BAYESIAN PREMIUM AND ASYMPTOTIC APPROXIMATION FOR A CLASS OF LOSS FUNCTIONS

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1 TH USHNG HOUS ROCDNGS OF TH ROMANAN ACADMY Series A OF TH ROMANAN ACADMY Volume 6 Number 3/005 pp AYSAN RMUM AND ASYMTOTC AROMATON FOR A CASS OF OSS FUNCTONS Roxana CUMARA Department of Mathematics Academy of conomic Studies ucharest; Roxana_Ciumara@hotmailcom We derie ayesian premium for a class of loss functions An exact calculation of ayesian premium is possible under restrictie circumstances only therefore we apply a theorem of Gatto [5] in order to derie an approximation for the premium This approximation is based on aplace's method is simply computed and allows for analytical interpretations Different forms of weighted quadratic loss functions are considered We proe that the aplace approximation of ayesian premium does not depend on the loss function inoled in exact ealuation of the ayesian premium if this loss function is of weighted quadratic type or exponentially scaled Key words: ayesian premium aplace approximation weighted quadratic loss function exponentially scaled loss function NTRODUCTON The determination of the premium for an insured risk is one of the important problems of risk theory y premium calculation it is meant the determination of the adequate premium for the risk assumed by an insured indiidual within a collectiity Moreoer if some claim experience for the gien indiidual is aailable we seek to determine the premium which exploits these actual claim amounts Therefore we compute ayesian premium which is a premium based on claim experience and it is defined as the alue which minimizes an expected loss with respect to a posterior distribution An exact calculation for the ayesian premium is possible under restrictie circumstances only regarding the prior distribution the likelihood function and the loss function That is why it would be helpful if we had a good approximation of the ayesian premium simply to compute and which allows for analytical interpretations n order to derie an approximation for the ayesian premium we employ aplace method by using a theorem of Gatto [5] We do this considering a weighted quadratic loss function mportant research concerning ayesian analysis in actuarial science is due to ailey [] uhlmann [4] Klugman [7] Mako [8] Schnieper [0] n what concerns the use of aplace approximation in actuarial science we hae to mention the work of leistein and Handelsman [3] Tierney and Kadane [] Gatto [5] AYSAN RMUM AND TS AAC AROMATON We consider random ariables n of claims in n consecutie periods for a gien indiidual Reccomended by Marius OSFSCU member of the Romanian Academy on a probability field { Ω K r} i takes positie real alues representing the size

2 et { ρ} Roxana CUMARA S be a measurable space (with measure ρ ) and Θ : Ω S a random ariable taking the alues S We now consider a probability measure Π on { S } which is absolutely continuous with respect to ρ and π is its corresponding density We say that π is the prior density of Θ The parameter (alues of Θ ) is the indiidual risk within a collectiity We assume that gien Θ the random ariables n are independently and identically distributed with common distribution function F Θ( x ) and density f Θ( x ) (for a fixed indiidual the claim amounts of consecutie periods are independently and identically distributed) et y ( y y y n ) be an obsered claim experience during the last n periods ie a realization of Θ Denote by the conditional random ector n l ( y y yn ) f Θ( yi ) the likelihood function Then the posterior distribution is gien by and the posterior density (from ayes' theorem) by Next define the loss function where ( x ) π n i ( y) r( Θ y) Π Θ ( y) l( y y yn ) π() l( y y yn ) π() S : R + R + is the loss incurred by a decision maker taking the action and facing the outcome x ( - the premium paid by the insured and x - the claim size) Definition The (indiidual) risk premium is a function : S R + which is measurable on S where { } or equialently () arg min ( ) d ( Θ ) f Θ () arg min ( x ) f Θ( x ) dx Ω if the expectation inoled in the aboe expression exists Similarly we define ayesian premium Definition The ayesian premium is the measurable function R n : gien by or equialently ( y) ( y) arg min ( ) π Θ min ( Θ ) S π ( ) y arg y d if the expectation inoled in the aboe expression exists

