On the Haezendonck-Goovaerts Risk Measure for Extreme Risks

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1 On the Haezendonc-Goovaerts Ris Measure for Extreme Riss Qihe Tang [a],[b] and Fan Yang [b] [a] Department of Statistics and Actuarial Science University of Iowa 241 Schaeffer Hall, Iowa City, IA 52242, USA [b] Applied Mathematical and Computational Sciences Program University of Iowa 14 MacLean Hall, Iowa City, IA 52242, USA September 21, 211 Abstract In this paper we are concerned with the calculation of the Haezendonc-Goovaerts ris measure, which is defined via a convex Young function and a parameter q, 1) representing the confidence level. We mainly focus on the case in which the ris variable follows a distribution function from a max-domain of attraction. For this case, we restrict the Young function to be a power function and we derive exact asymptotics for the Haezendonc-Goovaerts ris measure as q 1. As a subsidiary, we also consider the case with an exponentially distributed ris variable and a general Young function, and we obtain an analytical expression for the Haezendonc-Goovaerts ris measure. Keywords: Asymptotics; Haezendonc-Goovaerts ris measure; Max-domain of attraction; Regular/rapid variation; Young function Mathematics Subject Classification: Primary 62P5; Secondary 6G7, 91B82, 62E2 1 Introduction Throughout the paper, let X be a real-valued random variable, representing a ris variable in loss-profit style, with a distribution function F = 1 F on R =, ). Let ϕ ) be a non-negative and convex function on [, ) with ϕ) =, ϕ1) = 1 and ϕ ) =. This function is called a normalized Young function, which, due to the convexity of ϕ ), is continuous and strictly increasing. Recall that the Orlicz space associated with the Young function ϕ ) is defined as L ϕ = {X : E [ϕcx)] < for some c > } Corresponding author: Fan Yang; fan-yang-2@uiowa.edu; Cell: ; Fax:

2 and the Orlicz heart as L ϕ = {X : E [ϕcx)] < for all c > }. It is easy to see that L ϕ and L ϕ coincide with each other if lim sup x ϕ2x)/ϕx) < ; see also page 77 of Rao and Ren 1991). For a Young function ϕ ) and a ris variable X L ϕ, let H q [X, x] be the unique solution h to the equation [ )] X x) E ϕ =, q, 1), 1.1) h if F x) > and let H q [X, x] = otherwise, where and throughout the paper, denote by Y = Y 1 Y ) = Y the positive part of a random variable Y and by 1 A the indicator of an event A. The existence and uniqueness of the solution h to equation 1.1) for the case F x) > can be seen from the fact that the left-hand side of 1.1) is a continuous and strictly decreasing function of h > with limits as h and as h. For q, 1), the Haezendonc-Goovaerts ris measure for X is defined as H q [X] = inf x R x H q[x, x]). 1.2) This ris measure was first introduced by Haezendonc and Goovaerts 1982) and was named as the Haezendonc ris measure by Goovaerts et al. 24). Based on a recent conversation with Bellini and Rosazza Gianin during the 15th International Congress on Insurance: Mathematics and Economics in Trieste, we thin that it is more proper to call it the Haezendonc-Goovaerts ris measure in order to acnowledge the contribution of both authors in their seminal paper Haezendonc and Goovaerts 1982). This ris measure has recently been studied by Bellini and Rosazza Gianin 28a, 28b, 211), Nam et al. 211), Krätschmer and Zähle 211), Goovaerts et al. 211) and Ahn and Shyamalumar 211). We have followed the style of Bellini and Rosazza Gianin 28a, 28b) to define this ris measure. As pointed out by Bellini and Rosazza Gianin 28a, 211), the Haezendonc- Goovaerts ris measure H q [X] is a law invariant and coherent ris measure. We remar that, due to its very definition, an analytic expression for H q [X] is not possible in general. The simplest, yet interesting, special case is when ϕt) = t for t. In this case, H q [X] = inf x E [ ]) X x) = F q) E[X F q)) ], 1.3) x R where, and throughout the paper, F q) = inf{x R : F x) q} denotes the inverse function of F, also called the quantile of F or the Value at Ris of X. Thus, the Haezendonc- Goovaerts ris measure is reduced to CTE q [X], the well-nown Conditional Tail Expectation of X. 2

