A stochastic dynamic control approach for the claim reserving of a non – life insurance

1. 11th International Congress on Insurance Mathematics and Economics (IME), Piraeus 10 – 12 July 2007 A stochastic dynamic control approach for the claim reserving of a non – life insurance Athanasios A. Pantelous and Alexandros A. Zimbidis Department of Statistics, Athens University of Economics and Business

2. Summary • In this presentation, we discuss the various claim reserving methods for non – life insurance systems. Actually, we review the basic traditional reserving techniques and the proposed stochastic modifica-tions. • Furthermore, a dynamic control model with the associated stochastic differential equations is analytically designed describing the mecha-nisms of the payments and the reserving process, as well. A practical objective criterion is established measuring the performance of the pro-cess above. The theory of optimal stochastic control is effectively used for the solution of the system, establishing a dynamic formula for the determination of the reserve level. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

3. Introduction • The claim reserving process is a prospective estimation of what claims will cost to the insurance company. The reserves represent money that is set aside for the eventual payments of claims. From the company's point of view, claims are incurred when they happen; regardless of when in the future they are paid. • Since reserves represent future obligations, they are classified as liabilities on the insurance company's balance sheet. The accurate estimation of the claim reserving is profoundly important since it can be a clear measure of a company's financial strength. Improper reserves, either inadequate or excessive (under – reserving or over – reserving) can present a false picture of a company's financial position and lead to serious problems and misunderstandings, as well. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

4. Previous Results • The classical statistical approaches of claims reserving, as it is known the traditional deterministic methods, have a very long history. According to them the available information, often concerning hundred thousands of claims, may be reduced to very few numbers in the upper part of run – off triangles. In that direction, see Van Eeghen (1981), Taylor and Ashe (1983), De Jong and Zehnwirth (1983), Renshaw (1989, 1994), Verall (1989, 1990, 1991, 1993, 1994, 1996), Haberman and Renshaw (1996), and Renshaw and Verall (1998). • However, since the traditional deterministic methods do not take into consideration changes in future development pattern (for instance important changes in the business mix of portfolio of insurance company etc); it is not surprising that they might become inaccurate. In that undesirable direction and since the nature of the quantities, to which the claim reserving methods are applied, is pure stochastic, as well; several quite new approaches have been introduced to formulate the traditional techniques into a continuous (or discrete) stochastic framework. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

5. Moreover, since in stochastic modeling one of the main targets is the accurate description of the stochastic mechanism generating the data, two basic methodologies have been introduced. • The first methodology is based on classical statistical approaches of prediction or estimation which are appropriate for the claim reserving methods. Thus, the random variables which are observed into the model are estimated by certain probability distributions. In that direction, see Verall (2000), Mack and Venter (2000), Verall and England (2000), Hess and Schmidt (2002) and the unifying survey of Schmidt (2006) which has been presented recently, in CAS forum. In the second methodology, the Bayesian statistical approaches allow to the practitioners to intervene in the estimation of the development factors. The last decades, an important number of research works on Bayesian methods in claim reserving techniques have been introduced. In that direction, see Jewell (1989, 1990), Verall (1990), Makov, Smith and Liu (1996), Haastrup and Arjas (1996), De Alba, Juares and Moreno (1998), Scollnik (2001), England and Verall (2002, 2005), Verall (2004). A Stochastic dynamic control approach for the claim reserving of a non - life insurance

6. A new approach • In this presentation, the main contribution is the development of a structural stochastic model for managing the claim reserving process. This approach connects the claims reserving policy with continuous time stochastic control theory. Although optimal control theory was developed by engineers to investigate the properties of dynamic systems of difference or differential equations, it has also applied to financial problems. Tustin (1953) was the first to spot a possible analogy between the industrial and engineering processes and post – war macroeconomic policy – making (see Holly and Hallett, 1989, for further historical details). • In practice, this is not an easy task, since it does not exist a magic or a proven formula for setting individual case reserves. Although statistics helps an insurance company to determine whether or not there is a systematic flow in its reserving practice, the control theory can also be of use to define efficiently the reserve procedure. From this point of view, a method of controlling over time the claim reserving process is introduced buffering any kind of fluctuations, in order to absorb partially or completely the probable unexpectedness in micro or/and macro economic conditions, in external factors as competition, legal and regulatory requirements or other worsen random events. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

