Experience Rating in Group Life Insurance.

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1 Scand. Actuarial J. 1989: Experience Rating in Group Life Insurance. By Ragnar Norberg, University of Copenhagen Abstract Methods for experience rating of group life contracts are obtained as empirical Bayes or linear Bayes solutions in heterogeneity models. Each master contract is assigned a latent random quantity representing unobservable risk characteristics, which comprise mortality and possibly also age distribution and distribution of the sums insured, depending on the information available about the group. Hierarchical extensions of the set-up are discussed. An application of the theory to data from an authentic portfolio of groups revealed substantial between-group risk variations, hence experience rating could be statistically justified. Key words: group life insurance, proportional hazard model, Biihlmann-Straub model, hierarchical extensions. 1. Introduction A. Background. The existence of substantial mortality variations in the population is well recognized. In individual life insurance they are to some extent accounted for by select mortality tables, tariffication by sex, and special pricing of insurances for impaired life. Apart from this there will remain individual risk characteristics that are not observed by the insurer and that cannot be traced in the risk statistics since each insuree dies only once. Consequently, premiums in individual life insurance are fixed upon the settlement of the contracts and kept unaltered throughout the insurance period. If you should ask holders of individual life insurance policies if they find the premiums reasonable, the answers would typically be "I guess so" or "I don't know". They don't know and don't haggle over the price, simply because they have no access to statistics from which they could judge the fairness of the premiums. In group life insurance this is different. Each master contract is managed by a polidyholder who can compare premium payments with received benefits in the long run. Those policy holders who find that premiums exceed by far the benefits, will sooner or later call for a discount (the others will remain silent). This is precisely what has happened in Norway. Until recently all life insurance companies used the same technical bases (G79) in group life insurance. These were essentially based on the principles of individual life insurance, with no allowance of premium adjustments in regard of risk experiences as per group. Partly due to an increased awareness of costs on the part of the policyholders and partly due to increased competition in the life insurance business, the market now enforces experience rating of group life contracts. Scand. Actuarial J. 1989

2 Experience rating in group life insurance 195 B. Outline of the paper. Group life insurance presents problems of its own which require some special statistical engineeringaupon application of general theory of experience rating. In Section 2 we treat the case where complete individual records are available, just as in individual life insurance. Then only pure mortality differences remain to be uncovered by experience rating. They are accounted for by a proportional hazard extension of the traditional model framework of individual life insurance. With a gamma distribution on the latent mortality factors, we obtain results akin to those of the traditional Poissonlgamma model. Sections 3 and 4 treat cases where only more summary statistics are available, so that also age distribution and possibly the distribution of the sums insured become part of the hidden risk characteristics. Then standard methods of experience rating, viz. those of Biihlmann & Straub (1970), can be employed with minor modifications. Section 5 considers hierarchical extensions of the models, which come into play when an initial stratification of the groups seems relevant, e.g. according to occupational categories. Section 6 reports on the results of an analysis of an authentic data set by the methods developed in Sections 2-5. Section 7 sketches some ideas of how to incorporate additional benefits to dependents (spouse and children) in the analysis. 2. Experience rating based on complete individual policy and claim records A. The data. Consider a group life portfolio for which statistical records have been maintained during the period (z', f), where z" is the present moment. The portfolio comprises I master contracts, labeled by i= 1,..., I. Let (z:, 69 be the period during which contract i has been in force ($<z" if the contract has been terminated in its entirety). Let J; be the number of persons currently or formerly insured under the plan of contract i. They are labeled by (i, j), j= 1,..., J;. For each individual (i, J] introduce the following quantities, which are observable by time I": zi, the time of entrance into the group, xu, the age at entrance, Tu, the time exposed to risk as insured before time $, Kii, the number of times the coverage has been terminated on an individual basis before time 6, M", the number of deaths as insured before time fl. The pairs (Ku, M0) can only assume the values (0, O), (1,0), (0,l) (implying that participation in the group will not be resumed once it has been terminated). Define also Yu, the age at death, which is observed only for those who died before time fl. Clearly, Yi=xu +Tu if Mi=l. B. Model assumptions. The hidden mortality characteristics of group i Scand. Actuarial J. 1989

