Tail Distortion Risk and Its Asymptotic Analysis

Transcription

1 Tail Disorion Risk and Is Asympoic Analysis Li Zhu Haijun Li May 2 Revision: March 22 Absrac A disorion risk measure used in finance and insurance is defined as he expeced value of poenial loss under a scenario probabiliy measure. In his paper, he ail disorion risk measure is inroduced o assess ail risks of excess losses modeled by he righ ails of loss disribuions. The asympoic linear relaion beween ail disorion and Value-a-Risk is derived for heavy ailed losses wih he linear proporionaliy consan depending only on he disorion funcion and ail index. Various examples involving ail disorions for locaion, scale and shape invarian loss disribuion families are also presened o illusrae he resuls. JEL code and keywords: G32, disorion risk measure, regular variaion, ail risk, ail condiional expecaion. Classificaion codes: IM, IM54. Inroducion Le L be he convex cone consising of all he performance variables which represen loss of a financial porfolio a he end of a given period. Noe ha X, where X L, represens he ne worh of a financial posiion. To analyze righ ail risks of loss disribuions, one ofen uses he ail condiional expecaion (TCE): TCE p (X) := E(X X > VaR p (X)), (.) lzhu@mah.wsu.edu, Deparmen of Mahemaics, Washingon Sae Universiy, Pullman, WA 9964, U.S.A. lih@mah.wsu.edu, Deparmen of Mahemaics, Washingon Sae Universiy, Pullman, WA 9964, U.S.A. Research is suppored by NSF grans CMMI and DMS A subse L of a linear space is a convex cone if x L and x 2 L imply ha λ x + λ 2 x 2 L for any λ > and λ 2 >.

2 where VaR p (X) := sup{x R : P(X > x) > p} is known as he Value-a-Risk (VaR) wih confidence level p. VaR is he mos commonly used risk measure in finance and insurance, bu TCE focuses more han VaR does on assessing righ ail risk. Since we are mainly ineresed in analyzing exremal risk for righ ails of loss disribuions, we assume hroughou his paper ha loss variables Xs are nonnegaive. Observe from (.) ha in he case of coninuous loss disribuions, VaR and TCE can be easily expressed as follows. VaR p (X) = TCE p (X) = g VaR (P (X > x))dx, g TCE (P (X > x))dx where g VaR (x) := I {x> p} is he indicaor funcion of {x > p}, and g TCE (x) := min{x/( p), }. Tha is, boh VaR and TCE are he risk measures ha can be expressed in erms of Choque inegrals [3]. Definiion.. For a given nondecreasing funcion g : [, ] [, ] such ha g() = and g() =, he Choque inegral H g (X) of any nonnegaive random variable X wih disribuion F X is defined as follows: H g (X) = g( F X (x))dx = where F X (x) = F X (x) denoes he survival funcion of X. g(f X (x))dx, (.2) The funcion g in (.2) is called a disorion funcion, and if X is viewed as a loss variable, hen H g (X) is also called a disorion risk measure of X wih disorion funcion g. Le P g,f denoe he probabiliy measure induced by he disribuion funcion ḡ(f X (x)), where ḡ(u) = g( u), hen H g (X) = ( ḡ(f X (x)))dx = E Pg,F (X). Tha is, a disorion risk H g (X) measures he expeced value of X under he disored probabiliy measure P g,f ha may describe a possible scenario. The basic properies of Choque inegrals (see [3], chapers 5 and 6) are lised below. Lemma.2. Le g : [, ] [, ] be a disorion funcion, and hen he disorion risk measure H g ( ) saisfies following properies.. (monooniciy) For X, Y L wih X Y almos surely, H g (X) H g (Y ). 2