3 The parameter q in the definition of the Haezendonc-Goovaerts ris measure vaguely represents a confidence level. This is demonstrated by the special case above corresponding to ϕt) = t for t. In the post financial crisis era, ris managers become more and more concerned with the tail area of riss due to the excessive prudence of nowadays regulatory framewor. The majority of this paper focuses on the asymptotic behavior of H q [X] as q 1. For a ris variable X with a distribution function F on R, denote by ˆx = F 1) its upper endpoint and by ˆp = PrX = ˆx) the probability assigned to the upper endpoint. Note that ˆp = holds automatically if ˆx = or if F is continuous at ˆx. Theorem 3.1 of Goovaerts et al. 24) shows that F q) H q [X] ˆx. Thus, if < ˆp 1, then H q [X] = ˆx for all 1 ˆp < q < 1, while if ˆp =, then lim H q [X] = ˆx. q 1 We only need to consider the latter case with ˆp =. When ˆx = we shall establish exact asymptotics for H q [X] diverging to as q 1, while when ˆx < we shall establish exact asymptotics for ˆx H q [X] decaying to as q 1. Hence, the notion of extreme value theory becomes relevant. We shall assume that the ris variable X follows a distribution function F from the maxdomain of attraction of an extreme value distribution function. Due to the complexity of the problem, we shall only consider a power Young function, ϕt) = t for 1. We prove the following: If F is from the Fréchet max-domain of attraction, then H q [X] c 1 F q); If F is from the Gumbel max-domain of attraction, then H q [X] F 1 c 2 q) provided ˆx = or ˆx H q [X] ˆx F 1 c 2 q)) provided ˆx < ; If F is from the Weibull max-domain of attraction, then ˆx H q [X] c 3 ˆx F q)). In these assertions, the notation means that the quotient of both sides tends to 1 as q 1 and the coeffi cients c 1, c 2 and c 3 are explicitly given. As a subsidiary, we also consider the case with an exponentially distributed ris variable X and a general Young function ϕ ). For this case, we obtain an analytical expression for the Haezendonc-Goovaerts ris measure H q [X]. However, at this stage we cannot extend the study to the case of both a generally distributed ris variable X and a general Young function ϕ ). The rest of this paper consists of six sections. In Section 2 we establish a general result for the Haezendonc-Goovaerts ris measure with a power Young function. This result forms the 3

4 theoretical basis of our asymptotic analysis. Then after preparing some preliminaries on maxdomains of attraction and regular/rapid variation in Section 3, we derive exact asymptotic formulas for the Haezendonc-Goovaerts ris measure with a power Young function for the Fréchet, Gumbel and Weibull cases in Sections 4, 5 and 6, respectively. For each case, numerical studies are also carried out to examine the accuracies of the asymptotic formulas. In Section 7 we consider the case of an exponentially distributed ris variable and a general Young function. 2 General discussions with a power Young function Let X be a ris variable distributed by F with an upper endpoint ˆx. Assume a power Young function, ϕt) = t for some 1. For this case, the Orlicz space and Orlicz heart coincide with each other, both equal to { X : E[X] < }. As we mentioned before, if = 1, then the Haezendonc-Goovaerts ris measure for X is equal to CTE q [X] given by 1.3). Thus, we only consider > 1 in the following theorem in which we analyze the Haezendonc-Goovaerts ris measure for a general ris variable X: Theorem 2.1 Let the Young function be ϕt) = t for some > 1 and let X be a ris variable with E[X] <, ˆx and PrX = ˆx) =. Then the Haezendonc-Goovaerts ris measure for X is equal to [ ]) 1/ H q [X] = x, q, 1), 2.1) where x = xq), ˆx) is the unique solution to the equation [ ]) 1 [ ]) 1 =. 2.2) The proof of Theorem 2.1 relies on the following elementary result: Lemma 2.1 Consider the function gx) = E [ X x) ] for x R, where 1 is a constant and X is a random variable with E[X ] <. a) If > 1, then gx) is continuously differentiable with d dx gx) = E [ ] X x) 1, x R; 2.3) b) If = 1, then d dx g x) = F x) and d dx g x) = F x ) for each x R. 4

5 Proof. Our derivation for d g dx x) below is good for both > 1 and = 1. Observe that [ ] d E X x x)) dx g E [ ] X x) x) = lim x x = lim x E [ X x) x ] 1 x<x x x) X x x)) X x) 1 X>x x) x = lim x E [I 1 x) I 2 x)]. 2.4) Clearly, I 1 x) x) 1 1 x<x x x). Moreover, there is some ξ between X x x) and X x such that I 2 x) = ξ 1 1 X>x x). These estimates for I 1 x) and I 2 x) enable us to apply the dominated convergence theorem to interchange the order of the limit and expectation in 2.4). Hence, [ ] d dx g x) = E lim I 1 x) lim I 2 x) = E [ ] X x) 1. x x When = 1, this gives d dx g x) = F x). For d dx g x), we need to distinguish the cases > 1 and = 1. For > 1, going along the same lines as above we obtain d dx g x) = E [ ] X x) 1. For = 1, we have [ ] d dx g E X x x)) E [X x) ] x) = lim x x = lim x E [ X x x) x 1 x x<x<x) 1 X x) ]. Clearly, the first term after the expectation, denoted by I 3 x), satisfies I 3 x) 1 x x<x<x). Hence, by the dominated convergence theorem, [ ] d dx g x) = E lim I 3 x) 1 X x) x = F x ). Finally, for > 1, the expression for d gx) given by 2.3) is obviously continuous in dx x R. This ends the proof of Lemma

