MODELLING INCURRED BUT NOT REPORTED CLAIM RESERVE USING COPULAS

Transcription

1 MODELLING INCURRED BUT NOT REPORTED CLAIM RESERVE USING COPULAS Gaida Pettere Professor, Dr. Math. Banking Institution of Higher Education, Riga, Latvia In practice of an insurance company claims occur almost every day. But usually they are not reported the same day. Therefore when accounting period ends, many occurred claims are still not reported. To have a right profit and loss account, it is necessary to find reserves for such claims. Therefore the problem, how to calculate appropriate incurred but not reported claim reserves, has always been in the centre of attention of actuaries in insurance companies. There are many classical methods how to do that (see, for example, Benjamin (977), Hossack, Pollard, Zehnwirth (983)). All these methods are based on different coefficient calculations and deal with classical development triangle. Nowadays new stochastic methods have become actual to calculate the mentioned reserves (Charpentier, 24). Copula as a tool for modelling different dependence structures has been intensively used in different fields of application: finance, insurance, risk theory, environmental studies etc. Sklar introduced the copula theory in 959 but only in 997 Wang introduced copula models in insurance studies. Copulas are intensively used in both, life and non-life insurance. Most commonly copulas are used to model claim sizes and allocated loss adjusted expenses of an insurance company (Freez, Valdez (998) and Klugman, Parsa (999)), evaluate economic capital (Tang, Valdez (26), Jackie (26), Bagarry (26))), for combining different risks (Clemen, Reilly (998), Cherubini (24), Embrechts, Lindskog, McNeil (23)) or in ruin theory (Bregman, Kluppelberg (25)). We have studied how to use copulas to model outstanding claim reserves (Kollo, Pettere, 26). But there are some problems with outstanding claim reserves: some part of them is always known (reported but not jet paid claims) and if reported sums are larger, they are paid out in another part of reserves (incurred but not reported claim reserve). Therefore more important is to find direct method how to model incurred but not reported claim reserve (IBNR). We have started to do that in the paper Pettere, Kollo (26) but an algorithm was not worked out there because we concentrated on modelling claim sizes.

2 Aim in this article is different from the previous one. We are aimed to work out algorithm in detailed way for using copulas and to apply stochastic approach in direct calculation of IBNR reserves and to show on an example that method works even if portfolio is not very large. Additionally we show how the amount of calculated reserves is related to the probability that real paid out sum will not be larger than calculated reserve. The main idea, which gave us a possibility to calculate IBNR reserve directly without total outstanding reserve, is the idea about development time. In the classical triangle and in Kollo, Pettere (26) development is considered as the time from the moment of claim occurrence until payment. To calculate IBNR we consider development time from the moment of claim occurrence until its settlement. In our model each claim in non-life insurance is characterized by two random variables: the claim size and the development time. First step: Firstly the marginal univariate distributions are examined and approximated using different families of distributions. The very important thing in this step is to find appropriate marginal distributions. We have checked different families of distributions for different business lines and have came to conclusions that on Latvian data the best fit is received by lognormal distribution. Goodness of fit was tested by the Kolmogorov-Smirnov test-statistic. Second step: Next step is to find appropriate copula model for the given data. Several classes of Archimedean copulas as well as Gaussian copula have been applied. Different method for testing goodness of the obtained models is discussed in literature (Freez, Valdez (998), de Matteis (2)). We used univariate distribution function K (, t ) for Archimedean copulas introduced by Genest, Rivest (993) and applied then Kolmogorov-Smirnov test to find the best copula. Additionally different measures are calculated to compare simulated data with the original given data. Third step: After an appropriate copula model has been found, average claim size in each development time unit is found by simulation. Fourth step: The next step is to find number of claims occurred in each used time unit. That information is possible to get also from historical data but when using this