3 2. (posiive homogeneiy) For all X L and every λ >, H g (λx) = λh g (X). 3. (ranslaion invariance) For all X L and every l R, H g (X + l) = H g (X) + l. 4. (subaddiiviy) If g is concave, hen H g (X + Y ) H g (X) + H g (Y ). 5. (superaddiiviy) If g is convex, hen H g (X + Y ) H g (X) + H g (Y ). The monooniciy says a porfolio wih more poenial loss is riskier. The subaddiiviy is he propery of risk reducion by diversificaion. The homogeneiy describes how porfolio size direcly influences is risk, whereas he ranslaion invariance describes how incurring addiional (deerminisic) loss would affec porfolio risk. Remark.3.. In general, a risk measure ϱ is defined as a measurable mapping, wih some basic operaional axioms, from he space of all loss variables ino R, and hese operaional axioms reflec he risk percepion of agens (or regulaors) involved in he siuaion under consideraion [8]. The risk ϱ(x) for loss X corresponds o he amoun of exra capial requiremen ha has o be invesed in some secure insrumen so ha he resuling posiion ϱ(x) X is accepable o agens. 2. A risk measure relaed o disorion risk is he coheren risk measure inroduced in []. A mapping ϱ : L R is called a coheren risk measure if ϱ saisfies he economically coheren axioms of (), (2), (3), and (4) in Lemma.2. Under some regulariy condiions (such as he Faou propery, see [4]), a coheren risk measure ϱ(x) arises as he supremum of expeced values of loss X under various scenarios: ϱ(x) = sup Q S E Q (X) (.3) where E Q ( ) denoes he expecaion wih respec o he probabiliy measure Q, and S is a convex se of scenario probabiliy measures on saes, ha are absoluely coninuous wih respec o he underlying measure P. If he scenario se S = {P( A) : P(A) p}, hen ϱ(x) is known as he wors condiional expecaion, and in he case of coninuous losses, ϱ(x) equals o TCE p (X). I follows from Lemma.2 ha H g (X) is a coheren risk measure for any concave disorion funcion g, bu here are also non-coheren disorion risk measures. For example, boh VaR and TCE can be wrien in erms of Choque inegrals, hence, hey are wo disorion risk measures (see, e.g., [22, 23] for deails). In conras o TCE ha is a coheren and disorion risk measure, VaR violaes he subaddiiviy and hus is no coheren. Noe ha here are also non-disorion coheren risk measures (see [4]). 3

4 Early work on disorion risk measures and heir represenaion can be found in [2, 24] and he references herein. The connecion of disorion risk o coheren risk was firs esablished in [6, 7], and early work on maxmin ordering preferences and heir dual inegral represenaion can be found in [9, 6]. Goovaers e al. in a recen paper [8] explained he origin of disorion risk measures, and heir use in a risk managemen seing. In paricular, [8] highlighs he role of axiomaic characerizaions in specifying risk in he siuaion under consideraion a he modeling level, and in deriving a he dual level a decision principle based on opimizaion procedures o quanify he risk. Disorion risk measures were inroduced in he acuarial lieraure in [3, 9, 2]. The relaions among risk, enropy and disorion were discussed in [3]. The goal of his paper is o sudy he asympoic behavior of disorion risk measures focusing on he righ ail of a heavy-ailed loss disribuion. A non-negaive loss variable X wih disribuion funcion F has a heavy or regularly varying righ ail a wih ail index α > if is survival funcion is of he following form (see, e.g., [2] for deail), F (x) := P(X > x) = x α L(x), x >, α >, (.4) where L is a slowly varying funcion; ha is, L is a posiive funcion on (, ) wih propery lim x L(cx)/L(x) =, for every c >. The disribuions (.4) consiue a large semi-parameric family of loss disribuions, including he Pareo and Fréche disribuions, ha fi various daa in finance and insurance [4]. While he disorion measures for loss disribuions (.4) generally do no have closed-form expressions, heir asympoic behaviors, as we will show in his paper, can be described explicily in erms of he disorion funcion g and ail index α. To sudy he asympoic behavior of disorion risk, we inroduce ail disorion risk measures. Definiion.4. For a given nondecreasing funcion g : [, ] [, ] such ha g() = and g() =, he ail disorion risk measure H g ( ) of any nonnegaive random variable X is defined as follows: H g (X X > VaR p (X)) = where F X X> (x) = F X X> (x) = P(X x X > ). g(f X X>VaRp(X)(x))dx, (.5) Proposiion.5. The ail disorion risk of a coninuous loss X, as defined above, is a disorion risk measure. 4

