Sweeting, Paul

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  DISCUSSION PAPER PI-1012  Making the Most of Experience Data – an Augmented Beta-Binomial Approach   P.J.Sweeting  November 2010  ISSN 1367-580X   The Pensions Institute  Cass Business School  City University  106 Bunhill Row London  EC1Y 8TZ  UNITED KINGDOM   http://www.pensions-institute.org/  Making the Most of Experience Data - anAugmented Beta-Binomial ApproachP.J.SweetingCentre for Actuarial Science, Risk and InvestmentUniversity of KentCanterburyKent CT2 7NZp.j.sweeting@kent.ac.ukNovember 5, 20101AbstractThe popular beta-binomial approach to credibility offers an attractiveway of combining the results of experience and risk rating. However, whenapplied to a particular age the information available from surrounding agesis ignored. In this paper I propose an augmentation to the beta-binomialapproach which not only allows for the information contained in ages nearto the age under analysis, but also for variation in risk types across thedifferent ages.21 IntroductionWhen considering the appropriate mortality rate to use for a group of lives, twosources of data are available. The first is the mortality experience of that groupof lives. Calculating a mortality rate using this approach is known as experiencerating. However, the smaller the number of lives, the greater the extent to whichcalculated mortality rates reflect random volatility rather than the underlyingrate of mortality. It is therefore useful to consider the characteristics of the livesunder review. The mortality rate for the sub-section of a broader populationwith the same characteristics can then also be calculated and used instead. Thisis known as risk rating.Risk rating is, though, imperfect. For any group of lives there may well beidiosyncrasies that have an impact on mortality rates that will not be picked upthrough risk rating. Consider, for example, the mortality experience of a groupof pension scheme members. This mortality might be influenced by hard-to-measure factors such as a sense of community. Such factors will not be allowedfor in a risk rating approach using, say, socio-economic group as the key variable.In practice, therefore the results of experience and risk rating will be com-bined using a measure of credibility. This gives an increasing weight to theresult of experience rating the larger the number of observations is. However,importance of idiosyncratic factors means that it is important to make as muchuse of experience data as possible.Credibility can be expressed as a factor by which the results of experienceand risk rating are combined, as shown in Equation 1:qˆx = Zqex + (1− Z)qrx. (1)Here, the initial mortality rate for age x, qˆx, is estimated as a weightedaverage of the rate derived from mortality experience, qex, and the rate derivedfrom the risk factors, qrx. The term Z is the credibility factor, which rangesfrom zero (for no credibility) to one (for full credibility).It is also possible to combine the results of experience and risk rating througha single expression, although here it is still possible to calculate the credibilityfactor implied by the resulting mortality rate. This can be useful not only forfinding the most appropriate set of mortality assumptions, but also for calculat-ing margins for error as discussed by Hardy and Panjer (1998). One method thatis particularly relevant for mortality rating is the beta-binomial approach. ThisBayesian approach is discussed by Mayerson (1964) and others, with more detailbeing given in many standard textbooks on credibility, such as that by Herzog(1999). The mortality rate at age x derived from experience, qex is calculated asthe number of deaths at that age, dx divided by the initial number of lives, lx,with lx being adjusted to reflect any entrants to or exits from the populationthat might affect the level of exposure. However, it is assumed that underlyingmortality rate, qx, has a beta distribution with parameters β1,x and β2,x. Theexpected underlying mortality rate is qrx = E(qx) = β1,x/(β1,x + β2,x), and thevariance is V ar(qx) = β1,xβ2,x/(β1,x + β2,x)2(β1,x + β2,x + 1). This means that3if the first two moments of the distribution of the underlying rate are known,β1,x and β2,x can be determined using the method