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

2. 2 The basis risk of index-based reinsurance instrumentsOutline Introduction / background Introduction / background Defining basis risk Defining basis risk Calculating basis risk Calculating basis risk Optimal hedging strategies Optimal hedging strategies

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

4. 4 The basis risk of index-based reinsurance instruments General concept BuyerSeller Fixed premium Agree on an index Variable payout Actual loss

5. 5 The basis risk of index-based reinsurance instrumentsExamples Call option on an industry loss index Call option on an industry loss index Call spread on an industry loss index Call spread on an industry loss index W: index; S: strike; L: limit; P: payout; k: payout ratio

6. 6 The basis risk of index-based reinsurance instruments Examples (continued) Industry loss warranty (ILW) Industry loss warranty (ILW) Sometimes subject to an actual loss Index-linked cat bond Index-linked cat bond  P = Principal payment  I = Interest payments  X = Parameters related to natural disaster event(s)

7. 7 The basis risk of index-based reinsurance instruments Compared to traditional indemnity instruments Advantages Advantages  Simpler underwriting  Lower moral hazard  Potentially lower cost Challenges Challenges  Tax/reporting implications  Basis risk: mismatch between payout and actual loss

8. 8 The basis risk of index-based reinsurance instrumentsOutline Introduction / background Introduction / background Defining basis risk Defining basis risk Calculating basis risk Calculating basis risk Optimal hedging strategies Optimal hedging strategies

9. 9 The basis risk of index-based reinsurance instruments Example: Mismatching of a cat option payout and the actual excess loss

10. 10 The basis risk of index-based reinsurance instruments Example: Mismatching of a cat option payout and the actual excess loss Actual loss 50100150200250300 0 100 200 300 Payout factor * Index (K*W) retention Strike (K*S)

11. 11 The basis risk of index-based reinsurance instruments Actual loss 50100150200250300 0 100 200 300 Payout factor * index ( K*W) retention strike (K*S) Basis “gain” Basis risk

12. 12 The basis risk of index-based reinsurance instruments What is “basis risk”? Actual excess loss Payout of an “comparable” reinsurance policy Payout of an index-based instrument Basis risk  Basis risk  Basis risk 

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

14. 14 The basis risk of index-based reinsurance instrumentsDefinitions Symbols Symbols  L g = actual gross loss  r t = retention  L = max(0, L g - r t ) (excess loss)  P i = payout of the index-based instrument A  P r = payout of a “ comparable” traditional reinsurance policy B

15. 15 The basis risk of index-based reinsurance instruments Definitions (continued) An index-based instrument A and a traditional reinsurance policy B are comparable if 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. 16 The basis risk of index-based reinsurance instruments Quantification of basis risk Measures based on covariance and/or linear correlation between excess loss and payout 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. 17 The basis risk of index-based reinsurance instruments Example 1: payout vs. actual excess loss Actual excess loss ($100M) Payout ($100M)

18. 18 The basis risk of index-based reinsurance instruments Example 2: payout vs. actual excess loss Actual excess loss ($100M) Payout ($100M)

19. 19 The basis risk of index-based reinsurance instruments How to differentiate the two structures?

20. 20 The basis risk of index-based reinsurance instruments How to differentiate the two structures?

21. 21 The basis risk of index-based reinsurance instruments Better quantification of basis risk Conditional probability-based measures Conditional probability-based measures  Probability distribution of payout shortfall given an excess loss  Explicit actuarial implications

22. 22 The basis risk of index-based reinsurance instruments Basis risk for reinsurance instruments Basis risk type  the mismatch between actual excess loss and payout when L > 0 Basis risk type  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  the mismatch between index and indemnity instruments when L > 0 Basis risk type  the mismatch between index and indemnity instruments when L > 0  Probability distribution of  = P r - P i  Focus on probability of “regret”

23. 23 The basis risk of index-based reinsurance instruments Basis risk for reinsurance instruments Which measure to focus on? Which measure to focus on?  To develop an optimal reinsurance program,  should be used  To address existing bias towards traditional reinsurance,  should be used

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

25. 25 The basis risk of index-based reinsurance instruments Example 3 (continued) Measure of risk 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. 26 The basis risk of index-based reinsurance instruments Example 3 (continued) Measure of success 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. 27 The basis risk of index-based reinsurance instruments Example 3 (continued) Evaluate competing strategies 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. 28 The basis risk of index-based reinsurance instruments Probability of non exceedance

29. 29 The basis risk of index-based reinsurance instruments Gross loss Net after retro  Attached at 100-year loss  Cover up to 250-year loss Probability of non exceedance

30. 30 The basis risk of index-based reinsurance instruments 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. 31 The basis risk of index-based reinsurance instruments 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

32. 32 The basis risk of index-based reinsurance instruments

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

34. 34 The basis risk of index-based reinsurance instruments Cumulative probability distribution of  (ILW - retro payout) given L > 0  “worst case”  ~ 50% of cover limit Probability of non exceedance

35. 35 The basis risk of index-based reinsurance instruments Example 4 Reinsurer in a natural disaster area Reinsurer in a natural disaster area  10% market share  Not geographically diversified within the region Goal: Goal:  same as Example 3 Evaluate competing strategies Evaluate competing strategies  same as Example 3