5 Proof. Le = VaR p (X), < p <, and consider H g (X X > ) = g(p(x > x X > ))dx ( ) P(X > x, X > ) = g dx P(X > ) ( ) P(X > x) = g()dx + g dx P(X > ) ( ) P(X > x) = dx + g dx (.6) p Le s define a funcion as follows: { g p (u) = g ( u p) if u < p if p u, (.7) and hen i is easy o see ha g p is a nondecreasing funcion saisfying g p () =, g p () =. Combining (.6) and (.7), we obain H g (X X > VaR p (X)) = g p (P(X > x))dx, (.8) which is a disorion risk measure wih disorion funcion g p, < p <. I follows from (.8) ha he ail disorion H g (X X > VaR p (X)) of X can be viewed as he expeced value of X under a change of he underlying measure ha is deformed on he ail loss disribuion. If g =, hen H g (X X > VaR p (X)) = TCE p (X). I is known (see, e.g., []) ha for a heavy-ailed loss X wih disribuion (.4), he asympoic relaion of TCE p (X) and VaR p (X) can be derived as follows: for α >, TCE p (X) α α VaR p(x), as p. (.9) The main resul of his paper, o be deailed in Secion 2, esablishes an explici, asympoic relaion beween H g (X X > VaR p (X)) and VaR p (X), as p, which includes (.9) as a special case. Our mehod is based on a limiing resul for inegrals of disorion funcions wih respec o regularly varying loss disribuions. The ail disorions for locaion, scale, and shape invarian families are discussed in Secion 3 o illusrae our asympoic resul. Finally, some commens and an example involving insurance premium pricing are discussed in Secion 4 o conclude he paper. 5

6 2 Asympoic Properies of Tail Disorion Risk Le X denoe a loss wih regularly varying disribuion (.4). We use RV α in his paper o denoe he class of all nonnegaive loss variables whose survival funcions are regularly varying wih ail index α >. Roughly speaking, regularly varying funcions are hose funcions which behave asympoically like power funcions. The heory of regularly varying funcions is an essenial analyical ool for dealing wih heavy ails, long-range dependence and domains of aracion, and he deailed discussions on hese funcions can be found in [2, 4]. In paricular, he following uniform convergence for regularly varying funcions can be found in [4], Secion 2.. Lemma 2.. For any regularly varying funcion U(x) = x α L(x) for α R +, where L(x) is slowly varying, one has ha lim U(x)/U() = x α uniformly on inervals of he form (b, ), b >. Observe from (.6) ha H g (X X > ) = + g ( ) P(X > x) dx, P(X > ) where = VaR p (X). Le x = w, we obain ha ( ) P(X > w) H g (X X > ) = + g dw, P(X > ) ha is, for any >, H g (X X > ) = + g ( ) P(X > w) dw. (2.) P(X > ) Since = VaR p (X), as p, he asympoic relaion beween H g (X X > VaR p (X)) and VaR p (X), as p, boils down o he convergence of he inegral in (2.). We firs illusrae our idea using coninuous disorion funcions. Proposiion 2.2. If disorion funcion g( ) is coninuous, and a loss random variable X is regularly varying (i.e., X RV α, α > ), hen ( ) P(X > w) g g(w α ), as, P(X > ) uniformly on [, ). 6

