sets A a := { X > a} with a < ess sup( X). Let us now compare the coherent entropic risk measures ρ c to the familiar risk measures Value at Risk Average Value at Risk defined by VaR α (X) := inf{m R P [X + m < 0] α} AVaR α (X) := α α 0 VaR λ (X) dλ VaR α (X) for any α (0, ). Recall that VaR α is a monetary risk measure which is positively homogeneous but not convex, while AVaR α is a coherent risk measure which can also be written as AVaR α (X) = α E P [(q α X) + ] q α for any α-quantile q α of X; cf., e. g., [7], Section 4.4. decreasing right-continuous in α with left limits VaR α (X) := inf{m R P [X + m 0] α}. Proposition 3.2. For any α (0, ) any X L, where c(α) := log α > 0. Note also that VaR α (X) is VaR α (X) VaR α (X) AVaR α (X) ρ c(α) (X), (3) Proof. Clearly, we have VaR α (X) VaR α (X) AVaR α (X). In view of Corollary 3. it is enough to verify the inequality VaR α (X) ρ c(α) (X), since AVaR α is the smallest law-invariant coherent risk measure which is continuous from above dominates VaR α ; cf. [7], Theorem 4.6. For any γ > 0, P [X + m 0] e γm E P [e γx ], the right-h side is α if γm + log E P [e γx ] log α, i. e., if Thus, by Proposition 3., m c(α) γ + e γ (X). VaR α (X) inf { c(α) γ>0 γ + e γ (X)} = ρ c(α) (X). 6

7 Alternatively, we can check directly the last inequality in (3), using the robust representation AVaR α (X) = max Q Q α E Q [ X] with Q α := {Q M Q P, dq dp α }; cf. [7], Lemma 4.46 Theorem Indeed, any Q Q α satisfies log dq dp c(α), hence H(Q P ) c(α). 4 Capital requirements for i. i. d. portfolios Consider a homogeneous portfolio of n insurance contracts whose uncertain outcomes are described as i. i. d. rom variables X,..., X n on some probability space (Ω, F, P ). Let µ denote the distribution of X under P. To keep this exposition simple, we assume that µ is non-degenerate has bounded support; in fact it would be enough to require finite exponential moments e γx µ(dx) for any γ R. If ρ is a monetary risk measure, then ρ(x X n ) can be viewed as the smallest monetary amount which should be added to make the portfolio acceptable. This suggests to equate ρ(x X n ) with the portfolio s total premium, to use the fraction π n = n ρ(x X n ) as a premium for each individual contract X i, i =,..., n. For any γ > 0, the entropic risk measure e γ satisfies e γ (X X n ) = ne γ (X ). If e γ is used to calculate the premium π n, it yields π n = n e γ(x X n ) = e γ (X ) > E P [ X ]. Thus the exponential premium principle based on the convex entropic risk measure does not have the desirable property that the risk premium π n E P [ X ] decreases to 0 as n tends to ; cf., e. g., [6], Example 2.5. Remark For the coherent risk measure ρ c, however, the pooling of risks does have the desired effect. Corollary 4.. The premium π c,n := n ρ c(x X n ) (4) computed in terms of the coherent entropic risk measure ρ c satisfies π c,n > E P [ X ] lim n π c,n = E P [ X ]. Proof. In view of (4) we can choose for any ɛ > 0 some δ > 0 such that e δ (X ) E P [ X ] + ɛ. Thus, by (8), π c,n = n inf γ>0 { c γ + e γ(x X n )} = inf γ>0 { c γn + e γ(x )} c δn + E P [ X ] + ɛ, hence lim n π c,n E P [ X ]. Since π c,n E P [ X ], the conclusion follows. In fact we have π c,n > E P [ X ] since the distribution µ of X is non-degenerate. 7