6 Proof of Theorem 2.1. For ϕt) = t, it follows straightforwardly from 1.1) that [ ] H q [X, x] = ) 1/. By virtue of Minowsi s inequality, one can easily verify that H q [X, ] is convex over R and is strictly convex over, ˆx); see also Proposition 3 of Bellini and Rosazza Gianin 211). Write g x) = x [ ]) 1/, x R, so that H q [X] = inf x R g x). The function g ) inherits the convexity of H q [X, ] and g x) diverges to as x ±. Hence, its overall infimum is attainable. To obtain this infimum, we naturally consider the equation d dx g x) =. By Lemma 2.1a), [ ]) d 1/) 1 [ ] dx g x) = ) ) 1/ Thus, the equation d dx g x) = is equivalent to 2.2). For every q, 1), the existence of a solution x, ˆx) to 2.2) can be verified as follows. The left-hand side of 2.2) is a continuous function of x R. As x, applying the dominated convergence theorem to both the numerator and denominator yields that [ [ ]) 1 X x) 1 ]) E x) [ ]) 1 = 1 [ ]) X x) 1 1; E x) while as x ˆx, applying Hölder s inequality to the numerator yields that [ ]) 1 [ ]) 1 1 X>x) [ ]) 1 = [ ]) 1 E [ X x) ]) 1 E [ 1X>x) ] E [ X x) ]) 1 = F x). 2.6) Moreover, the uniqueness of the solution to equation 2.2), or, equivalently, to the equation d dx g x) =, is ensured by the strict convexity of the function g ) on, ˆx). This ends the proof of Theorem 2.1. When proceeding our asymptotic analysis we shall need the following lemma too: Lemma 2.2 Consider equation 2.2) in which 1 is a constant, X is a random variable with E[X ] <, ˆx and PrX = ˆx) =, and q, 1), x, ˆx) are two deterministic variables. Then q 1 if and only if x ˆx. 6

7 Proof. For = 1, equation 2.2) is simplified to F x) =. Thus, the equivalence of q 1 and x ˆx is obvious. Now consider > 1 only. The derivation in 2.6) shows that the left-hand side of 2.2) is bounded by F x). Thus, F x), from which we easily infer that x ˆx implies q 1. Conversely, the proof of Theorem 2.1 shows that the function g ) is strictly convex over, ˆx). Thus, d dx g x) given by 2.5) is strictly increasing in x, ˆx), or, equivalently, the left-hand side of 2.2) is strictly decreasing in x, ˆx). Thus, q 1 must lead to x ˆx. 3 Max-domains of attraction and regular variation In this section we highlight some basic concepts in extreme value theory. Monographs on extreme value theory in the context of insurance and finance are Resnic 1987, 27), Embrechts et al. 1997), McNeil et al. 25) and Malevergne and Sornette 26), among others. In this paper we follow the main methodology of Hashorva et al. 21) and Asimit et al. 211), who studied some insurance problems using extreme value theory. A distribution function F on R is said to belong to the max-domain of attraction of an extreme value distribution function G, written as F MDAG), if lim sup F n c n x d n ) Gx) = n x R holds for some norming constants c n > and d n R, n N = {, 1,...}. By the classical Fisher-Tippett theorem see Fisher and Tippett 1928) and Gnedeno 1943)), only three choices for G are possible, namely the Fréchet, Gumbel and Weibull distributions, which we denote by Φ γ, Λ and Ψ γ, respectively, with γ > indexing members of the Fréchet and Weibull max-domains of attraction. The Fréchet distribution function is given by Φ γ x) = exp { x γ } for x >. A distribution function F belongs to MDAΦ γ ) if and only if its upper endpoint ˆx is infinite and the relation holds; see Theorem of Embrechts et al. 1997). F xy) lim x F x) = y γ, y >, 3.1) The standard Gumbel distribution function is given by Λx) = exp { e x } for x R. A distribution function F with an upper endpoint ˆx belongs to MDAΛ) if and only if the relation F x yax)) lim x ˆx F x) = e y, y R, 3.2) holds for some positive auxiliary function a ) on, ˆx). Recall that a ) is unique up to asymptotic equivalence and a commonly-used choice for ax) is the mean excess function, 7

8 ax) = E [X x X > x] for x < ˆx. Recall also that { ax) = ox), if ˆx =, ax) = oˆx x), if ˆx <. 3.3) See Resnic 1987) and Embrechts et al. 1997) for more details. The following representation theorem is useful; see Balema and de Haan 1972), Proposition 1.4 of Resnic 1987) or relation 3.35) of Embrechts et al. 1997). For F MDAΛ) with ˆx, there is some x < ˆx such that { x } 1 F x) = bx) exp x ay) dy, x < x < ˆx, 3.4) where a ) is an auxiliary function, chosen to be positive and absolutely continuous with lim x ˆx d ax) =, and b ) is a positive measurable function with lim dx x ˆx bx) = b >. The Weibull distribution function is given by Ψ γ x) = exp { x γ } for x. A distribution function F belongs to MDAΨ γ ) if and only if its upper endpoint ˆx is finite and see Theorem of Embrechts et al. 1997). F ˆx xy) lim x F ˆx x) = yγ, y > ; 3.5) The following elementary result might be nown somewhere but we cannot suitably address a reference: Lemma 3.1 Let F on R belong to the max-domain of attraction of a non-degenerate distribution function. Then a) F x ) F x) as x ˆx; b) F F q)) as q 1. Proof. a) The result for the Gumbel case is nown in Corollary 1.6 of Resnic 1987). The result for the Fréchet and Weibull cases easily follows from the equivalent conditions 3.1) and 3.5), respectively. b) This follows from the two-sided inequality F F q)) F F q) ) and the result in a). It is often convenient and useful to restate the equivalent conditions for the max-domains of attraction in terms of regular or rapid variation. A positive measurable function r ) is said to be regularly varying at x = ± or ± with a regularity index α, ), denoted by r ) R α x ), if rxy) lim x x rx) = yα, y >. 8

