Aggregate Loss Modeling modelsforboththenumberofclaimspertimeperiod and the number of claims per insured. These models can then be used to compute the distribution of One of the primary goals of actuarial risk theory is aggregatelossesgivenbyequation (1)foraportfolio the evaluation of the risk associated with a portfo- of insurance risks. lio of insurance contracts over the life of the con- It should be noted that the analysis of claim tracts. Many insurance contracts (in both life and numbers depends, in part, on what is considered non-life areas) are short-term.Typically,automobile to be a claim. Since many insurance policies have insurance, homeowner’s insurance, group life and deductibles, which means that small losses to the health insurance policies are of one-year duration. insured are paid entirely by the insured and result in One of the primary objectives of risk theory is no payment by the insurer, the term ‘claim’ usually to model the distribution of total claim costs for only refers to those events that result in a payment portfolios of policies, so that business decisions can by the insurer. be made regarding various aspects of the insurance The computation of the aggregate claim distribu- contracts. The total claim cost over a fixed time tion can be rather complicated. Equation (1) indi- periodisoftenmodeledbyconsideringthefrequency cates that a direct approach requires calculating of claims and the sizes of the individual claims the n-fold convolutions. As an alternative, simula- separately. tion and numerous approximate methods have been LetX ,X ,X ,...beindependentandidentically developed. 1 2 3 distributed random variables with common distribu- Approximate distributions based on the first few tion function F (x). Let N denote the number of lowermomentsisoneapproach;forexample,gamma X claimsoccurring ina fixedtime period.Assumethat distribution. Several methods for approximating spe- thedistributionofeachX,i =1,...,N,isindepen- cific values of the distribution function have been i dent of N for fixed N. Then the total claim cost for developed; for example, normal power approxima- the fixed time period can be written as tion, Edgeworth series, and Wilson–Hilferty trans- form. Other numerical techniques, such as the Fast S =X1+X2+···+XN, Fourier transform, have been developed and pro- moted. Finally, specific recursive algorithms have with distribution function been developed for certain choices of the distribu- (cid:1)∞ tion of the number of claims per unit time period FS(x)= pnFX∗n(x), (1) (the ‘frequency’ distribution). n=0 Details of many approximate methods are given where F∗n(·) indicates the n-fold convolution of in [1]. X F (·). X Reference Thedistributionoftherandomsumgivenbyequa- tion (1) is the direct quantity of interest to actuaries [1] Beard,R.E., Pentikainen, T.& Pesonen, E.(1984). Risk for the development of premium rates and safety Theory,3rdEdition,Chapman&Hall,London. margins.Ingeneral,theinsurerhashistoricaldataon thenumberofevents(insuranceclaims)perunittime (Seealso Beekman’sConvolutionFormula;Claim period (typically one year) for a specific risk, such Size Processes; Collective Risk Theory; Com- as a given driver/car combination (e.g. a 21-year-old pound Distributions; Compound Poisson Fre- maleinsuredinaPorsche911).Analysisofthisdata quency Models; Continuous Parametric Distribu- generally reveals very minor changes over time. The tions; Discrete Parametric Distributions; Discrete insureralsogathersdataontheseverityoflossesper Parametric Distributions; Discretization of Dis- event (the X’s in equation (1)). The severity varies tributions; Estimation; Heckman–Meyers Algo- overtimeasaresultofchangingcostsofautomobile rithm; Reliability Classifications; Ruin Theory; repairs,hospitalcosts,andothercostsassociatedwith Severity of Ruin; Sundt and Jewell Class of Dis- losses. tributions; Sundt’s Classes of Distributions) Data is gathered