8 Let us now describe the decay of the risk premium π c,n E P [ X ] more precisely. Note that for any Q γ Q P,X+...+X n we have dq γ dp = e γ P n i= Xi (E P [e γx ]) n H(Q γ P ) = nh(µ γ µ), where µ γ is the distribution on R with density dµ γ dµ (x) := e γx ( e γx µ(dx)), mean m(γ) := ( x)µ γ (dx), variance σ 2 (γ) := (x + m(γ)) 2 µ γ (dx). We denote by σ 2 P (X ) := σ 2 (0) the variance of X under P. Proposition 4.. For a given level c > 0, the premium π c,n defined by (4) is given by π c,n = m(γ c,n ), where γ c,n is such that H(µ γc,n µ) = c n, we have lim n(πc,n E P [ X ]) = 2c σ P (X ). n Proof. Recall from Proposition 3. that ρ c (X X n ) = E Qγc,n [ (X X n )], where Q γc,n Q P,X+...+X n, where the parameter γ c,n > 0 is taken such that H(Q γc,n P ) = c. Thus, the individual premium π c,n can be rewritten as c = H(Q γc,n P ) = nh(µ γc,n µ), (5) π c,n = n ρ c(x X n ) = m(γ c,n ). (6) The smooth function f defined by f(γ) := log E P [e γx ] = log e γx µ(dx) satisfies f (γ) = m(γ) f (γ) = σ 2 (γ), so we have for some γ [0, γ]. The condition clearly implies lim n γ c,n = 0, hence Since for some γ c,n [0, γ c,n ], we have H(µ γ µ) = γm(γ) f(γ) = 2 f ( γ)γ 2 c n = H(µ γ c,n µ) = 2 f ( γ c,n )γ 2 c,n lim n nγ2 c,n = 2c(f (0)). (7) f(γ c,n ) = f(0) + f (0)γ c,n + 2 f ( γ c,n )γ 2 c,n = m(0)γ c,n + 2 f ( γ c,n )γ 2 c,n π c,n = m(γ c,n ) = γ c,n (H(µ γc,n µ) + f(γ c,n )) = γ c,n ( c n + m(0)γ c,n + 2 f ( γ c,n )γ 2 c,n). 8

9 Due to (7), we finally obtain lim n(πc,n E P [ X ]) = lim n n ( c nγc,n = 2c σ P (X ). ) + 2 f ( γ c,n ) γ2 c,n n c Let us now fix a premium π such that E P [ X ] < π < ess sup( X ), let us determine the maximal tolerance level c π,n := max{c > 0 n ρ c(x X n ) π} at which the portfolio X,..., X n is made acceptable by the total premium nπ. Corollary 4.2. Take γ(π) > 0 such that m(γ(π)) = π. Then Proof. At level c π,n we have c π,n = nh(µ γ(π) µ). In view of (5) (6) this is the case iff n ρ c π,n (X X n ) = m(γ(π)). c π,n = nh(µ γ(π) µ). Remark 4.. Due to (3), Corollary 4.2 implies VaR απ,n (X X n ) nπ for α π,n := exp( c π,n ). But this translates into so we obtain P [ n (X X n ) π] α π,n, n log P [ n (X X n ) π] cπ,n n = H(µ γ(π) µ). In other words, the combination of Corollary 4.2 with the estimate (3) simply amounts to a reformulation, in the language of risk measures, of the classical proof of Cramér s upper bound for the large deviations of the averages n (X X n ); see, e. g., Dembo & Zeitouni [5]. 5 Model ambiguity robust large deviations So far we have fixed a probability measure P which is assumed to be known. Let us now consider a situation of model ambiguity where P is replaced by a whole class P of probability measures on (Ω, F). Assumption 5.. We assume that all measures P P are equivalent to some reference measure R on (Ω, F), that the family of densities is convex weakly compact in L (R). Φ P := { dp dr P P} 9