9 The class R x ) consists of functions slowly varying at x. Moreover, a positive measurable function r ) is said to be rapidly varying at x = ± or ±, denoted by r ) R x ) or 1/r ) R x ), if { rxy), for y > 1, lim x x rx) =, for < y < 1. Relation 3.1) shows that F MDAΦ γ ) if and only if F ) R γ ), while relation 3.5) shows that F MDAΨ γ ) if and only if F ˆx ) R γ ). Furthermore, for F MDAΛ), it easily follows from relations 3.2) and 3.3) that F ) R ) provided ˆx = or F ˆx ) R ) provided ˆx <. Recall Potter s bounds as summarized by Theorem of Bingham et al. 1987): Lemma 3.2 Let r ) be a function regularly varying at x = or with index α, ). It holds for arbitrary < ε < 1 and all x, y suffi ciently close to x that y αε y ) ) α ε 1 ε) x) ry) y αε y ) ) α ε 1 ε). x rx) x) x The following lemma is motivated by Proposition 1.1 of Davis and Resnic 1988): Lemma 3.3 Let F MDAΛ) with the representation given by 3.4). Then, for arbitrary < ε < 1, there is some x < ˆx such that, for all x < x < ˆx and all y, F x yax)) F x) 1 ε)1 εy) 1/ε. Proof. Since lim x ˆx d ax) =, there is some x dx < ˆx such that the inequality ax zax)) ax) εzax) holds for all x < x < ˆx and all z. It follows that Hence, for all x < x < ˆx and all y, F x yax)) F x) This proves Lemma 3.3. = = ax) ax zax)) 1 1 εz. bx yax)) bx) bx yax)) bx) 1 ε) exp exp { { { exp y = 1 ε)1 εy) 1/ε. 9 xyax) x y 1 1 εz dz } 1 az) dz } ax) ax zax)) dz }

10 4 The Fréchet case with a power Young function Our asymptotic analysis in the next three sections is based on Theorem 2.1. In this section we consider the Fréchet case. As usual, denote by B, ) the beta function, namely, Ba, b) = 1 x a 1 1 x) b 1 dx, a, b >. Theorem 4.1 Let ϕt) = t for some 1 and let F MDAΦ γ ) for some γ >. Then, as q 1, H q [X] We first prepare an elementary result: γ γ )/γ) 1 1)/γ B γ, )) 1/γ F q). 4.1) Lemma 4.1 If F MDAΦ γ ) for some γ >, then it holds for all < < γ that E [ ] X x) lim = Bγ, ). 4.2) x x F x) Proof. Since F ) R γ ), by Lemma 3.2, for arbitrary < ε < γ, there is some x > such that, for all x > x and all y, ) γ ε x y 1 ε) x F x y) F x) ) γε x y 1 ε). x By the second inequality above, it holds for all x > x that E [ ] X x) F x y) = dy F x) F x) ) γε x y 1 ε) dy x = 1 ε)x z 1) γε dz. By the arbitrariness of ε, it follows that E [ ] X x) lim sup x x F x) z 1) γ dz. In the same way we can establish the corresponding inequality for the lower limit. In addition, using the change of variables u = z 1) 1 we have Thus, relation 4.2) holds. z 1) γ dz = 1 u γ 1 1 u) 1 du = Bγ, ). 1

11 Proof of Theorem 4.1. In the proof below we shall tacitly use this equivalence. Recall Lemma 2.2, which shows that q 1 if and only if x ˆx. We distinguish the cases = 1 and > 1. For = 1, applying Lemmas 4.1 and 3.1b) to relation 1.3) we have This proves relation 4.1) for = 1. H q [X] F q) F q) F F q)) γ 1 γ γ 1 F q). Now turn to the case > 1. Starting from Theorem 2.1 we need to approximate the optimal value of x that solves equation 2.2). By Lemma 4.1, = [ ]) 1 E [ X x) ]) 1 or, equivalently, 1)Bγ 1, 1)) F x) Bγ, )) 1 = γ ) Bγ, ) F x), 1 F x) 1 ). 4.3) γ ) Bγ, ) By Proposition.8V) of Resnic 1987), it is easy to verify that F ) R γ ) if and only if F 1 ) R 1/γ ). It follows from 4.3) that ) ) x = F 1 o1)) 1 γ ) 1/γ Bγ, ) 1 ) F q). 4.4) γ ) Bγ, ) 1 Now, substituting 4.2), 4.3) and 4.4) into 2.1) yields that H q [X] = x E [ X x) ] Bγ, ) x Bγ, ) x = γ γ x γ γ )/γ) 1 1)/γ This proves relation 4.1) for > 1. ) 1/ ) 1/ x F x) ) 1 1/ ) x γ ) Bγ, ) B γ, )) 1/γ F q). 11

