Experience Rating in Group Life Insurance.


 Sharyl Bates
 1 years ago
 Views:
Transcription
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 setup are discussed. An application of the theory to data from an authentic portfolio of groups revealed substantial betweengroup risk variations, hence experience rating could be statistically justified. Key words: group life insurance, proportional hazard model, BiihlmannStraub 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 25. 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 GompertzMakeham 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 dtxi 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 NewtonRaphson 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 whitecollars 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 costsaving 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), andthey 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) representson the averagea 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 BiihlmannStraub 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 nonexperimental 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 betweengroup 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 nonhierarchical 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 setup 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 nonhierarchical 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 setup 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 5CE 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 103 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
Identities for Present Values of Life Insurance Benefits
Scand. Actuarial J. 1993; 2: 100106 ORIGINAL ARTICLE Identities for Present Values of Life Insurance Benefits : RAGNAR NORBERG Norberg R. Identities for present values of life insurance benefits. Scand.
More informationA credibility method for profitable crossselling of insurance products
Submitted to Annals of Actuarial Science manuscript 2 A credibility method for profitable crossselling of insurance products Fredrik Thuring Faculty of Actuarial Science and Insurance, Cass Business School,
More informationSome Observations on Variance and Risk
Some Observations on Variance and Risk 1 Introduction By K.K.Dharni Pradip Kumar 1.1 In most actuarial contexts some or all of the cash flows in a contract are uncertain and depend on the death or survival
More informationChapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal
More informationFurther Topics in Actuarial Mathematics: Premium Reserves. Matthew Mikola
Further Topics in Actuarial Mathematics: Premium Reserves Matthew Mikola April 26, 2007 Contents 1 Introduction 1 1.1 Expected Loss...................................... 2 1.2 An Overview of the Project...............................
More information99.37, 99.38, 99.38, 99.39, 99.39, 99.39, 99.39, 99.40, 99.41, 99.42 cm
Error Analysis and the Gaussian Distribution In experimental science theory lives or dies based on the results of experimental evidence and thus the analysis of this evidence is a critical part of the
More informationLOGNORMAL MODEL FOR STOCK PRICES
LOGNORMAL MODEL FOR STOCK PRICES MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD 1. INTRODUCTION What follows is a simple but important model that will be the basis for a later study of stock prices as
More informationA Coefficient of Variation for Skewed and HeavyTailed Insurance Losses. Michael R. Powers[ 1 ] Temple University and Tsinghua University
A Coefficient of Variation for Skewed and HeavyTailed Insurance Losses Michael R. Powers[ ] Temple University and Tsinghua University Thomas Y. Powers Yale University [June 2009] Abstract We propose a
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationSolution. Let us write s for the policy year. Then the mortality rate during year s is q 30+s 1. q 30+s 1
Solutions to the May 213 Course MLC Examination by Krzysztof Ostaszewski, http://wwwkrzysionet, krzysio@krzysionet Copyright 213 by Krzysztof Ostaszewski All rights reserved No reproduction in any form
More informationIt is important to bear in mind that one of the first three subscripts is redundant since k = i j +3.
IDENTIFICATION AND ESTIMATION OF AGE, PERIOD AND COHORT EFFECTS IN THE ANALYSIS OF DISCRETE ARCHIVAL DATA Stephen E. Fienberg, University of Minnesota William M. Mason, University of Michigan 1. INTRODUCTION
More informationPREMIUM AND BONUS. MODULE  3 Practice of Life Insurance. Notes
4 PREMIUM AND BONUS 4.0 INTRODUCTION A insurance policy needs to be bought. This comes at a price which is known as premium. Premium is the consideration for covering of the risk of the insured. The insured
More informationVilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis
Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................
