# Approximation of Aggregate Losses Using Simulation

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1 Journal of Mathematics and Statistics 6 (3): , 2010 ISSN Science Publications Approimation of Aggregate Losses Using Simulation Mohamed Amraja Mohamed, Ahmad Mahir Razali and Noriszura Ismail Statistics Program, School of Mathematical Sciences, Faculty of Science and Technology, University Kebangsaan Malaysia, UKM Bangi, Selangor, Malaysia Abstract: Problem statement: The modeling of aggregate losses is one of the main objectives in actuarial theory and practice, especially in the process of making important business decisions regarding various aspects of insurance contracts. The aggregate losses over a fied time period is often modeled by miing the distributions of loss frequency and severity, whereby the distribution resulted from this approach is called a compound distribution. However, in many cases, realistic probability distributions for loss frequency and severity cannot be combined mathematically to derive the compound distribution of aggregate losses. Approach: This study aimed to approimate the aggregate loss distribution using simulation approach. In particular, the approimation of aggregate losses was based on a compound Poisson-Pareto distribution. The effects of deductible and policy limit on the individual loss as well as the aggregate losses were also investigated. Results: Based on the results, the approimation of compound Poisson-Pareto distribution via simulation approach agreed with the theoretical mean and variance of each of the loss frequency, loss severity and aggregate losses. Conclusion: This study approimated the compound distribution of aggregate losses using simulation approach. The investigation on retained losses and insurance claims allowed an insured or a company to select an insurance contract that fulfills its requirement. In particular, if a company wants to have an additional risk reduction, it can compare alternative policies by considering the worthiness of the additional epected total cost which can be estimated via simulation approach. Key words: Aggregate losses, compound Poisson-Pareto, simulation INTRODUCTION Let X 1, X 2, X N denote the amount of loss of an insurance portfolio that recorded N losses over a fied time period. If N is a random variable independent of X i, i = 1,2,,N, which are identical and independently distributed (i.i.d.), then the aggregate losses is N calculated as S = X. Due to numerical difficulties, i = 1 i approimation methods of the eact cumulative distribution function (c.d.f.) of S, F S (s), have been suggested and tested namely the Fast Fourier Transform, inversion method, recursive method, Heckman-Meyers method and Panjer method (Heckman and Meyers, 1983; Von Chossy and Rappl, 1983; Pentikainen, 1977; Jensen, 1991). All of these approaches are based on the assumption that the distributions of loss frequency and severity are available separately. In other cases, due to incomplete information on separate frequency and severity distributions, only aggregate loss distribution is available for further statistical estimation and inference. Nevertheless, several researches on aggregate loss distributions have been carried out and such eamples can be found in Dropkin (1964) and Bickerstaff (1972) who showed that the Lognormal distribution closely approimates certain types of homogeneous loss data, Pentikainen (1977) who improved the results of Normal approimation by suggesting Normal Power method, Seal (1977) who compared Normal Power method with Gamma approimation and concluded that the Gamma provides a generally better approimation, Venter (1983) who suggested transformed Gamma and transformed Beta distributions for approimating aggregate losses, Chaubey et al. (1998) who proposed Inverse Gaussian distribution for approimation of aggregate losses and Papush et al. (2001) who compare Gamma distribution with Normal and Lognormal distributions and concluded that the Gamma provides a better fit for aggregate losses. Corresponding Author: Noriszura Ismail, School of Mathematical Sciences, Faculty of Science and Technology, University Kebangsaan Malaysia, UKM Bangi, Selangor, Malaysia Tel: Fa:

2 Another approach that can be used for approimating aggregate losses is the method of simulation. The main advantage of simulation is that it can be performed for several mitures of loss frequency and severity distributions, thus producing approimated results for the compound distribution of aggregate losses. In addition, the effects of deductible and policy limit on individual loss as well as aggregate losses can also be investigated and the simulation programming can be implemented in a fairly straightforward manner. MATERIALS AND METHODS Collective risk model: In a collective risk model, the c.d.f. of aggregate losses, S, can be calculated numerically as: F () = Pr(S ) = S *n Pr(S N = n)p n 0 n = F n 0 ()P = = n (1) where, F () = Pr(X ) denotes the common c.d.f. of each of X 1, X 2,, X N, P n = Pr[N = n] the probability mass function (p.m.f.) of N and *n F () = Pr = (X Xn ) the n-fold convolutions of F (.). The distribution of S resulted from this approach is called a compound distribution. If X is discrete on 0,1,2,..., the k-th convolution of F (.) is: I{ 0 }, k = 0 *k F () = F (), k = 1 *(k 1) F y 0 ( y)f (y), k = 2,3,... = (2) Based on Eq. 1, a direct approach for calculating the c.d.f. of S requires the calculation of the n-fold convolutions of F (.) implying that the computation of the aggregate loss distribution can be rather complicated. In many cases, realistic probability distributions for loss frequency and severity cannot be combined mathematically to derive the distribution of aggregate losses. such instance is the Inverse Gaussian and the Gamma mitures suggested by Chaubey et al. (1998). This study aims to approimate the aggregate loss distribution using simulation approach. The following steps summarize the algorithm for such approach: The distributions for loss frequency and severity are chosen, if possible, based on the analysis of historical data Accordingly, the number of losses, N, is generated based on either Poisson, or Binomial, or Negative Binomial distributions For i = 1 to i = N, the individual loss amount, X i, is generated based on Gamma, or Pareto, or Lognormal, or other positively skewed distributions The aggregate losses, S, is obtained by using the sum of all X i s The same steps are then repeated to provide the estimate of the probability distribution of aggregate losses In addition to the approimation of aggregate losses, the effects of deductible and policy limit on individual loss amount can also be investigated using simulation approach. Let X denotes the random variable for individual amount of loss. When an insurer introduces a deductible policy, say at the value of d, the loss endured or retained by the insured can be represented by the random variable Y: Y = X, X d = d, X > d (3) whereas the loss covered or paid as claim by the insurer can be represented by the random variable Z: Z = 0, X d = X d, X > d (4) so that X = Y+Z. When an insurer introduces a policy limit in its coverage, say at the value of u, the loss retained by the insured can be represented by the random variable Y: Simulation: Based on the actuarial literature, several Y = 0, X u (5) methods can be applied for the approimation of = X u, X > d aggregate loss distribution. For eamples, the approimations of Normal and Gamma are fairly easy whereas the loss paid as claim by the insurer can be to be used since they are pre-programmed in several represented by the random variable Z: mathematical and statistical softwares. However, other Z = X, X u distributions are not as convenient to be computed but (6) may produce a more accurate approimation and one = u, X > u 234

3 Therefore, if an insurance contract contains deductible, d and policy limit, u, the loss retained by the insured can be represented by the random variable Y: Y = X, X < d = d, d X d + u = X u, X > d + u J. Math. & Stat., 6 (3): , 2010 (7) whereas the loss paid as claim by the insurer can be represented by the random variable Z: Z = 0, X < d = X d, d X d + u = u, X > d + u The implementation of deductible and policy limit may not be limited to the basis of individual loss as they can also be etended to the basis of aggregate losses. The main advantage of simulation approach is that it allows the implementation of deductible and policy limit not only on individual loss but also on N aggregate losses. Let S = i = X 1 i denotes the random variable for aggregate losses. If an insurance contract contains both aggregate deductible, d * and aggregate policy limit, u *, the aggregate losses retained by the insured can be defined by the random variable V: V = S, S < d = d, d S d + u * * * * = S u, S > d + u * * * * (8) (9) whereas the aggregate losses paid as claims by the insurer can be represented by the random variable W: V = 0, S < d = S d, d S d + u * * * * = u, S > d + u * * * * (10) so that S = V+W. The following steps can be used for approimating the distributions of individual loss and aggregate losses of an insurance contract containing deductible and policy limit and also for an insurance contract containing aggregate deductible and policy limit: The distributions for loss frequency and severity distributions are chosen, if possible, based on the analysis of historical data Accordingly, the number of losses, N, is generated based on either Poisson, or binomial, or negative binomial distributions 235 For i = 1 to i = N, the individual loss amount, X i, is generated based on gamma, or Pareto, or lognormal, or other positively skewed distributions The aggregate losses, S, is obtained by using the sum of all X i s For an insurance contract containing deductible and policy limit, the conditions specified by Eq. 7 and 8 are applied on all X i s, i = 1,,N, producing the retained loss, Y i, i = 1, N and the loss paid as claim, Z i, i = 1,,N If deductible and policy limit on the basis of aggregate claims are to be included in the insurance contract, the conditions specified by Eq. 9 and 10 are applied also on all X i s, i = 1,,N, producing the retained loss, Y i, i = 1, N and the loss paid as claim, Z i, i = 1,,N The same steps are then repeated to provide the approimated distributions of loss, X, retained loss, Y, claim, Z and aggregate losses, S RESULTS This section presents several results from the approimation of aggregate loss distribution via simulation approach based on a compound Poisson- Pareto distribution which is often a popular choice for modeling aggregate losses because of its desirable properties. Detailed discussions of the compound models and their applications in actuarial and insurance areas can be found in Klugman et al. (2006) and Panjer (1981). In this eample, the loss frequencies are generated from Poisson distribution whereas the loss severities are generated from Pareto distribution. The p.m.f., mean and k-th moment about zero for Poisson distribution are: n λ λ e Pr(N = n) =, n! E(N) = λ, k E(N ) = n = 0 n k λ e n n! λ (11) whereas the probability density function (p.d.f.), mean and k-th moment about zero for Pareto distribution are: α αβ β f () =,E(X) =, α+ 1 ( β + ) α 1 k! β ( α 1)( α 2)...( α k) k k E(X ) =, α > k (12)