for the entire insurance industry in many countries. This data can be used to develop HARRY H. PANJER Approximating the There are two main types of approximation. The first matches the moments of the aggregate claims Aggregate Claims to the moments of a given distribution (for exam- ple, normal or gamma), and then use probabili- Distribution ties from the approximating distribution as estimates of probabilities for the underlying aggregate claims distribution. Introduction Thesecondtypeofapproximationassumesagiven The aggregate claims distribution in risk theory is distributionforsometransformationoftheaggregate the distribution of the compound random variable claims random variable. (cid:1) We will use several compound Poisson–Pareto S = (cid:2) 0 N =0 (1) distributions to illustrate the application of the Nj=1Yj N ≥1, approximations. The Poisson distribution is defined by the parameter λ=E[N]. The Pareto distribution where N represents the claim frequency random has a density function, mean, and kth moment variable,andY istheamount(orseverity)ofthejth j about zero claim. We generally assume that N is independent of {Y }, and that the claim amounts Y >0 are αβα β j j f (y)= ; E[Y]= ; independent and identically distributed. Y (β+y)α+1 α−1 Typical claim frequency distributions would be k!βk Poisson, negative binomial or binomial, or modified E[Yk]= for α >k. versions of these. These distributions comprise the (α−1)(α−2)...(α−k) (a,b,k) class of distributions, described more in (2) detail in [12]. The α parameter of the Pareto distribution deter- Except for a few special cases, the distribution mines the shape, with small values corresponding to function of the aggregate claims random variable is a fatterrighttail. The kth moment of the distribution not tractable. It is, therefore, often valuable to have exists only if α >k. reasonably straightforward methods for approximat- The four random variables used to illustrate the ing the probabilities. In this paper, we present some aggregateclaimsapproximationsmethodsaredescri- of the methods that may be useful in practice. bed below. We give the Poisson and Pareto param- We do not discuss the recursive calculation of the eters for each, as well as the mean µ , variance distribution function, as that is covered elsewhere. S σ2, and γ , the coefficient of skewness of S, that However, it is useful to note that, where the claim S S is,E[(S−E[S])3]/σ3],forthecompounddistribution amount random variable distribution is continuous, S for each of the four examples. the recursive approach is also an approximation in so far as the continuous distribution is approximated using a discrete distribution. Example1 λ=5,α =4.0,β =30;µS =50;σS2 = Approximating the aggregate claim distribution 1500, γS =2.32379. using the methods discussed in this article was crucially important historically when more accu- Example 2 λ=5, α =40.0, β =390.0; µS =50; rate methods such as recursions or fast Fourier σS2 =1026.32, γS =0.98706. transforms were computationally infeasible. Today, approximation methods are still useful where full Example 3 λ=50, α =4.0, β =3.0; µS =50; individual claim frequency or severity information σS2 =150, γS =0.734847. is not available; using only two or three moments of the aggregate distribution, it is possible to apply Example 4 λ=50, α =40.0, β =39.0; µS =50; most of the methods described in this article. The σS2 =102.63, γS =0.31214. methods we discuss may also be used to pro- vide quick, relatively straightforward methods for Thedensity functionsof thesefour randomvariables estimating aggregate claims probabilities and as a were estimated using Panjer recursions [13], and are check on more accurate approaches. illustrated in Figure 1. Note a significant probability 2 Approximating the Aggregate Claims Distribution 5 0 0. Example 1 4 Example 2 n 0 o 0. cti Example 3 n u on f 03 Example 4 uti 0. b stri di 2 y 0 bilit 0. a b o Pr 01 0. 