Research Article When Inflation Causes No Increase in Claim Amounts

Probability an Statistics Volume 2009, Article ID 943926, 10 pages oi:10.1155/2009/943926 Research Article When Inflation Causes No Increase in Claim Amounts Vytaras Brazauskas, 1 Bruce L. Jones, 2 an Ričaras Zitikis 2 1 Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA 2 Department of Statistical an Actuarial Sciences, University of Western Ontario, Lonon, ON, Canaa N6A 5B7 Corresponence shoul be aresse to Bruce L. Jones, jones@stats.uwo.ca Receive 27 January 2009; Accepte 26 June 2009 Recommene by Tomasz J. Kozubowski It is well known that when re insurance coverages involve a euctible, the impact of inflation of loss amounts is istorte, an the changes in claims pai by the re insurer cannot be assume to reflect the rate of inflation. A particularly interesting phenomenon occurs when losses follow a Pareto istribution. In this case, the observe loss amounts those that excee the euctible are ientically istribute from year to year even in the presence of inflation. Nevertheless, in this paper we succee in estimating the inflation rate from the observations. We evelop appropriate statistical inferential methos to quantify the inflation rate an illustrate them using simulate ata. Our solution hinges on the recognition that the istribution of the number of observe losses changes from year to year epening on the inflation rate. Copyright q 2009 Vytaras Brazauskas et al. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. 1. Introuction A number of challenges arise when an insurance policy covers only loss amounts that excee a threshol known as the euctible. The insurer typically oes not know about losses that are less than this amount, making appropriate characterization of the loss istribution impossible. This can even give rise to misleaing an/or paraoxical observations about the istribution. An interesting example of this has been observe in actuarial practice. A reinsurer esire to unerstan the impact of inflation on loss amounts. However, upon exploring the losses that were reporte to the reinsurer, it was foun that no inflation was present. The losses reporte to the reinsurer were only those that exceee a fixe euctible, which i not change over time as is typically the case. The losses reporte in ifferent years ha near ientical istributions. Specifically, the reinsurer foun that the istribution of reporte losses in each year coul be accurately escribe by the same Pareto istribution. Moreover,

2 Probability an Statistics attempts to moel inflation by employing various macroeconomic inexes e.g., consumer price inex also faile to yiel satisfactory results as the reinsurance ata was inustry specific. The etails of this problem were obtaine through personal communications with reinsurance inustry practitioners. The Pareto istribution arises quite often in moelling insurance losses. This istribution uniquely possesses a property that gives rise to the reinsurer s observation regaring the inflation of loss amounts. To examine this phenomenon statistically, we simulate losses corresponing to 10 successive years. The numbers of losses in these years is assume to be inepenent Poisson ranom variables with mean 1000, an all loss amounts are inepenent. These are common assumptions in insurance loss moelling. The losses occurring uring the jth year have a Pareto istribution with scale parameter θ 1.05 j 1 an shape parameter α 2. These parameter choices were arbitrary but reflect the phenomenon that has been observe. Throughout the paper, we will use the shorthan Y Pareto θ, α to inicate that a ranom variable Y has the Pareto istribution function F Y ( y ) 1 ( θ y ) α, y > θ, θ,α > 0 1.1 with corresponing probability ensity function ( ) ( ) α θ α f Y y, y > θ, 1.2 y y mean given by an meian given by E Y αθ, α > 1, 1.3 α 1 meian Y 2 1/α θ. 1.4 So, losses uring the jth year are istribute as Pareto 1.05 j 1, 2. We assume that the insurer will pay only the amount of losses that excee 5 an therefore will be unaware of any losses that are less than 5. The simulate ata are summarize in Figure 1. The left-han graph shows box-an-whisker plots of loss amounts in each year. Each box extens from the first quartile to the thir quartile, with the meian inicate by the line insie the box. The whiskers exten to the most extreme observations that are not more than 1.5 times the interquartile range outsie the box. We see very clearly from the left-han graph the impact that inflation has on the loss istribution. The right-han graph in Figure 1 summarizes the istribution of losses in each year that are greater than 5. These box-anwhisker plots o not show any signs of inflation of loss amounts. Table 1 provies some aitional information about the simulate loss ata. The table shows that while the average loss amount increases with inflation, the average observe loss amount oes not appear to increase. We also see that the number of observe losses