10 For a probability measure Q on (Ω, F), the extent to which it differs from the measures in the class P will be measured by the relative entropy of Q with respect to the class P, defined as H(Q P) := inf H(Q P ). Our assumption implies that for each Q such that H(Q P) < there is a unique measure P Q P, called the reverse entropic projection of Q on P, such that H(Q P Q ) = H(Q P); cf. [6], Remark 2.0 Proposition 2.4. Let us denote by M (R) the class of all probability measures on (Ω, F) which are absolutely continuous with respect to R. From now on we write L = L (Ω, F, R); note that L L (Ω, F, P ) for any P M (R). We also use the notation e P,γ ρ P,c for the convex the coherent entropic risk measures defined in terms of the specific measure P. In this context of model ambiguity, we define the robust version e P,γ of the (convex) entropic risk measure by e P,γ (X) := sup e P,γ (X) = γ sup log E P [e γx ], X L. Assumption 5. implies that the supremum is actually attained. Clearly, e P,γ is again a convex risk measure, its robust representation takes the form e P,γ (X) = Lemma 5.. We have e P,γ (X) is increasing in γ with Proof. The functions sup {E Q [ X] γ H(Q P)}, X L. Q M e P,γ (X) max E P [ X], lim e P,γ (X) = max E P [ X]. (8) γ 0 f γ (ϕ P ) := e P,γ (X) E P [ X] with ϕ P := dp dr are weakly continuous on Φ P they decrease pointwise to 0, due to (4). Since Φ P is weakly compact, the convergence is uniform by Dini s lemma, this implies (8). Note that the maximum in (8) is actually attained since ϕ P E R [( X)ϕ P ] is continuous on the weakly compact set Φ P. From now on we focus on the robust extension ρ P,c of the coherent entropic risk measure defined by ρ P,c (X) := sup Q M :H(Q P) c E Q [ X] (9) for any X L. Lemma 5.2. The supremum in (9) is attained, i. e., for any X L there is a pair (Q c, P c ) M (R) P such that H(Q c P c ) c ρ P,c (X) = E Qc [ X]. In particular, ρ P,c (X) = max ρ P,c(X) max E P [ X]. (20) 0

11 Proof. Since any Q such that H(Q P) < admits a reverse entropic projection P Q P, we can write where we define ρ P,c (X) = sup E Q [ X] Q M (R):H(Q P) c = sup E Q [ X] Q M (R),:H(Q P ) c = sup (ϕ,ψ) C c E R [( X)ϕ], C c := {(ϕ, ψ) Φ Φ P E R [h(ϕ, ψ)] c} with Φ := { dq dp Q M (R)} h(x, y) := x log x y for y > 0 x 0, h(0, 0) := 0, h(x, 0) = for x > 0. The functional F (ϕ, ψ) := E R [( X)ϕ] is weakly continuous on Φ Φ P, the set C c is weakly compact in L (R) L (R); cf. the proof of Lemma 2.9 in Föllmer & Gundel [6]. This shows that the supremum in (9) is actually attained that (20) holds. Recall that for X L P P we denote by Q P,X the exponential family introduced in (6). Proposition 5.. For we have c < log max P [X = ess inf X] ρ P,c (X) = max min γ>0 { c γ + e P,γ(X)} = c γ c + e Pc,γ c (X) = E Qc [ X], where Q c denotes the measure in the exponential family Q Pc,X with parameter γ c, γ c > 0 is such that H(Q c P) = H(Q c P c ) = c. (2) If c log max P [X = ess inf X], then ρ P,c (X) = ess sup( X). (22) Proof. ) If c log P [X = ess inf X] for some P P, then ρ P,c (X) = ess sup( X) due to Proposition 3., this implies (22). 2) The proof of Lemma 5.2 shows that for any X L there exists a pair (Q c, P c ) M (R) P such that H(Q c P c ) c ρ P,c (X) = E Qc [ X]. Let us first show that Q c belongs to the exponential family Q Pc,X, that H(Q c P c ) = c. To this end, we take γ c > 0 such that H(Q Pc,γ c P c ) = c, we show that Q c = Q Pc,γ c. Indeed, Q Pc,γ c satisfies the constraint H(Q Pc,γ c P) H(Q Pc,γ c P c ) = c,