3 196 R. Norberg are represented by a latent quantity Oi. We shall view the Y,, T, K,, Mu, and O, as random variables and the remaining quantities asfixed, and make the following basic assumptions. (i) Variables belonging to different groups are stochastically independent and 01,..., OI are iid (independent and identically distributed). (ii) Variables belonging to different persons within one and the same group i are conditionally independent, given Oi. (iii) All persons in one and the same group follow the same pattern of mortality and termination. More specifically, it is assumed that the Oi are positive and that, conditional on Oi=Oi, a person who entered group i at age x and is still a member of the group at age x+t(x, DO), then has a force of termination and a force of mortality of the form Assumption (i) corresponds to the idea that the groups are independent random selections from a population of groups that are comparable, but not entirely similar. It is this assumption, in conjunction with (2.2), that establishes a relationship between the groups and forms the rationale of combining portfoliowide mortality experience with the mortality experience of a given group in an assessment of the mortality in that group. The "proportional hazard" assumption (2.2) represents, perhaps, the simplest possible way of modelling mortality variations between groups. It states that the risk characteristics specific of a group act on the force of mortality only through a multiplicative factor, implying that the mortality pattern is basically the same for all groups. Such an assumption is not apt for describing more complex mortality differences, e.g. that a group may have a mortality below the average at early ages and above the average towards the end of life. For example, it is thinkable that such hazardous occupations as blast furnace operation and mining attract only physically fit and healthy applicants and that those who are employed quickly,get worn out by the severe working conditions. The statistical data presently available from group i are the individual entrance times, zb, entrance ages, xi, and histories as insurees, q,= {(K U' M--, v T-);j 'J = 1,..., J,). The conditional distribution of (K,, M,, T,), when Oi=Oi, is given by Scand. Actuarial J. 1989

4 Experience rating in group life insurance 197 = x,(x,, t,) dt, exp [-1" t)+rip(x,, I)) dt [-1 I I.. = exp u {%(xi. t)+ f3ip(ry. 0) dt, ti = g r;. From these expressions we gather the following formula for the conditional likelihood of (Kc, M", T,), j= 1,..., Ji: K, M ) ( 0 r; or (K,, M,)=(O, 0) and T..=<-T!., r~ j= 1, an., J,. For each person (i, j) introduce the cumulative basic force of mortality 1 (1,O)) and O<r,<<- It is seen from (2.3) that a set of sufficient statistics for group i are Ji z Mi = M,, the total number of deaths, (2.5) j= I Ji Wj = z W,, the sum of cumulative basic intensities, (2.6) j= 1 and that the conditional likelihood, considered as a function of B,, is proportional to w,. * (2.7) The expression in (2.7) is of gammoid shape, and so the gamma distributions are the natural conjugates that give particularly simple analysis. Therefore, we assume that the common distribution of the latent Oi is the gamma distribution %(y, 6) with density The conditional density of Oi, given (Mi, Wi), is proportional to the product of the expressions in (2.7) and (2.8), hence Scand. Actuarial.I. 1989

5 198 R. Norberg the %(Mi+y, W;+6) density Using the easy result valid for m>- y and w>-6, we find The conditional mean in (2.9) is the Bayes estimator 8; (say) of Qi with respect to squared loss. It can be cast as where bi = M,IWi is the maximum likelihood estimator of 0; in the conditional model, given O;=Oi, and The expression in (2.12) is a credibility weighted mean of the sample estimator 6; and the unconditional mean, EOi=y/6. The credibility 5; is an increasing function of the exposure times Tij, confer (2.4), and of the "coefficient of variation",.. 6-' = Var Oi /EOi. (2.15) C. Net premiums. The set of master contracts in force at the present moment is 4= {i: e= 2")' and for each group i E 4 the set of persons presently covered under the plan of the group is $(= {j:t!.+t..=f). u u Scand. Actuoriol J. 1989

