We revisit the optimal bonus scales introduced by Norberg (Norberg, R., 1976, Scandinavian Actuarial Journal (2): 92-107), Borgan, Hoem, and Norberg (Borgan, O., J. Hoem, and R. Norberg, 1981, Scandinavian Actuarial Journal (2): 165-178), and Gilde and Sundt (Gilde, V., and B. Sundt, 1989, Scandinavian Actuarial Journal (1): 13-22) and underline some potential problems of the linear scales. As a possible solution we propose the use of geometric scales..info:eu-repo/semantics/publishedVersio

Centeno, M. de Lourdes

Silva, João Andrade e

English

UTL Repository

C© The Journal of Risk and Insurance, 2005, Vol. 72, No. 4, 601-607A NOTE ON BONUS SCALESJoão Manuel Andrade e SilvaMaria de Lourdes CentenoABSTRACTWe revisit the optimal bonus scales introduced by Norberg (Norberg, R.,1976, Scandinavian Actuarial Journal (2): 92-107), Borgan, Hoem, and Norberg(Borgan, O., J. Hoem, and R. Norberg, 1981, Scandinavian Actuarial Journal (2):165-178), and Gilde and Sundt (Gilde, V., and B. Sundt, 1989, ScandinavianActuarial Journal (1): 13-22) and underline some potential problems of thelinear scales. As a possible solution we propose the use of geometric scales.INTRODUCTIONBonus-malus systems, namely Markovian systems, are widely used in automobileinsurance. As it is well known, see Lemaire (1995), the basic idea of these systemsis to divide the policies into a finite number of classes, numbered from 1 to K, toplace each policy in one of these classes during each insurance period, supposed ofconstant length (usually a year) and to determine the class where the policy will be inthe next year based only on the present class and on the number of claims reportedto the insurer. We assume that the classes are numbered from the least to the mostdangerous one.In 1976, Norberg developed optimal credibility premium scales for Markovian bonus-malus systems, with given transition rules, under an infinite horizon approach andassuming the minimization of the expected squared difference between the true netpremium and the premium paid by the policyholder. In 1981, Borgan, Hoem, andNorberg generalized this result to a finite horizon approach.These approaches, as mentioned by the authors, can both lead to a sequence ofnon-monotonic premiums, which is unacceptable from the practical point of view,since a policyholder who moves to a better class should not have to suffer an increasein his premium. Gilde and Sundt, a few years later, tackled this problem by constrain-ing the optimization problem, assuming that the increase between consecutive classesof the system is always the same, i.e., the premium sequence is linear. However, asJoão Manuel Andrade e Silva and Maria de Lourdes Centeno are with ISEG, Technical Univer-sity of Lisbon, Rua do Quelhas, 2, 1200-781 Lisboa, Portugal. The authors can be contacted viae-mail: joaoas@iseg.utl.pt. This research has been supported by Fundação para a Ciência e aTecnologia-FCT/POCTI.601602 THE JOURNAL OF RISK AND INSURANCEwe will show in the present article, this approach can lead to another flaw: in somesituations, not unrealistic, premiums given by the linear scale become negative, whichis, obviously, unacceptable.To solve this problem we propose the use of a geometric scale, i.e., the premiumincrease between consecutive classes is always given by the same percentage of thepremium.Using a similar framework, one could use other constraints that lead to other premiumscales. Amongst the various alternatives, the geometric scale presents two qualities:its interpretation is very simple and it can be very easily combined with the otherrating factors when using generalized linear models with a logarithmic link function.However, the premiums given by the geometric scale are not financially balanced and,consequently, we have to correct the results given by the constrained minimization.Note that the correction is straightforward.The structure of the article is as follows: We start by presenting the model and theresults obtained by Norberg (1976), Borgan, Hoem, and Norberg (1981), and Gilde andSundt (1989). Then, we introduce the geometric scales. In the last section, we presenttwo examples to illustrate our approach.THE MODELLet us consider a Markovian bonus system with K classes. After a year with r claims inclass j the policy is transferred to class T(j, r). This function represents the transitionrules associated to the bonus system and is assumed to be nondecreasing in its secondargument. Let π be a vector whose components π ( j), j = 1, 2, . . . , K , represent thepremium in class j and let k be the initial class. In this context we can represent thebonus system