of moments.Since the number of deaths occurring at each age has a binomial distribution,the Bayesian probability of there being dx deaths from a group of lx lives canbe found by using E(qx) = β1,x/(β1,x + β2,x) in the calculation of the binomialprobability formula. The result is that the posterior distribution of the expectednumber of deaths also has a beta distribution with parameters dx + β1,x andlx − dx + β2,x. This means that the expected number of deaths is therefore:E(qx|dx) =dx + β1,xlx + β1,x + β2,x. (2)If qˆx is substituted for E(qx|dx), then Bailey (1950) showed that this can berewritten as:qˆx =lxlx + β1,x + β2,xqex +(1−lxlx + β1,x + β2,x)qrx= Zqex + (1− Z)qrx, (3)where the credibility factor Z = lx/(lx + β1,x + β2,x).Whilst this technique gives an intuitively attractive way of combining theresults of experience and risk rating, it has a serious drawback. This is thatthe information on the underlying mortality rate at age x that can be gleanedfrom the mortality experience from ages either side of age x is ignored. This isimportant, as even the most robust risk rating approach can overlook importantunderlying factors that are important to a particular group of lives, as discussedabove. These factors can only be detected through experience rating. The solu-tion is to augment the credibility approach such that information on mortalityfor ages x± 1, x± 2 and so on can be used to supplement the information avail-able on mortality at age x, thus reducing the reliance on risk rating. In thispaper, I therefore propose an approach to make better use of the informationavailable from the full mortality experience.2 Assumptions and NotationThe objective is to arrive at a single initial mortality rate that can be applied toa group of lives aged x that adequately reflects their average level of mortalityEach individual is assumed to be of some homogeneous mortality group gwhere g = 1, 2, . . . , G, as described by Richards (2008). Each group refers toa combination of markers, such as whether the individual is a smoker or not,or the socio-economic group to which the individual belongs. The number ofdeaths at age x for group g is therefore denoted dx,g, with the correspondingnumber of lives being lx,g and the initial mortality rate derived from experiencetherefore being:4qex,g =dx,glx,g. (4)The above information relates to the estimate derived from experience rating.For the risk rating estimate, the mortality rate for individuals of group g andage x in the more general population – or as calculated from some other externalsource – is qrx,g.As implied above, it is assumed that the mortality experience is calculatedover a period of one calendar year to give an annual rate. For a longer periodof investigation, rates would need to be scaled accordingly.3 ApproachThe philosophy behind the approach proposed here is that the underlying pat-tern of mortality rates combined with the rates of mortality at a range of agescan provide information on the rate of mortality at a particular age.The simplest approach is to assume that, for older ages, the natural loga-rithm of mortality rates is approximately linear so, ignoring all subscripts apartfrom x:ln qx = a+ bx, (5)where a and b are constants. It is not essential that this relationship holdsfor the full range of ages considered, only that it is approximately true for thea range of a few years around the age being analysed. This means, therefore,that to remain valid, this approach can be used only for a range of a few yearseither side of the age for which mortality rates are being calculated.The relationship in Equation 5 can then be used, with some adjustment, toartificially increase the number of deaths observed in each group, g, as follows.First, choose the number of years either side of the age for which the mortalityrate is being estimated that will be used to give additional information. Letthis bandwidth be h years. Then, fit Equation 5 to the ages x− h to x+ h.There are a number of ways in which Equation 5 can be fitted. The moststraightforward is to use ordinary least squares regression to find a and b, sothat the estimated mortality rate is given directly as ea+bx. However, such anapproach ignores important information, in particular the size of the populationat each age. For consistency with the binomial model used in the calculationof credibility, it is possible to use a binomial maximum likelihood approach tofit Equation 5. This involves first defining the probability that the number ofdeaths will be as observed for a particular age x as:Pr (Dx,g = dx,g) =lx,g!dx,g!