More informationPricing Alternative forms of Commercial Insurance cover
Pricing Alternative forms of Commercial Insurance cover Prepared by Andrew Harford Presented to the Institute of Actuaries of Australia Biennial Convention 2326 September 2007 Christchurch, New Zealand
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More informationPart 2: Oneparameter models
Part 2: Oneparameter models Bernoilli/binomial models Return to iid Y 1,...,Y n Bin(1, θ). The sampling model/likelihood is p(y 1,...,y n θ) =θ P y i (1 θ) n P y i When combined with a prior p(θ), Bayes
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationAuxiliary Variables in Mixture Modeling: 3Step Approaches Using Mplus
Auxiliary Variables in Mixture Modeling: 3Step Approaches Using Mplus Tihomir Asparouhov and Bengt Muthén Mplus Web Notes: No. 15 Version 8, August 5, 2014 1 Abstract This paper discusses alternatives
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationModels for Count Data With Overdispersion
Models for Count Data With Overdispersion Germán Rodríguez November 6, 2013 Abstract This addendum to the WWS 509 notes covers extrapoisson variation and the negative binomial model, with brief appearances
More information1 Prior Probability and Posterior Probability
Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which
More informationThe Basics of Interest Theory
Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest
More informationMixing internal and external data for managing operational risk
Mixing internal and external data for managing operational risk Antoine Frachot and Thierry Roncalli Groupe de Recherche Opérationnelle, Crédit Lyonnais, France This version: January 29, 2002 Introduction
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a realvalued function defined on the sample space of some experiment. For instance,
More informationA Model of Optimum Tariff in Vehicle Fleet Insurance
A Model of Optimum Tariff in Vehicle Fleet Insurance. Bouhetala and F.Belhia and R.Salmi Statistics and Probability Department Bp, 3, ElAlia, USTHB, BabEzzouar, Alger Algeria. Summary: An approach about
More information11. Analysis of Casecontrol Studies Logistic Regression
Research methods II 113 11. Analysis of Casecontrol Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:
More informationREINSURANCE PROFIT SHARE
REINSURANCE PROFIT SHARE Prepared by Damian Thornley Presented to the Institute of Actuaries of Australia Biennial Convention 2326 September 2007 Christchurch, New Zealand This paper has been prepared
More informationEDUCATION AND EXAMINATION COMMITTEE SOCIETY OF ACTUARIES RISK AND INSURANCE. Copyright 2005 by the Society of Actuaries
EDUCATION AND EXAMINATION COMMITTEE OF THE SOCIET OF ACTUARIES RISK AND INSURANCE by Judy Feldman Anderson, FSA and Robert L. Brown, FSA Copyright 25 by the Society of Actuaries The Education and Examination
More information4. Life Insurance. 4.1 Survival Distribution And Life Tables. Introduction. X, Ageatdeath. T (x), timeuntildeath
4. Life Insurance 4.1 Survival Distribution And Life Tables Introduction X, Ageatdeath T (x), timeuntildeath Life Table Engineers use life tables to study the reliability of complex mechanical and
More informationThe CUSUM algorithm a small review. Pierre Granjon
The CUSUM algorithm a small review Pierre Granjon June, 1 Contents 1 The CUSUM algorithm 1.1 Algorithm............................... 1.1.1 The problem......................... 1.1. The different steps......................
More informationRandom variables, probability distributions, binomial random variable
Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that
More informationJANUARY 2016 EXAMINATIONS. Life Insurance I
PAPER CODE NO. MATH 273 EXAMINER: Dr. C. BoadoPenas TEL.NO. 44026 DEPARTMENT: Mathematical Sciences JANUARY 2016 EXAMINATIONS Life Insurance I Time allowed: Two and a half hours INSTRUCTIONS TO CANDIDATES:
More informationStatistics in Retail Finance. Chapter 6: Behavioural models
Statistics in Retail Finance 1 Overview > So far we have focussed mainly on application scorecards. In this chapter we shall look at behavioural models. We shall cover the following topics: Behavioural
More informationPortfolio Using Queuing Theory
Modeling the Number of Insured Households in an Insurance Portfolio Using Queuing Theory JeanPhilippe Boucher and Guillaume CouturePiché December 8, 2015 Quantact / Département de mathématiques, UQAM.
