Hedging catastrophe risks using index-based instruments CAS reinsurance seminar New York

1. Hedging catastrophe risks using index-based instruments CAS reinsurance seminar New York Feb. 28, 2002 Lixin Zeng, Ph.D. Willis Re

2. Outline • Introduction / background • Defining basis risk • Calculating basis risk • Optimal hedging strategies

3. Index-based risk management instrument • Index types • Industry losses • Geophysical parameters • Instruments • Cat options • Industry loss warranty (ILW) • Index-linked cat bonds • Other index-linked instruments (yield guarantee, index-based WC products, etc.)

4. Fixed premium Actual loss Variable payout General concept Buyer Seller Agree on an index

5. Examples • Call option on an industry loss index • Call spread on an industry loss index W: index; S: strike; L: limit; P: payout; k: payout ratio

6. Examples (continued) • Industry loss warranty (ILW) Sometimes subject to an actual loss • Index-linked cat bond • P = Principal payment • I = Interest payments • X = Parameters related to natural disaster event(s)

7. Compared to traditional indemnity instruments • Advantages • Simpler underwriting • Lower moral hazard • Potentially lower cost • Challenges • Tax/reporting implications • Basis risk: mismatch between payout and actual loss

8. Outline • Introduction / background • Defining basis risk • Calculating basis risk • Optimal hedging strategies

9. Example: Mismatching of a cat option payout and the actual excess loss

10. Example: Mismatching of a cat option payout and the actual excess loss 300 Strike (K*S) 200 Payout factor * Index (K*W) 100 retention 0 50 100 150 200 250 300 Actual loss

11. Basis “gain” Basis risk 300 strike (K*S) 200 Payout factor * index (K*W) 100 retention 0 50 100 150 200 250 300 Actual loss

12. What is “basis risk”? Actual excess loss Basis risk a Basis risk g Payout of an “comparable” reinsurance policy Payout of an index-based instrument Basis risk b

13. Why do we care about basis risk? • Type a • How effective is the index-based instrument in reducing the risk of the underlying portfolio • Type b • How does the index-based instrument compare to the traditional reinsurance policy • Type g • Probability of exhausting the limit, counter-party credit risk, contract dispute, etc.

14. Definitions • Symbols • Lg = actual gross loss • rt = retention • L = max(0, Lg - rt) (excess loss) • Pi = payout of the index-based instrument A • Pr = payout of a “comparable” traditional reinsurance policy B

15. Definitions (continued) • An index-based instrument A and a traditional reinsurance policy B are comparable if • The strike of A and the attachment of B have similar probabilities of attaching • A and B have similar payout limit • The costs of A and B are similar

16. Quantification of basis risk • Measures based on covariance and/or linear correlation between excess loss and payout • Easy to calculate • Commonly used • Actuarial meaning not clear • Can be misleading

17. Example 1: payout vs. actual excess loss Payout ($100M) Actual excess loss ($100M)

18. Example 2: payout vs. actual excess loss Payout ($100M) Actual excess loss ($100M)

19. How to differentiate the two structures?

20. How to differentiate the two structures?

21. Better quantification of basis risk • Conditional probability-based measures • Probability distribution of payout shortfall given an excess loss • Explicit actuarial implications

22. Basis risk for reinsurance instruments • Basis risk type a: the mismatch between actual excess loss and payout when L > 0 • Focus on how the net loss probability will change with different reinsurance strategies • Basis risk type b: the mismatch between index and indemnity instruments when L > 0 • Probability distribution of b = Pr - Pi • Focus on probability of “regret”

23. Basis risk for reinsurance instruments • Which measure to focus on? • To develop an optimal reinsurance program, a should be used • To address existing bias towards traditional reinsurance, b should be used

24. Example 3 • Reinsurer in a natural disaster area • 15% market share • Geographically diversified within the region • Goal: • Reduce probability of default from 1% to 0.4% • Enhance risk/return profile • Reduce earning volatility