6 Experience rating in group life insurance 199 For each person (i, j) presently insured let Mh and S; denote, respectively, the number of deaths and the sum payable by death in the next year, (z", t"+l). To prevent technicalities from obscuring the main points, let us disregard interest and assume that all the S; will remain constant throughout the year. For a group i 4 the net annual premium based on the available information Oi is The expected values appearing in (2.16) are which can be calculated by formula (2.11): defining and we find Substituting (2.19) into (2.16) yields with Qi,,.! defined by (2.17) and (2.18). Since 6nly deaths as insured are covered by the contract, it might be argued that the annual premium should be I,' I Oi,u(x9 T,+t)exp{- (Oi,u(xu, T,j+t)+x,(xu, T,+t)) dt) dt(oi. This premium is smaller than the one in (2.16)-(2.20), which is obtained by formally putting xi=o. However, the expression above is not an appropriate premium since it disregards the possibility that termination may be Scand. Actuarial J. 1989

7 200 R. Norberg deliberately postponed until the end of the premium payment period to gain a coverage corresponding to the premium (2.20). To break'this dependence of the termination mechanism on the periodicity of premium payments, one should charge the full premium (2.20) and in case of termination repay the unearned premium as a return. As an alternative to the premium (2.20), which is exact on an annual basis, one could use the "instantaneous net premium" per time unit at time I" 9 where Mi(Az) is the number of deaths of person (i, j) in the time interval (I", I"+As). Now, the last passage being a consequence of (2.9) and (2.12). Combining (2.21) and (2.22), we get f'! = S;p(x,, T,) 6,. j A (2.23) To see that f'! is an approximation to P;, apply the first order Taylor expansion (1 +x)-"=l -ax to the second term on the right of (2.18) and then approximate w; in (2.17) by p(xij, TU), which gives Using (2.24) in (2.20), we get and using (2.25), we get P;'=f'! given by (2.23). It follows that the expressions in (2.23) and (2.26) approximate the one in (2.20). Which one to use of the three, depends on the quality of the approximations; (2.26) is good if the w; are << Wi, which is the case for Scand. Actuarial J. 1989

8 Experience rating in group life insurance 201 groups with a reasonable large risk exposure in the past, and (2.23) is good if the p(x,, T,+t) are nearly constant for O<t< 1. D. Thejluctuation reserve. A measure of the uncertainty associated with the annual result for group i is the conditional variance, (M'$ is equal to its square since it is 0 or 1). The expected values appearing in the second sum in (2.27) are = E[{l --exp(-oi,,;i> = Qi,,+Q;,,,-Q;, w';+".$' (2.28) confer (2.1 1) and (2.18). Now, insert the expressions (2.19) and (2.28) into (2.27) to obtain where Qi, w: and e are given by (2.17). (2.18), and (2.20). By use of the approximation (2.24), we easily obtain Charging each group i its net premium e would only secure expected equivalence of premium incomes and benefit payments for the portfolio as a whole. (At any time groups with low mortality will subsidize those with high mortality, but as time passes and risk experience accrues, these transfers will diminish: eventually each group will be charged its true risk premium.) To meet unfavourable random fluctuations in the results, the company should provide a reserve for the entire portfolio. By approximation to the normal distribution, which is reasonable for a portfolio of some size, a fluctuation reserve given by Scand. Actuarial J. 1989

