Insuance Picing unde Ambiguity Alois Pichle a,b, a Univesity of Vienna, Austia. Depatment of Statistics and Opeations Reseach b Actuay. Membe of the Austian Actuaial Association Abstact Stating fom the equivalence pinciple vaious picing methods have been consideed in insuance, each aiming at finding a fai pice and eflecting the intinsic isk of an insuance contact at an appopiate level. In this pape we conside fist isk measues, in paticula the Conditional Tail Expectation in ode to establish a isk avese pice fo an insuance contact. We elaboate explicit fomulas fo impotant types of life insuance contacts and obseve that distoted pobability distibutions play an essential ole. In ode to detemine a isk avese pemium the actuay is moeove inteested in computing the pemium unde elated pobability distibutions. We elaboate that a suitable concept to qualify elated pobability distibutions is povided by the Wassestein distance. Fo this distance the actuay often is able to contol and quantify the distance to the eal pobability distibution in an explicit way. We elaboate the picing issues in this context and give eliable, shap bounds fo a isk avese pice, which ae easily available fo the actuay. Keywods: Risk Measues, Continuity popeties, Wassestein and Kantoovich Distance, Distotion Function 21 MSC: 62P5, 6B5, 62E17 1. Intoduction How should one pice an insuance contact? An impotant and fai answe to this question is povided by the equivalence pinciple, that is to say the fai lump sum pemium fo an insuance contact is just the expected value of its possible losses. Besides the fact that othe loadings have to be added to the pemium fo, say, administation and acquisition of the contact, the ealized loss of a concete, individual potfolio will diffe fom the pemiums, the eceipts and expenditues of the entie company natually diffe. Although many othe financial poducts othe than insuance contacts ae piced by the same method as the equivalence pinciple, the vaiance of potential outcomes was investigated significantly ealie in insuance than in banking: Hattendoff pesented his esult on the vaiance of the ealized loss aleady in 1868 cf. [Hat68], [GS3] on Hattendoff s Theoem. This esult natually genealizes to the loss distibution of an entie potfolio, and insuance companies absob individual losses, as long as they do not exceed a specified amount: The cental limit theoem justifies descibing the loss distibution of an insuance company appoximately. To account fo individual insuance contacts with high individual exposue o fo einsuance contacts altenative techniques o methods have been elaboated in the liteatue to establish an adjusted pice, a isk-adjusted, isk-avese o isk-based pice. Examples include utility functions o appopiate tansfoms, cf. [Den89] and [Wan]. Othe methods ovevalue potential losses, fo example by employing isk measues, which have been intoduced in [ADEH99] fo geneal financial contacts. The most pominent among those isk measue is the Conditional Tail Expectation CTE 1 Coesponding Autho UR: http://iso.univie.ac.at/ Alois Pichle 1 The Pepint submitted to Elsevie June 14, 212
2 cf. [KRS9] fo its statistical popeties, which is now pat of the US and Canada insuance industy egulations. The loss distibution is an essential ingedient fo all of these computations. Howeve, the loss distibution is not known pecisely enough in many situation of actuaial elevance, available ae typically a few empiical obsevations o the associated empiical distibution up to some pecision. Fo this eason it is essential fo an actuay to undestand the impact of the loss distibution on the pemium and on the eseve fo both, picing and eseving thooughly and caefully enough, that is to say with actuaial due diligence. In this pape we study the impact of pobability distibutions on pices and isk functions. We shall ecall the notion of isk measues in Section 2 and povide explicit evaluations fo impotant types of life insuance contacts in Section 3. Section 5 intoduces the Wassestein distance to measue the distance of pobability distibutions, and the following section addesses picing issues isk measues unde ambiguity. Concete examples thoughout the pape and a numeical discussion at its end illustate the esults. 2. Risk Measues We conside R valued andom vaiables, in the context of insuance a andom vaiable is typically associated with loss. Risk measues, which have been intoduced [ADEH99] and discussed in a flood of scientific papes since then, assign a eal numbe to any potential loss. In the following we conside the loss functions on spaces such as = p Ω, Σ, P fo 1 p < o = Ω, Σ, P, although thee ae some topological diffeences. This does not matte in the pesent context of insuance, as any loss distibution of pactical impotance satisfies p 1 ; howeve, we shall addess the topological diffeences at the appopiate place in the pape. Definition 1 Risk measues. A isk measue ρ: R } is a function assigning a eal numbe o ρ to a R valued andom vaiable. A isk measue has the following popeties i iv: i Monotonicity: If 1 2, then ρ 1 ρ 2 2 1, 2 ; ii Convexity: ρ λ 1 + 1 λ λρ 1 + 1 λρ wheneve λ 1, 1 ; iii Tanslation Equivaiance: If y R, then ρ + y = ρ + y; 3 iv Positive Homogeneity: Fo λ >, ρ λ = λ ρ. v Vesion Independence: A isk functional ρ is vesion independent 4, if ρ 1 = ρ 2 wheneve 1 and 2 shae the same law, that is P 1 y = P 2 y fo all y R. In this definition we allow ρ to evaluate to. Howeve, thoughout this pape we shall assume that ρ is pope, that is thee is at least one fo which ρ <. Conditional Tail Expectation CTE is sometimes also called Aveage Value-at-Risk, o conditional Value-at-Risk, expected shotfall, tail value-at-isk o newly supe-quantile. 2 Fo 1 and 2 andom vaiables we wite 1 2 wheneve 1 k 2 k fo all samples k. 3 The andom vaiable + y is + y 1 whee 1 is the constant andom vaiable, 1 k = 1. 4 sometimes also law invaiant o distibution based.