More informationUnobserved heterogeneity; process and parameter effects in life insurance
Unobserved heterogeneity; process and parameter effects in life insurance Jaap Spreeuw & Henk Wolthuis University of Amsterdam ABSTRACT In this paper life insurance contracts based on an urnofurns model,
More informationInstitute of Actuaries of India Subject CT3 Probability and Mathematical Statistics
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models  part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK2800 Kgs. Lyngby
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationMATHEMATICS OF FINANCE AND INVESTMENT
MATHEMATICS OF FINANCE AND INVESTMENT G. I. FALIN Department of Probability Theory Faculty of Mechanics & Mathematics Moscow State Lomonosov University Moscow 119992 g.falin@mail.ru 2 G.I.Falin. Mathematics
More informationMaking use of netting effects when composing life insurance contracts
Making use of netting effects when composing life insurance contracts Marcus Christiansen Preprint Series: 2113 Fakultät für Mathematik und Wirtschaftswissenschaften UNIVERSITÄT ULM Making use of netting
More informationNONRANDOM ACCIDENT DISTRIBUTIONS AND THE POISSON SERIES
NONRANDOM ACCIDENT DISTRIBUTIONS AND THE POISSON SERIES 21 NONRANDOM ACCIDENT DISTRIBUTIONS AND THE POISSON SERIES BY JOHN CARLETON In recent years several papers have appeared in the Proceedings in
More informationTABLE OF CONTENTS. 4. Daniel Markov 1 173
TABLE OF CONTENTS 1. Survival A. Time of Death for a Person Aged x 1 B. Force of Mortality 7 C. Life Tables and the Deterministic Survivorship Group 19 D. Life Table Characteristics: Expectation of Life
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationSupplement to Call Centers with Delay Information: Models and Insights
Supplement to Call Centers with Delay Information: Models and Insights Oualid Jouini 1 Zeynep Akşin 2 Yves Dallery 1 1 Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290
More informationOptimal hedging of demographic risk in life insurance
Finance and Stochastics manuscript No. (will be inserted by the editor) Optimal hedging of demographic risk in life insurance Ragnar Norberg the date of receipt and acceptance should be inserted later
More informationThe Notebook Series. The solution of cubic and quartic equations. R.S. Johnson. Professor of Applied Mathematics
The Notebook Series The solution of cubic and quartic equations by R.S. Johnson Professor of Applied Mathematics School of Mathematics & Statistics University of Newcastle upon Tyne R.S.Johnson 006 CONTENTS
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationThe Basics of Graphical Models
The Basics of Graphical Models David M. Blei Columbia University October 3, 2015 Introduction These notes follow Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan. Many figures
More informationLife Settlement Pricing
Life Settlement Pricing Yinglu Deng Patrick Brockett Richard MacMinn Tsinghua University University of Texas Illinois State University Life Settlement Description A life settlement is a financial arrangement
More informationErrata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page
Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 1 Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Practice exam 1:9, 1:22, 1:29, 9:5, and 10:8
More information2. Simple Linear Regression
Research methods  II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationMultivariate Normal Distribution
Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #47/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues
More informationHeriotWatt University. BSc in Actuarial Mathematics and Statistics. Life Insurance Mathematics I. Extra Problems: Multiple Choice
HeriotWatt University BSc in Actuarial Mathematics and Statistics Life Insurance Mathematics I Extra Problems: Multiple Choice These problems have been taken from Faculty and Institute of Actuaries exams.
More informationThe Treatment of Insurance in the SNA
The Treatment of Insurance in the SNA Peter Hill Statistical Division, United Nations Economic Commission for Europe April 1998 Introduction The treatment of insurance is one of the more complicated parts
More informationInequality, Mobility and Income Distribution Comparisons
Fiscal Studies (1997) vol. 18, no. 3, pp. 93 30 Inequality, Mobility and Income Distribution Comparisons JOHN CREEDY * Abstract his paper examines the relationship between the crosssectional and lifetime
More informationReview of Basic Options Concepts and Terminology
Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some
More informationA GENERALIZATION OF AUTOMOBILE INSURANCE RATING MODELS: THE NEGATIVE BINOMIAL DISTRIBUTION WITH A REGRESSION COMPONENT