9 with V; defined by (2.29) or (2.30), will be sufficient to cover claim expenses in excess of the total net premium, zi,,e, with 99% probability. To establish the reserve in (2.31), it may be necessary to charge each group an initial loading in addition to the net premium. Thereafter the reserve can be maintained by transfer of surplus in years with favourable results for the whole portfolio. (Charging the insurees a total premium loading equal to FA each year is, of course, not necessary: that would create an unlawful profit on the part of the insurer.) The loadings can be determined in several ways. One reasonable possibility is to let each group i contribute to FA by an amount F; proportional to the standard deviation (v$, that is, Upon termination of a master contract, the group should be credited with the amount (2.32). E. Estimation of parameters. At time f the observations that can be utilized in parameter estimation are q, i= 1,..., I. It is, of course, only in this connection that the data from terminated master contracts come into play. Assume now that the basic mortality law is of Gompertz-Makeham type and is aggregate, that is, p(x, t)=p(x+t), where (In group life insurance there is actually no reason to expect selectional effects since eligibility is not made conditional on the insuree's health or other risk characteristics.) From (2.3), (2.8) and (2.33) one gathers the following expression for the unconditional likelihood of the data: xexp{-z ~Tuq.xi, tl dt-xi j= 1 do,]. The forces of termination do not appear in any of the expressions for premiums and reserves, and so one can concentrate on the estimation of y, 6, a, 8, C. The essential part of the likelihood is Scand. Actuarial J. 1989

10 Experience rating in group life insurance 203 The maximum likelihood estimators y*, d*, a*,b*, c* have to be determined by numerical methods, e.g. steepest ascenc or Newton-Raphson techniques. Note that the number of parameters is essentially only four since a scale parameter in Oi can be absorbed in p in (2.2). One should, therefore, put y=d or a= 1 or P= 1. Reliable estimation of all parameters requires a substantial amount of risk exposure spread over a not too small number of groups. In the worked example of Section 6 below the baseline mortality intensity was determined exogeneously, hence only y and 6 had to be estimated. 3. Experience rating based on summary data on risk exposures and complete registration of sum functions, numbers of deaths and ages at death A. The data. Suppose now that for each contract i only the risk exposures, numbers of deaths and the ages at death have been recorded on an annual basis throughout the period of existence of the policy. More specifically, suppose that contract i has been in force for Ji years (the meaning of J; now being different from what it was in the previous section) and that for each year j= 1,..., Ji the following data are recorded: p", the number of years exposed to risk, M", the number of deaths, Yik, k=l,..., M", the ages at death (when Mii>O). The group insurance programme specifies the benefits by S(y), the face amount payable by death at age y. A reduction of costs can be gained by not having to keep track of all the details of the individual histories, as was required in the previous section. The price one has to pay for this advantage is, of course, a loss of information. But here experience rating may serve to retrieve knowledge of substantial differences between the groups as regards occupational hazards and age distribution. Actually, there are reasons to believe that intergroup mortality differences are not predominant nowadays: groups of employees include white-collars as well as Korkers exposed to hazards specific of the industry, and, furthermore, with improved safety procedures, better plant methods, and the advent of automation, the variation in mortality by industry has gradually narrowed. In these circumstances it might well be that the merits of experience rating in group life insurance is not so much to uncover hidden mortality characteristics, but rather to compensate the loss of information incurred by the introduction of cost-saving schemes based on summary data. B. Model assumptions. The basic assumptions (i) and (ii) in Paragraph 2B are retained, whereas assumption (iii) is replaced by the following. Scand. Acruorial J. 1989

11 204 R. Norberg (iii') Conditionally, the Mu are independent, each Mu following a Poisson distribution with parameter p"a(oi), and-they are independent of the Yuk, which are conditionally iid. This assumption is appropriate for large groups with a fairly steady composition, and for small groups it can still serve as an approximation. Anyway, only the moment structure inherited from this assumption will be essential in what follows. For a motivation of (iii'), see Norberg (1987). In the present model the Oi account for differences with respect to mortality and age composition. Thus, O, is now a more complex quantity than in the previous section. In particular, no assumption like (2.2) is made here. Under the present assumptions the total annual claim amounts, M:. (to be defined as 0 when Mv=O) are conditionally independent, given Oi, and the conditional distribution of Xi is generalized Poisson with mean and variance given by confer e.g. Beard et al. (1984). It is convenient to work with the annual loss ratios, From (3.2) and (3.3) it is seen that the 6" have conditional first and second order moments of the form ~(6~10;) = b,, (3.5) where and vi=u(oi) for some functions b and v that do not depend on i. C. Netpremiums. The situation fits into the classical model of Biihlmann and Straub (1970), and so the credibility estimator of bi based on the observed Xil,..., XiJi is where Scand. Actuarial