2.1 The Conditional Tail Expectation, CTE α 3 2.1. The Conditional Tail Expectation, CTE α The most pominent example of a isk measue is the Conditional Tail Expectation, CTE. The α-level CTE α < 1 is given via the equivalent epesentations CTE α : = 1 1 α 1 α F p dp 1 = inf q + 1 q R 1 α E q + 2 } = sup E Z : Z 1 α, EZ = 1 3 dq = sup E Q : dp 1 } 4 1 α whee F α = q α = inf q : P q α} is the left-continuous α quantile sometimes also the Value-at-Risk o lowe invese cdf at level α. In case that F q α = α which is cetainly the case if F is continuous at q α the additional fomula CTE α = E [ q α ] 5 is often useful. The equivalence of these diffeent epesentations 1 5 can be found, e.g., in [RU] and [PR7]. The function α 1 α CTE α by 1 is concave. It is moeove linea between neighboing values of α F x : x R}, which is an essential obsevation when evaluating CTE α by means of 5 fo discete distibutions. The CTE at level 1 is defined as 2.2. Intepetation in insuance CTE 1 := lim α 1 CTE α = ess sup. In the context of insuance is a andom vaiable associated with loss. It follows fom the definition that CTE = E CTE α CTE 1 = ess sup, which shows that CTE is the pice of the insuance contact accoding the equivalence pinciple. CTE α is a highe lump sum pemium fo the insuance contact, incopoating a cetain loading by judging the insuance contact to be isky fo the company: Accoding to 1 outcomes with highe losses > q α } ae ove-weighted with 1 1 α > 1, wheeas outcomes with smalle losses < q α} ae not even consideed, they ae ignoed by CTE α. Natually the contact s pice cannot exceed CTE 1, as this is the maximum potential loss anyhow and no downside isk can be associated with the pemium CTE 1 fo the insuance company. 3. Applications in life insuance An insuance contact often has a monotone payoff function: if the insued lives longe, then she/ he will eceive moe cash monotone inceasing o less monotone deceasing depending on the contact. This monotonicity popety allows to compute the Conditional Tail Expectation fo some impotant life insuance contacts in an explicit way, the esults often come along with a helpful intepetation, as changing the measue esults in a simple modification of the life table in an appopiate way. Moeove the esults allow evaluating the CTE by employing simple and standad actuaial tools involving just standad pesent values. Fo this we list the esults fo fou elementay insuance types. The payoff, epesenting the insue s loss, depends on the lifetime K. As usual in actuaial science we assume the lifetime K intege valued and wite k := k when convenient in this situation. We
3.1 Inceasing payoff functions 4 employ the intenational, standad actuaial notation basically following [Ge97], such as k p x = P K k and assume thoughout the pape that P K < K x = k= kp x q x+k = 1 x N. 6 3.1. Inceasing payoff functions Fo an inceasing payoff function it holds that k} = K k} by monotonicity, and hence P k = P K k = k q x fo any fixed k, 1... }: The quantile fo α = k q x is kq x = k. F i Pue endowment The andom vaiable fo the payoff of a pue endowment is K := v n 1 K n}, which is an inceasing function in K a step function. The lump sum fo the endowment insuance, accoding the equivalence pinciple, is E = n E x. Obviously F p = if p < n q x v n, such if p n q x that the Conditional Tail Expectation evaluates to CTE α = np x 1 α vn v n if α n q x if α n q x by 1. It follows immediately fo α = k q x that CTE k q x = np x kp x v n v n if k n if k n = v k n k E x+k if k n v n if k n, 7 which is the pice fo an endowment with sum insued due given the insued suvives n yeas, o dies within the fist k yeas. The CTE α of a geneal level α can be found by appopiately intepolating 7. ii ife Annuities The payoff fo an whole life annuity-due is = ä K+1 = K j= vj, its pesent value is E ä K+1 = ä x. The payoff function k ä k+1 is inceasing, as fo the pue endowment. Hence, by 5, ] ] CTE k q x = E [ ä k+1 = E [ä K+1 K k 1 = jp x q x+j ä j+1 = 1 j+kp x q x+k+j ä kp x kp k+j+1 x j=k j= = jp x+k q x+k+j ä k + v k ä j+1 = ä k + v k ä x+k. 8 j= This is the pemium fo an annuity with modified motality, the insued is assumed immotal in the fist k yeas. By the elementay elation A x = 1 d ä x this is equivalent to the expession CTE k q x = ä k + v k ä x+k = ä x+k + A x+k ä k. Fo the moe geneal n yea tempoay life annuity-due with pesent value ä x:n = E ä minn,k+1} the CTE evaluates to äk + v k ä CTE k q x = x+k:n k if k n ä n if k n.