0 0. 0 50 100 150 200 Aggregate claims amount Figure1 Probabilitydensityfunctionsforfourexamplerandomvariables mass at s =0 for the first two cases, for which the sum is itself random. The theorem still applies, and probability of no claims is e−5 =0.00674.The other the approximation can be used if the expected num- interestingfeatureforthepurposeofapproximationis ber of claims is sufficiently large. For the example thatthechangeintheclaimfrequencydistributionhas random variables, we expect a poor approximation a much bigger impact on the shape of the aggregate for Examples 1 and 2, where the expected number claims distribution than changing the claim severity of claims is only 5, and a better approximation for distribution. Examples 3 and 4, where the expected number of When estimating the aggregate claim distribution, claims is 50. we are often most interested in the right tail of the In Figure 2, we show the fit of the normal distri- loss distribution – that is, the probability of very butiontothefourexampledistributions.Asexpected, large aggregate claims. This part of the distribution the approximation is very poor for low values of has a significant effect on solvency risk and is key, E[N],butlooksbetterforthehighervaluesofE[N]. for example, for stop-loss and other reinsurance However, the far right tail fit is poor even for these calculations. cases. In Table 1, we show the estimated value that aggregateclaimsexceed themean plus fourstandard The Normal Approximation (NA) Table1 Comparison of true and estimated right tail Using the normal approximation, we estimate the (4 standard deviation) probabilities; Pr[S >µ +4σ ] S S distribution of aggregate claims with the normal dis- usingthenormalapproximation tribution having the same mean and variance. True Normal Thisapproximationcanbejustifiedbythecentral Example probability approximation limittheorem,sincethesumofindependentrandom variables tends to a normal random variable, as the Example 1 0.00549 0.00003 Example 2 0.00210 0.00003 number in the sum increases. For aggregate claims, Example 3 0.00157 0.00003 we are summing a random number of independent Example 4 0.00029 0.00003 individual claim amounts, so that the number in the Approximating the Aggregate Claims Distribution 3 5 1 0.0 Actual pdf Actual pdf Estimated pdf 15 Estimated pdf 0 0. 5 0 0 0. 5 0 0 0. 0 0 0. 0. 0 50 100 150 200 0 50 100 150 200 Example 1 Example 2 5 5 0.0 Actual pdf 0.0 Actual pdf Estimated pdf Estimated pdf 3 3 0 0 0. 0. 1 1 0 0 0. 0. 0 0 0. 0. 0 20 40 60 80 100 0 20 40 60 80 100 Example 3 Example 4 Figure2 Probabilitydensityfunctionsforfourexamplerandomvariables;trueandnormalapproximation deviations, and compare this with the ‘true’ value – The translated gamma distribution has an identical that is, the value using Panjer recursions (this is shape, but is assumed to be shifted by some amount actually an estimate because we have discretized the k, so that Pareto distribution). The normal distribution is sub- stantially thinner tailed than the aggregate claims (x−k)a−1e−(x−k)/θ distributions in the tail. f(x)= x,a,θ >0. (4) θa(cid:7)(a) The Translated Gamma Approximation So, the translated gamma distribution has three para- The normal distribution has zero skewness, where meters,(k,a,θ)andwefitthedistributionbymatch- most aggregate claims distributions have positive ing the first three moments of the translated gamma skewness. The compound Poisson and compound distribution to the first three moments of the aggre- negativebinomialdistributionsarepositivelyskewed gate claims distribution. for any claim severity distribution. It seems natural Themomentsofthetranslatedgammadistribution thereforetouseadistributionwithpositiveskewness given by equation (4) are as an approximation. The gamma distribution was proposedin [2]andin [8],andthetranslatedgamma Mean=aθ +k distributionin [11].Afullerdiscussion,withworked examples is