7. In the following figure, the actual claim reserving process is presented. Since, the insurance company knows the exact amount of claims’ liabilities at the end of the first year (i.e. an accumulated paid claims’ amount F), it should reserve an unknown, at advance, percentage of the accumulated paid claims’ amount with respect to events occurred within the 1st year (i.e. incident year), but obviously being paid to the very next years. That percentage can obtain either positive or instantly negative values; in the case that money is being returned to the insurance company. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

8. Furthermore, as it is shown in next figure, the claims run – off pattern which measures the difference between the reserve procedure and the actual paid amount is also constructed. That pattern can take positive or negative value, as well; in case of over or under estimation. The more specific information the adjuster obtains about the loss, the more accurate the reserve will be. With the necessary information, the actuarial department can make a fairly accurate assessment of the company's exposure and decide upon a monetary figure that represents the ultimate cost of the claim. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

9. The framework model First of all, we consider that uncertainty is modeled by a probability space . The flow of information is given by the natural filtration , i.e. the aug-mentation of a one dimensional Brownian filtration. Without loss of generality, we assume that . is the accumulated amount of paid claims at the end of the 1st year. Furthermore, we assume that the actual amount of claim payment is driven by classical Brownian motion, see the following expression (1) (1) Assume that the previous years of observations are large enough to determine the deterministic functions , and , which represent the drift and the volatility respectively of the specific actual amount of claim payment. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

10. The reserve procedure is determined by the following equation: or in differential form (2) where the t-continuous function is the instantaneous percen-tage of claims run – off pattern. We take a positive function for the percentage, since we want to increase or decrease the evolution of reserve only by the values of . Moreover, this function is the controller such that minimizes an appropriate quadratic cost function. The claims run off pattern is determined by the following equation: (3) or in differential form (4) Without loss of generality, is supposed to be equal to 1, and so our equations may be further simplified. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

11. The system of equations Under the assumptions above and the respective relationships (1) – (2), we obtain the following equation which describe the evolution of the claims run – off. (5) The manager controls the proportion where is a given Borel set at any instant time t in order to control the process , keeping in mind two basic constraints: • Reduce the fluctuations of and . The manager requires a small (as possi-ble) claims run – off result , and small proportions of which can be accepted both by the share – holders, the competition of the insurance market, etc and • Target to a certain value of claims run – off equal to at the end of the control period (at time t = T). Thus, the managerial office should make an effort to spread effectively any reserving proceedings, and not adopt a very conservative strategy for during the interval . A Stochastic dynamic control approach for the claim reserving of a non - life insurance

12. Moreover, the above controller should be also stochastic process, and since our decision at each instant time t must be based upon what has previously happened up to that specific time – moment, they must (at least) be measurable with respect to , i.e. the should be - measurable. In addition, we minimize the following performance function, which in our case is a quadratic cost criterion: (6) where, t < T, T > 0, R = β and G = (1-β), β is a weighting factor i.e. . Furthermore, the given interest rate is fixed. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

13. The closed form solution of the model Now, the problem is to choose the appropriate controller such that it minimizes (6). Thus, the stochastic system described above may be solved using standard tools of stochastic analysis and specifically the Hamilton – Jacobi – Bellman (HJB) equation. Firstly, we define the value function which corresponds to the optimal value of the expression (6) at time with claims run – off value equal to R. Thus, according to the relevant stochastic optimal control theory the value function satisfies the HJB equation i.e. (7) where, (8) We also define that with the additional terminal condition: (9) A Stochastic dynamic control approach for the claim reserving of a non - life insurance