3.2 Deceasing payoff functions 5 Afte these illustating examples it is evident that the geneal patten fo inceasing payoff function is CTE k q x = E [ k] = Ẽ, whee Ẽ is the usual pesent value with espect to the measue which assumes that the insued is immotal within the fist k yeas: the modified life table q to compute Ẽ as in 7 and 8 is Fo this modified life table it should be noted that q =,,..., q } } x+k, q x+k+1,.... 9 k times P K < = k= k p x q x+k = 1 1 by 6, which means that any outcome will die with pobability 1 unde the modified measue as well. 3.2. Deceasing payoff functions If the payoff function is deceasing then k} = K k} by monotonicity, and consequently P k = P K k = k p x fo any fixed k, 1... }. The quantile fo α = k p x thus is given by F kp x = k. i ife insuance The payoff fo a whole life insuance contact is := v K+1, its pesent value is E = A x. The payoff k v k+1 is deceasing. Then the quantile fo α = k p x is q α = v k+1, such that CTE k p x = E [ v k+1] = E [ v K+1 K k ] = k A x kq x 11 by employing 5. Fo the moe geneal tem life insuance = v K+1 1 K n} one deduces the closed fom CTE k p x = na x kq x if k n, if k n fo the Conditional Tail Expectation. ii Endowment Endowment insuance combines life insuance and a pue endowment by use of the deceasing payoff = v mink+1,n}. Its CTE genealizes 11 and 12 by CTE k p x = ka x kq x na x+ ne x k n q x+n kq x if k n, ka x kq x if k n. To descibe the geneal patten fo a deceasing payoff function conside the modified life table 12 q = q x, q x+1,... q x+k,,,..., 13 fo which P K < = k q x 1, which is in notable contast to 1. It follows that P = k k p x q x+k k q x is a pobability measue fo which CTE k p x = Ẽ. δ k
3.3 Geneal insuance contacts 6 3.3. Geneal insuance contacts The methods developed in the latte Sections 3.1 and 3.2 can be applied to geneal insuance contacts, although closed foms ae moe involved o simply not available in many situations. Just conside an endowment insuance with egula pemium Π. Its payoff mink,n} Π = v n 1 K n} Π is neithe de-, no inceasing. Howeve, the CTE α Π can be easily computed by appopiately odeing the potential payoffs and applying 5: Denote P = k kp x q x+k δ k =: k p kδ k the pobability measue and the pemutation such that 1 2 3.... Define Z k := j= 1 1 α if p 1 +... p k 1 α, 1 if p 1 +... p k 1 α = 1 α if P k 1 α, if P 14 k 1 α with the supplementay equiement that E Z = k p k Z k = 1 fo the emaining index. Then, by 3, v j CTE α = E Z = k kp x q x+k k Z k. 15 Equivalently, by employing the modified measue P := k kp x q x+k Z k δ k it holds in accodance with 4 that CTE α = Ẽ. The measue P can be incopoated in a life table q again such that P = k kp x q x+k Z k δ k = k k p x q x+k δ k, but in this geneal situation the patten is not as stiking as in 9 and 13. 3.4. Picing and adapted eseving stategies Risk avese picing A lump sum pemium Π α can be chosen such that CTE α Π α =, 16 whee Πα := Π α is the loss function with incopoated pemium payment at the vey beginning of the contact. It is evident fom the axiom on tanslation equivaiance in Definition 1 that Π α = CTE α. Fo egulaly paid pemiums the annual pemium Π α as well can be chosen such that CTE α Π α =, 17 whee Πα = Π α ä mink,n}. It follows then that = CTE α Π α E Π α = E Π α ä x:n, i.e. Π α E ä x:n = Π, which demonstates that the annual pemium Π α accoding 17 is highe than the annual pemium Π poposed by the equivalence pinciple. The equations 16 and 17 epesent a isk-avese picing stategy, a genealization of the equivalence pinciple, which is discussed in many places of actuaial liteatue.