12 Experience rating in group life insurance 205 the total loss ratio (dropping a subscript in Xu or pg signifies summation over that subscript, confer (2.5) and (2.6)), ci = P~L~(P~L+$), (3.9) the credibility weight, and /3 = Eb,, L = Var b,, $ = Ev,. (3.10) The annual net premium of a contract i 9 is approximated by where Ji+l represents the year (f, f+1) and 6; is given by (3.7)-(3.10). D. The fluctuation reserve. In accordance with the reasoning in Paragraph 2D the fluctuation reserve should be calculated by formula (2.31), now with Vf equal to A linear approximation of this conditional variance can be arranged, but is complicated, see Norberg (1986a). A simpler measure Vt of the uncertainty associated with XirJi+, is the unconditional variance, which by use of (3.4)-(3.6) and (3.10) is found to be Pi. J,+I L+P;, Since VarX,, J,+ >E Var(X,, J,+l IDi), (3.13) represents--on the average-a more prudent assessment of the uncertainty than (3.12). We shall, however, recommend that (2.31) be based on the very simple obtained from (3.13) by dropping the first term on the right. The rationale of this solution is that the expression in (3.14) is just the mean value of which would 'tie the appropriate choice of V: were known. Moreover, (3.14) is the mean of the limit of (3.12) as the amount of experience Oi increases. One might also argue that the second term on the right of (3.13) is the genuine fluctuation part of the variance. The first term is due to differences in risk premiums between groups. E. Parameter estimation. The parameters in (3.10) are estimated by standard methods for the Biihlmann-Straub model, confer Paragraph 5E below. Note that the parameters /3, L, $ now depend on the sum function S and have to be estimated separately for each S that occur in the portfolio. F. An approximative analysis based on complete individual data. Under certain conditions the situation in Section 2 can be dealt with by the linear Scand. Acruorial J. 1989

13 206 R. Norberg method developed in the present section. The analysis will be based on the occurrencelexposure rates 6; defined by (2.5), (2.6) and (2.'13) under the approximative assumptions The justification of (3.15) and (3.16) rests on the fact that 6; maximizes the likelihood in (2.7) or, equivalently, its logarithm Li (8,) = Mi log 8;-8, W;. (3.17) By a standard result on maximum likelihood estimators, 6; is asymptotically normally distributed as J; increases, with asymptotic expected value and variance given by as. ~(6~18;) = B;, (3.18) as. ~ar(6;( 8;) = {-~(~1"(8,)1 Bi)}-I, (3.19) for where LY'(8,) is the second order derivative of L;(Bi) in (3.17). We find = a%c-*; ei j=, [ eip(xv, l I t) exp - {xi(x,, S) + B;~(x,, s)) ds dt. Integration by parts yields Combining (3.19), (3.20), and (3.2 l), we obtain as. ~ar(g~l8,) = ~ile(wi18,). Scand. Actuarial J. 1989