given in [5]. Variance>aθ2 The gamma distribution has two parameters 2 and a density function (using parameterization as Coefficient of Skewness> √ . (5) a in [12]) xa−1e−x/θ This gives parameters for the translated gamma f(x)= x,a,θ >0. (3) θa(cid:7)(a) distribution as follows for the four example 4 Approximating the Aggregate Claims Distribution distributions: Table2 Comparison of true and estimated right tail probabilities Pr[S >E[S]+4σ ] using the translated S a θ k gammaapproximation Example 1 0.74074 45.00 16.67 Example 2 4.10562 15.81 −14.91 Translated True gamma Example 3 7.40741 4.50 16.67 Example probability approximation Example 4 41.05620 1.58 −14.91 Example 1 0.00549 0.00808 The fit of the translated gamma distributions for the Example 2 0.00210 0.00224 four examples is illustrated in Figure 3; it appears Example 3 0.00157 0.00132 Example 4 0.00029 0.00030 thatthe fitis notvery goodfor the firstexample,but looks much better for the other three. Even for the first though, the right tail fit is not bad, especially provided that the area of the distribution of interest when compared to the normal approximation. If we is not the left tail. reconsiderthefourstandarddeviationtailprobability from Table 1, we find the translated gamma approx- imation gives much better results. The numbers are Bowers Gamma Approximation given in Table 2. So, given three moments of the aggregate claim Notingthatthegammadistributionwasagoodstart- distributionwecanfitatranslatedgammadistribution ingpointforestimatingaggregateclaimprobabilities, to estimate aggregate claim probabilities;the left tail Bowers in [4] describes a method using orthogonal fit may be poor (as in Example 1), and the method polynomials to estimate the distribution function of may give some probability for negative claims (as in aggregate claims. This method differs from the pre- Example 2), but the right tail fit is very substantially vious two, in that we are not fitting a distribution to better than the normal approximation. This is a very the aggregate claims data, but rather using a func- easy approximation to use in practical situations, tional form to estimate the distribution function. The 0 0 3 3 0.0 Actual pdf 0.0 Actual pdf Estimated pdf Estimated pdf 0 0 2 2 0 0 0. 0. 0 0 1 1 0 0 0. 0. 0 0 0. 0. 0 50 100 150 200 0 50 100 150 200 Example 1 Example 2 5 5 0.0 Actual pdf 0.0 Actual pdf Estimated pdf Estimated pdf 3 3 0 0 0. 0. 1 1 0 0 0. 0. 0 0 0. 0. 0 20 40 60 80 100 0 20 40 60 80 100 Example 3 Example 4 Figure3 Actualandtranslatedgammaestimatedprobabilitydensityfunctionsforfourexamplerandomvariables Approximating the Aggregate Claims Distribution 5 first term in the Bowers formula is a gamma dis- the probability function for X would simply be tribution function, and so the method is similar to F (x;α). The subsequent terms adjust this to match G fitting a gamma distribution, without translation, to the third, fourth, and fifth moments. the moments of the data, but the subsequent terms Aslightlymoreconvenientformoftheformula is adjust this to allow for matching higher moments of F(x)≈F (x;α)(1−A+B−C)+F (x;α+1) the distribution. In the formula given in [4], the first G G fivemomentsoftheaggregateclaimsdistributionare ×(3A−4B+5C)+F (x;α+2) G used; it is relatively straightforward to extend this to ×(−3A+6B−10C)+F (x;α+3) evenhighermoments,thoughitisnotclearthatmuch G benefit would accrue. ×(A−4B+10C)+F (x;α+4) G Bowers’ formula is applied to a standardized ran- dom variable, X =βS where β =E[S]/Var[S]. The ×(B−5C)+FG(x;α+5)(C). (8) mean and variance of X then are both equal to Wecannotapplythismethodtoallfourexamples,as E[S]2/Var[S].Ifwefitagamma(α,θ)distributionto the kth moment of the Compound Poisson distribu- the transformed random variable X, then α =E[X] tion exists only if the kth moment of the secondary and θ =1. distribution exists. The fourth and higher moments Let µ denote the kth central moment of X; note k of the Pareto distributions with α =4 do not exist; that α =µ =µ . We use the following constants: 1 2 it is necessary for α to be greater than k for the kth µ −2α momentoftheParetodistributiontoexist.Sowehave A= 3 appliedtheapproximationtoExamples 2and 4 only. 3! Theresults areshowngraphicallyin Figure 4;we µ −12µ −3α2+18α B= 4 3 alsogivethefourstandarddeviationtailprobabilities 4! in Table 3. µ −20µ −(10α−120)µ +6−α2−144α Usingthismethod,weconstrainthedensitytopos- C= 5 4 3 . 