7 The eseves coesponding to isk avese pemiums It is tempting to define the balance sheet eseves net pemium eseve afte yeas in line with the picing fomulae 16 and 17 as V = CTE α Π K, whee the measue is esticted to the futue lifetime K } and the payoff Π is adjusted to account fo the futue cash flow. This is just in line with the net pemium eseve fo the equivalence pinciple. But this is misleading, the setting violates impotant time-consistency constaints [Sha12], which may cause undesiable jumps in the company s annual balance sheets, and consecutive pofit and loss accounts. The pope adjustment has to accept an annually adjusted level α. The eseves, adjusted fo time consistency and in line with 16 and 17, ae V := CTE α Π K = Ẽ Π K, whee P is the modified measue. Fo a compehensive discussion and justifications of these settings, in paticula fo a time consistent choice of the sequence α we efe to [PP12b] and [PP12a]. 4. Futhe isk functionals It is obvious that aveaging isk functionals, fo example ρ λ := 1 λ CTE α + λ CTE α1, whee λ, α, α 1 [, 1] ae isk functional again, and so is ρ G := sup ρ g, g G whee ρ g is a isk measue fo evey g G. Kusuoka s theoem states that evey isk functional can be obtained by combining the latte two opeations, povided it is vesion independent. Theoem 1 Kusuoka. Any law invaiant, positively homogeneous and lowe semi-continuous isk functional ρ on obeys the epesentation ρ = sup µ M 1 whee M is a set of pobability measues on [, 1]. Moeove ρ = sup h H 1 CTE α µ dα, F α h α dα, 18 whee H is a set of positive, continuous, bounded and inceasing functions h satisfying 1 h α dα = 1. Fo positive loss functions P = 1 the epesentation ρ = sup h H holds in addition, whee H h α = 1 h p dp. α H h F q dq, Poof. As fo the poof cf. [Kus1] and [Pfl6]. The latte statement follows fom 18 by means of integation by pats and substitution. Kusuoka s epesentation demonstates that the Conditional Tail Expectation is indeed cental, as all othe isk measues can be expessed by employing the CTE at diffeent levels α. This obsevation
8 fostes the cental ole of the CTE. Moeove it is inheited fom the Conditional Tail Expectation that E ρ ess sup, which makes ρ a candidate fo a moe involved insuance pice: As above, the pice accoding the equivalence pinciple, E, is a lowe bound fo the isk-adjusted pice. Moeove any of these isk measues ρ oveestimate the impact of expensive losses in elation, and in contast to compaably cheap losses. Example. A compehensive list of Kusuoka epesentations fo impotant isk functionals is povided in [PR7]. A compelling example is the absolute semi-deviation isk measue fo some fixed c [, 1], ρ c := E [ + c E + ], assigning an additional loading of c to any loss exceeding the pice E, computed accoding the equivalence pinciple. Its Kusuoka epesentation is cf. [Sha12, SDR9] ρ c = sup 1 c λ E + c λ CTE 1 λ. λ 1 5. Ambiguity In the pevious sections we have poposed ρ, in paticula CTE α as a isk avese pemium fo an insuance contact. It was an impotant obsevation, which is aleady intinsic in 4, that the pobability measue can be exchanged in ode to evaluate the Conditional Tail Expectation of a loss distibution. The following point on s distibution function has to be made as well: To compute E, CTE α o ρ the distibution function F x = P x has to be known pecisely: In ode to pice a contact the actuay has to be sue about the loss distibution F. In a eal wold situation this is, howeve, not often the case. The loss distibution may esult fom empiical obsevations, may have eos in measuement o may be deived fom othe obsevations elated to the isk of inteest. In exteme situations the loss distibution may even be deived by scientific aguments without any obsevations. In ode to pice a contact the esponsible actuay thus will compute the pices fo othe possible loss distibutions F, say, as well and compae the esulting pemiums. The actuay is inteested in the question: What is the pice CTE α fo the given loss function, but subject to a diffeent pobability distibution? To investigate this question a notion of distance of pobability distibutions is advisable. Vaious distances have been poposed in the mathematical liteatue, [Rac91, RSF11] povide an impessive oveview and aggegation applications can be found in [PP11] and in [DSS12]. In the pesent actuaial context the Wassestein distance is appopiate, as it obeys a sequence of popeties one would intuitively equest fom a distance of pobability measues having applications in insuance in mind. Definition 2 Wassestein distance. et Ξ, Σ, P and Ξ, Σ, P be pobability spaces and d: Ξ Ξ R a distance function. The Wassestein distance 1 is d P, P := inf d ξ, ξ π dξ, d ξ, π Ξ Ξ whee the infimum is among all pobability measues π which have maginals P and P, that is π A Ξ = P A and π Ξ B = P B wheneve A Σ and B Σ. Mathematical and topological details fo the Wassestein distance can be found in [Villani Vil3]. In the Russian liteatue cf. [Ve6] the Wassestein distance is usually called Kantoovich distance. We adopt the concept with actuaial applications in mind.