25. Example 3 (continued) • Measure of risk • Probability of default • Probable maximum loss or Value at Risk with a 0.4% exceeding probability: a proxy of risk capital • Tail Value at Risk (TVaR): a coherent risk measure • Semi-deviation of underwriting profit (i.e. standard deviation of negative underwriting profit): related to earning volatility

26. Example 3 (continued) • Measure of success • Return on equity (ROE) expected profit / company equity • Return on Risk Capital (RORC) expected profit / PML • Modified Sharpe ratio expected profit / semi-deviation

27. Example 3 (continued) • Evaluate competing strategies • Traditional retro policy • retention: 100-year PML • limit: 250-year PML - 100-year PML • ILW (i.e. a binary call option) • trigger: 100-year industry loss • limit: same as above • Industry loss index call option (ICO) • strike: 90% of 100-year industry loss • limit: same as above

28. Probability of non exceedance

29. Probability of non exceedance Gross loss Net after retro • Attached at 100-year loss • Cover up to 250-year loss

30. Gross loss Net after retro Net after ILW • Attached at industry 100-year loss • Same limit as the indemnity contract above Probability of non exceedance

31. Gross loss Net after retro Net after ILW Net after Index Call Option • Attached at 90% of industry 100-year loss • Same limit as the indemnity contract above Probability of non exceedance

33. Probability density of b (ILW - retro payout) given L > 0 Basis “gain” Basis risk

34. Cumulative probability distribution of b (ILW - retro payout) given L > 0 “worst case” b ~ 50% of cover limit Probability of non exceedance b

35. Example 4 • Reinsurer in a natural disaster area • 10% market share • Not geographically diversified within the region • Goal: • same as Example 3 • Evaluate competing strategies • same as Example 3

37. Probability density of b given L > 0 Basis risk Basis “gain” b

38. Probability distribution of b given L > 0 worst case b = 100% of cover limit Probability of non exceedance b

39. Evaluating pros and cons of using index-based instruments: Factors to consider • Lower margin than a comparable retro • At the same premium, it offers greater reduction of expected loss • Basis risk • Reasonably small for geographically diversified exposures • Potential for negative surprise for concentrated portfolio • Don’t count on the “basis gain”

40. Index-based or indemnity: which one to use? • No universally applicable answer • Depends on financial objective and risk tolerance • A combination of subjective judgment and objective analysis • Quantitative analyses facilitate consistent decision making • Consistent objective • Optimal position at the risk/return curve • Explicit monitoring of portfolio risk

41. Outline • Introduction / background • Defining basis risk • Calculating basis risk • Optimal hedging strategies

42. How to calculate conditional loss distributions • Representation of probability distributions in cat models • Cat model provides loss distributions of gross and net losses • For basis risk type a: calculate probability distribution of annual aggregate loss • For basis risk type b: derive Fbbased on cat model output

43. Event-based representation of loss probability in a cat model • Cat model output Loss due to simulated event #k Rate of event #k (average number per year)

44. Event-based representation of loss probability in a cat model • Assumptions • n is large enough for the set to contain nearly all possible natural disaster events • NkNumber of occurrences of event #k ~ Poisson Process with • ? • Events are independent

45. For basis risk type a • Probability distribution of annual aggregate loss after reinsurance or index-based instrument • Available approaches • Simulation based on per event losses • FFT (e.g. Wang, 1998)

46. For basis risk type b • Probability distribution of per event loss X • may be any losses e.g. Pr, Pi, b, L, etc.

47. Number of times event k occurs CDF of Xk

48. Event-based representation of loss probability in a cat model • Loss probability distribution Xk Frequency for event k Probability that the loss exceeds x given event k occurs

49. Event-based representation of loss probability in a cat model • Probability distribution of X

50. Event-based representation of loss probability in a cat model • A frequently used simplification • Assuming Xkis deterministic, i.e. • Then