WORKSHOP A GENERALIZATION OF AUTOMOBILE INSURANCE RATING MODELS: THE NEGATIVE BINOMIAL DISTRIBUTION WITH A REGRESSION COMPONENT BY GEORGES DIONNE and CHARLES VANASSE Universit~ de MontrEal, Canada * ABSTRACT
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 PrincipalAgent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More information4. Introduction to Statistics
Statistics for Engineers 41 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation
More informationRatemakingfor Maximum Profitability. Lee M. Bowron, ACAS, MAAA and Donnald E. Manis, FCAS, MAAA
Ratemakingfor Maximum Profitability Lee M. Bowron, ACAS, MAAA and Donnald E. Manis, FCAS, MAAA RATEMAKING FOR MAXIMUM PROFITABILITY Lee Bowron, ACAS, MAAA Don Manis, FCAS, MAAA Abstract The goal of ratemaking
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationRating Methodology for Domestic Life Insurance Companies
Rating Methodology for Domestic Life Insurance Companies Introduction ICRA Lanka s Claim Paying Ability Ratings (CPRs) are opinions on the ability of life insurance companies to pay claims and policyholder
More informationThe Elasticity of Taxable Income: A NonTechnical Summary
The Elasticity of Taxable Income: A NonTechnical Summary John Creedy The University of Melbourne Abstract This paper provides a nontechnical summary of the concept of the elasticity of taxable income,
More informationInfinitely Repeated Games with Discounting Ù
Infinitely Repeated Games with Discounting Page 1 Infinitely Repeated Games with Discounting Ù Introduction 1 Discounting the future 2 Interpreting the discount factor 3 The average discounted payoff 4
More informationGuidance Note on Actuarial Review of Insurance Liabilities in respect of Employees Compensation and Motor Insurance Businesses
Guidance Note on Actuarial Review of Insurance Liabilities in respect of Employees Compensation and Motor Insurance Businesses GN9 Introduction Under the Insurance Companies Ordinance (Cap. 41) ( Ordinance
More informationNovember 16, 2015. Interpolation, Extrapolation & Polynomial Approximation
Interpolation, Extrapolation & Polynomial Approximation November 16, 2015 Introduction In many cases we know the values of a function f (x) at a set of points x 1, x 2,..., x N, but we don t have the analytic
More informationA nonparametric twostage Bayesian model using Dirichlet distribution
Safety and Reliability Bedford & van Gelder (eds) 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 551 7 A nonparametric twostage Bayesian model using Dirichlet distribution C. Bunea, R.M. Cooke & T.A. Mazzuchi
More informationIncome and the demand for complementary health insurance in France. Bidénam KambiaChopin, Michel Grignon (McMaster University, Hamilton, Ontario)
Income and the demand for complementary health insurance in France Bidénam KambiaChopin, Michel Grignon (McMaster University, Hamilton, Ontario) Presentation Workshop IRDES, June 2425 2010 The 2010 IRDES
More informationChi Square Tests. Chapter 10. 10.1 Introduction
Contents 10 Chi Square Tests 703 10.1 Introduction............................ 703 10.2 The Chi Square Distribution.................. 704 10.3 Goodness of Fit Test....................... 709 10.4 Chi Square
More informationINTERNATIONAL COMPARISON OF INTEREST RATE GUARANTEES IN LIFE INSURANCE
INTERNATIONAL COMPARISON OF INTEREST RATE GUARANTEES IN LIFE INSURANCE J. DAVID CUMMINS, KRISTIAN R. MILTERSEN, AND SVEINARNE PERSSON Abstract. Interest rate guarantees seem to be included in life insurance
More informationMath 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 2 Solutions
Math 70/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 2 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,
More informationPoisson Models for Count Data
Chapter 4 Poisson Models for Count Data In this chapter we study loglinear models for count data under the assumption of a Poisson error structure. These models have many applications, not only to the
More informationStock valuation. Price of a First period's dividends Second period's dividends Third period's dividends = + + +... share of stock
Stock valuation A reading prepared by Pamela Peterson Drake O U T L I N E. Valuation of common stock. Returns on stock. Summary. Valuation of common stock "[A] stock is worth the present value of all the
More informationMaster s Theory Exam Spring 2006
Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationA POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More information**BEGINNING OF EXAMINATION** The annual number of claims for an insured has probability function: , 0 < q < 1.