14. The minimization of may be obtained by the optimal functional (or for simplicity ) which, satisfies the following necessary conditions: (10) (11) (12) The equations (11) may be solved as below (13) Since we are not interesting about the trivial solution , the (14) expression is derived as follows: (14) So, equation (14) provides the rules for controlling the proportion of claims run – off at time t inserted to the reserve procedure, see expression (2). Before going further and find explicitly the form of the optimal controller, let us develop the sufficiency condition described in (12). A Stochastic dynamic control approach for the claim reserving of a non - life insurance

15. Firstly, we calculate the partial derivative of second order i.e. So, the optimal described by the equation (14) is actually a minimum for our problem when simply (15) The condition (15) is always true and it may be checked later. Now, we proceed with the calculation of the value function . We substitute the optimal functional (14) into equation (8) and after some algebra we obtain (16) with terminal condition For the solution of the partial differential equation (16), we try a quadratic value function of the following form (17) Then, we derive respectively (18) and because of the terminal condition (9) we obtain that and (19) A Stochastic dynamic control approach for the claim reserving of a non - life insurance

16. Substituting equations (18) into partial differential equation (16), we obtain after some algebra the following result (20) The last equation holds for any value of R so, we conclude the following system of differential equations (by putting equal to zero the coefficients of , and the constant term of equation (20)). (21) (22) We may go further by solving the above differential equations system. The solutions of the (21) – (22) may be easily expressed in close format. Although, we may easily solve the differential equation (22) and obtain , actually we do not fully calculate it as the optimal control variables depends only upon . Furthermore, we may also recall the condition described by (19) and confirm that for every . A Stochastic dynamic control approach for the claim reserving of a non - life insurance

17. The solution of initial differential equation (21) is provided by the following expression: (22) Moreover, by substituting the expressions (22) into the optimal functional (14), the following format is obtained: (23) Remark 1 For , we obtain the limiting expression for (23) controller Remark 2 For a great horizon, i.e. , we can also obtain an asymptotic result A Stochastic dynamic control approach for the claim reserving of a non - life insurance

18. Sensitivity Analysis for the optimal controller A key element of the new proposal is the sensitivity analysis of the optimal controller (23). The numerical application is subject to the basic parameters r and σset out in the following table and the other subsidiary variables β, Τ ,t also shown as below. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

19. In this analysis, we obtain a stable volatility pattern (see table 1), and for different values of interest rate r, the above interesting figure derives. According to that figure the instantaneous percentage of claim run – off pattern, i.e. for is a pure decreasing continuous function both of time t and the interest rate r, which is an apparently interesting result. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

20. Conclusions – Further Results The paper investigates the classical problem of claim reserving process for Non – Life insurance companies. We enhance and empower the view of the “traditional” approaches by inserting the optimal dynamic control methodology to describe the mechanisms of the payments and the reserving process, as well. The derived optimal controller formulae provide deep insight into the mechanisms of the different parameter values and the effect in the management decisions with respect to the volatility and the interest rate. Finally, it is in progress, an illustrative numerical application with real data from a major Greek insurance company. Some more insightful comments may derive when the first results are obtained. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

21. THE END (Thank you a lot for your patience) Have a nice time in Greece A Stochastic dynamic control approach for the claim reserving of a non - life insurance

22. References (selected) • De Alba, E., Juares, M. and Moreno, T. M., 1998. Bayesian estimation of IBNR Reserves. Transaction 26th International Congress of Actuaries 4, 255 – 273. • De Jong, D. and Zehnwirth, B., 1983. Claims Reserving, state space models and Kalman filter. The Journal of the Institute of Actuaries 110, 157 – 181. • England, P. D. and Verall, R. J., 2002. Stochastic Claims Reserving in general insurance (with discussion). British Actuarial Journal 8, 443 – 544. • Haastrup, S. and Arjas, E., 1996. Claims Reserving in continuous time; a nonparametric Bayesian approach. ASTIN Bulletin 26, 139 – 164. • Haberman, S. and Renshaw, A., 1996. Generalized linear models and Actuarial Science. The Statistician 45, 407 – 436. • Hess, K. T and Schmidt, K. D., 2002. A comparison of models for the Chain – Ladder method. Insurance: Mathematics and Economics 31, 351 – 364. • Holly, S. and Hallett H. A., 1989. Optimal Control, Expectations and Uncertainty, Cambridge University Press, U.K.. • Jewell, W. S., 1989. Prediction INNYR events and delays I: continuous time, ASTIN Bulletin 19, 25 – 55. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