Probability an Statistics 3 5 20 Amount 4 3 2 Amount 15 10 1 1 2 3 4 5 6 7 8 9 10 year 5 1 2 3 4 5 6 7 8 9 10 year a All loss amounts b Observe loss amounts Figure 1: Box-an-whisker plots of all loss amounts a an observe loss amounts b. Year Number of losses Average loss Table 1: Summary of simulate loss ata. Number of observe losses Average of observe losses Sum of observe losses 1 1004 1.9813 37 9.8732 365.3071 2 971 2.1358 43 10.6640 458.5501 3 1029 2.1206 44 9.4408 415.3972 4 1063 2.3359 56 9.8994 554.3648 5 1026 2.3554 62 8.2097 509.0030 6 1030 2.5579 78 9.2125 718.5715 7 1003 2.7498 75 10.7216 804.1190 8 955 2.7866 71 10.0545 713.8679 9 982 3.1771 89 12.1130 1078.0582 10 1029 3.0533 92 9.8543 906.5962 tens to increase over time, an this is how the information about inflation is capture. The sum of observe losses also increases over time. However, the increases reflect the socalle leveraging effect of the euctible see 1, page 189 an o not properly represent the increases ue to inflation. This is because, if the euctible is kept unchange, then total observe losses will not increase by the inflation rate because losses that were previously below the euctible may, with inflation, excee the euctible. The rest of the paper is organize as follows. In Section 2, we provie some backgroun information an erive two methos for estimation of the inflation rates. In Section 3, numerical illustrations base on our simulate ata are presente. 2. Estimating Inflation Rates If we were observing every loss, then we woul have a realization of the following array of ranom variables: } {Y j,1,...,y j,nj Pareto ( θ j,α ), j 1,...,J, 2.1

4 Probability an Statistics where J represents the total number of years for which losses are observe, an N j is the number of losses that occur in the year j. All ranom variables in array 2.1 are assume inepenent an, row-wise, have Pareto istributions with the specifie parameters, which are unknown an thus nee to be estimate from available ata. The ata consist of only those losses whose amounts Y j,k excee a specifie threshol, as the insurer is not informe of the losses which are less than this euctible. Hence, our ata set is a realization of the following array: {X j,1,...,x j,mj }, j 1,...,J, 2.2 which is a subarray of 2.1. Obviously, the observe M j o not excee the unobserve N j for every 1 j J. All ranom variables in array 2.2 are inepenent an every X j,k Pareto, α. The latter fact can be seen by noting that the X j,k s are copies of a ranom variable X j,anthey j,k s are copies of a ranom variable Y j.nowx j Y j Y j >. Therefore, for all x, P [ X j >x ] P [ Y j >x Y j > ] P[ Y j >x ] P [ Y j > ] 2.3 ( ) α. x The fact that this istribution oes not epen on j is unique to the Pareto loss istribution an is reflecte in the title of this paper. The property ientifie in the above equations raises the question of how to estimate the rate of inflation given the observe losses X j,k.wenote in passing that this property has been note an utilize in a number of contexts incluing econometrics an engineering sciences see 2, 3. Suppose that the annual inflation rates for the observation perio are represente by r 2,...,r J, where these rates are relate to the Pareto-scale parameters by the equation θ j θ j 1 1 r j. 2.4 Equation 2.4 arises from the very reasonable requirement that if r j is the rate of loss inflation as one goes from year j 1toyearj, then Y j 1 r j Y j 1. Note that if the Pareto istributions have finite first moments i.e., α > 1, then the ratio θ j /θ j 1 in 2.4 can be replace by E Y j /E Y j 1. However, we o not require the finiteness of first moments in this paper. We first present a simple an intuitively appealing approach to estimating the inflation rate when we assume that it is the same in each year. We can also view this as a metho of estimating the average inflation rate uring the observation perio. That is, the inflation rate r is such that θ j θ 1 r j 1, 2.5

Probability an Statistics 5 with θ θ 1. This metho allows us to estimate α an r recognizing that most of the information about α is provie by the X j,k s, an given α, most of the information about r is provie by the M j s. We assume that N 1,...,N J are inepenent Poisson ranom variables, an for each j, N j has mean λ j such that λ j λe j, where e j represents the known number of exposure units in year j an λ is a parameter representing the claim rate per exposure unit. In other wors, the e j values inicate the amount of insurance in force in year j, an it is appropriate that the claim rate is proportional to e j. The assumption that the number of losses has a Poisson istribution is common in actuarial science, though our first metho generalizes easily to mixe Poisson istributions. Now since the number of losses N j has a Poisson istribution with mean λe j,the number of observe losses M j has a Poisson istribution with mean λe j θ j / α.thus, E [ ( ) α ] θj M j λej, log E [ M j ] log λ log ej α log α log θ j 2.6 log λ log e j α log α log θ { α log 1 r }( j 1 ). Therefore, log ( [ ]) E Mj e j log λ α log α log θ { α log 1 r }( j 1 ). 2.7 Notice that the right-han sie of 2.7 is a linear function of j with the slope α log 1 r. We coul therefore estimate r by first estimating α by maximum likelihoo using the conitional likelihoo of the X j,k s, an then fit a linear function to the points j, log m j /e j, j 1,...,J, by orinary least squares an estimate r using the estimate of the slope along with the MLE of α. This gives α m j mj log( x j,k / ), 12 r exp j log( ) m j /e j 6 J 1 log( ) m j /e j αj J 2 1 1, 2.8 2.9 where x j,k is the realize value of X j,k,anm j is the realize value of M j. This approach allows us to estimate r without estimating the parameters λ an θ, which we consier nuisance parameters in our problem.