**BEGINNING OF EXAMINATION** 1. You are given: (i) The annual number of claims for an insured has probability function: 3 p x q q x x ( ) = ( 1 ) 3 x, x = 0,1,, 3 (ii) The prior density is π ( q) = q,
More informationPEER REVIEW HISTORY ARTICLE DETAILS VERSION 1  REVIEW. Elizabeth Comino Centre fo Primary Health Care and Equity 12Aug2015
PEER REVIEW HISTORY BMJ Open publishes all reviews undertaken for accepted manuscripts. Reviewers are asked to complete a checklist review form (http://bmjopen.bmj.com/site/about/resources/checklist.pdf)
More informationApplication of Credibility Theory to Group Life Pricing
Prepared by: Manuel Tschupp, MSc ETH Application of Credibility Theory to Group Life Pricing Extended Techniques TABLE OF CONTENTS 1. Introduction 3 1.1 Motivation 3 1.2 Fundamentals 1.3 Structure 3 4
More information171:290 Model Selection Lecture II: The Akaike Information Criterion
171:290 Model Selection Lecture II: The Akaike Information Criterion Department of Biostatistics Department of Statistics and Actuarial Science August 28, 2012 Introduction AIC, the Akaike Information
More informationJoint Exam 1/P Sample Exam 1
Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More information4. Joint Distributions of Two Random Variables
4. Joint Distributions of Two Random Variables 4.1 Joint Distributions of Two Discrete Random Variables Suppose the discrete random variables X and Y have supports S X and S Y, respectively. The joint
More informationAP Physics 1 and 2 Lab Investigations
AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks
More informationPayment streams and variable interest rates
Chapter 4 Payment streams and variable interest rates In this chapter we consider two extensions of the theory Firstly, we look at payment streams A payment stream is a payment that occurs continuously,
More informationTRANSACTIONS OF SOCIETY OF ACTUARIES 1952 VOL. 4 NO. 10 COMPLETE ANNUITIES. EUGENE A. RASOR* Ann T. N. E. GREVILLE
TRANSACTIONS OF SOCIETY OF ACTUARIES 1952 VOL. 4 NO. 10 COMPLETE ANNUITIES EUGENE A. RASOR* Ann T. N. E. GREVILLE INTRODUCTION I N GENERAL, a complete annuity of one per annum may be defined as a curtate
More information7: The CRR Market Model
Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The CoxRossRubinstein
More informationHighway Loss Data Institute Bulletin
Highway Loss Data Institute Bulletin Helmet Use Laws and Medical Payment Injury Risk for Motorcyclists with Collision Claims VOL. 26, NO. 13 DECEMBER 29 INTRODUCTION According to the National Highway Traffic
More informationUsing simulation to calculate the NPV of a project
Using simulation to calculate the NPV of a project Marius Holtan Onward Inc. 5/31/2002 Monte Carlo simulation is fast becoming the technology of choice for evaluating and analyzing assets, be it pure financial
More informationWhat Does the Correlation Coefficient Really Tell Us About the Individual?
What Does the Correlation Coefficient Really Tell Us About the Individual? R. C. Gardner and R. W. J. Neufeld Department of Psychology University of Western Ontario ABSTRACT The Pearson product moment
More informationLecture notes for Choice Under Uncertainty
Lecture notes for Choice Under Uncertainty 1. Introduction In this lecture we examine the theory of decisionmaking under uncertainty and its application to the demand for insurance. The undergraduate
More informationThe Cost of Financial Frictions for Life Insurers
The Cost of Financial Frictions for Life Insurers Ralph S. J. Koijen Motohiro Yogo University of Chicago and NBER Federal Reserve Bank of Minneapolis 1 1 The views expressed herein are not necessarily
More informationImpact of Genetic Testing on Life Insurance
Agenda, Volume 10, Number 1, 2003, pages 6172 Impact of Genetic Testing on Life Insurance he Human Genome project generates immense interest in the scientific community though there are also important
More informationMath 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 2 Solutions
Math 370, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the
More informationA LOGNORMAL MODEL FOR INSURANCE CLAIMS DATA
REVSTAT Statistical Journal Volume 4, Number 2, June 2006, 131 142 A LOGNORMAL MODEL FOR INSURANCE CLAIMS DATA Authors: Daiane Aparecida Zuanetti Departamento de Estatística, Universidade Federal de São
More informationOverview of Monte Carlo Simulation, Probability Review and Introduction to Matlab
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?
More informationSOME ASPECTS OF GAMBLING WITH THE KELLY CRITERION. School of Mathematical Sciences. Monash University, Clayton, Victoria, Australia 3168
SOME ASPECTS OF GAMBLING WITH THE KELLY CRITERION Ravi PHATARFOD School of Mathematical Sciences Monash University, Clayton, Victoria, Australia 3168 In this paper we consider the problem of gambling with
More information