5! itive values only for the aggregate claims. Although (6) this seems realistic, the result is a poor fit in the leftsidecomparedwiththetranslatedgammaforthe Then the distribution function for X is estimated as more skewed distribution of Example 2. However, follows,whereFG(x;α)representstheGamma(α,1) the right side fit is very similar, showing a good tail distribution function – and is also the incomplete approximation. For Example 4 the fit appears better, gammagammafunctionevaluatedatx withparam- and similar in right tail accuracy to the translated eter α. gammaapproximation.However,theuseoftwoaddi- (cid:3) xα tional moments of the data seems a high price for FX(x)≈FG(x;α)−Ae−x (cid:7)(α+1) little benefit. (cid:4) (cid:3) 2xα+1 xα+2 xα − + +Be−x Normal Power Approximation (cid:7)(α+2) (cid:7)(α+3) (cid:7)(α+1) (cid:4) The normal distribution generally and unsurprisingly 3xα+1 3xα+2 xα+3 − + − offersapoorfittoskeweddistributions.Onemethod (cid:7)(α+2) (cid:7)(α+3) (cid:7)(α+4) (cid:3) xα 4xα+1 6xα+2 Table3 Comparison of true and estimated right tail +Ce−x − + probabilities Pr[S>E[S]+4σ ] using Bower’s gamma (cid:7)(α+1) (cid:7)(α+2) (cid:7)(α+3) S approximation (cid:4) 4xα+3 xα+4 − + . (7) True Bowers (cid:7)(α+4) (cid:7)(α+5) Example probability approximation Obviously, to convert back to the original claim Example 1 0.00549 n/a distribution S =X/β, we have F (s)=F (βs). Example 2 0.00210 0.00190 S X Example 3 0.00157 n/a Ifweweretoignorethethirdandhighermoments, Example 4 0.00029 0.00030 andfitagammadistributiontothefirsttwomoments, 6 Approximating the Aggregate Claims Distribution 0.020 Actual pdf 0.05 Actual pdf Estimated pdf Estimated pdf 4 15 0.0 0 0. 3 0 0. 0 1 0 0. 2 0 0. 5 0 0.0 01 0. 0 0 0. 0. 0 50 100 150 200 0 20 40 60 80 100 Example 2 Example 4 Figure4 ActualandBower’sapproximationestimatedprobabilitydensityfunctionsforExample2andExample4 of improving the fit whilst retaining the use of the Note that the density function is notdefined at all normal distribution function is to apply the normal parts of the distribution. The normal power approx- distribution to a transformation of the original ran- imation does not approximate the aggregate claims domvariable,wherethetransformationisdesignedto distribution with another, it approximates the aggre- reduce the skewness. In [3] it is shown how this can gate claims distribution function for some values, betakenfromtheEdgeworthexpansionofthedistri- specifically where the square root term is positive. bution function. Let µ , σ2 and γ denote the mean, Thelackofafulldistributionmaybeadisadvantage S S S variance,andcoefficientofskewnessoftheaggregate for some analytical work, or where the left side of claims distribution respectively. Let (cid:8)() denote the thedistributionisimportant –forexample,insetting standard normal distribution function. Then the nor- deductibles. malpowerapproximationtothedistributionfunction is given by Haldane’s Method (cid:5)(cid:6)(cid:7) (cid:8) (cid:9) F (x)≈(cid:8) 9 +1+ 6(x−µS) − 3 . (9) Haldane’s approximation [10] uses a similar theo- S γ2 γ σ γ retical approach to the normal power method – that S S S S is, applying the normal distribution to a transforma- provided the term in the square root is positive. tion of the original random variable. The method is In [6]itisclaimedthatthenormalpowerapproxi- described more fully in [14], from which the follow- mationdoesnotworkwherethecoeffientofskewness