Hedging Catastrophe Risk Using Index-Based Reinsurance Instruments Lixin Zeng

1. Hedging Catastrophe Risk Using Index-Based Reinsurance Instruments Lixin Zeng 2003 CAS Seminar on Reinsurance June 1-3, 2003 Philadelphia, Pennsylvania

2. Presentation Highlights • Index-based instruments can play a key role in managing catastrophe risk and reducing earnings volatility • The issue of basis risk • Possible solutions

3. Fixed premium Variable payout Index-based instruments: general concept Buyer Seller Index

4. General concept(continued) • Instrument types • Index-based catastrophe options • Industry loss warranty (ILW) a.k.a. original loss warranty (OLW) • Index-linked cat bonds • Index types • Weather and/or seismic parameters • Modeled losses • Industry losses

5. limit Industry loss trigger Payoff* Industry loss Industry loss warranty (ILW) * Payoff XI might not exceed actual loss, depending on accounting treatment

6. Industry loss warranty (ILW) • Simple • Can be combined to replicate other payoff patterns • Different regional industry loss indices • Different triggers • Used as examples in this presentation

7. Some advantages of index-based instruments • Simplified disclosure and underwriting • Practically free from moral hazard • Opens additional sources of possible capacity (e.g. capital market) • Potentially lower margin and cost • Attractive asset class for capital market investors • Selected background references: Litzenberger et. al. (1996), Doherty and Richter (2002), Cummings, et. al. (2003)

8. Potential drawbacks of index-based instruments • Form (reinsurance or derivative) may affect accounting • Basis risk – the random difference between actual loss and index-based payout • The term “basis risk” came from hedging using futures contracts

9. An illustration of basis risk Index-based recovery Indemnity-based recovery Reinsured’s loss recovery Reinsured’s incurred loss

10. Our tasks • Quantify/measure basis risk • Reduce basis risk • Optimize an index-based hedging program

11. Measures of basis risk • Rarely are 100% of incurred losses are hedged; instead, we usually hedge large losses only • Index-based payoff vs. a benchmark payoff • Benchmark • Indemnity-based reinsurance contract, e.g., a catastrophe treaty • Other types of risk management tools

12. XI= Index-based payoff L*I= L -XI = loss net of index-based payoff XR = Benchmark payoff L*R= L - XR = loss net of benchmark payoff Measures of basis risk (cont.) L = Incurred loss L*I vs. L*R Basis risk

13. Measures of basis risk (cont.) ComparingL*I and L*R Calculate risk measures of L, L*I and L*R(denoted yg, yi andyr) Compare the differences among yg, yi andyr Define DL = L*R - L*I = XI - XR Analyze the conditional probability distribution of DL Type-I basis risk(a) Related to hedging effectiveness Type-II basis risk(b) Related to payoff shortfall

14. Type-I basis risk (a) • Hedging effectiveness • Basis risk a • Related references: Major (1999), Harrington and Niehaus (1999), Cummins, et. al. (2003), and Zeng (2000)

15. Type-II basis risk (b) • Based on the payoff shortfall DL • DL is a problem only when a large loss occurs • We are primarily concerned about negative DL • Calculate the conditional cumulative distribution function (CDF) of DL:

16. Type-II basis risk (b, cont.) • Basis risk bis measured by • The quantile (sq) of the conditional CDF • Scaled by the limit of the benchmark reinsurance contract (lr)

17. Example 1 • Regional property insurance company wishes to reduce probability of default (POD)* from 1% to 0.4% at the lowest possible cost • Benchmark strategy: catastrophe reinsurance Retention = 99th percentile probable maximum loss (PML) Limit = 99.6th percentile PML – 99th percentile PML * Default is simply defined as loss exceeding surplus

18. Example 1 (cont.) • Alternative strategy: ILW Index = industry loss for the region where the company conducts business Trigger = 99th percentile industry loss Limit = 99.6th percentile company PML – 99th percentile company PML (same as the benchmark) • Next: show the two measures of basis risk (a and b) for this example

19. Type-I basis risk (a) • Hedging effectiveness • Basis risk a

20. Example 1 (cont.)

21. Type-II basis risk (b) • Based on the payoff shortfall DL • DL is a problem only when a large loss occurs • We are primarily concerned about negative DL • The conditional cumulative distribution function (CDF) of DL: • Basis risk bis measured by the quantile (sq) of the conditional CDF scaled by the limit of the benchmark reinsurance contract (lr)