7 Proof. Since funcion g( ) is coninuous on [, ], hen by he Heine-Canor heorem, g( ) is uniformly coninuous on [, ]. Thus, for all ε >, here exiss δ >, such ha g(x) g(x ) < ε, x, x [, ] wih x x < δ. (2.2) Since X RV α wih P(X > ) = α L(), >, α >, where L is a slowly varying funcion, i follows from Lemma 2. ha P(X > w) P(X > ) = (w) α L(w) α L() w α, as, (2.3) uniformly on [, ). The uniform convergence of (2.3) implies ha for he seleced δ > in (2.2), here exiss an N >, whenever > N, P(X > w) P(X > ) w α < δ, w [, ). Since boh P(X>w) and w α are in [, ] for w, we have from (2.2) ha whenever > N, P(X>) ( ) P(X > w) g g(w α ) P(X > ) < ε, w [, ). The uniform convergence holds. Applying Proposiion 2.2, we obain ha as p, ( ) H g (X X > VaR p (X)) + g(w α )dw VaR p (X), for any coninuous disorion funcion g. The assumpion of coninuiy can be weakened by using he following deeper analysis. Theorem 2.3. If a loss random variable X RV α, and g( ) is any disorion funcion wih g(w α+δ )dw < for some < δ < α, hen for any b, lim Proof. Since g is nondecreasing, b b g ( P(X > w) P(X > ) for any ε >, here exiss an N such ha Since X RV α, we have ) dw = b g(w α )dw. g(w α )dw g(w α+δ )dw < for any b, hen b N g(w α )dw < ε 4. (2.4) P(X > w) P(X > ) = (w) α L(w) α L() 7 = w α L(w), L()

8 where L is a slowly varying funcion. By he represenaion heorem of slowly varying funcions (see, e.g., Theorem.3. in [2] for deail), we know ha { x } L(x) = c(x) exp ε(u)du/u, x a a for some a >, where c( ) is measurable and bounded, and c(x) c > as x, and ε(x) as x. Therefore, P(X > w) P(X > ) = w α L(w) L() { α c(w) w } = w c() exp ε(u)du/u, for w b. For given δ >, we have ha for sufficienly large, δ < ε() < δ, so ha w ε(u)du/u δ w du/u = δ log w. Since c(w)/c() is asympoically bounded by, say, c, as, here exiss a such ha whenever >, { P(X > w) α c(w) w } = w P(X > ) c() exp ε(u)du/u w α c exp{δ log w} = c w α+δ, for w b. (2.5) Since g( ) is nondecreasing, he upper bound in (2.5) implies ha here exiss an N 2 such ha whenever >, ( ) P(X > w) g dw g(c w α+δ )dw < ε P(X > ) N 2 4. (2.6) N 2 Le N = max{n, N 2 }. Since g( ) is non-decreasing and bounded, he se of disconinuiy poins of g( ) is a mos counable (Froda s heorem) and hus has Lebesgue measure zero. Tha is, lim g ( ) P(X > w) = g(w α ), (2.7) P(X > ) almos everywhere on [, N]. By he hird Lilewood s principle (see, e.g., Proposiion 24, [5], page 73), here exiss a se A [, N] such ha µ(a) ɛ/8, where µ( ) denoes he Lebesgue measure, and he convergence in (2.7) is uniform on [, N]\A. Tha is, here is a, such ha whenever >, ( ) P(X > w) g g(w α ) P(X > ) ɛ, w [, N]\A. 4(N b) 8