6 Probability an Statistics A more general approach involves estimating the parameters α an r j, j 1,...,J,by maximum likelihoo estimation using the full likelihoo function. That is, L [ ( J λej θj / ) α] mj { ( exp λej θj / ) α} m j! m j! ( mj ( ) ) f Xj xj,k [ ( J λej θj / ) α] mj { ( exp λej θj / ) α} ( mj α x j,k ( x j,k ) α ). 2.10 Note that we have an ientifiability problem because λ coul be replace by λ cλ an θ j by θ j θ j/c 1/α, an the likelihoo is unchange. So, while we can etermine estimates of λ an θ 1,...,θ J that maximize the likelihoo, these estimates are not unique. However, this is not a concern because we are not intereste in λ, an we are intereste in θ 1,...,θ J only to the extent that they tell us the year-to-year inflation rates. We procee with this in min. By cancelling multiplicative constants in the likelihoo function an taking logs, we have l [ ( J θj m j log λ αm j log θ j αm j log λe j ( ) ] log α α log α log xj,k. m j ) α 2.11 Differentiating with respect to θ j, we have l αm j αλejθα 1 j θ j θ j α. 2.12 Therefore, ( ) α m j λe θj j 0, 2.13 ) 1/ α mj θ j (. 2.14 λe j This allows us to obtain the MLE of the inflation rate in year j, r j θ j /θ j 1 1, j 2,...,J. That is, ( ) 1/ α mj e j 1 r j 1. 2.15 m j 1 e j

Probability an Statistics 7 Differentiating the log-likelihoo with respect to α, we have [ l J α m j log ( θj ) λe j log ( )( θj θj ) α m j ( ) ] 1 α log log x j,k. 2.16 Replacing the parameters in 2.16 by their MLE s an using 2.14, we have m J j ( ) 1 α log log x j,k 0, 2.17 which leas to 2.8, the same estimate we obtaine using the first metho. The latter approach oes not assume any structure between r 2,...,r J. However, as we i earlier, it might be reasonable to assume that r 2 r J, in which case we enote the inflation rate by r. Hence, as before, θ j θ 1 r j 1,withθ θ 1. In this case, we have only four unknown parameters, λ, α, θ,anr, an the log-likelihoo function is l [ ( J m j log λ αm j log {θ 1 r j 1} θ 1 r j 1 αm j log λe j ( ) ] log α α log α log xj,k. m j ) α 2.18 Our ientifiability problem remains. However, we can eliminate the problem by letting φ λ θ/ α. Then l [ J m j log φ m j α ( j 1 ) m j log 1 r e j φ 1 r α j 1 ( ) ] log α α log α log xj,k, 2.19 an we can etermine the unique MLE s of α, φ, anr. Differentiating with respect to φ we have l J φ [ mj ] φ e r α j 1, 2.20 an hence, φ m j e r α j 1. 2.21

8 Probability an Statistics Next we ifferentiate with respect to r an obtain [ ( ) l J α j 1 r mj 1 r φα ( j 1 ) e j 1 r α j 1 1 ], 2.22 which leas to J [ (j ) 1 mj φ ( j 1 ) e j 1 r α j 1 ] 0. 2.23 Finally, ifferentiating with respect to α, we have [ l J (j α ) 1 mj log 1 r φ ( j 1 ) e j 1 r α j 1 log 1 r m j ( ) ] 1 α log log x j,k. 2.24 Replacing the parameters by their MLE s, setting the right-han sie of 2.24 equal to 0, an using 2.23, weobtain 2.8, as before. Substituting 2.21 into 2.23 an iviing by the numerator of 2.21, we have ( ) j 1 mj m j ( j 1 ) ej 1 r α j 1 e r α j 1 0. 2.25 Since 2.8 provies an explicit expression for α, we can obtain r by solving 2.25. In practice, rather than simply assuming that all r j s are equal, we shoul perform a hypothesis test with the null hypothesis H 0 : r 2 r J. This can be accomplishe by employing the well-known likelihoo ratio test LRT whose test statistic is given by 2 { maximum of l uner H 0 [ maximum of l over full parameter space ]}. 2.26 As follows, for example, from Casella an Berger 4, Section 10.3, the asymptotic istribution of the statistic given by 2.26 is chi-square with J 1 3 egrees of freeom. 3. Numerical Illustrations In this section we provie numerical illustrations of the methos presente in Section 2. We use the simulate ata iscusse in Section 1. However, assume we o not know the number of losses an average loss amounts shown in the secon an thir columns of Table 1. We o know the number of observe losses given in the fourth column as well as the amount of each observe loss that occurre in each year. Also, it is reasonable for us to assume that we know that the exposure is the same each year. The same Poisson parameter was use to generate the number of losses in each year. Therefore, suppose that e j 1forj 1,...,10.

