Distort The Function And Insurance Pricing

The traditional mathematical expectation (i.e.the expectation about additive measure) takes an important role in risk pricing . But in many markets,the price functions are not satisfy the additive property. In insurance market and some finance markets , the price of the sum of two risks are usually smaller than the sum of the price of the two risks.The traditional mathematical expectation also results in the problem which puzzled the economist many years such as the Allais paradox and the Ellesberg paradox in utility theory.After the concept of capacity was introduced by Choquet(1953),Choquet intrgral(i.e. the expectation about non-additive measure) as an alternative to traditional mathematical expectation are introduced to economical theories.Choquet integral as a kind of non-linear expectation,its theory has been fully developed because of its characters suit for pricing financial assets. At present it has been widely used in the pricing of insurance (Wang et al(1996,1997) and finacial assets (such as option pricing) (see,e.g.Chateauneuf et al(1996)).Wang et al(1997)introduced a axiom system of insurance pricing. In this paper,I mainly study the premium principle and distortion function g bases on the results of Wang et al(1997) and Young(1998),etc. The paper is composed of five sections:In section 1, I want to introduct the paper's background and why I want to sdudy the questions.In section 2, I want to introduct some preliminary knowledge,including some definitions such as capacity,survival fuction,distortion function and distorted probability,and the contents of comonotonity and correlation orders.Theorem2.7 Assume premium principle keeps stop-loss order,if (X₁ ,Y₁), (X₂, Y₂) ∈ R(F_X,F_Y),and (X₁,Y₁) ≤_c (X₂,Y₂),thenH[X₁ + Y₁]≤H[X₂ + Y₂ ].Corollary2.9 If premium principle H[·]preserves stop-loss order and is additive for comonotonic risks.then it is sub-additive: H[X + Y] ≤ H[X] + H[Y]ior all risks X and Y.The section 3 , 4 and 5 is the emphases of my paper,and aslo my primary results. In section 3, I introduct the Choque represention theorem of risk premium prin-ciple.At first,I give some primarily property of the premium principle should satisfied:Conditional state independence,Monotonity,Comonotonic additive,Continuity and Normalization.Following that,I give the theorem of the existence and uniqueness of the distorted fuction g.Theorem 3.1 For any given X G X,non-negative numbers a,ai,a2)ai < a^such that aX € X, min(X, a2) - min(X, ax) G X, and for any X G X,we havea G R+,I{x>a} 6 AT premium principle H : X-> [0, +oo] satisfies (Pi) - (P5),then there exists a uniquedis-torted function g,satisfiesg(0)=0,and g(l)=l,such that for any given X G X,we haveCorollary3.3 Suppose the conditions in theorem 3.1 are satisfied.The premium of X = /{Z>t}Yand I{Z>t}H\Y)dj% equal,where/{Z>(}and Fare indepedent, then H[-\ can be ■epjesented as H[X] = /0+oo[Sx(z)]rcfo,r > 0In the end,I give the sufficient and necessary condition of H[?] is a coiheFeni risk meassure:g is a concave distortion function.In section 4,1 introduct the concept and the property of conditional capacity^and the pioperty of according Choquet integral:Theorem4.1 If C, B G T,vc\b defines oni\)vC\B{A) = l,for sttA D Bf]Cc,A G 7 holds.(2)^c|b(^4) is monotony.(3If /x is an other capacity on ^",then 1 =? vc\b < (4)If v is co.ncaBfla^flfSftd vc\b is also concave.Theoreom 4.2 ^he modified price / XdvC\B of risk X has the following pioper-tyies: (l)If v — goP, g is concasre distortion function ,P is probability measure.Then / XdPc\B [0,1] is continuous , twice diflferentiabte , increasing , concave distortion function, with g$) = l,g(l) — l,th#n %e conditkwial distortion function gsit) ■ $, 1} -* |0,1] satisfies:(1) 9e(M k &tSHtiK)&uad concave with gB(0) = 0, gB{$ = 1.(2) 4,(0) ￡ 0WfrAO9 *4tt-(3) ga{t) > g{t),t<&% 1^0ben -Pfffl = l,we tike " =".In section 5, I dfclss fbe lite opftmd of stop-loss reinsurance. Theorem 5.2 The optimal reinsurane contract /* € Ip has the f?m of I*(x) = (x - d)+. wkew d is fteapchise , deckles by E[(x - d)+] = P , g is a concave distortion...