9 Remak 1. If P = i p iδ ξi and P = j p jδ ξj ae discete measues, then the Wassestein distance can be computed by the linea pogam P minimize in π subject to i,j d i,j π i,j = E π d j π i,j = p i, i π i,j = p j, π i,j, 19 whee d i,j is the matix with enties d i,j = d ξ i, ξ j. As any linea pogam it has a dual pogam, which can be stated as maximize in λ, µ i p iλ i + j p jµ j = E P λ + E P µ 2 subject to λ i + µ j d i,j. In all situations elevant fo insuance we have that Ξ = Ξ is a vecto space, and d is a nom, that is d ξ, ξ = ξ ξ. In this situation the following holds tue. emma 1. et Ξ = Ξ be a vecto space and the distance d a nom. Then whee e P := xp dx is the baycente unde P. e P e P d P, P, 21 Poof. By Jensen s inequality, e P e P = = xp dx y P dy x y π dx, dy x y π dx, dy, wheneve π has the ight maginals accoding to Definition 2. Taking the infimum with espect to all possible measues π eveals 21 fo = 1. The geneal case deduces fom d P, P d P, P, which is a consequence of Hölde s inequality. Intepetation of the Wassestein distance in life insuance In life insuance the pobability measue P = k kp x q x+k δ k is usually given via a life table. To compae life tables it is natual to employ the function d k, k = k k, which measues the diffeence in age of two sample-individuals. Fo this choice 21, i.e. e P e P d P, P, elates the aveage life expectancy unde P and P, as e P = k k kp x q x+k is the aveage life expectancy unde P : The aveage life expectancy unde diffeent measues will not diffe moe than thei Wassestein distance. The following example futhe suppots the intepetation that d P, P eflects the aveage diffeence of life expectancies of diffeent geneations. Example. The life table of a diffeent geneation is in pactical situations often appoximated by an age-shift of, say, yeas age-adjusted life expectation. The shifted pobability measue is P = k kp x q x+k δ k+. Then the Wassestein distance ecoves the age-shift, d P, P =, 22
1 when employing the distance function d k, k = k k epesenting the natual diffeence in age. p k if k To accept 22 choose π k, k := k = and obseve that π is feasible fo 19 with the else objective E π d = k, k π k, k k k = k p k =, so d P, P. On the othe side choose λ k := k and µ k := 1 k. Then λ k + µ k = + k k k k by convexity of y y. The dual vaiables λ and µ thus ae feasible fo the dual pogam 2. The objective fo the dual is E P λ + E P µ = k kp x q x+k k + k kp x q x+k 1 k + =, which shows that d P, P : the duality gap thus vanishes and d P, P =. In the context of insuance the loss function is always R valued, such that the image measue pushfowad measue P := P is a measue on R. The following lemma to compute the Wassestein distance fo measues on the eal line facilitates evaluating the Wassestein distance in many situations of pactical actuaial impotance. emma 2. Fo measues P and P on R it holds that 1 d P, P = F α F α dα, whee F x = x P dp, F x = x P dp and F α =inf q : F q α} F α = inf q : F q α }, esp. ae the quantiles. Poof. The poof is contained in [Vil3, Theoem 2.18 and the following emak] o [AGS5] in a moe geneal context. This initial and pepaing discussion of the Wassestein distance on pobability measues enables a compehensive teatment of actuaial picing unde distoted pobability measues, whee the potential distance is a model paamete which can be intepeted and estimated by the actuay. The next section addesses a thoough mathematical teatment of this appoach. 6. Ambiguity isk measues We have poposed in the pevious Section 2 ρ as a pice of an individual insuance contact, whee ρ is a vesion independent isk measue. We ae eady now to addess the question of ambiguity: The loss function is known pecisely and descibed in the insuance contact. But F, its distibution, is not known entiely, but only up to some gap which is specified in tems of the Wassestein distance. Definition 3. et K and ρ = sup h H F α h α dα be a vesion independent isk measue. The K ambiguity isk measue is ρ K := sup ρ Q, d P,Q K 1 whee ρ Q = sup h H F Q; α h α dα and FQ; α = inf q : Q q α}. We have the following theoem. Theoem 2. Fo any K, ρ K is a vesion independent isk measue. Poof. et 1 2, then Q 1 q Q 2 q, and F Q; 1 α F Q; 2 α fo evey measue Q, fom which the monotonicity popety follows.