4 Where: α = Shape parameter β = Scale parameter The parameter of α in Pareto distribution determines the shape, with small values corresponding to a heavy right tail. The k-th moment of the distribution eists only if α>k. The moments of the compound distribution of S can be obtained in terms of the moments of N and X. In particular, the mean and variance of S are: E(S) = E(N) E(X), Var(S) = E(N)Var(X) + Var(N)(E(X)) 2 (13) The frequency, severity and aggregate distributions based on 1,000 simulations are illustrated in Fig The simulated loss and aggregate losses are assumed to be in the currency of Ringgit Malaysia (RM). Figure 1 shows the frequency of loss assuming a Poisson distribution with an epected value of 20, i.e., λ = 20. Note that the distribution is bell shaped for an epected number of loss this large. The distribution would be more skewed for lower epected number of loss. Figure 2 shows the amount of loss or loss severity from a Pareto distribution with an epected severity of RM15,000 and a standard deviation of RM16,771, i.e., α = 10 and β = 135,000. Note that most losses are less than RM10,000, but some are much larger. Figure 3 shows the aggregate losses from a compound Poisson-Pareto distribution with λ = 20, α = 10 and β = 135,000. Note that the aggregate loss distribution is also positively skewed but not as skewed as the severity distribution and most aggregate losses are around RM200,000- RM350,000. Table 1-3 show the statistics summary of simulated aggregate losses based on 1,000 simulations of compound Poisson-Pareto distribution. In particular, we assume a fied value for α and β and increase the value of λ in Table 1, in Table 2 we assume a fied value for λ and β and increase the value of α and in Table 3 we assume a fied value for λ and α and increase the value of β. In general, the results show that the mean and standard deviation of aggregate losses increase when the values of λ or β increase and the mean and standard deviation of aggregate losses decrease when the values of α increase. Thus, the simulated results of the compound Poisson-Pareto distribution agree with the mean and variance of each of the loss frequency, N, loss severity, X and aggregate losses, S, shown in Eq Fig. 1: Simulated loss frequency (Poisson distribution, λ = 20) Table 1: Simulated aggregate losses (Poisson-Pareto compound distribution, λ, α = 10, β = ) λ 1st quarter (RM) Median (RM) 3rd quarter (RM) Mean (RM) Std. dev. (RM) , , , ,406 71, , , , , , , , , , , , , , , , ,343,541 1,495,566 1,650,632 1,498, ,403 Table 2: Simulated aggregate losses (Poisson-Pareto compound distribution, λ = 30, α, β = ) α 1st quarter (RM) Median (RM) 3rd quarter (RM) Mean (RM) Std. dev. (RM) 5 809, ,186 1,195,972 1,020, , , , , , , , , , ,399 55, ,213 82,343 95,402 83,109 21, ,841 40,736 47,185 41,130 10,461 Table 3: Simulated aggregate claims (Poisson-Pareto compound distribution, λ = 30, α = 10,β) β 1st quarter (RM) Median (RM) 3rd quarter (RM) Mean (RM) Std. dev. (RM) 90, , , , ,920 80, , , , , ,466 89, , , , , , , , , , , , , , , , , , ,

7 Panjer, H., Recursive evaluation of a family of compound distributions. ASTIN Bull., 12: no1/22.pdf Papush, D.E., G.S. Patrik and F. Podgaits, Approimations of the aggregate loss distribution. Casualty Actuar. Soc. Forum, Winter, pp: pdf Pentikainen, T., On the approimation of the total amount of claims. ASTIN Bull., 9: o3/281.pdf Seal, H., Approimations to risk theories F(X, t) by the means of the gamma distribution. ASTIN Bull., 9: o1and2/213.pdf Venter, G., Transformed beta and gamma distributions aggregate losses. Proc. Casualty Actuar. Soc., 70: pdf Von Chossy, R. and G. Rappl, Some approimation methods for the distribution of random sums, insurance. Math. Econ., 2: DOI: / (83)

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