14 Under the assumption that Experience rating in group life insurance 207 wi E( W;I 8;) 1 in probability for fixed Oi=Bi, (3.22) is equivalent to as. ~ar(6~18;) = 8, l W,. (3.24) The relations (3.18) and (3.24) motivate (3.15) and (3.16) as approximative model assumptions for groups that are not too small. The asymptotic results (3.18), (3.22), and (3.23) are valid under mild assumptions about the ages and times of entrance. Roughly speaking, they should satisfy the requirements xii<xo<m and 4'-z''>rO>O for infinitely many j. It is clearly sufficient to assume that the (xii, tk), j= 1,2,..., are outcomes of iid random pairs that are independent of Oi, which seems appropriate in a non-experimental context like group life insurance. Assuming (3.15) and (3.16), we. are back in the situation (3.4)-(3.6), with * Ji=l, Xi,=Xi=Mi, p,,=pi=wi, bi=oi, and b,=v,=o,, hence From (3.7)-(3.10) we compile that the credibility estimator of Oi is where and B = EO,, A = Var O,. (3.28) It is noteworthy that we have now anived at (2.12) by another route, freed from the assumption that the Oi's are gamma distributed and utilizing only large sample properties of the occurrence/exposure rates. Note that in the., gamma case P=y/d and A= y/d2, so that (2.12H2.15) is really a specialization of (3.26)-(3.28). Approximative annual premiums and reserves are given by (2.26), (2.30), and (2.31), with 6; in the place of 8,. Finally, we mention an alternative to the approximative assumption (3.16), which consists in replacing Wi by From (3.20) it is seen that this amounts to putting Oi=l in the exponent. By judicious choice of p, the distribution of 8; will be centered about 1, and if Scand. Acruarial J. 1989

15 208 R. Norberg this distribution is not too dispersed, the approximation Bi=1 is reasonable with high probability. Moreover, the mortality is small in thk ages that are typical of participants in group life schemes, so that the term Bip(xii, S) ds is of minor importance, especially if the <-ti are of moderate extension. Anyway, a misspecification of the volumes p, does not affect the overall unbiasedness of estimators of the type (3.7). The good thing about using (3.16) based on the observed Wi is that it does not require estimation of the forces of termination, x;(x, t). 4. Experience rating based on summary data on risk exposures, sum functions, and total amounts of claims paid A. The data. Suppose now that the only data available from contract i in year j are pij, the number of years exposed to risk, fij, the average sum insured, X6, the total amount of claims paid. Here fij may be the simple average of the individual face amounts for those who are insured under the plan of contract i at the beginning of year j, or it may be a more refined quantity, e.g. the weighted average of the individual face amounts for those who are insured under that plan during the whole or a part of year, j, with weights equal to the individual times exposed to risk. The total claim amounts Xu are of the form where M" denotes the number of deaths in group i in year j, just as in the previous section, and Siik, k= 1,2,..., Mij, are the individual benefits paid (when Mii>O). If a sum function SA.) is stipulated for contract i in year j, then the SSik are of the form Siik=S(Yijk), where the Yiik are the ages at death. Neither So(.) nor the Yck need to be observed. The present specification of the data covers also the case where the face amounts are not laid down by a sum function, but are chosen on an' 'individual basis by each member, possibly subject to certain constraints specified by the master contract. In the present case the observations are even more aggregate or summary than those encountered in the previous section. This means, on the one hand, that more information is sacrificed to save administration expenses, and, on the other hand, that further between-group risk differentials are introduced, which call for an adaption of the premiums to individual group experiences. B. Model assumptions. Just as in the previous section, the basic assumptions (i) and (ii) in Paragraph 2B are retained, whereas assumption (iii) is replaced by the following. Scand. Actuarial J. 1989