9 which, ogeher wih he fac ha g is bounded by, imply ha, whenever >, N ( ) P(X > w) N g dw g(w α )dw b P(X > ) b ( ) ( ) P(X > w) g g(w α ) A P(X > ) dw + P(X > w) g g(w α ) [b,n]\a P(X > ) dw 2 dw + ε 4 ε 2. (2.8) A Combining (2.4), (2.6) and (2.8), we have ha for given ε >, whenever >, ( ) Pr(X > w) g dw g(w α )dw b Pr(X > ) b N ( ) Pr(X > w) N g dw g(w α )dw b Pr(X > ) b ( ) + Pr(X > w) g dw Pr(X > ) + g(w α )dw The desired limi follows. N < ε 2 + ε 4 + ε 4 = ε. Since a disorion funcion g( ) is uniformly coninuous on [, ] excep perhaps for a subse wih small Lebesgue measure, he deailed represenaion of regular variaion has o be used o esimae he ails of he involved inegrals. Applying Theorem 2.3 o (2.) yields our main resul. Theorem 2.4. If X RV α and g( ) is any disorion funcion wih g(w α+δ )dw < for some < δ < α, hen, H g (X X > VaR p (X)) lim p VaR p (X) = + N g(w α )dw. Remark If g( ) is coninuous, hen by Proposiion 2.2, g(w α )dw mus be finie in order o use our esimae for ail disorion. For example, if g() k as, where k > /α, hen g(w α )dw <. 2. In he case ha disconinuiy exiss, by Theorem 2.4, g(w α+δ )dw mus be finie in order o use our esimae for ail disorion. This is slighly sronger han requiring ha g(w α )dw <. For example, if g() k as, where k > /(α δ) (> /α), hen g(w α+δ )dw <. 9

10 Noe ha if g = and α >, hen he asympoic relaion in Theorem 2.4 reduces o (.9). I is also easy o see ha he raio of H g (X X > VaR p (X)) over VaR p (X) is asympoically decreasing o as ail index α increases o. Example 2.6. (Proporional Hazards Disorion) The proporional hazards ransform is obained by consraining he expeced inegraed hazard rae under he deformed disribuion. This disorion has he following form: g(x) = x k, x, k. (2.9) This disorion funcion is concave if and only if k, and convex if and only if k. Hence, he disorion risk measure is coheren if and only if k. This disorion risk measure has been exensively sudied in insurance applicaions (see [9], [2] and [22]). For a loss random variable X RV α, where α > /k, we have, as, H g (X X > ) + = + αk = g(w α )dw = + αk αk. w αk dw If we choose = VaR p (X), hen we have ha as p, H g (X X > VaR p (X)) αk αk VaR p(x). (2.) Noe ha if < α, hen TCE of X does no exis. In conras, he ail disorion risk of X wih disorion funcion wih k > /α can sill be esimaed via (2.). Example 2.7. (Exponenial Disorion) The exponenial disorion is simply he cumulaive disribuion funcion of an exponenial random variable consrained o he uni inerval. Here we have: g(x) = e x/c, x. (2.) e /c The exponenial disorion funcion is always concave, herefore, he corresponding disorion risk measure is coheren. Consider an exponenial disorion funcion g(x) = e x, x [, ] and a loss random e variable X RV α, α >. As = VaR p (X), we have H g (X X > ) + g(w α )dw = + e /wα e dw.

11 For various ail index values, we abulae he asympoics as follows: α e /wα H dw lim g(x X>) e Tail Disorion for Locaion, Scale and Shape Invarian Families In his secion, we discuss hree disorion funcion families, resuling he disored disribuions ha are invarian, respecively, under locaion, scale and shape ransformaions. In each case, we calculae he ail disorion via Theorem Locaion invarian disorion Le F l denoe he family of disribuions on R ha are invarian under locaion ransforms; ha is, if he disribuion of a random variable W is in F l, hen he disribuion of W λ is also in F l for any λ R. Define g F,λ (x) = F (F (x) + λ), x [, ], (3.) where λ R. Obviously, (3.) is a well defined disorion funcion, and if F is sricly increasing, hen F F l implies ha g F,λ (F (x)) F l for any λ. The examples of (3.) include he following Wang ransform, inroduced in [2] o develop a universal pricing mehod: F (x) = Φ[Φ (F (x)) + λ], (3.2) where Φ is he sandard normal cumulaive disribuion. The parameer λ is called he marke price of risk, reflecing he level of sysemaic risk, and F (x) is he cumulaive disribuion of a financial asse value X. For a liabiliy wih loss variable X, he Wang ransform has an equivalen represenaion. F (x) = Φ[Φ ( F (x)) + λ], (3.3)

