Loss Modeling: Introducing Simulation using a Simplified Real World Problem

1. Loss Modeling: Introducing Simulation using a Simplified Real World Problem Domingo Castelo Joaquin Illinois State University dcjoaqu@ilstu.edu Klugman, Stuart et al.(2004), Loss Models: From Data to Decisions, 2nd Ed., New York: Wiley, 618-621. Joaquin, Domingo Castelo (2007), Loss Modeling using Spreadsheet-based Simulation, Risk Management and Insurance Review, Vol. 10, No. 2, 283-297. http://www.blackwell-synergy.com/loi/RMIR

2. Target: PV of Payments for Losses covered by a one-year policy Loss Payments

3. Simulating the PV of Loss Payments covered by a one-year Policy

4. Cj = time of the jth loss, Cj-Cj-1 ~ i.i.d. Exponential(0.2), Co=0 Xj = amount of loss associated with loss event jXj ~ i.i.d. Pareto(1000,3)

5. Inter-arrival Times: Exponential

6. Loss Severity: Pareto

7. Lj = time from occurrence to payment for the jth loss Lj ~ Weibull(1.5,LN(Xj+1)/6)Tj= date of payment for the jth loss, Tj = Cj+Lj Claim Processing Time: Weibull

8. Claim Processing Time: Weibull

9. Target: PV of Payments for Losses covered by a one-year policy Tj= date of payment for the jth loss, Tj = Cj+Lj

10. RΔT = required log return over ΔT years, RΔT ~ N(0.06 ΔT,0.02√ΔT) V(T*n) = EXP(-RΔT*1) x EXP(-RΔT*2) x…x EXP(-RΔT*n) V(T*n)= EXP(-Σi=1,..,n RΔT*i)

12. Frequency and Severity of Loss

13. Claim Settlement

14. Discount Rates and Present Values

15. PV of Specific Payments

16. Output Cells

17. Simulation Settings

18. Summary Output

19. If there are no payments, then ΔT is undefined and the standard deviation for the log return also will be undefined. A discount rate calculation will not be carried out and an “error” will be registered. A Note on Error Counts

20. Beyond the error counts… • Do the numbers seem too small or too large or have the wrong sign?For example, loss figures should be non-negative. • Do the numbers make sense relative to other numbers?For example, retained loss should not exceed aggregate loss • Does a statistic look right by itself or in relation to other statistics?For example, the average number of losses per year should be close the variance if the inter-arrival times are exponentially distributed. • Problems? Review the underlying cell formulas for typos.

21. Loss severity Loss frequency Claim processing time Discount rates The one-year policy period usually would have elapsed after six loss events Most of the payments are made in a short time There is not much opportunity for the power of compounding to assert itself Tornado Diagram for PV of Aggregate Loss Payments

22. References Klugman, Stuart et al.(2004), Loss Models: From Data to Decisions, 2nd Ed., New York: Wiley, 618-621. Joaquin, Domingo Castelo (2007), Loss Modeling using Spreadsheet-based Simulation, Risk Management and Insurance Review, Vol. 10, No. 2, 283-297. http://www.blackwell-synergy.com/loi/RMIR dcjoaqu@ilstu.edu