Approximation methods for distributions of aggregate insurance losses

Convolutions of probability distributions have long been used in collective risk theory to determine the distribution of aggregate claims. Two examples of such a distribution are given. The paper then presents results for the distribution of aggregate claims for the family of claim distributions in which claim values are equi-spaced, and equi-probable. The distribution of claims may be either the Poisson or negative binomial law. Examples and tables are included. A convolution type series for the infinite time ruin function is examined, including approximations and their error analyses.

Estimation of the operational risk capital under the Loss Distribution Approach requires evaluation of aggregate (compound) loss distributions which is one of the classic problems in risk theory. Closed-form solutions are not available for the distributions typically used in operational risk. However with modern computer processing power, these distributions can be calculated virtually exactly using numerical methods. This paper reviews numerical algorithms that can be successfully used to calculate the aggregate loss distributions. In particular Monte Carlo, Panjer recursion and Fourier transformation methods are presented and compared. Also, several closed-form approximations based on moment matching and asymptotic result for heavy-tailed distributions are reviewed.

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 fixed time period is often modeled by mixing 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 approximate the aggregate loss distribution using simulation approach. In particular, the approximation 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 approximation 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 approximated 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 expected total cost which can be estimated via simulation approach.

Having discussed models for claim frequency and claim severity separately, we now turn our attention to modeling the aggregate loss of a block of insurance policies. Much of the time we shall use the terms aggregate loss and aggregate claim interchangeably, although we recognize the difference between them as discussed in the last chapter. There are two major approaches in modeling aggregate loss: the individual risk model and the collective risk model. We shall begin with the individual risk model, in which we assume there are n independent loss prospects in the block. As a policy may or may not have a loss , the distribution of the loss variable in this model is of the mixed type. It consists of a probability mass at point zero and a continuous component of positive losses. Generally, exact distribution of the aggregate loss can only be obtained through the convolution method. The De Pril recursion, however, is a powerful technique to compute the exact distribution recursively when the block of policies follow a certain set-up. On the other hand, the collective risk model treats the aggregate loss as having a compound distribution, with the primary distribution being the claim frequency and the secondary distribution being the claim severity. The Panjer recursion can be used to compute the distribution of the aggregate loss if the claim-frequency distribution belongs to the ( a, b , 0) class and the claimseverity distribution is discretized or approximated by a discrete distribution.

Aggregate loss is the total loss suffered by an insured in a certain period. The aggregate loss depends on the claim frequency and the amount of the claim each time the insured makes a claim. The distribution of aggregate losses must be known to calculate motor vehicle insurance premiums. In general, there are two methods that can be used in determining the distribution of aggregate losses, namely exact and numerical. When an exact solution is difficult to find, numerical methods such as Monte Carlo, Panjer Recursion, and Fast Fourier Transform can be used. This research will discuss the determination of the distribution of aggregate losses through the numerical inverse of the characteristic function using the trapezoidal quadrature rule, on the data of motor vehicle insurance category 7 in Indonesia. The estimated cumulative distribution function for the largest aggregate loss is 0.999993. When x=0, it means that if someone does not file a claim, the estimated value of the cumulative distribution function is 0.9293. This value is close to the percentage of the number of insured, which is 0.9241.

Determining the distribution of aggregate loss is an important issue for insurers. Basically, the distribution of aggregate loss can be determined using n-fold convolution of the probability density function of severity distribution. However, problems in computation for this method causes findings of new methods to approximate aggregate loss distribution. One of the methods which is widely used and claimed to give a good approximation is Panjer recursion. Panjer introduced a recursion formula which can be used to compute aggregate loss probabilities. This method requirements are discrete severities distribution and (a, b, 0) class frequency distribution. The Panjer recursion method could not be applied if those two requirements are not met, so a discretization process is needed for continuous severity cases. This paper explored the use of Panjer recursion method for continuous severity cases. The method of rounding is used to discretize the continuous severity random variable with span h. It means that random variables which are the discretized version of severity random variables have probabilities in the span of h (h, 2h, 3h, and so on). The result improves when the discretization span is small enough. Beside Panjer recursion method, there is a new method, called moment-based, which can be used to approximate aggregate loss distribution using its moments. This method presents an approximation formula of aggregate loss distribution probability density function, which contain coefficients which can be determined by matching its moments with aggregate loss moments. Both of these methods tend to give relatively similar results when the span used in the recursive method is small enough and moments used in moment-based is adequate. The span of discretization for the recursive method is said to be small enough if no jump is seen in cumulative distribution function of severity random variables. t moments used in moment-based is said to be adequate if the result using t + 1 moments give no significant difference.