12 Wang ransform no disorion lambda=.6 lambda=-.7 lambda=.25 Scale invarian disorion no disorion normal wih sigma=.5 normal wih sigma= x x Figure : Graphs for Wang Transform and Scale Invarian Disorion which can be used in he se-up of Definiions. and.4. Figure shows he Wang ransform wih various parameers. The disorion funcion (3.) and is dynamic exension were proposed in [7] under he guise of Esscher-Girsanov ransform. The connecion beween he Esscher-Girsanov ransform and he Wang ransform is highlighed in []. Example 3.. Suppose a loss random variable X RV α, α >. funcion g(x) = Φ[Φ (x) + λ], x [, ], as, we have For he disorion H g (X X > ) + Φ[Φ (w α ) + λ]dw. (3.4) For various values of ail index α and marke price λ, we abulae he corresponding ail esimaes as follows. H α lim g(x X>) H, λ =.7 lim g(x X>), λ = Table : Tail Esimaions for Wang Transform 2

13 3.2 Scale invarian disorion Le F s denoe he disribuion family invarian under scale ransforms; ha is, if he disribuion of W is in F s, hen he disribuion of W/σ is also in F s for any σ >. Define g F,σ (x) = F (F (x) σ), x [, ], (3.5) where σ R +. The funcion in (3.5) is also a well defined disorion funcion, and if F is sricly increasing, hen F F s implies ha g F,σ (F (x)) F s for any σ >. For example, he family of he normal disribuions are scale invarian. Figure shows scale invarian disorions for normal disribuions. Example 3.2. Suppose a loss random variable X RV α, α >. funcion g(x) = Φ[Φ (x) σ], x [, ], as, we have For he disorion H g (X X > ) + Φ[Φ (w α ) σ]dw. (3.6) For various ail index values, Table 2 shows ail esimaions for scale invarian disorion wih differen parameers. H α lim g(x X>) H, σ = 2 lim g(x X>), σ = Table 2: Tail Esimaions for Scale Invarian Disorion 3.3 Power disorion For any nonnegaive, regularly varying loss variable X RV α, i follows from Proposiion 2.6 in [4] ha X k RV αk. Define, for he disribuion F of a nonnegaive random variable, g F,k (x) = F ((F (x)) k ), x [, ], (3.7) where k >. I is easy o see ha (3.7) is a well defined disorion funcion, and if F is sricly decreasing and has a regularly varying righ ail wih ail index α, hen g F,k (F (x)) also has a regularly varying righ ail wih ail index kα. 3

14 Power Disorion wih shape parameer no disorion k=. k=.5 k=.5 k=2 Power Disorion wih shape parameer no disorion k=. k=.5 k=.5 k= x x Figure 2: Graphs for Power Disorion wih Pareo Disribuion Tail Condiional Expecaion k =.99 k =.97 k =.95 k =.93 Wihou disorion E(X X > VaR.95 (X)) $32882 $32882 $32882 $32882 Wih disorion H g (X X > VaR.95 (X)) $495 $974 $ $ Risk Adjusmen $623 $58822 $2265 $22342 Loading Percenage 2.% 44.3% 92.3% 67.8% Table 3: Summary of Risk Adjusmens If we choose F as he Pareo disribuion, Figure 2 shows power disorions wih Pareo disribuions for various k s and differen shape parameers (ail indexes). Figure 3 shows ail esimaions for power disorions wih Pareo disribuions (k = 2). I is worh menioning ha he power disorion wih Pareo disribuion wih shape parameer α = 2 resembles he Pareo disorion discussed in [5, 8]. Example 3.3. As in [2], consider a ground-up liabiliy risk X wih a Pareo survival (severiy) disribuion ( 2 ).2, S(x) = for x >. 2 + x To compare he risk loading, we apply he power disorion g S,k (x) = S((S (x)) k ) wih various values for k and we calculae he risk loading wih and wihou disorion. The resuling summary of he expecaions of he excess of loss amoun is provided in Table 3. The risk adjusmen and loading percenage increase as he power of disorion funcion decreases. This can be explained by he fac ha here is greaer uncerainy in he ails of 4