22. Example 1 (cont.) conditional CDF DL

23. Which basis risk measure to use? • They view basis risk from different angles • Which one to use as the primary measure depends on the objective • to structure a reinsurance program with optimal hedging effectiveness, a should be the primary measure • to address the bias toward traditional indemnity-based reinsurance, b should be the primary measure

24. Ways to reduce basis risk (Example 1, cont.) Cost=95M* Cost=70M* POD=0.2% Cost=45M* Limit ($M) POD=0.4% Cost=20M* POD=0.6% POD=0.8% * technical estimates Trigger ($M)

25. Ways to reduce basis risk (Example 1, cont.) Cost=95M* Cost=70M* POD=0.2% Cost=45M* Limit ($M) POD=0.4% Cost=20M* POD=0.6% POD=0.8% * technical estimates Trigger ($M)

26. Keys to reducing basis risk • Cost/benefit analysis • Should be an integral part of the process of building an optimal hedging program • Accomplish specific risk management objectives at the lowest possible cost • Maximize risk reduction given a budget • Objective: building an optimal hedging program using index-based instruments

27. Building an optimal hedging program • Specify constraints For Example 1: POD≤ 0.4% • Define an objective function For Example 1: cost of ILW = f( ILW trigger, limit, …) • Search for the hedging structure such that • The objective function is minimized or maximized • The constraints are satisfied For Example 1: find the ILW that costs the least such that POD≤ 0.4% • References: Cummins, et. al. (2003) and Zeng (2000)

28. Improvement to a (Example 1, cont.)

29. Improvement to b (Example 1, cont.) conditional CDF DL

30. Building an optimal hedging program (cont.) • Real-world problem • Exposures to various perils in several regions • Multiple ILWs and other index-based instruments are available • Same optimization principle but requires a robust implementation • Challenges to traditional optimization approach • Non-linear and non-smooth objective function and constraints • Local vs. global optimal solutions

31. Building an optimal hedging program (cont.) • A viable solution based on the genetic algorithm (GA) • Less prone to being trapped in a local solution • Satisfactory numerical efficiency • More robust in handling non-linear and non-smooth constraints and objective function • GA reference: Goldberg (1989)

32. Example 2 • Objective: maximize r = expected profit / 99%VaR • Constraints: 99%VaR < $30M

33. Example 2 (cont.) • Available ILWs

34. Example 2 (cont.) • GA-based vs. exhaustive search (ES) solutions

35. Example 2 (cont.) • Results of optimization

36. Summary: basis risk may not be a problem… • If the buyer is willing to accept some uncertainty in payouts in exchange for the advantages of an index based structure. • If basis risk does not pose an impediment to achieving the buyer’s objectives. • If the effects of basis risk can be minimized at the optimal cost (our topic today).

37. Areas for ongoing and future research • Appropriate constraints and objective functions for optimal hedging • The choice of risk measure • Bias toward using traditional reinsurance • Parameter uncertainty • The sensitivity of the loss model results to parameter uncertainty (e.g., cat model to assumption of earthquake recurrence rate) • The sensitivity of the optimal solution to the choice of risk measures and objective function

38. References • Artzner, P., F. Delbaen, J.-M. Eber and D. Heath, 1999, Coherent Measures of Risk, Journal of Mathematical Finance, 9(3), pp. 203-28. • Cummins, J. D., D. Lalonde, and R. D. Phillips, 2003: The basis risk of catastrophic-loss index securities, to appear in the Journal of Financial Economics. • Doherty, N.A. and A. Richter, 2002: Moral hazard, basis risk, and gap insurance. The Journal of Risk and Insurance, 69(1), 9-24. • Goldberg, D.E., 1989: Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley Pub Co, 412pp. • Harrington S. and G. Niehaus, 1999: Basis risk with PCS catastrophe insurance derivative contracts. Journal of Risk and Insurance, 66(1), 49-82. • Litzenberger, R.H., D.R. Beaglehole, and C.E. Reynolds, 1996: Assessing catastrophe reinsurance-linked securities as a new asset class. Journal of Portfolio Management, Special Issue Dec. 1996, 76-86. • Major, J.A., 1999: Index Hedge Performance: Insurer Market Penetration and Basis Risk, in Kenneth A. Froot, ed., The Financing of Catastrophe Risk (Chicago: University of Chicago Press). • Meyers, G.G., 1996: A buyer's guide for options and futures on a catastrophe index, Casualty Actuarial Society Discussion Paper Program, May, 273-296. • Zeng, L., 2000: On the basis risk of industry loss warranties, The Journal of Risk Finance, 1(4) 27-32.