16 Experience rating in group life insurance 209 (iii") Conditionally, given Oi, the total claim amounts Xii are independent with 1st and 2nd order moments of the form ' This assumption can be motivated in vatious ways. One possibility is the following analogy to (iii') in Paragraph 3B. Conditionally, given Oi, the M, are independent, each Mu having a Poisson distribution with parameter p,a(0j, they are independent of the Stk, and the ratios S,dS, are iid. The last item in this list of assumptions implies that the shape of the distribution of the Siik remains unchanged in the course of time for a fixed group i, but that the amounts are allowed to undergo proportional changes from one year to another. This assumption is appropriate when the sum function is currently adjusted by a scaling factor in accordance with the development of some economic key quantity like the basic amount of the national social insurance scheme or the price index for consumers' goods. At any rate, Assumption (iii") may serve as a working hypothesis. Introduce the loss ratios that is, the loss measured in units of the total sum at risk. From (4.2) and (4.3) it follows that the first two conditonal moments of 6, are given by (3.5) and (3.6). Thus, formulas (3.7H3.10) carry over to the present case. C. Net premiums. By virtue of (4.2), the annual net premium of a contract i 9 is estimated by with bi defined by (3.7H3.10). D. The fluctuation reserve. Adapting the ideas of Paragraph 3D to the present case, we propose that the reserve be calculated from (2.31) and, by virtue of (4.3) vf = Pi, ji+l s:j ji+, V, E. Parameter estimation. We refer to the non-hierarchical specialization of the results in Paragraph 5E below. 5. Hierarchical extensions A. The notion of hierarchy. The iid assumption (i) in Paragraph 2B implies that the groups are viewed as independent random selections from a population of basically similar groups. Often, however, it seems relevant to perform an initial classification of the insured groups according to certain Scand. Actuarial J. 1989

17 210 R. Norberg observable risk characteristics. If, for instance, occupational hazards are judged to be of possible importance, then the model ought td be extended to allow for a dependence between groups within one and the same industry. Such dependencies are accomodated in the hierarchical model framework, which was introduced in the credibility context by Jewel1 (1975) and later studied by a number of authors. The results cited here are picked from Norberg (1986 b), which can be consulted for proofs and further references. Initially, the groups are divided into H classes labeled by h= 1,..., H. The hidden risk characteristics of class h are represented by a latent quantity Oh. Within each class h there are Ih groups labeled by (h, i), i= 1,..., Ih. The risk characteristics specific of group (h, i) are represented by a latent quantity Oh;. Thus, the risk characteristics form a hierarchical structure. B. Model assumptions for summary data. The following assumption expresses that the classes are viewed as independent random selections from a "hyperpopulation" of classes. (h) Variables belonging to different classes are stochastically independent, and 01,..., Oh are iid. The hierarchical model is made up of the hierarchy assumption (h) as marginal model for the Oh and the former assumptions (i), (ii), and (iii) or (iii') or (iii") as conditional model for the Oh, and the observations of the individual groups within each class h (add the qualification "conditianal on Oh" and replace all subscripts i= 1,..., I by (h, i), i= 1,..., Ih). We shall here consider a hierarchical extension of the model in Section 3 (with minor modifications it carries over to the set-up of Section 4). Thus, let phij and XhD. denote the exposure and the total claim amount of group (h, i) in year j, and introduce the annual loss ratios The conditional first and second moments of bhij, given Oh and Ohir are the hierarchical variations of (3.5) and (3.6), with bh,=b(oh, 0,;) and vh,=v(oh, Oh;). Introduce the latent class h mean, bh = ~ ( 6 Oh) ~ ~ = E(bh,l 1 Oh), (5.4) and the parameters B = Ebhi = Eb,, g, = Ev,,, A = EVar(bhiJ Oh), x = Var 6,. (5.5) Roughly speaking, the quantities in (5.2)-(5.4) have the following interpretations: bhi is the risk premium or mean claim amount per unit exposure Scond. Actuarial J. 1989

18 Experience rating in group life insurance 21 1 of group (h, i); vhi measures the variation of claim amounts between time intervals with unit risk exposure for group (h, i); bi is the risk premium of class h (this is the net premium per unit exposure for new groups that are assigned to the class); /3 is the portfoliowide risk premium (this is the net premium per unit exposure for new classes that are entered into the portfolio); x measures the variation of risk premiums between classes; A measures the average variation of risk premiums between groups within one and the same class; q, is the mean value of the vhi. C. Net premiums. The credibility estimators of bhi and bh are given by (Norberg, 1986 b) with Numerical computations are performed by a simple recursive algorithm: calculate - all 6hi and chi by (5.8) and (5.9); - all 6 h and by (5.10) and (5.11); - a11 6, by (5.7); - all bhi by (5.6). The net premium of group (h, i) in year Jhi+ 1 is given by the hierarchical analogue of (3.11) or (4.3, whichever applies (insert subscripts (h, i) instead of 11. D. Thefluctuation reserve. The reasoning leading to the use of (3.14) or (4.6) in the fluctuation reserve (2.31) carries over to the hierarchical variations of the models in Sections 3 and 4. The only difference is that the indices i are to be replaced by (h, i). E. Parameter estimation. Consider first the hierarchical extension of the model in Section 3. Define the summary statistics (recall the convention Xhi=CjXhV, Xh=CiXhi, X=ChXh, etc.) (5.12) Scand. Actuarial