(lx,g − dx,g)!(q˜ex,g)dx,g (1− q˜ex,g)lx,g−dx,g , (6)where Dx,g is the random number of deaths for age x and group g, andq˜ex,g is the regression estimate of qex,g. In the above example the calculation of5q˜ex,g is trivial. However, it becomes more interesting if a likelihood function isconstructed from probabilities spanning ages from x−h to x+h, and if ln q˜ex,g isa linear function of age. Adding the sub- and superscripts back to Equation 5,the linear relationship estimated can be defined as:ln q˜ex+h,g = a+ b(x+ h). (7)This can be rearranged to give q˜ex+h,g in terms of the other variables asfollows:q˜ex+h,g = ea+b(x+h). (8)The parameters a and b are then chosen to maximise the likelihood function,L:L =h∏i=−hlx+i,g!dx+i,g!(lx+i,g − dx+i,g)!(q˜ex+i,g)dx+i,g(1− q˜ex+i,g)lx+i,g−dx+i,g . (9)A more common alternative is Poissonmaximum likelihood estimation based.The standard approach is described by Brouhns et al (2002) and others. Underthis approach, the probability that the number of deaths will be as observed fora particular age x is:Pr (Dx,g = dx,g) =e−λx,g (λx,g)dx,gdx,g!, (10)where λx,g = q˜ex,glx,g. This means that the likelihood function to be max-imised is:L =h∏i=−he−λx+i,g(λx+i,g)dx+i,gdx+i,g!. (11)Whatever approach is used, the result is a series of regression estimates,q˜ex+h,g. This means that the number of deaths that would have been observedin group g at age x from lives actually aged x+ h had those lives actually beenaged x can be calculated as:dx+h,glx+h,g q˜ex+h,glx+h,g q˜ex,g =q˜ex,gq˜ex+h,gdx+h,g. (12)At this stage, it is possible simply to aggregate the number of lives and thenumber of deaths. However, this gives an equal weight to the experience at eachage x + h. Since a stable population will have experience data that decreaseswith age, and since the relevance of mortality data will decrease the furtheraway the analysis moves from the age of interest, it makes sense to weigh theinformation somehow. One approach is to use a weighting system based on akernel function.600.20.40.60.8−3−2−1 0 1 2 3uk|h|Uniform1.000.20.40.60.8−3−2−1 0 1 2 3uk|h|Triangular1.000.20.40.60.8−3−2−1 0 1 2 3uk|h|Epanechnikov1.000.20.40.60.8−3−2−1 0 1 2 3uk|h|Normal1.0Figure 1: Various Kernel Functions – Unit BandwidthA kernel function is a function that weights observations in relation to theirdistance from the point of interest. These functions are typically used to smoothdata. The area under a kernel function is equal to one, but an adjustment isneeded when a kernel is applied to discrete data. In particular, this means thatif a bandwidth of h is chosen and the kernel weight is given by k|h|, the kernelis scaled such that∑hi=−h k|i| = 1.However, following this approach would do nothing to increase the effectivenumber of lives used. Instead, the kernel weights should be scaled such thatk0 = 1. This means that the deaths – and lives – at age x for group g areincluded at their full weight, whilst the contribution of the deaths – and lives– for ages x ± h are given by k|h|, which can be anything from zero to onedepending on the form of kernel used. The total scaled number of deaths forage x and group g given a bandwidth of h, dx(h),g can be defined as:dx(h),g =h∑i=−hq˜ex,gq˜ex+i,gdx+i,gk|i|. (13)Similarly, the number of lives can be defined as:7lx(h),g =h∑i=−hlx+i,gk|i|. (14)This means that for each group, g, the expected initial mortality rate condi-tional on the number of deaths experienced and a bandwidth of h can be derivedfrom the beta-binomial credibility formula in Equation 2 as:E(qx,g|dx(h),g) =dx(h),g + β1,x,glx(h),g + β1,x,g + β2,x,g, (15)where β1,x,g and β2,x,g are the parameters from the beta distribution forgroup g.It is worth discussing further the derivation of β1,x,g and β2,x,g. Theseparameters define the distribution of the underlying mortality rate for eachgroup g at each age x. This mortality rate might be calculated from, say, thepopulation mortality rate for a particular socio-economic group at each age.The mean population mortality