3 number later when calculating IBNR reserves one has to be very careful because number of claims in each time unit is a function of total portfolio and therefore has to be adjusted by inflation coefficient. By our method we find number of claims occurred in a day as a function of the exposure coefficient. Usually, if portfolio is large, number of claims happening in one day (time unit) has a normal distribution. But if portfolio is not very large, this distribution can be different. Fifth step: After the number of claims in time unit is estimated, average number of claims reported in each development time unit has to be found. This we calculate by multiplying value of the density function of the development time to the length of the time interval and to the average number of claims happening in time unit plus one, two or three standard deviations. Sixth step: At the final step we calculate reserves by multiplying average claim size in development time unit to the average number of claims reported in development time unit and to the number of time units. The next example is given to demonstrate step by step the described method on real data. EXAMPLE We use accident insurance claims in Latvian lats (LTL) of one Latvian insurance company. The development time unit has length one day and in the data set we have all claims, which happened in the first quarter of 24. In this case it is possible to compare calculated IBNR reserve with the real amount of money paid out until the end of first quarter 26. First step: Descriptive statistics were calculated for both random variables. They are shown in Table. The best fir was obtained by lognormal distribution. The lognormal distribution is used in the following parameterisation: random variable X is lognormally distributed with parameters µ andσ, if ln X = Y N ( µ, σ ), where EY = µ and DY 2 = σ. The best fit to the claim size was obtained by the lognormal distribution with µ = 3.74 andσ =.2. Corresponding value of the Kolmogorov- Smirnov test statistic was (5% critical value.9362). Development time

4 has lognormal distribution with µ = 3.85 and σ =.7. Kolmogorov-Smirnov test statistic was equal to.3849 with the same critical value. Table. Characteristics of claim size and development time. Characteristics Claim size Development time Mean Median Mode 5. Standard Deviation Sample Variance Kurtosis Skewness Range Minimum 6. Maximum Sample size 2 2 Second step: The location of data is shown in Figure where R is development time and R 2 is claim size R2 i R i Figure. Location of reported claims in development time Kendall s tau between the variables equals.4 (linear correlation coefficient is.4). Gumbel, Clayton, Frank and Survival copulas, with parameters estimated using Kendall s tau, were examined to find the best approximation. All used formulas are given in Table 2 (see for detailed explanation Nelsen (999), for example). As fitting measure the maximal difference between distribution functions K (, t ) calculated for each copula and its sample estimate found from data was found. This measure was denoted by LM. Additionally we used as a measure the sum n K t FZ t 2. i= Ψ ( ) = ( (, ) ( ))

5 Table 2. Used copula formulas. Family Copula Cuv (, ) Gumbel exp{ [( ln u) + v + ( ln ) ] } Clayton ( u + v ) Frank ln( + u v ( e )( e ) + ) e Relation between Function K ( t, ) Kendall s τ and ln t = t t τ 2τ = t + t t τ t 4 4 t e τ = + dt ln 2 t e e ( t t e ) Survival uve τ = ln uln v 4 ( ln tt ) ln( ln td ) t + ( ln tt ) ln( ln t) + The function K (, t ) was introduced by Genest, Rivest (993). Used values of for each copula and numerical values of measures LM and Ψ are presented in Table 3. Table 3. Values of and fitting measures for copulas. Copula Gumbel Clayton Frank Survival Values of LM ( ) Ψ ( ) Graphical fitting of data with chosen copulas is shown in Figures 2,3,4,5. Cop_Gumbel F i ( t i, ) t i, i Figure 2. Comparison of K ( t, ) with its empirical estimate in the case of Gumbel copula.

6 Cop_Clayton F i ( t i, ) t i, i Figure 3. Comparison of K ( t, ) with its empirical estimate in the case of Clayton copula. Cop_Frank F i ( t i, ) t i, i Figure 4. Comparison of K ( t, ) with its empirical estimate in the case of Frank copula. Cop_Survival R i ( t i, ) t i, i Figure 5. Comparison of K ( t, ) with its empirical estimate in the case of Survival copula.