11 As fo convexity notice that evey measue ρ h;q = 1 and h H. Thus ρ K 1 λ + λ 1 = sup d P,Q K h H sup d P,Q K 1 λ F Q; sup ρ h;q 1 λ + λ 1 sup 1 λ ρ h;q + λρ h;q 1 h H sup d P,Q K sup h H = 1 λ ρ K + λ ρ K 1. Tanslation equivaiance follows fom F Q;+c α = λ FQ; α wheneve λ >. F Q;λ ρ h;q + λ α h α dα is convex fo evey Q sup d P,Q K sup ρ h;q 1 h H = c + F Q;, and positive homogeneity follows fom It emains to show that ρ K is vesion independent. But this is evident by the equivalent expession 1 ρ K = sup which holds by means of emma 2. F α h α dα h H, F an inceasing and l.s.c. quantile with F α F α dα K 1 }, 23 The K ambiguity measue ρ K depends on, the paamete of the Wassestein distance. To investigate the impact of this paamete we shall wite explicitly as ρ K; in the following emma. It tuns out that the highest deviation is to be expected fo the paamete = 1. emma 3 The ole of the paamete. It holds that ρ K; ρ K;. Poof. Recall that d P, Q d P, Q wheneve, hence which is the assetion. ρ K; = sup ρ Q sup ρ Q = ρ K;, d P,Q K d P,Q K The following theoem paticulaly holds fo the Conditional Tail Expectation. Theoem 3. If ρ is geneated by a single function h, that is ρ = 1 is concave. Poof. Recall fom 23 that K ρ K F α h α dα, then 1 ρ 1 λk+λk 1 = sup F α h α dα: F F } 1 λ K + λk 1 whee we assume hee and in what follows implicitly that F needs to be a quantile function. et two quantiles satisfying F F K and F F 1 K 1 be chosen and define F λ := 1 λ F + λ F 1. Then F F 1 λ K + λk 1, λ
12 such that 1 ρ 1 λk+λk 1 sup Now the assetion is immediate. = sup 1 λ F λ α h α dα: F 1 F F α h α dα: F 1 + sup λ F 1 α h α dα: F K, F } F K } F K 1. ρ 1 λk+λk 1 1 λ ρ K + λρ K1 1 } F K 1. It tuns out that the smoothness of the loss function is of impotance when evaluating isk measues with diffeent pobability distibutions. Fo this ecall the following definition. Definition 4. A function is Hölde continuous with exponent β if thee is constant C such that x y C d x, y β. The smallest of these constant is denoted H β, such that x y H β d x, y β. Fo β = 1 the notion of Hölde continuity coincides with ipschitz continuity and H 1 is s ipschitz constant. In addition it should be noted that β > 1 makes sense wheneve the space is discete any β continuous function is constant on [, 1] wheneve β > 1. Example. The Hölde constant fo an annuity k = k+1 j= vk is H β = v < 1, and fo a pue life insuance with loss function = v k+1 the Hölde constant is H β = v v 2 = v d. Theoem 4 Continuity of isk measues with espect to changing the measue. et P and P be pobability measues and Hölde-continuous with exponent β < and constant H β. Then whee β β. Poof. Fist let β = β ρ P ρ P h H β d P, P β β and define β := β ρ P ρ P = 1 F 1 F, such that 1 β + 1 β p h p F p h p dp p F p β = 1. Then, by Hölde s inequality, 1 β 1 1 dp h p β β dp. Note now that P = P and P = P ae measues on the eal line R fo which, by emma 2, Moeove d β P, P 1 β = F p F p β dp. d β P, P β = x y β π dx, dy = x y β π dx, dy H β d x, y β β π dx, dy = H β β d x, y π dx, dy, 1