12 as in the proof of Proposition 3. we see that E Qc [ X] = H(Q c P c ) H(Q c Q Pc,γ c ) γ c c γ c + e Pc,γ c (X) = E QPc,γc [ X]. + e Pc,γ c (X) But E Qc [ X] is maximal under the constraint H(Q P) c, so we must have equality. This implies H(Q c Q Pc,γ c ) = 0, hence Q c = Q Pc,γ c H(Q c P c ) = H(Q Pc,γ c P c ) = c. 3) We have ρ P,c (X) = ρ Pc,c(X) = sup Q:H(Q P c) c E Q [ X]. Indeed, is clear since P c P. Conversely, if H(Q P) c then, since Q c Q Pc,X H(Q c P c ) = c, Proposition 3. implies E Q [ X] E Qc [ X] = ρ Pc,c(X), this yields. 4) In view of 2) Proposition 3., we have sup E Q [ X] = ρ P,c (X) = sup ρ P,c (X) Q:H(Q P) c = sup inf { c γ>0 γ + e P,γ(X)}. The argument in part ) shows that the second the third supremum are attained by P = P c, the first by Q = Q c, the infimum by γ = γ c. 5) If (2) does not hold, then we have H(Q c P ) < c for some P P. But then the proof of Proposition 3. shows that there exists some Q such that H(Q P) H(Q P ) = c E Q [ X] > E Qc [ X], contradicting the definition of Q c. Corollary 5.. The robust versions VaR P,α (X) := sup VaR P,α (X) = inf{m R sup P [X + m < 0] α}, VaR P,α (X) := sup VaR P,α (X) = inf{m R sup P [X + m 0] α}, AVaR P,α (X) := sup AVaR P,α (X) of Value at Risk Average Value at Risk with respect to the class of prior models P satisfy with c(α) := log α > 0. VaR P,α (X) VaR P,α (X) AVaR P,α (X) ρ P,c(α) (X) (23) Proof. This follows from Proposition 5. Proposition 3.2. Let us now look at the asymptotic behavior of the robust premium for a portfolio which satisfies the following π c,n := n ρ P,c(X X n ) (24) 2

13 Assumption 5.2. For any P P, the rom variables X,..., X n are i. i. d. non-degenerate under P. Thus, model ambiguity only appears in the multiplicity of the distributions µ P of X under the various measures P P. As in Section 4 we assume for simplicity that X belongs to L. Corollary 5.2. The robust premium π c,n defined by (24) satisfies π c,n max E P [ X ] lim π c,n = max E P [ X ]. (25) n Proof. Due to (8) we can choose δ > 0 such that e P,δ (X ) max E P [ X ] + ɛ for a given ɛ > 0. In analogy to the proof of Corollary 4., Proposition 5. yields the estimate π c,n c δn + max E P [ X ] + ɛ, this implies (25). Remark 5.. While the pooling of risks has the desired effect if premiums are computed in terms of ρ P,c, this is not the case if we use the robust version e P,γ of the convex entropic risk version. Indeed, it is easy to check that the above homogeneity Assumption 5.2 implies e P,γ (X X n ) = ne P,γ (X ). We conclude with the robust extension of Corollary 4.2 Remark 4.. Proposition 5.2. For a fixed premium π such that the corresponding tolerance level max E P [ X ] < π < ess sup( X ), c π,n := max{c > 0 n ρ P,c(X X n ) π} (26) is given by where c π,n = ni P (π) = nλ P(π), (27) I P (π) := min H(Q P) (28) Q:E Q [ X ]=π Λ P(π) = sup{γπ sup log E P [e γx ]}. (29) γ>0 In particular, I P coincides with the convex conjugate Λ P of the convex function Λ P defined by Λ P (γ) := sup log E P [e γx ], γ > 0. 3

