費(fèi)率厘定平行四邊形方法的數(shù)學(xué)模型_英文_.pdf

北京大學(xué)學(xué)報(bào) 自然科學(xué)版 網(wǎng)絡(luò)版 預(yù)印本 2008 年 第 1 期 2008 03 30 Acta Scientiarum NaturaliumUniversitatis Pekinensis On Line First No 1 Mar 30 2008 論文編號(hào) Paper Code pkuxbw2008001 http P P bdxbw chinajournal net cn 收稿日期 2007 08 26 修回日期 2007 12 26 A Mathematical Model for the Parallelogram Method in Ratemaking ZHANG Bo School of Economics Peking University Beijing 100871 E mail bozhang pku edu cn Abstract A rigorous mathematical model for the wel l known practical parallelogram method for insurance premium rate adjustment is set up A modified parallelogram method and an evenmore general triangle method are presented which can be seen as new tools for ratemaking Key words ratemaking parallelogram method triangle method earned premium 費(fèi)率厘定平行四邊形方法的數(shù)學(xué)模型 張博 北京大學(xué)經(jīng)濟(jì)學(xué)院 北京 100871 E mail bozhang pku edu cn 摘要 為在保險(xiǎn)費(fèi)率調(diào)整實(shí)踐中著名的平行四邊形方法建立了嚴(yán)格的數(shù)學(xué)模型 提出了修正平行四邊形方法和更 一般的三角形方法 其皆可被看成是費(fèi)率厘定的新工具 關(guān)鍵詞 費(fèi)率厘定 平行四邊形方法 三角形方法 已掙保費(fèi) 中圖分類(lèi)號(hào) O211 In the propertyP casualty insurance industry premium ratemaking is done by the rule of thumb to a large extent The practical approaches devised by the actuaries include a wide range of techniques that have not yet been formulated into a precise science One of the most popular methods of ratemaking is the loss ratio method which can be stated roughly as follows new rate current rate experience loss ratio P target loss ratio The target loss ratio is determined by the subjective opinions of the actuaries And experience loss ratio experience lossPon level earned premium The experience loss refers to the estimated ultimate loss for all earned exposures in our concerned experience period The earned exposure is the exposure unit actually exposed to loss during our concerned period Theestimationoftheultimatelossneeds sophisticated techniques which we do not consider in this paper The wel l known parallelogram method has been usedwidely in the industry to estimate the on level earned premium However this method lacks a solid theoretical basis which will be set up in this paper For the description of parallelogram method see Ref 1 5 Our modeling methodology is inspired by the idea in the basic model for reserving in Ref 6 1 The Model Framework Consider a propertyP casualty insurance company which begins to issue an insurance product of one year term from the time t 0 on The accounting years are 0 1 1 2 t t 1 1 Let N t be the number of policies written in period 0 t and suppose that N t t 0 is a counting processwith N 0 0 and let K t E N t and suppose that K t is a strictly increasing function in 0 And assume that at any time s 0 the premium rate is h s For convenience we define N 1 0 and K t 0 for all t I 1 0 The time point at which the n th policy is written as Sn inf t N t n n 0 1 2 we have that N Sn n for each n For any t 0 denote the written premium unearned premium and earned premium in period t t 1 as Wt Utand Rtrespectively By the accounting meaning we know that Wt sum of premiums of all policies written in t t 1 Ut sum of unearned premiums of all policies written in t t 1 Rt Ut 1 Wt Ut where the unearned premium for a policy is the portion of the premium that is not yet actually exposed to losswithin the accounting year at which the policy is written More precisely if a policy is written in some accounting year with the premium A the time span from its effective date to the end of this accounting year is DI 0 1 then the unearned premium of this policy at the end of this accounting year for the insurer is A 1 D Therefore Wt E N t 1 k N t 1 h S k Ut E N t 1 k N t 1 h S kSk t Rt E N t k N t 1 1 h S kSk t 1 E N t 1 k N t 1 h S k t 1 Sk E N t 1 k N t 1 1 h S kftSk where ft x x t 1 I t 1 x t t 1 x I t x t 1 We need also define the number of earned exposures in period t t 1 as Mt Vt 1 Nt Vt where Vt is the number of unearned exposures in period t t 1 which is the sum of the unearned exposures0 for any individual exposure units the idea is the same as in the unearned premium that is Vt E N t 1 k N t 1 Sk t and hence Mt E N t 1 k N t 1 1f t Sk Now we formally state the lossratio method Suppose the current rate is C and we have experienced n years among which we select the data from the latest n m 1 years where 0 m n since the old data are too far from the current situation then theoretically the rate should be adjusted to the new level C as follows C 1 Q E E n t mL t E E n t mM t 1 where Q is the target loss ratio and Ltis the ultimate loss for all earned exposures in period t t 1 We see that C is not concerned with Cat all The