12 Let us use R to numerically examine the accuracy of the asymptotic relation 4.1). Assume that F is a Pareto distribution given by ) α θ F x) = 1, x, α, θ >. x θ Thus, F MDAΦ γ ) with γ = α. We use uniroot to find the root x to 2.2) and then compute 2.1) to get the exact value of the Haezendonc-Goovaerts ris measure H q [X]. Moreover, we compute the asymptotic formula given by 4.1). In both graphs below, we compare the asymptotic estimate to the exact value on the left and show their ratio on the right. For Graph 4.1, we set = 1.1 and 1.2, α = 1.6, and θ = 1. Apparently, the ratio converges to 1 as q 1. We also find that the accuracy improves as decreases. Graph 4.1 is inserted here. Similarly, for Graph 4.2, we set = 1.1, α = 1.5 and 1.6, and θ = 1. We find that the ratio converges to 1 as q 1 and that the accuracy improves gradually as α decreases. Graph 4.2 is inserted here. 5 The Gumbel case with a power Young function Now we consider the Gumbel case. Note that, for a random variable X distributed by F MDAΛ), the requirement E [ X ] < for all 1 holds automatically since either F R ) or ˆx <. As usual, denote by Γ ) the gamma function, namely, Γa) = x a 1 e x dx, a >. Theorem 5.1 Let ϕt) = t for some 1 and let F MDAΛ) with an upper endpoint < ˆx. Then, as q 1, i) when ˆx = we have ii) when ˆx < we have H q [X] F 1 ˆx H q [X] ˆx F 1 To prove Theorem 5.1, we first prepare an elementary result: ) ) ; 5.1) Γ 1) ) ). 5.2) Γ 1) 12

13 Lemma 5.1 If F MDAΛ) with an upper endpoint < ˆx, then it holds for all > that [ ] lim = Γ 1), 5.3) x ˆx a x)f x) where a ) is the auxiliary function appearing in the representation given by 3.4). Proof. When ˆx =, we have E [ ] X x) = F x y)dy = a x)f x) a x)f x) F x zax)) dz F x) e z dz, where in the last step we applied the dominated convergence theorem justified by Lemma 3.3. Similarly, when ˆx <, recalling 3.3) we have E [ ] ˆx x X x) = F x y)dy = a x)f x) a x)f x) Thus, for both cases, relation 5.3) holds. ˆx x)/ax) e z dz. F x zax)) dz F x) Proof of Theorem 5.1. In the proof below, we shall tacitly use the equivalence of q 1 and x ˆx, as shown by Lemma 2.2. The auxiliary function a ) appearing below corresponds to the one in the representation given by 3.4). i) We distinguish the cases = 1 and > 1. For = 1, applying Lemma 5.1 to relation 1.3) and then applying Lemma 3.1 and relation 3.3), we have This proves relation 5.1) for = 1. H q [X] F q) af q)) F F q)) For > 1, applying Lemma 5.1 to relation 2.2) leads to F q). = [ ]) 1 [ ]) 1 Γ)a 1 x)f x) ) Γ 1)a x)f x) ) 1 = Γ 1) F x). 5.4) Then, substituting 5.3) and 5.4) to 2.1) yields that [ ]) 1/ H q [X] x = ax), 5.5) 13

14 which implies that H q [X] x. By Proposition.8V) of Resnic 1987), F ) R ) implies F 1 ) R ). By this and 5.4) we have ) ) x = F 1 1 o1)) ) F 1 ). 5.6) Γ 1) Γ 1) This leads to relation 5.1) for > 1. ii) As before, we still distinguish the cases = 1 and > 1. For = 1, applying Lemma 5.1 to relation 1.3) and then applying Lemma 3.1 and relation 3.3), we have This proves relation 5.2) for = 1. ˆx H q [X] = ˆx F q)) E[X F q)) ] ˆx F q)) af q)) F F q)) ˆx F q). Next consider > 1. For this case the relations in 5.4) still hold. Then, substituting 5.3) and 5.4) to 2.1) yields that ˆx H q [X]) ˆx x) = [ ]) 1/ ax), 5.7) which implies that ˆx H q [X] ˆx x. By Proposition.8V) of Resnic 1987), F ˆx ) R ) implies ˆx F 1 ) R ). By this and 5.4) we have ) ) ˆx x = ˆx F 1 1 o1)) ) ˆx F 1 ). 5.8) Γ 1) Γ 1) This proves relation 5.2) for > 1. In the proof above, the asymptotics for x given by relation 5.6) can actually be changed to x F 1 c)) for any c > because F 1 ) R ). Similarly, the asymptotics for ˆx x given by relation 5.8) can be changed to ˆx x ˆx F 1 c)) for any c >. However, the most rational choice for c in both places should be c = /Γ 1). We would lie to point out that relations 5.5) and 5.7) give second-order asymptotics for H q [X]. They become more powerful than 5.1) and 5.2) provided that it is possible to get the exact value of x solving 2.2) or a good approximation for x. and Recall that a distribution function F belongs to the class Lλ) for some λ > if ˆx = F x y) lim x F x) = e λy, y R. 5.9) The class Lλ) contains many well-nown light-tailed distributions such as the exponential, gamma and inverse Gaussian distributions. Relation 5.9) directly shows that F MDAΛ) with a ) 1/λ. Thus, by 5.5) we arrive at the following: 14