19 212 R. Norberg and h i j By use of (d,, is the Kronecker delta), it is easily verified that the following functions of the total loss ratio and the quantities in (5.12)-(5.17) are unbiased estimators of the parameters in (5.5): Estimators of the parameters in the non-hierarchical model are obtained by putting H= 1, whereby (5.22) becomes void and all subscripts h and sums over h can be dropped from the remaining expressions. F. An approximative analysis based on complete individual data. We now consider a hierarchical extension of the set-up in Section 2. Norberg (1989) proposes a joint distribution of the latent quantities that allows for a full posterior analysis. The appropriate specification is to let Oh, h= 1,..., H, be iid according to %(p, v) and to let Oh,, i=l,..., Ih, be iid according to %(Oh, 6) for fixed Oh=$h. Then it turns out that, conditional on the data, the Oh; remain gamma variates for fixed Oh, and the Oh become mixed gamma variates. We shall, however, take the simple approach of Paragraph 3F, extending

20 Experience rating in group life insurance 213 the model to a hierarchy as described in general terms in Paragraph 5A. Without loss of generality, Oh). All formulas in Paragraphs 5C-E are inherited, with Jhi= 1, Xhil = Mhi, phi = W,,, 6,; = dhi, bhi = vhi = Oh, b, = Oh, and = p, in (5.5) so that (5.14) and (5.20) can now be dropped from the estimation procedure. 6. Applications to real data A. The data. The theories developed above have been applied to group life risk statistics from a major Norwegian insurance company. Data were available for 1125 groups insured through the whole or parts of the period The groups were divided into 72 classes representing different occupational categories. The total group exposures phi ranged from 0 to 11536, with mean value 144. The data analysis was performed on an IBM personal computer. Copies of the programs, written in Turbopascal, will be delivered upon request. B. Hierarchical analyses based on summary data. We report here on the results obtained for the hierarchical extension in Section 5 of the model in Section 3. Table 1 in Paragraph F below shows an excerpt of the results obtained with constant sum function S(y)=l. In this case the loss ratios and risk premiums are simply mortality rates. The estimated portfoliowide mortality rate is /3*=2.91 x 10-3 or 2.91 per mille. Note that, by virtue of (3.2) and (3.3) and the fact that in the present case and so we estimate p, The estimated coefficients of variation, -//3*=0.517, fi//3*= 0.436, and -ID* =0.676, measure the relative magnitudes of differences in mortality rates between classes, between groups within one and the same class, and between groups in the entire portfolio, respectively. We conclude that the data reveal substantial variations in risk premiums between classes as well as between groups within one and the same class. This circumstance is reflected by the estimated values of the credibility weights 5h and chi. On the whole they are significantly greater than 0, and so experience rating of classes" and groups is justifiable. The ratios ll*/x*=0.711 and p,*/ll*= 1810 are key figures in this respect, confer formulas (5.9) and (5.11). It is seen that chi=0.5 for phi=~/a. Thus, the estimate p*/ll* tells us that after 1810 years exposed to risk, which corresponds to 1810~ =5.27 expected number of deaths, a group is rated with 50 % weight attached to its own risk experience. It is instructing to examine the contents of Table 1 and consider the dependence of credibility weights and risk premiums on the exposures for the given values of the parameter estimates. The reader is invited to contemplate the following findings (explanations are indicated in parentheses). Scand. Actuarial J. 1989

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