ing description is taken. γ >1.0, but this is a qualitative distinction rather The transformation is S (cid:7) (cid:8) thanatheoreticalproblem –theapproximationisnot S h very good for very highly skewed distributions. Y = , (10) µ In Figure 5, we show the normal power density S function for the four example distributions. where h is chosen to give approximately zero skew- The right tail four standard deviation estimated ness for the random variable Y (see [14] for probabilities are given in Table 4. details). Approximating the Aggregate Claims Distribution 7 Actual pdf Actual pdf 5 Estimated pdf 5 Estimated pdf 1 1 0 0 0. 0. 5 5 0 0 0 0 0. 0. 0 0 0. 0. 0 50 100 150 200 0 50 100 150 200 Example 1 Example 2 5 5 0.0 Actual pdf 0.0 Actual pdf Estimated pdf Estimated pdf 3 3 0 0 0. 0. 1 1 0 0 0. 0. 0 0 0. 0. 0 20 40 60 80 100 0 20 40 60 80 100 Example 3 Example 4 Figure 5 Actual and normal power approximation estimated probability density functions for four example random variables Table4 Comparisonoftrueandestimatedright and variance tail probabilities Pr[S >E[S]+4σ ] using the S normalpowerapproximation (cid:7) (cid:8) σ2 σ2 True Normalpower sY2 = µS2h2 1− 2µS2(1−h)(1−3h) . (12) Example probability approximation S S Example 1 0.00549 0.01034 Example 2 0.00210 0.00226 We assume Y is approximately normally distributed, Example 3 0.00157 0.00130 so that Example 4 0.00029 0.00029 (cid:7) (cid:8) (x/µ )h−m F (x)≈(cid:8) S Y . (13) S s Y Usingµ ,σ ,andγ todenotethemean,standard S S S deviation, and coefficient of skewness of S, we have For the compound Poisson–Pareto distributions the γ µ parameter h=1− S S, 3σ S γ µ (α−2) (α−4) h=1− S S =1− = (14) and the resulting random variable Y =(S/µ )h has 3σ 2(α−3) 2(α−3) S S mean (cid:7) (cid:8) andsodependsonlyontheα parameterofthePareto σ2 σ2 m =1− S h(1−h) 1− S (2−h)(1−3h) , distribution. In fact, the Poisson parameter is not Y 2µ2 4µ2 involved in the calculation of h for any compound S S (11) Poisson distribution. For examples 1 and 3, α =4, 8 Approximating the Aggregate Claims Distribution giving h=0. In this case, we use the limiting equa- rather than the five used in the Bowers method, the tion for the approximate distribution function results here are in factmore accurate for Example 2,  (cid:7) (cid:8)  andhavesimilaraccuracyforExample 4.Wealsoget log x + σs2 − σS4  betterapproximationthananyoftheotherapproaches FS(x)≈(cid:8) µS (cid:6) 2µ2S 4µ4S . (15) forAEnxoatmheprleHsa1ldaannde3tr.ansformationmethod,employ-  σ σ2  S 1− S ing higher moments, is also described in [14]. µ 2µ2 S S TheseapproximationsareillustratedinFigure 6,and Wilson–Hilferty the righttailfourstandard deviationprobabilitiesare given in Table 5. The Wilson–Hilferty method, from [15], and des- Thismethodappearstoworkwellintherighttail, cribed in some detail in [14], is a simplified version comparedwiththeotherapproximations,evenforthe ofHaldane’smethod,wherethehparameterissetat first example. Although it only uses three moments, 1/3. In this case, we assume the transformed random variable as follows, where µ , σ and γ are the S S S Table5 Comparisonoftrueandestimatedright mean,standarddeviation,andcoefficientofskewness tail probabilities Pr[S >E[S]+4σ ] using the S of the original distribution, as before Haldaneapproximation (cid:7) (cid:8) True Haldane S−µ 1/3 Y =c +c S , (16) Example probability approximation 1 2 σ +c S 3 Example 1 0.00549 0.00620 Example 2 0.00210 0.00217 where Example 3 0.00157 0.00158 γ 6 Example 4 0.00029 0.00029 c = S − 1 6 γ S Actual pdf Actual pdf 5 5 Estimated pdf 1 Estimated pdf 1 0 0 0. 0. 5 5 0 0 0 0 0. 0. 0 0 0. 0. 0 50 100 150 200 0 50 100 150 200 Example 1 Example 2 5 5 0.0 Actual pdf 0.0 Actual pdf Estimated pdf Estimated pdf 3 3 0 0 0. 0. 1 1 0 0 0. 0. 0 0 0. 0. 0 20 40 60 80 100 0 20 40 60 80 100 Example 3 Example 4 Figure6 ActualandHaldaneapproximationprobabilitydensityfunctionsforfourexamplerandomvariables