15 Corollary 5.1 Let ϕt) = t for some > 1 and let F Lλ) for some λ >. Then lim H q [X] x) = q 1 λ, ) where x is determined by 2.2) and satisfies x F 1 ). Γ1) Finally, we use R to numerically examine the accuracy of the asymptotics given by 5.5). Assume that F is a lognormal distribution given by ) ln x µ F x) = Φ, x >, < µ <, σ >, σ where Φ ) denotes the standard normal distribution function. Note that ax) = Φσ 1 ln x µ))σx φσ 1 ln x µ)), where φ is the standard normal density function; see, e.g. page 15 of Embrechts et al. 1997). For Graph 5.1, we set = 1.5 and 2, µ = 2, and σ =.5. We compare the asymptotic estimate x ax) given by 5.5) to the exact value of H q [X] on the left and show their ratio on the right. Apparently, the ratio converges to 1 as q 1. We find that the accuracy is particularly good when σ is smaller than 1. Graph 5.1 is inserted here. 6 The Weibull case with a power Young function In the last section of asymptotic analysis we consider the Weibull case. For this case the requirement E [ X ] < for all 1 holds automatically since ˆx <. Theorem 6.1 Let ϕt) = t for some 1 and let F MDAΨ γ ) with γ > and < ˆx <. Then, as q 1, ) ˆx H q [X] γ 1/γ 1 ˆx F q)). 6.1) γ B γ 1, ) γ ) We first prepare an elementary result: Lemma 6.1 If F MDAΨ γ ) with γ > and < ˆx <, then it holds for all > that [ ] X x) lim = B γ 1, ). 6.2) ˆx x) F x) x ˆx E 15

16 Proof. Similarly as in the proof of Lemma 5.1, for x < ˆx, E [ ] ˆx x X x) = F x y)dy = F x) ˆx x F ˆx ˆx x y)) dy. 6.3) F ˆx ˆx x)) Since F ˆx ) R γ ), by Lemma 3.2, for arbitrary < ε < 1, there is some x < ˆx such that, for all x < x < ˆx and all < y < ˆx x, ) γε ) γ ε ˆx x y F ˆx ˆx x y)) ˆx x y 1 ε) 1 ε). ˆx x F ˆx ˆx x)) ˆx x Applying the second inequality above to 6.3), it holds for all x < x < ˆx that E [ ] ˆx x ) γ ε ˆx x y X x) 1 ε)f x) dy ˆx x 1 = 1 ε)f x) ˆx x) 1 z) γ ε dz. By the arbitrariness of ε, it follows that E [ ] X x) lim sup x ˆx ˆx x) F x) 1 1 z) γ dz. A corresponding lower bound can be obtained similarly. Thus, relation 6.2) holds. Proof of Theorem 6.1. In the proof below, we shall tacitly use the equivalence of q 1 and x ˆx, as shown by Lemma 2.2. We distinguish the cases = 1 and > 1. For = 1, starting from relation 1.3) and then applying Lemmas 6.1 and 3.1, we have ˆx H q [X] = ˆx F q)) E[X F q)) ] ˆx F q)) 1 γ 1 ˆx F q)) F F q)) γ γ 1 ˆx F q)). This proves relation 6.1) for = 1. Now consider > 1. Applying Lemma 6.1 to relation 2.2), we have = [ ]) 1 E [ X x) ]) 1 1)B γ 1, 1) ˆx x) 1 F x) ) 1 B γ 1, ) ˆx x) F x) ) = B γ 1, ) γ ) 1 F x). 6.4) 16