13 and thus d β P, P β Hβ β d P, P when passing to the infimum. Hence β ρ P ρ P h H β d P, P. β The same inequality can be deived with the oles of P and P exchanged, fom which the assetion follows fo β = β. The geneal situation β β deives fom the fact that h h wheneve, completing the poof. The latte theoem allows the following helpful estimate fo the K ambiguity isk measue. Coollay 1. et P and P be pobability measues and Hölde-continuous with exponent β < and constant H β. Then ρ ρ K ρ + K β h H β β whee β β. Fo the Conditional Tail Expectation the spectal function is h = 1 1 α 1 [α,1], fo which h β can be given explicitly by h = β 1 α 1/ β 1 1 β β dλ = 1 α β = 1 α cf. [PW9] fo an initial statement in this diection. The Conditional Tail Expectation allows the following impovement. Theoem 5. Fo the Conditional Tail Expectation, CTE α;k CTE α 1 1 α β/ K 1 α H max, q α }, whee max, q α } is the function x max x, q α } and q α = F α is the α quantile unde P. Poof. et P be chosen such that d P, P K. Recall 2 fo the Conditional Tail Expectation, thus CTE α; P CTE α;p = inf q = inf q q + 1 1 α E P q + inf q + 1 q 1 α E P q + q + 1 1 α E P q + q α 1 1 α E P q α + whee we have used that the infimum in 2 is attained at the quantile q α. Hence CTE α; P CTE α;p q α + 1 1 α E P q α + q α 1 1 α E P q α + = 1 EP q α 1 α + E P q α +. By Hölde continuity y q α + x q α + H qα + d x, y and in view of the duality elation cf. 2 it follows that such that E P q α + E P q α + H qα + d P, P, CTE α; P CTE α;p 1 1 α H qα + d P, P. Finally note that H qα + = H max, q α }, which completes the poof.
14 The next theoem finally elaboates that the bound in Theoem 5 is the best possible bound, at least fo discete measues and fo small petubations K. Theoem 6. Fo a discete pobability measue P = p x δ x with p x > an a discete set X thee is K > such that CTE α;k = CTE α;p + K 1 α H max, q α } fo evey K K. Poof. Choose x, x 1 agmax x x 1 maxx,q α} maxx 1,q α} dx,x 1 Define the measues P x 1 and and obseve that x x 1 d x, x 1 H max, q α }. P x + if y = x, P y := P x 1 if y = x 1, P y else if x = x 1 and y = x, P x 1 if x = x 1 and y = x 1, π x,y := P x if x x = y, else such that P x = y π x,y and P y = x π x,y. Moeove d P, P x,y π x,y d x, y = d x 1, x 24 P x 1 d x 1, x by the choice of. Next, let the dual vaiable Z EZ = 1, Z 1 1 α be chosen such that CTE α = E Z. } Without loss of geneality one may assume P Z = 1 1 α = α, as othewise the atom < Z < 1 1 α can be split into two atoms with the desied popety. As x x 1 q α it follows hence that Z x = Z x 1 = 1 1 α, and consequently ẼZ = 1. Next, CTE α; P CTE α;p = E P Z E P Z = In combination with 24 it follows that that is = H max, q α } 1 α CTE α; P CTE α;p H max, q α } 1 α 1 α x x 1 d x, x 1. CTE α;k CTE α;p H max, q α } 1 α povided that K P x 1 d x 1, x =: K. By Theoem 5 hence CTE α;k CTE α;p = H max, q α } 1 α d P, P, K, K.