23. Jewell, W. S., 1990. Prediction INNYR events and delays II: discrete time, ASTIN Bulletin 20, 93 – 111. • Mack, Th. and Venter, G., 2000. A comparison of stochastic models that reproduce Chain – Ladder Reserve estimates. Insurance: Mathematics and Economics 26, 101 – 107. • Makov, U. E., Smith, A. F. M. and Liu, Y. – H., Bayesian methods in actuarial science, The Statistician 45, 503 – 515. • Renshaw, A., 1989. Chain Ladder and interactive modeling. The Journal of the Institute of Actuaries 116, 559 – 587. • Renshaw, A., 1994. On the second moment properties and the implementation of certain GLIM based stochastic Claims Reserving models. Actuarial Research paper 65. Department of Actuarial and Statistics, City University, London, UK. • Renshaw, A. and Verall, R., 1998. A stochastic model underlying the Chain Ladder technique. British Actuarial Journal 4, 903 – 926. • Schmidt, K. D., 2002. Methods and models of loss reserving based on run – off triangles – A unifying survey. Casualty Actuarial Society forum, Fall 2006, 269 – 317. • Taylor, G. C. and Ashe, F. R., 1983. Second moments of estimates of outstanding claims, Journal of Econometrics 23, 37 – 61. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

24. Tustin A., 1953. The Mechanism of Economic Systems: an Approach to the Problem of Economic Stabilisation from the Point of View of Control – System Engineering, Harvard University Press, Cambridge, Massachusetts, U.S.A.. • Van Eeghen, J., 1981. Loss reserving methods: surveys on actuarial studies I. Nationale – Nederlanden, Rotterdam. • Verall, R. J., 1989. State space representation of the Chain – Ladder linear model. The Journal of the Institute of Actuaries 116, 589 – 610. • Verall, R. J., 1990. Bayes and Empirical Bayes estimation for the Chain Ladder model. ASTIN Bulletin 20, 217 – 243. • Verall, R. J., 1990. Statistical aspects of outstanding Claims Reserving. Department of Actuarial Science and Statistics, City University, London, UK. • Verall, R. J., 1991. Chain Ladder and Maximum Likelihood. The Journal of the Institute of Actuaries 118, 489 – 499. • Verall, R. J., 1993. Negative incremental claims: the Chain Ladder and linear models. The Journal of the Institute of Actuaries 120, 171 – 183. • Verall, R. J., 1994. A method of modeling Varying – off evolutions in Claims Reserving. ASTIN Bulletin 24, 325 – 332. A Stochastic dynamic control approach for the claim reserving of a non - life insurance

25. Verall, R. J., 1996. Claims Reserving and generalized additive models. Insurance: Mathematics and Economics 19, 31 – 43. • Verall, R. J., 2000. An investigation into stochastic Claims Reserving model and the Chain – Ladder Technique. Insurance: Mathematics and Economics 26, 91 – 99. • Verall, R. J. and England, R. D., 2000. Comments on: A comparison of stochastic models that reproduce Chain – Ladder Reserve estimates, by Mack, Th. and Venter, G.. Insurance: Mathematics and Economics 26, 109 – 111. • Verall, R. J., 2004. A Bayesian generalized linear model for Bornhuetter – Ferguson method of Claims Reserving. North American Actuarial Journal 8, 67 – 89. • Verall, R. J. and England, P. D., 2005. Incorporating expert opinion into a stochastic model for the Chain – Ladder technique. Insurance: Mathematics and Economics 37, 355 – 370. A Stochastic dynamic control approach for the claim reserving of a non - life insurance