The aim of this paper is to analyse two functions that are of general interest in the collective risk theory, namely F, the distribution function of the total amount of claims, and II, the Stop Loss premium. Section 2 presents certain basic formulae. Sections 17-18 present five claim distributions. Corresponding to these claim distributions, the functions F and II were calculated under various assumptions as to the distribution of the number of claims. These calculations were performed on an electronic computer and the numerical method used for this purpose is presented in sections 9, 19 and 20 under the name of the C-method which method has the advantage of furnishing upper and lower limits of the quantities under estimation. The means of these limits, in the following regarded as the “exact” results, are given in Tables 4-20. Sections 11-16 present certain approximation methods. The N-method of section 11 is an Edgeworth expansion, while the G-method given in section 12 is an approximation by a...

The distribution of aggregate claims is of importance in evaluating risks both from the insurer’s and the insured’s point of view. The paper presents a procedure for developing an expression of the distribution of aggregate claims which is applicable under a wide variety of conditions. The procedure is illustrated by a sample problem and the results are compared with Mereu’s enumerative approach (TSA, 1973).

Applications of the GB2 family of distributions in modeling insurance loss processes

Credit contagion refers to the propagation of economic distress from one firm to another. This article proposes a reduced-form model for these contagion phenomena, assuming they are due to the local interaction of firms in a business partner network. We study aggregate credit losses on large portfolios of financial positions contracted with firms subject to credit contagion. In particular, we provide an explicit Gaussian approximation of the distribution of portfolio losses. This enables us to quantify the relation between the volatility of losses and the determinants of credit contagion. We find that contagion processes have typically a second order effect on portfolio losses. They induce additional fluctuations of losses around their averages, whose size depends on the denseness of the business partner network.

Credit contagion and aggregate losses

The focus of this book is on the two major areas of risk theory: aggregate claims distributions and ruin theory. For aggregate claims distributions, detailed descriptions are given of recursive techniques that can be used in the individual and collective risk models. For the collective model, the book discusses different classes of counting distribution, and presents recursion schemes for probability functions and moments. For the individual model, the book illustrates the three most commonly applied techniques. Beyond the classical topics in ruin theory, this new edition features an expanded section covering time of ruin problems, Gerber–Shiu functions, and the application of De Vylder approximations. Suitable for a first course in insurance risk theory and extensively classroom tested, the book is accessible to readers with a solid understanding of basic probability. Numerous worked examples are included and each chapter concludes with exercises for which complete solutions are provided.

Loss distribution plays an influential role in evaluating risks from policyholders’ claims. Nevertheless, the auto insurance market in Ghana pays little attention to policyholders’ claims distribution, resulting in the market’s inefficiency. This study investigates the type of loss distribution function that best approximates policyholders’ claims in Ghana. We applied the Kullback-Leibler divergence, Kolmogorov Smirnov, Anderson-Darling statistical tests and maximum likelihood estimation (MLE) to estimate policyholders’ claims. The results suggest that Ghana’s auto policyholder’s claims are better approximated using the lognormal probability distribution. Through the lognormal distribution, the industry can adequately evaluate policyholders’ claims to minimize potential loss. Additionally, this distribution could enable the market reach decisions on premiums and expected profits theoretically.

The purpose of this paper is to estimate county-level aggregate crop insurance and reinsurance losses under systematic risk. The effect of dependence risk on losses assessment and insurance pricing is quantified by establishing joint distribution functions between county-level yields using different forms of multivariate copulas. The research also stresses the importance of selecting the appropriate copula form for estimating losses. This article highlights several significant findings. The estimated aggregate losses for related counties are not significantly different between the model assuming dependence (copula-based) and the model assuming independence (individual) that adheres to the equivalence principle. On the other hand, the copula-based model has a discernible effect on the estimated Value-at-Risk and Expected Shortfall for related counties. Additionally, for the different layers of the Standard Reinsurance Agreement policy, the copula-based model can measure the aggregate losses more accurately than the individual models. Furthermore, when there is obvious tail dependence in the related counties’ yields, the vine copula function form, which provides a more flexible description of the dependence, is more suitable for quantifying tail risk. As a result, insurers and governments should conduct a more comprehensive risk assessment of yield dependence when rate-making and allocating subsidies.

The following problem in risk theory is considered. An insurance company, endowed with an initial capital a>0, receives insurance premiums and pays out successive claims. The losses occur according to renewal process. At any moment, the company may broaden or narrow down the offer, what entails the change of the parameters. This change concerns the rate of income, the intensity of renewal process and the distribution of claims. After the change, the management wants to know the moment of the maximal value of the capital assets. Therefore, our goal is finding two optimal stopping times: the best moment of change the parameters and the moment of maximal value of the capital assets. We will use a dynamic programming method to calculate the expected capital at that times.