3 3 ayesian premium and asymptote approximation for a class of loss functions Note that the ayesian premium is the real number ( y) ( ) y π oer where Θ is the indiidual risk premium Remark n fact the ayesian premium is where ( y) min ( ) minimizing the posterior expected loss ( y) arg π Θ ( Θ ) () arg min ( ) and : are loss functions not necessarily identical f nothing else is mentioned we will consider that n order to compute the ayesian premium we need to ealuate the integral () y d R S This could be soled analitically under rather restrictie circumstances only which concerns the choice of the loss function and the prior density and the likelihood function giing the posterior density Anyway this choice should be based on objectie criteria possibly related to the real enironment of the insurer rather than on the feasibility of the ensuing calculations Of course numerical integration (wheneer it is possible) proides the most general solutions but they do not yield an analytical result which would allow for interpretations n this context the aplace method gies an asymptotic approximation of the integral aboe with a small asymptotic relati error and this allows to obtain an accurate asymptotic approximation of the ayesian premium (Gatto [5]) The results obtained in this paper are based on the result below alue of Theorem (Gatto [5]) et ( y y ) y y n denote a realization of n gien a fixed Θ ( n Θ ) and suppose that Π Θ ( y) is an absolutely continuous π Θ y which is positie oer an interal ( a b) f π Θ ( y) is differentiable distribution with density oer ( a b) has a unique maximum at ( ab) ( y) π Θ exists and is strictly negatie then (ie is the only point satisfying ( π ) 0 arg min ( ( ) ) y as n y and ( y) ( () ) ) min n Theorem (leistein Handelsman 986) et g :( a b) R + be a differentiable function oer ( ab) arg such that it has a unique minimum at an interior point which is the only point satisfying g () 0 and () g exists and is strictly positie f h ( a b ) R argument on ( a b) then : is differentiable with respect to the first ng() ng ( arg min h e d arg min h e ) as n

4 Roxana CUMARA 4 ng() ng n fact these authors proe that ( ) arg min h e d arg min h e + O t should be n mentioned that the error term contains otherwise the approximation would be an exact one The aboe result is based on aplace's method of approximation riefly this means that g and h are approximated by a Taylor polynomial of order around and then their product is integrated on a small neighborhood of Remark 3 f the loss function inoled in the calculation of ayesian premium is ( x ) different than the one inoled in indiidual risk premium ( x ) then under the assumptions of Theorem we get y arg min as n ( ) Remark 4 from Theorem is the posterior mode of Θ gien y Gatto [5] proposed an alternatie approximation of ayesian premium as y arg min ˆ as n ( ) where ˆ is the maximum likelihood estimate of the risk parameter (ie the solution of the likelihood equation l ( y y y n ) 0 ) The main disadantage of this approximation is that the prior information is lost Denote ( ( y arg min ) ) or ( y ) arg min ( ( ) ) (if ) and ( y ) arg min ( ( ˆ ) ) or arg min ( ( ˆ ) ) y (if ) n what follows we will consider a loss function of weighted quadratic type and suppose that the conditions of Theorem are fulfilled Theorem et : R+ R + be the loss function gien by ( x ) w( x)( x ) where w : R is a measurable function Then the indiidual risk premium is while the ayesian premium is gien by ( y) ( ) ( w Θ ) ( w Θ ) ( ( ) y) π w( ) y π w Furthermore an approximation of the ayesian premium is (! ) y where is the posterior mode of Θ assuming that all expectations aboe exist roof: Apply Theorem for the weighted quadratic loss function roposition Under the assumptions of Theorems and and assuming that w is differentiable of order k at 0 the indiidual risk premium is

5 5 ayesian premium and asymptote approximation for a class of loss functions () while the ayesian premium is gien by which could be approximated by ( y) k 0 k 0 k 0 k 0 ( k ) k+ ( Θ ) w ( k ) k ( Θ ) w ( k w ) π ( k w ) k 0 k + ( y) k ( y) ( k ) k+ f ( Θ ) w Θ k 0 y k w k f ( Θ ) Θ assuming that all expections inoled exist roof: Apply Theorem We will next analyze what happens if the loss functions inoled in calculations of risk premium and ayesian premium are both of weighted quadratic type but they are not identical Theorem 3 et R : be the loss functions gien by w ()( x x ) i i i where w i : R are measurable functions Suppose that the risk premium is computed from and the ayesian premium is based on loss function Then the indiidual risk premium is the ayesian premium is gien by () and an approximation of the ayesian premium is ( w ( ) Θ ) w ( ) Θ f ( ) Θ ( w ( ( Θ ) y) w ( ) y ( y) ( w ( ) Θ ) w ( ) Θ y () where is the posterior mode of Θ assuming that all expectations aboe exist roof: Apply Theorem and use the fact that w ()( x x ) i i for i Remark 5 f the loss function inoled in the calculation of the ayesian premium is of weighted quadratic type and different from the loss function inoled in ealuating the risk premium the approximation of ayesian premium does not depend on The weighted quadratic loss function inoled in the calculation of ayesian premium is not the only loss function that leads to an approximation of ayesian premium that depends only on the loss function