by the triplet S = (T , π, k).Let Mn be the number of claims reported in period n. We assume that M1, M2, . . . areconditionally independent given the value of an unknown risk parameter θ , regardedas the outcome of a positive random variable  with distribution function U(.). Finally,ZS,n denotes the bonus class in period n for a given policy.Norberg (1976), assuming that {Mn}n=1,2,... are identically distributed, shows that thepremium scale leading to the minimization of the expected quadratic error using thelong-run distribution of the Markov chain, i.e., which minimizesQ0(S) = E[E(M | ) − π (ZS,0)]2, (1)where the index 0 stands for the long-run distribution and M has the same distributionas Mn, is given byπ0( j) = E[E[M | ] | ZS,0 = j] =∫E[M | θ ]pT,θ ( j) dU(θ )pT ( j), j = 1, . . . , K , (2)where pT,θ = [pT,θ ( j)] is the conditionally long-run distribution and pT = [pT ( j)] isthe long-run distribution.A NOTE ON BONUS SCALES 603Borgan, Hoem, and Norberg (1981) generalize this approach. They construct anonasymptotic criterion, by minimizingQ(S) =∞∑n=0wn Qn(S) =∞∑n=0wnE[E(Mn | ) − π (ZS,n)]2, (3)where w = [w0, w1, . . .] is a vector of nonnegative weights adding 1 and where w0 isthe weight given to the long-run distribution. The solution, named Bayes scale, isπB( j) =∞∑n=0wn∫E[Mn | θ ]p(n)T,θ ( j) dU(θ )pS( j), j = 1, . . . , K , (4)where p(n)S,θ = [p(n)S,θ ( j)] is the probability that, given θ , the policy stands in class j inperiod n andpS( j) =∞∑n=0wn∫p(n)T,θ ( j) dU(θ ), j = 1, . . . , K . (5)Note that the last scale generalizes the former, given by (2), considering that {Mn} areindependently and identically distributed (i.i.d.) and that w0 = 1.From the statistical point of view, these scales are optimal for a quadratic loss. How-ever, in some practical circumstances they can lead to some annoyances: the variationof the premiums in consecutive classes of the system can be quite irregular and, insome cases, a more dangerous class of the system can pay less than a less dangerousone. Even if the situation could be understood in mathematical terms, it would beunacceptable by the policyholders.As a solution to these annoyances Gilde and Sundt (1989) introduce linear scales, i.e.,they minimize (3) subject to the constraints that π ( j) = a + bj, j = 1, . . . , K .The solution is now πL( j) = aL + bL j = 1, . . . , K , withbL =K∑j=1jπB( j)pS( j) −K∑j=1j pS( j)K∑j=1πB( j)pS( j)K∑j=1j2 pS( j) −(K∑j=1j pS( j))2 ,aL =K∑j=1πB( j)pS( j) − bLK∑j=1j pS( j).(6)604 THE JOURNAL OF RISK AND INSURANCEAs we have numbered the classes of the bonus system from the least to the mostdangerous one, we expect bL to be positive. This would be the case ifK∑j=1jπB( j)pS( j) −K∑j=1j pS( j)K∑j=1πB( j)pS( j) > 0, (7)that is to say, if the covariance betweenπB and ZS, where ZS =∑∞n=0 wn ZS,n, is positive.This must be the case for all the acceptable systems and we will assume that (7) holds.But even assuming (7) these scales can, for some very heterogeneous situations, leadto a big setback: negative premiums for the best classes of the system can be obtained,i.e., aL + bL can be negative.GEOMETRIC SCALESThe basic idea of the geometric scales is similar to the linear scales. We minimize (3)under constraints of the form π ( j) = abj, j = 1, . . . , K , with a and b positive.Note that using (4) and (5), we can write (3) asQ(S) =∞∑n=0wn∫ K∑j=1[E(Mn | θ ) − π ( j)]2 p(n)S,θ ( j) dU(θ )=∞∑n=0wn∫ K∑j=1E2(Mn | θ )p(n)S,θ ( j) dU(θ )+K∑j=1(π ( j))2 pS( j) − 2K∑j=1πB( j)π ( j)pS( j)= Constant +K∑j=1(π ( j) − πB( j))2 pS( j).(8)From the last expression in (8) it is clear that the linear scale is nothing more than theresult of a weighted least square regression between the Bayes scale and the indices j.To calculate the geometric scale we will now minimizeQG (S) =K∑j=1(ab j − πB( j))2 pS( j) (9)that corresponds to a weighted nonlinear least squares regression. Differentiating (9)with respect to a and b and equating to zero we get the first order conditionsa =K∑j=1b jπB( j)pS( j)K∑j=1b2 j pS( j)(10)A NOTE ON BONUS SCALES 605andf (b) = 0, (11)withf (b) =K∑j=1b jπB( j)pS( j)K∑j=1jb2 j pS( j) −K∑j=1jb jπB( j)pS( j)K∑j=1b2 j pS( j). (12)Note that f (b) is a polynomial of order 3K − 1 and that the coefficient of b3K−1 isnegative, namely −πB (K − 1)pS (K − 1)pS (K ). Furthermore f (1) > 0, according to(7). Consequently, as f (1) > 0 and lim b→∞ f (b) = −∞, f (b) must have at least oneroot greater than 1. We were not able to prove the uniqueness of the real solution forb > 1, but we could not find a counterexample. Anyway, if the root is unique it is aminimum, otherwise we would compare the relative minima. Let the optimal solutionbe (aG, bG) and the optimal scale be πG.Note that the expected value of the premiums (relative to pS) under both the Bayesscale and the linear scale is equal to∑∞n=0 wnE[Mn], i.e., both systems are balanced.This does not happen with the geometric scale. If we require a balanced system weshould minimize (9) subject toK∑j=1ab j pS( j) =K∑j=1πB( j)pS( j). (13)In this case the first order conditions area =K∑j=1πB( j)pS( j)K∑j=1b j pS( j),K∑j=1[ab j − πB( j)] jb j−1 pS( j)K∑j=1b j pS( j) =K∑j=1[ab j − πB( j)]b j pS( j)K∑j=1jb j−1 pS( j).