15 Tail esimaion shape parameer=.5 shape parameer= Tail index Figure 3: Tail Esimaions for Power Disorion he disribuions as ail index decreases (ha is, he ail becomes heavier). 4 Conclusion In his paper, we have inroduced he ail disorion risk for exreme loss, and derived he explici, asympoic expression of he ail disorion for losses ha have regularly varying righ ails. In conras o disorion risks, he ail disorion risk measures for heavy ailed losses admi asympoically linear relaions wih VaR wih he proporionaliy consan depending only on he disorion funcion and ail index. The well-known asympoic relaion beween TCE and VaR serves as a special case of he resul we derived. We conclude his paper by illusraing an applicaion of our resuls o insurance premium pricing. The calculaion of premium is usually affeced by various elemens: pure risk premium, risk loading, adminisraive expenses, and loading for invesmen risk, credi risk, operaional risk, ec. We consider here a simplified premium pricing model ha consiss of he pure risk premium and risk loading only. Le X denoe he oal claim incurred for one insurance porfolio, having a regularly varying loss disribuion wih ail index α. The pure risk premium is he mean E(X) of loss X, and he risk loading depends on he excess of a risk measuremen over he mean loss. If we are ineresed in he ail of he underlying loss disribuion and adop a ail disorion measure H g (X X > VaR p (X)) for p near, hen he risk loading depends on he excess H g (X X > VaR p (X)) E(X) of 5

16 he ail risk measuremen over he pure risk premium. Tha is, if we hold he risk capial H g (X X > VaR p (X)) VaR p (X), we know we wouldn have ruin wih probabiliy p. If an insurance company receives he excess H g (X X > VaR p (X)) E(X) from invesors and invess he capial a he risk-free rae r (e.g., in governmen bonds), he company needs o pay he invesors a a higher rae r > r, because heir invesmen is exposed o risk. Thus, he premium paid by he policyholder in his simple pricing model is he sum of he pure risk premium and risk loading: E(X)+(r r )(H g (X X > VaR p (X)) E(X)) = ( r+r )E(X)+(r r )H g (X X > VaR p (X)). Since p is close o, Theorem 2.4 implies ha he premium is approximaely equal o ( ) ( r + r )E(X) + (r r ) + g(w α )dw VaR p (X). The loss disribuion and model parameers can be fied from claim daa, and he calculaion of he premium is hen sraighforward. Acknowledgmens: The auhors would like o sincerely hank a referee for his/her insighful commens, which led o significan improvemens of he resuls and presenaion of his paper. References [] Arzner, P., Delbaen, F., Eber, J.M. and Heah, D. (999). Coheren measures of risks. Mahemaical Finance 9: [2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (987). Regular Variaion. Cambridge Universiy Press, Cambridge, UK. [3] Denneberg, D. (994). Non-Addiive Measure and Inegral. Kluwer Academic Publishers, Dordrech, The Neherlands. [4] Delbaen, F. (22). Coheren risk measure on general probabiliy spaces. Advances in Finance and Sochasics-Essays in Honour of Dieer Sondermann, Eds. K. Sandmann, P. J. Schönbucher, Springer-Verlag, Berlin, 37. [5] Frees, E. W. and Valdez, E. A. (998). Undersanding relaionships using copulas. Norh American Acuarial Journal, 2:-25. [6] Gilboa, I. and Schmeidler, D. (989). Maxmin expeced uiliy wih non-unique prior. Journal of Mahemaical Economics 8:

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