6 Roxana CUMARA 6 inoled in ealuating the risk premium n the next theorem we consider an exponentially scaled loss function Theorem 4 et R x : be the exponentially scaled loss function gien by ( x ) ( e e ) where > 0 Suppose that all conditions of Theorem are fulfilled Then the indiidual risk premium is while the ayesian premium is gien by () ln ( e Θ ) π ( y) ln e ( y) Furthermore an approximation of the ayesian premium is y where is the posterior mode of Θ assuming that all expectations aboe exist x roof: Apply Theorem for ( x ) ( e e ) The next propositions analyze the cases when the loss functions inoled in ealuating risk premium and ayesian premium are different and at least one of them is exponentially scaled roposition et i : x ( x ) ( e e ) R x w x x i be loss functions gien by where w : R is a measurable function and > 0 Suppose that the risk premium is computed from and the ayesian premium is based on loss function Then the indiidual risk premium is the ayesian premium is gien by () ( w ( ) Θ ) w ( ) Θ f ( ) Θ π ( y) ln e ( y) and an approximation of the ayesian premium is y where is the posterior mode of Θ assuming that all expectations aboe exist roof: Apply Theorem for the two loss functions mentioned aboe roposition 3 et R i : ( x ) w ( x)( x ) x i be loss functions gien by x e e where w : R is a measurable function and > 0 Suppose that the risk premium is computed from and the ayesian premium is based on loss function Then the indiidual risk premium is the ayesian premium is gien by ( y) () ln ( e Θ ) ( w ( ( Θ ) y) w ( ) y

7 7 ayesian premium and asymptote approximation for a class of loss functions and an approximation of the ayesian premium is ( ) y where is the posterior mode of Θ assuming that all expectations aboe exist roof: Apply Theorem for the two loss functions mentioned aboe roposition 4 et R i : βx β ( x ) ( e e ) x i be loss functions gien by x e e where β > 0 Suppose that the risk premium is computed from and the ayesian premium is based on loss function Then the indiidual risk premium is the ayesian premium is gien by and an approximation of the ayesian premium is () ln ( e Θ ) π β ( y) ln e β ( ) y ( y) where is the posterior mode of Θ assuming that all expectations aboe exist roof: Apply Theorem for the two loss functions mentioned aboe Remark 6 n all the cases presented in ropositions through 4 the approximation of the ayesian premium does not depend on the loss function inoled in the calculation of ayesian premium RFRNCS AY A Credibility procedures aplace's generalizations of ayes rule and the combination of collateral knowledge with obsered data roceedings of the Casualty Actuarial Society 37 pp RGR J O Statistical Decision Theory and ayesian Analysis New York Springer STN N HANDSMAN RA Asymptotic xpansions of ntegrals nd dition Doer UHMANN H Mathematical Methods in Risk Theory Springer GATTO R An accurate asymptotic approximation for experience rated premiums Astin ulletin 34 pp OSFSCU M MHOC G THODORSCU R Teoria robabilităţilor şi Statistica Matematică ucureşti ditura Tehnică KUGMAN S ayesian Statistics in Actuarial Science Kluwer 99 8 MAKOV U oss robustness ia Fisher-weighted squared-error loss function nsurance: Mathematics and conomics 6 pp RDA V Statistical Decision Theory ucharest Romanian Academy ubl House 99 0 SCHNR R Robust ayesian experience rating Astin ulletin 34 pp TRNY KADAN J Accurate approximations for posterior moments and marginal densities Journal of the American Statistical Association 8 pp ZĂGANU G Metode matematice în teoria riscului şi actuariat ucureşti ditura Uniersităţii 004 Receied h September 9 005