estimation of Ltis not considered in this paper and we suppose that this has been already done Since Mtis not recorded in any accounting statements therefore we need to estimate E Mt by somewhat indirect way To this end put rt Rt Mt C t E R t E M t Since Rtis recorded in the income statement and hence we can turn to estimate rtand Ct Suppose that h s is a positive step function defined on 0 such that h s hSkfor all s ISk Sk 1 where 0 S0 S1 Sn are the moments of rate adjustment Without loss of any generality we may assume that k k 1 2 ASk k 1 2 Denote t k SkI t t 1 and suppose that for any t 1 tis a finite set Denote k Sk Sk 1 and Htk Q k ft x dK x Q t 1 t 1 ft x dK x 2 北京大學(xué)學(xué)報(bào) 自然科學(xué)版 網(wǎng)絡(luò)版 預(yù)印本 第 3 卷 第 1 期 Clearly for any t 1 Htk 0 Pk 0 Htk 0 if kI t 1G t and E k Htk 1 And let Atkbe the Lebesgue measure of x y t x t 1 0 y 1 Sk y x Sk 1 in the x y plane 2 Results At first we provide two lemmas which will be used in the proofs of our results Lemma 1 Assume that Z t t 0 is a counting processwith Z 0 0 and denote G t E Z t and Tn inf t Z t n n 0 1 2 Then for any non negative continuous function g x on 0 and any 0 s t 0 then Ct E k hSkAtk for all t 1 4 and Ct 2E k hSkAtk for t 0 Theorem2 SupposethatN t F K U t whereF t t 0 is the Poisson process with intensity parameter 1 U is a strictly increasing function on 0 with U 0 0 and K 0 Then for any t 0 when Ky we have that Rt E Rt p 1 Mt E Mt p 1 rt Ct L1 1 and hence RtPCt E M t p 1 rt Ct p 1 E r t Ct y 1 5 Proof Using Lemma 2 and the ordinary techniques in probability limit theory one can easily completes the proof 3 Remarks Remark 1 Formula 2 has an interesting geometric presentation In the x yplane the x axis represents time and the y axis represents the proportion of premium For any t 2 in order to get Ct we only need to focus our attention to the triangle with three vertices t 1 0 t 1 and t 1 0 And across each point Sk 0 draw a vertical line these lines cut every such triangle to slices Then Ctis just the sum of all possible products of premium in each slice and the corresponding weight According to its geometric meaning wepd like to call this method the triangle method Remark 2 Formula 4 has another form of 3 pkuxbw2008001張博 費(fèi)率厘定平行四邊形方法的數(shù)學(xué)模型 geometric presentation In fact setting a x yplane coordinate system where the x axis represents time and the y axis represents the proportion of premium we focus our attention only to the area x 1 0 y 1 Across each of the pointsSk 0 draw a line with slop 1 These parallel lines cut each square with four vertices t 0 t 1 t 1 0 t 1 1 into slices The slice between lines y x Skand y x Sk 1should be labeled a premium index C Sk Now to get C t we only need to sum up all possible products of the premium indices and the Lebesgue measures of the corresponding slices This is just the wel l known parallelogram method Formula 3 has a similar geometric meaning as 4 and can be called generalized parallelogram method We can see that in general the wel l known parallelogram method will no longer holds if the intensities of policieswritten in different accounting years are not even and that is so often just the case we encounter in practice Remark 3 Under the assumption of Theorem 2 if Kis sufficiently large and hence according to 5 we can say that RtPCtis sufficiently close to EMtand hence can be used as an substitution of EMt Then returning to 1 we can finally get an approximation of the new rate as follows C U C 1 Q E n t mL t E n t mR t 6 where L tis the estimation of ELt and R t Rt C Ct Formula 6 is just the loss ratio method in practice and R tis called the on level earned premium in period t t 1 If we assume that the intensities of policies in all accounting years are even that is for all t 0 K t S K t for some K 0 and 0 m n then for any t 1 we have that R t Rt C E k C S kAtk 7 This is just the wel l known parallelogram method of the adjustment of real earned premium to the on level earned premium The parallelogram method stated in Ref 1 is exactly of the form 7 If we consider the difference of the intensities of policieswritten in different accounting years but maintain the assumption that the intensity of policies within each accounting year is even that is we assume that K t is a polyline as described in Corollary 1 and all the numbers of policies written in any accounting years are sufficiently large then 7 can be modified to the following form R t Rt C E k hSkB ctk 8 where B ctk nt 1Ik I t 1 ntIk I t nt 1 nt2 Atk and ntis the number of policies written in t t 1 Formula 8 can be calledmodified parallelogram method for the rate adjustment which can treat more