14 Proof. ) Let us first show the identity I P = Λ P. Indeed, for any Q M such that E Q [ X ] π, for any P P, for any γ > 0, (0) implies H(Q P ) γπ log E P [e γx ] with equality iff Q = Q γ Q P,X γ > 0 is such that E Qγ [ X] = π. This yields the classical identity I P (π) := min Q:E Q [ X ]=π H(Q P ) = Λ P (π), where Λ P denotes the convex conjugate of the convex function Λ P defined by Λ P (γ) := log E P [e γx ]; cf., e. g., [7], Theorem Thus I P (π) = min H(Q P) = inf Q:E Q [ X ]=π min H(Q P ) Q:E Q [ X ]=π = inf Λ P (π). In order to identify the right-h side with Λ P (π), we apply a minimax theorem, for example Terkelsen [7], Corollary 2, to the function f on Φ P (0, ) defined by f( dp dr, γ) = γπ log E R[e γx dp dr ]. This yields inf Λ P (π) = inf sup γ>0 {γπ Λ P (γ)} = sup inf {γπ Λ P (γ)} γ>0 = sup{γπ Λ P (γ)} γ>0 = Λ P(π), hence I P (π) = Λ P (π). 2) In order to verify the first equality in (27), recall from Proposition 5. that n ρ P,c(X X n ) = n E Q c [ n X i ] = E Qc [ X ], where Q c := Q γc Q Pc,X +...+X n γ c is such that c = H(Q c P c ) = H(Q c P). Our assumptions imply that ρ P,c (X X n ) is strictly increasing continuous in c. Thus the tolerance level c π,n is determined by i. e., by the two conditions i= n ρ P,c π,n (X X n ) = π, c π,n = H(Q cπ,n P) = H(Q cπ,n P cπ,n ) E Qcπ,n [ X ] = π. Using part ), we thus see that c π,n = inf H(Q c π,n P ) = inf {E Q cπ,n [ γ cπ,n n X i ] n log E P [exp( γ cπ,n X )]} i= = n inf {γ c π,n π log E P [exp( γ cπ,n X )]} = n(γ cπ,n π log sup E P [exp( γ cπ,n X )]) nλ P(π) = ni P (π). 4

15 On the other h, the same arguments applied to P cπ,n yield c π,n = H(Q cπ,n P cπ,n ) = n(γ cπ,n π log E Pcπ,n [exp( γ cπ,n X )]) = n sup{γπ log E Pcπ,n [e γx ]} γ>0 = ni Pcπ,n (π) ni P (π) since the supremum in the second line is attained by γ = γ cπ,n. As in Remark 4., (27) translates into the upper bound of the following extension of Cramér s theorem to our present context of model ambiguity. For related results on robust large deviations we refer to Sadowsky [5], Pit & Meyn [4], Hu [0]. Corollary 5.3. For any π > max E P [ X ] we have n log( sup P [ n (X X n ) π]) I P (π), lim n n log( sup P [ n (X X n ) > π]) = I P (π), (30) where the rate function I P is given by (28) coincides with (29). Proof. For α π,n := exp( c π,n ), (23) (26) imply i. e., hence VaR P,απ,n (X X n ) ρ P,cπ,n (X X n ) nπ, sup P [X X n + nπ 0] α π,n, n log( sup P [ n (X X n ) π]) I P (π). In order to verify (30), simply recall that Cramér s theorem yields lim n for any P P, hence lim n log( sup n n log P [ n (X X n ) > π] min H(Q P ) Q:E Q [ X ]=π P [ n (X X n ) > π]) sup ( min H(Q P )) Q:E Q [ X ]=π = min Q:E Q [ X ]=π H(Q P) = I P (π). Acknowledgements Thanks are due to Shigeo Kusuoka for a stimulating discussion to an anonymous referee for several helpful comments. In part the results were obtained while the first author was visiting LMU Munich, funded by LMU Excellent; this support is gratefully acknowledged. For both authors, it is a special pleasure to dedicate this paper to Peter Imkeller on the occasion of his 60th birthday. 5

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