17 Substituting 6.2) and 6.4) into 2.1) yields that [ ]) 1/ H q [X] x = ˆx x). 6.5) γ By Proposition.8V) of Resnic 1987), F ˆx ) R γ ) implies ˆx F 1 ) R 1/γ ). Thus, after rewriting 6.4) as we see that F ˆx ˆx x)) ) 1 B γ 1, ) γ ), ) ˆx x = ˆx F 1 o1)) 1 1 ) B γ 1, ) γ ) ) 1/γ 1 ˆx F q)). 6.6) B γ 1, ) γ ) Finally, by 6.5) and 6.6), ˆx H q [X] = ˆx x) H q [X] x) γ ˆx x) γ ) γ 1/γ 1 ˆx F q)). γ B γ 1, ) γ ) This proves relation 6.1) for > 1. Similarly as before, we use R to numerically examine the accuracy of relation 6.1). Assume that F is a beta distribution with probability density function given by fx) = xa 1 1 x) b 1, < x < 1, a, b >. Ba, b) Thus, F MDAΨ γ ) with γ = b. We compare the asymptotic estimate for ˆx H q [X] given by relation 6.1) to its exact value on the left and show their ratio on the right. For Graph 6.1, we set = 3 and 6, a = 2, and b = 6. Apparently, the ratio converges to 1 as q 1. We also find that the varying value of does not affect much the convergence rate. Graph 6.1 is inserted here. For Graph 6.2, we set = 3, a = 2 and b = 6 or 1. The same as before, the ratio converges to 1 as q 1. We also find that the accuracy slightly improves as b increases. Graph 6.2 is inserted here. 17

18 7 The exponential case with a general Young function In this section we consider the case in which the ris variable X is exponentially distributed and the Young function is general. We see an analytical expression for the Haezendonc- Goovaerts ris measure. Theorem 7.1 Let ϕ ) be a general Young function such that e st dϕt) < for all s > and let X follow an exponential distribution function with rate λ >, namely, F x) = e λx for x. Then H q [X] = 1 λ ln e λht dϕt) h, q, 1), 7.1) where h, ) is the unique solution to the equation e λht dϕt) = te λht dϕt). 7.2) The condition on the Young function ϕ ) ensures that the exponential ris variable X belongs to the Orlicz heart of ϕ ). It is interesting to notice that the solution h to equation 7.2) is invariant with both q and x. In order to prove Theorem 7.1, we first prepare an elementary result below: Lemma 7.1 Let ϕ ) be a general Young function such that e st dϕt) < for all s >. Then te st dϕt) te st dϕt) lim = and lim s e st =. 7.3) dϕt) s e st dϕt) Proof. With arbitrary M > we derive te st dϕt) e st dϕt) M te st dϕt) M e st dϕt) M M e st dϕt) ϕm) M e st dϕt) M e st dϕt) M = M, as s, ϕm) 1 M e st dϕt) where the last step is due to the monotone convergence theorem and ϕ ) =. Hence, the first relation in 7.3) holds by the arbitrariness of M. Notice that, with arbitrary < ε < 1, Similarly, ε ε te st dϕt) te st dϕt) ε ε te st dϕt) ε/2 te st dϕt) lim s ε ε te st dϕt) ε ε 2 e sε ϕε) ϕ ε 2 e st dϕt) e st dϕt) =. 18 )), as s.

19 It follows that te st ε dϕt) lim sup = lim sup te st dϕt) s e st ε dϕt) s e st dϕt) ε. Hence, the second relation in 7.3) holds by the arbitrariness of ε. Proof of Theorem 7.1. Goovaerts ris measure in the following way. First of all, let us modify the definition of the Haezendonc- We thin of x in 1.1) as x = x q [X, h], a function of h and q, and introduce g h) = x h. Then we can rewrite 1.2) as H q [X] = inf <h< g h). 7.4) Our idea is to find the optimal value of h at which the infimum in 7.4) is attained. Relation 1.1) is rewritten as e λht dϕt) = ) e λx, which implies that x = 1 λ ln e λht dϕt). 7.5) Under the condition on ϕ ), it is easy to see that e λht dϕt) is infinitely differentiable with respect to h, ). By 7.5), dx dh = te λht dϕt) e λht dϕt). Furthermore, by Hölder s inequality, d 2 x dh = λ t 2 e λht dϕt) e λht dϕt) te λht dϕt) ) 2 2 e λht dϕt) ) 2 >, where the strict inequality is due to the non-degeneracy of the Young function ϕ ). This means that the function x = x q [X, h] and, hence, the function g h) = x h are strictly convex over h, ). Thus, the infimum in 7.4) is attained at h such that d te λht dϕt) dh g h) = e λht dϕt) 1 = 7.6) provided that this last equation admits a solution. Equations 7.6) and 7.2) are equivalent. The existence of a solution h, ) to equation 7.6) is justified by Lemma 7.1, while the uniqueness of the solution h to equation 7.6) results from the strict convexity of the function g ) over, ). Finally, substituting this optimal value h and the expression for x given by 7.1) into 7.4), we obtain the desired expression for H q [X] in 7.1). By Theorem 7.1, the calculation of H q [X] is reduced to solving equation 7.2). a general Young function ϕ ), it is not possible to solve this equation analytically. consider the following special cases: For We 19