(14)The discussion about the uniqueness of the solution of (14) is similar to the discussionabout the uniqueness of (7), being now the degree of the polynomial 3K − 2. Let theoptimal solution be (ag, bg) and the optimal scale be πg.It is straightforward to generalize the geometric scales to bonus systems with path-dependent rules or to models where the policyholders can move from one companyto another; see Centeno and Andrade e Silva (2001, 2002) for the presentation of thosesystems.606 THE JOURNAL OF RISK AND INSURANCEEXAMPLESIn the following examples we consider a system with K = 15, k = 10, and transitionrules given byT( j, r ) ={max( j − 1, 1) if r = 0,min( j + 3r, K ) if r > 0,that is a policy goes up three classes for each reported claim and goes down one classfor each claim free year.The number of claims {Mn} are, conditionally on  = θ , i.i.d. and Poisson distributedwith mean θ . The structural distribution U(θ ) is given by a Gamma distribution andw =[0.30 0.12 0.10 0.09 0.08 0.07 0.07 0.06 0.06 0.05].In the first example we consider a very heterogeneous situation: E[] = 0.12 andVar[] = 0.039. The Bayes scale is given in column 3 of Table 1. As we can see πB (4) >πB (5), which is not acceptable. For the linear scale we obtained aL = −0.0404 andbL = 0.0247, which implies that πL(1) < 0. The geometric scale (balanced) is given inthe fourth column.In this situation only the geometric scale is acceptable. The main difference betweenthe Bayes and the geometric scales is found in the first classes of the system. Theexpected quadratic loss of both systems is very similar. Q(S) = 0.02229 for the Bayesscale while Q(S) = 0.02283 for the geometric (balanced) scale.TABLE 1Optimal Bonus Scales When E() = 0.12 and Var() =0.039j pS ( j) πB ( j) πg ( j)1 0.21120 0.03807 0.017762 0.03968 0.05126 0.022883 0.04833 0.05392 0.029474 0.05116 0.05992 0.037955 0.05347 0.05727 0.048886 0.06416 0.07145 0.062957 0.07634 0.07914 0.081088 0.08805 0.08853 0.104439 0.10551 0.10607 0.1345010 0.13892 0.13409 0.1732311 0.02076 0.25703 0.2231112 0.02196 0.30592 0.2873613 0.02319 0.37630 0.3701114 0.02192 0.48007 0.4766815 0.03536 0.61228 0.61394A NOTE ON BONUS SCALES 607TABLE 2Optimal Bonus Scales When E() = 0.12 and Var() =0.00085j pS( j) πB( j) πL( j) πg( j)1 0.18430 0.08122 0.07039 0.070302 0.03929 0.09510 0.07923 0.076583 0.04835 0.09751 0.08807 0.083434 0.05343 0.10223 0.09691 0.090895 0.05035 0.10131 0.10575 0.099026 0.06825 0.10879 0.11459 0.107877 0.08207 0.11247 0.12343 0.117528 0.09389 0.11603 0.13227 0.128039 0.11373 0.12210 0.14111 0.1394810 0.14929 0.12944 0.14995 0.1519511 0.02975 0.17816 0.15879 0.1655412 0.02795 0.19207 0.16763 0.1803413 0.02458 0.21288 0.17646 0.1964714 0.01662 0.25541 0.18530 0.2140315 0.01816 0.28162 0.19414 0.23317Table 2 shows the results for a more homogeneous case: E[] = 0.12 and Var[] =0.00085.In this case the Bayes scale continues to suffer from the same problem. The linearand the geometric scales are both acceptable, but the expected quadratic error of thegeometric scale is smaller: 0.00687, 0.00733, and 0.00714 for the Bayes, linear, andgeometric scales, respectively.REFERENCESBorgan, O., J. Hoem, and R. Norberg, 1981, A Nonasymptotic Criterion for the Eval-uation of Automobile Bonus Systems, Scandinavian Actuarial Journal, (2): 165-178.Centeno, M. L., and J. M. Andrade e Silva, 2001, Bonus Systems in an Open Portfolio,Insurance: Mathematics and Economics, 28(3): 341-350.Centeno, M. L., and J. M. Andrade e Silva, 2002, Optimal Bonus Scales Under Path-Dependent Bonus Rules, Scandinavian Actuarial Journal, (2): 129-136.Gilde, V., and B. Sundt, 1989, On Bonus Systems With Credibility Scales, ScandinavianActuarial Journal, (1): 13-22.Lemaire, J., 1995, Bonus Malus Systems in Automobile Insurance (Boston, MA: KluwerAcademic Publishers).Norberg, R., 1976, A Credibility Theory for Automobile Bonus Systems, ScandinavianActuarial Journal, (2): 92-107.

A note on bonus scales

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A note on bonus scales

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