15 As an immediate application we mention the paticula case = 1, as it can be used to give immediate uppe bounds fo the pice of insuance, allowing distotions in the Wassestein distance up to a value of K. The ingedients of the following coollay ae often easily available fo the actuay and povide a quick bound fo the lump sum pemium. Coollay 2. It holds that sup CTE α;k CTE α;p + K d 1P, P K 1 α H 1 max, q α }, with equality fo K in a neighbohood of and H 1 max, q α } denoting the ipschitz constant. In paticula we have sup Ẽ E + K H 1 max, q α } d 1P, P K fo the pice accoding the equivalence pinciple. 7. Numeical analysis and teatment An explicit fomula fo the CTE α is available just fo selected types of insuance contacts, and a moe geneal pocedue to compute CTE α is povided by 15. The CTE can be computed equally well by the linea pogam P whee we have employed the measues maximize in Z subject to k p k k Z k k p kz k = 1, Z k, Z k 1 1 α, P := k kp x q x+k δ k = k p k δ k p k = P K = k = k p x q x+k as above. To illustate the esults we have computed the Conditional Tail Expectation fo impotant types of insuance in line with 3.1 and 3.2, they ae depicted in Figue 1. Equally well can the isk measue ρ h be natually evaluated by ρ h = F α h α dα = E Z = p k k Z k, k whee the optimal dual vaiable Z is povided by Z k := 1 p k 1 p1...p k 1 p 1...p k h p dp. It is significantly moe difficult to evaluate the K ambiguity isk measue even fo CTE α;k, the K ambiguity Conditional Tail Expectation. This poblem can be stated as maximize in π and Z subject to i,j π i,j j Z j Ẽ Z j π i,j = p i, π A X = P A π i,j, π a pobability measue i,j π i,jd i,j K d P, P K Z j 1 1 α Z 1 1 α i,j π i,jz j = 1 ẼZ = 1, 25
16 a The CTE α small values of α fo the pue endowment 2 E 4 = 64.6 % left and the annuity ä 4:2 = 16.6 % ight. b The CTE α fo the life insuance 2 A 4 = 6.8 % left and the endowment A 4:2 = 71.4 % ight. c The Conditional Tail Expectation at level α =.3, CTE.3, fo the life insuance 2 A 4 left and the annuity ä 6 ight; notice, that ä 4 29. Displayed is the concave map K CTE.3;K fo values of K up to 5 fo the espective loss functions. The ode fo the Wassestein distance is = 1 and = 2. Figue 1: Conditional Tail Expectation CTE α. The computations ae made fo the Geman life table 28T M/F fo which 2 q 4 = 8.6 %, and an inteest ate of 1.75 %.
17 whee we have indicated the conditions on the ight again, subject to the new measue P = j p jδ j p j = i π i,j, which is at a distance of d P, P K. It should be noted that 25 is not a linea pogam, as it involves multiplications as π i,j Z j in its constaints as well as in the objective the poblem is bilinea. This fact significantly complicates the computations. To escape this difficulty and to obtain easonable computing times we popose to handle the poblem in the same way, as bilinea poblems ae usually teated, that is by iteatively impoving 25: In an iteation step fix Z in 25 and compute π, next keep π fixed in 25 and compute Z o even quicke compute Z by employing 14. Any of these poblems then is a linea pogam, which impoves the objective. The descibed pocedue can be poved to convege, povided that, what is the case fo a typical payoff in insuance. 8. Summay and outlook In this pape we addess the poblem of picing insuance contacts unde ambiguity. An ambiguous situation is an evey day situation fo an actuay, as the distibution of a loss function is not known pecisely in a typical situation. In a fist pat of the pape we addess geneal isk measues, in paticula the Conditional Tail Expectation as a picing stategy. These isk measues incopoate designed loadings fo the pemium of an insuance contact. To illustate the impact of the design paamete fo the CTE the computations ae made explicit fo usual types of life insuance contacts, as, supisingly, simple explicit expessions ae available in these situations by changing the life table. The second pat of the pape is dedicated to evaluate these isk measues in ambiguous situations. Closed foms ae not available hee, but shap bounds allow to pecisely estimate the esults. These bounds ae designed in a way to seve as a quick guideline fo an actuay when picing a contact in a situation, whee the distibution of the loss function is known only at an insufficient level. The concept to have a notion of distance on pobability measues available is the Wassestein distance. Finally some elevant examples ae used to demonstate the impact of these picing methodologies in concete situations. Ambiguity was studied in a diffeent context in finance ecently, it is an essential ingedient to pove key popeties of the 1/N investment stategy [PPW12]. These fundamental esults demonstate the paticula impotance of consideations on ambiguity and justify futhe eseach in the diection, they eveal vey natual, useful and capable estimates. 9. Acknowledgment We wish to expess ou gatitude to Pof. Pflug fo his continual advice and thank the efeees fo thei constuctive citicism. The plots have been poduced by use of Mathematica and Matlab. 1. Bibliogaphy [ADEH99] Philippe Atzne, Feddy Delbaen, Jean-Mac Ebe, and David Heath. Coheent Measues of Risk. Mathematical Finance, 9:23 228, 1999. 1, 2 [AGS5] uigi Ambosio, Nicola Gigli, and Giuseppe Savaé. Gadient Flows in Metic Spaces and in the Space of Pobability Measues. Bikhäuse Velag AG, Basel, Switzeland, 2nd edition, 25. 1 [Den89] D. Dennebeg. Distoted pobabilities and insuance pemiums. Poceedings of the 14th SOR, 1989. 1 [Ge97] Hans U. Gebe. ife Insuance Mathematics. Spinge-Velag Belin and Heidelbeg GmbH&Co. K, 3d edition, 1997. 4
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