7 Finally we choose Gumbel copula for further calculations as the best model. Third step: From the Gumbel copula data points were simulated 5 times and average claim size reported in each development day was found. As an example average claim amounts in LVL in first eight days are shown in Table 4. Table 4. Average claim amounts reported in first eight days after their occurring. Development time (days) Average reported claim (LVL) Fourth step: All 9 days of the first quarter of 24 were examined and descriptive statistics was calculated for the number of claims happening in one day. Results are shown in Table 5. Table 5. Characteristics of number of claims happening in one day. Characteristics Number of claims Mean. Median Mode Standard Deviation.93 Sample Variance 3.73 Kurtosis 4.8 Skewness 2.6 Range Minimum Maximum Sample size 9 Fitting of claim sizes to lognormal distribution reached values µ =.6 andσ =.8. The corresponding value of Kolmogorov-Smirnov test-statistic is with the 5% critical value Fifth step: To find development of average number of claims occurred in each day plus one or several standard deviations we multiply value of the lognormal density function by the average number of claims occurred in each day plus standard deviation and by the length of interval (it is one in our case). Results are shown in Table 6.

8 Table 6. Development of 4.96 ( µ + 2 σ = ) calculation example for first 8 days. Development (day) Value of lognormal density Development of Sixth step: Finally reserve is calculated by multiplying average claim size in development time unit to the average number of claims reported in development time unit and to the number of time units. We have calculated reserves by using the average number of claims in one day, the average number of claims in one day plus standard deviation, the average number of claims in one day plus two standard deviations and the average number of claims in one day plus three standard deviations. The results are shown in Table 7. Table 7. Calculated IBNR reserves and related probabilities. Calculated reserve (LVL) Used number of claims happening in one day Probability that necessary paid out sum will be less Probability that necessary paid out sum will be larger There was one very large and not traditional paid sum 5 LVL and the real paid out amount of claims was LVL. If this large amount would not had occurred the amount would have been LVL. REFERENCES. Benjamin, B., General Insurance, Heinemann, Bregman, Y., Kluppelberg, C., Ruin estimates in multivariate models with Clayton dependence structure, Munich University of Technology, unpublished manuscript, Bagarry, M., Economic Capital: a plea for the Student Copula, 26.

9 4. Charpentier A. Advanced Statistical Methods in Non-Life Insurance, Solvency and simulation based methods. Lecture notes based on joint work with Michel Denuit (UCL), 3 rd Conference in Actuarial Science&Finance in Samos, September Cherubini, U., Luciano, E., Vecchiato, W. Copula Methods in Finance. Wiley, Chichester, Clemen, R. T., Reilly, T. Correlations and copulas for decision and risk analysis. Management Science, 999, vol. 45, Embrechts, P., Lindskog, F., McNeil, A. Modelling dependence with copulas and applications to risk management. In: Handbook of Heavy Tailed Distributions in Finance, Ed. E. Rachev, 23. Elsevier, Frees, E. W., Valdez, E. A. Understanding relationships using copulas. North American Actuarial Journal, 998, vol. 2, Genest, C., Rivest, L. P. Statistical inference procedures for bivariate Archimedean copulas. JASA, 993, vol. 88, Hossack I. B., Pollard J. H., Zehnwirth B. Introductory statistics with applications in general insurance. Cambridge University Press, Jackie, Li, Modelling Dependency between different Lines of Business with Copulas, Klugman, S. A., Parsa, R. Fitting bivariate loss distributions with copulas. Insurance: Mathematics and Economics, 999, vol. 24, Kollo, T., Pettere, G. Copula Models for Estimating Outstanding Claim Provisions. In: Festschrift for Tarmo Pukkila on his 6 th Birthday. Eds. E.P. Liski, J. Isotalo, J. Niemelä, S. Puntanen, G.P.H. Styan. Department of Mathematics, Statistics and Philosophy, University of Tampere, Vammala, 26, de Matteis R. Fitting Copulas to Data. Diploma Thesis, Institute of Mathematics of the University of Zurich, Nelsen, R. B. An Introduction to Copulas. Springer-Verlag, New York, Pettere, G., Kollo, T. Modelling Claim Size in Time via Copulas Sklar, A. Fonctions de repartition a n dimensions etleurs marges. Publ. Inst. Statist. Univ. Paris 8, 959,

10 8. Tang, A., Valdez, E., A., Economic Capital and the aggregation of Risks Using Copulas, Wang, S. PH-Measure of Portfolio Risks. Chapter 9 of unpublished lecture notes, Risk Measures with Applications in Insurance Ratemaking and Actuarial Valuation. Waterloo, Ont.: University of Waterloo, 997.