rate for each group g and age x has alreadybeen defined as qrx,g. If the lives at each age in each group are assumed to behomogeneous, then the variance the underlying mortality rate can be calculatedas qrx,g(1 − qrx,g)/Lx,g, where Lx,g is the number of lives aged x from group gin the whole population, adjusted for entrants and exits (probably immigrationand emigration in this case) as appropriate.If the underlying rate of mortality is assumed to have beta distribution, thenmean and variance of these rates can be used to calculate β1,x,g and β2,x,g ateach age and for each group using the method of moments.One drawback of the by-group approach is that it requires separate mortalityrates to be calculated for each group g. This means that the overall level ofcredibility is reduced, particularly at older ages where data is limited. However,a review by Guilley et al (2010) concludes that for extreme old ages, where thedata is most sparse, the difference in mortality between socio-economic groupsis significantly reduced as genetic factors become more important. This meansthat a single group can be used.Another potential issue is the choice of bandwidth. As alluded to above,a large bandwidth would stretch the assumption that mortality rates are log-linear with respect to age. However, using this approach with a bandwidth ofonly three years either side of the age of interest can give result in almost fourtimes the information on deaths if an Epanechnikov kernel form is used.The fact that a bandwidth must be chosen is also an issue at the upper agerange. However, this can be dealt with by reducing the bandwidth as the upperage limit is approached, so that at the highest age only information from thatage is used. The practical impact of this approach is that an increasing weightis given to the risk rating mortality estimate, which in the absence of any betterdata seems appropriate.84 ConclusionCredibility is an important way of combining the results of experience and riskrating. However, it is important that as much information as possible is used inthe calculation of the mortality rates, including information from rates eitherside of the rate under analysis.This technique can be particularly helpful at older ages where the numberof lives may be small. Indeed, where the number of lives varies significantlyfrom age to age, as is often the case for very high ages, this approach can allowexperience data to provide a useful contribution at an age where the actualexperience is limited, providing there is greater experience at surrounding ages.The proposed approach does have limitations. In particular, the division ofthe mortality experience into homogeneous groups reduces the degree of credi-bility, although the absence of grouping at older ages limits the impact of thisproblem. The choice of bandwidth must also be made carefully to ensure thatthe increase in useful data does not mean that log-linearity ceases to be a realisticapproximation. However, this approach does offer the prospect that experiencedata might be able to play a bigger role in the mortality rates assumed.ReferencesBailey, A.L.: Credibility Procedures: LaPlace’s Generalization of Bayes’ Ruleand the Combination of Collateral Knowledge with Observed Data, Proceed-ings of the Casualty Actuarial Society, 37, (1950), 7-23.Brouhns, N., Denuit, M. and Vermunt, J.K.: A Poisson Log-bilinearRegression Approach to the Construction of Projected Life Tables, Insurance:Mathematics and Economics, 31(3), (2002), 373-393.Guilley, E., Bopp, M., Faeh, D. and Paccaud, F.: Socioeconomic Gradi-ents in Mortality in the Oldest Old: a Review, Archives of Gerontology andGeriatrics, 51, (2010), e37-e40.Hardy, M.R. and Panjer, H.H.: A Credibility Approach to Mortality Risk,ASTIN Bulletin, 28(2), (1998), 269-283.Herzog, T.N.: Introduction to Credibility Theory (3rd ed.), (1999), ACTEXPublications, Winsted.Mayerson, A,L.: A Bayesian View of Credibility, Proceedings of the CasualtyActuarial Society, 51, (1964), 85-104.Nielson, J.P. and Sandqvist, B.L.: Credibility Weighted Hazard Estima-tion, ASTIN Bulletin, 30(2), (2000), 405-417.Richards, S.C.: Applying Survival Models to Pensioner Mortality Data,British Actuarial Journal, 14(2), (2008), 257-303.9

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