36. 36 The basis risk of index-based reinsurance instruments

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

38. 38 The basis risk of index-based reinsurance instruments Probability distribution of  given L > 0  worst case  = 100% of cover limit Probability of non exceedance

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

40. 40 The basis risk of index-based reinsurance instruments Index-based or indemnity: which one to use? No universally applicable answer 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 Quantitative analyses facilitate consistent decision making  Consistent objective  Optimal position at the risk/return curve  Explicit monitoring of portfolio risk

41. 41 The basis risk of index-based reinsurance instrumentsOutline Introduction / background Introduction / background Defining basis risk Defining basis risk Calculating basis risk Calculating basis risk Optimal hedging strategies Optimal hedging strategies

42. 42 The basis risk of index-based reinsurance instruments How to calculate conditional loss distributions Representation of probability distributions in cat models Representation of probability distributions in cat models Cat model provides loss distributions of gross and net losses Cat model provides loss distributions of gross and net losses For basis risk type  : calculate probability distribution of annual aggregate loss For basis risk type  : calculate probability distribution of annual aggregate loss For basis risk type  derive F  based on cat model output For basis risk type  derive F  based on cat model output

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

44. 44 The basis risk of index-based reinsurance instruments Event-based representation of loss probability in a cat model Assumptions Assumptions  n is large enough for the set to contain nearly all possible natural disaster events  N k Number of occurrences of event # k ~ Poisson Process with  ?  Events are independent

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

46. 46 The basis risk of index-based reinsurance instruments For basis risk type  Probability distribution of per event loss X Probability distribution of per event loss X  may be any losses e.g. P r, P i, , L, etc.

47. 47 The basis risk of index-based reinsurance instruments Number of times event k occurs CDF of X k

48. 48 The basis risk of index-based reinsurance instruments Event-based representation of loss probability in a cat model Loss probability distribution X k Loss probability distribution X k Frequency for event k Probability that the loss exceeds x given event k occurs

49. 49 The basis risk of index-based reinsurance instruments Event-based representation of loss probability in a cat model Probability distribution of X Probability distribution of X

50. 50 The basis risk of index-based reinsurance instruments Event-based representation of loss probability in a cat model A frequently used simplification A frequently used simplification  Assuming X k is deterministic, i.e.  Then

51. 51 The basis risk of index-based reinsurance instruments Validity of the simplification Loss Prob of Non Exc. 020406080100120 0.990 0.992 0.994 0.996 0.998 1.000 Per-event loss standard deviation / mean ___________ 0 ___________ 10% ___________ 25% ___________ 50% ___________ 75% ___________ 100%

52. 52 The basis risk of index-based reinsurance instruments How to calculate F  F  ( b | L > 0 ) = Prob(  0) = Prob(P r - P i 0) = Prob(P r - P i 0) / Prob(L > 0)

53. 53 The basis risk of index-based reinsurance instruments The problem of deriving F  ( b | L > 0 ) = Prob(P r - P i 0) / Prob(L > 0) consists of two components: (1) Prob(P r - P i < b) (2) Prob(P r - P i 0)

54. 54 The basis risk of index-based reinsurance instruments Problem 1: Prob(Z < z), where Z = X - Y

55. 55 The basis risk of index-based reinsurance instruments Problem 2: Prob(P r - P i <b) - Prob(P r - P i < b & L < 0) solution is simple only if independent or bivariate normal If independent then:

56. 56 The basis risk of index-based reinsurance instruments Problem 2 (continued) : If bivariate normal then:

57. 57 The basis risk of index-based reinsurance instrumentsOutline Introduction / background Introduction / background Defining basis risk Defining basis risk Calculating basis risk Calculating basis risk Optimal hedging strategies Optimal hedging strategies

58. 58 The basis risk of index-based reinsurance instruments Example 3 (continued) Goal: design an ILW structure such that Goal: design an ILW structure such that  Probability of default reduced from 1.0% to 0.4%  Maintaining highest possible ROE Select the optimal ILW parameters Select the optimal ILW parameters  Trigger  Limit  Exhaustive search of combinations

59. 59 The basis risk of index-based reinsurance instruments How different trigger / limit combinations affect ROE

60. 60 The basis risk of index-based reinsurance instruments How different trigger / limit combinations affect ROE and RORC

61. 61 The basis risk of index-based reinsurance instruments How different trigger / limit combinations affect ROE, RORC, and prob. of default *

62. 62 The basis risk of index-based reinsurance instruments

63. 63 How the optimal ILW structure improves the risk/return profile

64. 64 The basis risk of index-based reinsurance instruments Summary: Index-based instruments Are becoming an increasingly important risk management tool Are becoming an increasingly important risk management tool Change the risk/return profile in a different manner than traditional reinsurance Change the risk/return profile in a different manner than traditional reinsurance Require the buyer to thoroughly analyze the basis risk to Require the buyer to thoroughly analyze the basis risk to  Avoid surprise or regret  Take full advantage of index-based instruments

65. 65 The basis risk of index-based reinsurance instruments Summary: Basis risk There is no universally applicable definition of basis risk There is no universally applicable definition of basis risk  Depends on the financial objective and risk tolerance of the buyer Calculating basis risk is a nontrivial task Calculating basis risk is a nontrivial task The goal of this presentation The goal of this presentation  Promote a discussion on this topic among actuaries  Not intended to provide the “right” answer or demonstrate the “right” method