Probability an Statistics 9 Table 2: Maximum likelihoo estimates of r j for j 2,...,10. j 2 3 4 5 6 7 8 9 10 r j 0.0786 0.0116 0.1291 0.0526 0.1226 0.0196 0.0272 0.1205 0.0168 Table 3: Point estimates an approximate 95% confience intervals of r an α using the full likelihoo an using the first approach. Note: the true parameter values are r 0.05, α 2. Full likelihoo approach First approach Parameter Estimate Asymptotic CI Estimate Bootstrap CI r 0.0503 0.0353; 0.0654 0.0526 0.0375; 0.0702 α 1.9858 1.8328; 2.1389 1.9858 1.8246; 2.1495 Applying the first metho, we can estimate α an then r using 2.8 an 2.9. We obtain the estimates 1.9858 an 0.0526, respectively. Recall that the true parameter values are α 2anr 0.05. In practice, we o not know that the loss inflation rate is the same each year, an our full maximum likelihoo approach allows us to estimate the iniviual inflation rates. The estimates reporte in Table 2 were obtaine using 2.15, with α obtaine from 2.8. If we then impose the restriction that the inflation rate is the same each year, we can obtain the maximum likelihoo estimate of r by solving 2.25. Alternatively, rather than solving 2.25, the estimates can be obtaine by numerically maximizing the loglikelihoo function using, for example, the optim function in R see 5. This approach has the avantage of allowing one to obtain the Hessian matrix as a by-prouct of the maximization. Since the Hessian matrix equals minus the observe information matrix evaluate at the maximum likelihoo estimates, an estimate variance-covariance matrix for the parameter estimators can be foun by matrix inversion. This approach was use to obtain the point estimates an approximate 95% confience intervals presente in Table 3. The estimates obtaine using the first approach are also provie for comparison. In this case, the approximate confience intervals were constructe by proucing 1000 parametric bootstrap samples. Having maximize the log-likelihoo with an without the restriction that the inflation rate in each year is the same, we can perform a likelihoo ratio test of the hypothesis that the inflation rates are the same. Using the LRT statistic in 2.26, we fin that its value is 4.5741. Base on a chi-square istribution with 8 egrees of freeom we fin that the P-value is.8020 an conclue that the r j s are statistically equal. Acknowlegments The authors sincerely thank Eitor Tomasz J. Kozubowski an two anonymous referees for queries an suggestions that have guie them in revising the paper. The first author gratefully acknowleges the stimulating scientific atmosphere at the 38th ASTIN Colloquium in Manchester, Unite Kingom, an Michael Fackler in particular for posing the problem whose solution makes the contents of the present paper. The secon an thir authors are grateful to the University of Wisconsin-Milwaukee for the most prouctive an pleasant stay uring which results of the present paper ha evolve to fruition.

10 Probability an Statistics References 1 R. V. Hogg an S. A. Klugman, Loss Distributions, Wiley Series in Probability an Mathematical Statistics: Applie Probability an Statistics, John Wiley & Sons, New York, NY, USA, 1984. 2 B. C. Arnol, Pareto Distributions, vol. 5 of Statistical Distributions in Scientific Work, International Cooperative Publishing House, Burtonsville, M, USA, 1983. 3 N. L. Johnson, S. Kotz, an N. Balakrishnan, Continuous Univariate Distributions. Vol. 1, Wiley Series in Probability an Mathematical Statistics: Applie Probability an Statistics, John Wiley & Sons, New York, NY, USA, 2n eition, 1994. 4 G. Casella an R. L. Berger, Statistical Inference, Duxbury, Pacific Grove, Calif, USA, 2n eition, 2001. 5 R Development Core Team, R: A Language an Environment for Statistical Computing, R Founation for Statistical Computing, Vienna, Austria, 2008, http://www.r-project.org.

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