20 i) Let ϕt) = t for some 1. Then equation 7.2) has a unique positive solution h = /λ. Thus, H q [X] = 1 λ ln Γ 1) ) λ, which is consistent with Corollary 5.1. ii) Let ϕt) = n =1 a t for some n N and some real-valued coeffi cients a 1,..., a n fulfilling a certain condition such that ϕ ) is a normalized Young function. Then equation 7.2) becomes n 1 a 1 λh) n 1)a 1 a ) Γ 1)λh) n na n Γn 1) =. 7.7) =1 We can always numerically solve this polynomial equation 7.7) in R. For example, let ϕt) = 2t 5 3t 4 2t 3 3t 2 t)/7, q =.95 and λ = 1. Equation 7.7) becomes h 5 5h 4 24h 3 18h 2 48h 12 =. We use polyroot to find its unique positive solution h = Plugging it in 7.1) gives H q [X] = Acnowledgments. The wor was supported by the Centers of Actuarial Excellence CAE) Research Grant ) from the Society of Actuaries. References [1] Ahn, J. Y.; Shyamalumar, N. D. Asymptotic theory for the empirical Haezendonc ris measure. Woring Paper 211), University of Iowa. [2] Asimit, A. V.; Furman, E.; Tang, Q.; Vernic, R. Asymptotics for ris capital allocations based on conditional tail expectation. Insurance Math. Econom ), no. 3, [3] Balema, A. A.; de Haan, L. On R. von Mises condition for the domain of attraction of exp e x ). Ann. Math. Statist ), [4] Bellini, F.; Rosazza Gianin, E. On Haezendonc ris measures. J. Ban Finance 32 28a), no. 6, [5] Bellini, F.; Rosazza Gianin, E. Optimal portfolios with Haezendonc ris measures. Statist. Decisions 26 28b), no. 2, [6] Bellini, F.; Rosazza Gianin, E. Haezendonc-Goovaerts ris measures and Orlicz quantiles. Insurance Math. Econom. 211), to appear. [7] Davis, R.; Resnic, S. Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution. Stochastic Process. Appl ), no. 1,

21 [8] Embrechts, P.; Klüppelberg, C.; Miosch, T. Modelling Extremal Events for Insurance and Finance. Springer-Verlag, Berlin, [9] Fisher, R. A.; Tippett, L. H. C. Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math. Proc. Cambridge Philos. Soc ), no. 2, [1] Gnedeno, B. Sur la distribution limite du terme maximum d une série aléatoire. French) Ann. of Math ), no. 2, [11] Goovaerts, M. J.; Kaas, R.; Dhaene, J.; Tang, Q. Some new classes of consistent ris measures. Insurance Math. Econom ), no. 3, [12] Goovaerts, M. J.; Linders, D.; Van Weert, K.; Tan, F. On the interplay between distortion -, mean value - and Haezendonc ris measures. Insurance Math. Econom. 211), to appear. [13] Haezendonc, J.; Goovaerts, M. A new premium calculation principle based on Orlicz norms. Insurance Math. Econom ), no. 1, [14] Hashorva, E.; Paes, A. G.; Tang, Q. Asymptotics of random contractions. Insurance Math. Econom ), no. 3, [15] Kortscha, D.; Albrecher, H. Asymptotic results for the sum of dependent nonidentically distributed random variables. Methodol. Comput. Appl. Probab ), no. 3, [16] Krätschmer, V.; Zähle, H. Sensitivity of ris measures with respect to the normal approximation of total claim distributions. Insurance Math. Econom ), no. 3, [17] Malevergne, Y.; Sornette, D. Extreme Financial Riss. From Dependence to Ris Management. Springer-Verlag, Berlin, 26. [18] McNeil, A. J.; Frey, R.; Embrechts, P. Quantitative Ris Management. Concepts, Techniques and Tools. Princeton University Press, Princeton, NJ, 25. [19] Nam, H. S.; Tang, Q.; Yang, F. Characterization of upper comonotonicity via tail convex order. Insurance Math. Econom ), no. 3, [2] Rao, M. M.; Ren, Z. D. Theory of Orlicz Spaces. Marcel Deer, Inc., New Yor, [21] Resnic, S. I. Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New Yor, [22] Resnic, S. I. Heavy-tail Phenomena. Probabilistic and Statistical Modeling. Springer, New Yor,

22 Haezendonc Goovaerts ris measure Asymptotic Exact =1.2 =1.1 Ratio of asymptotic estimate to exact value =1.1 = q q Graph 4.1 Young function with =1.1 and 1.2, and Pareto distribution with α=1.6 and θ=1

23 Haezendonc Goovaerts ris measure Asymptotic Exact α=1.5 α=1.6 Ratio of asymptotic estimate to exact value α=1.5 α= q q Graph 4.2 Young function with =1.1, and Pareto distribution with α=1.5 and 1.6 and θ=1

24 Haezendonc Goovaerts ris measure Asymptotic Exact =2 =1.5 Ratio of asymptotic estimate to exact value =1.5 = q q Graph 5.1 Young function with =1.5 and 2, and lognormal distribution with µ=2 and σ=.5

25 Haezendonc Goovaerts ris measure Asymptotic Exact =6 =3 Ratio of asymptotic estimate to exact value =3 = q q Graph 6.1 Young function with =3 and 6, and beta distribution with a=2 and b=6

26 Haezendonc Goovaerts ris measure Asymptotic Exact b=6 b=1 Ratio of asymptotic estimate to exact value b=6 b= q q Graph 6.2 Young function with =3, and beta distribution with a=2 and b=6 and 1