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

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

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. Reinsured’s loss recovery
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. Measures of basis risk (cont.)
L = Incurred loss
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
L*I vs. L*R
Basis risk

13. Related to hedging effectiveness Related to payoff shortfall
Measures of basis risk (cont.)
Comparing L*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 b is 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. Net of benchmark reinsurance
Example 1 (cont.)
Underlying portfolio
Net of benchmark reinsurance
Net of ILW
POD (risk measure)
yg=1.00%
yr=0.40%
yi=0.60%
Hedging effectiveness
hr=60.0%
hi=40.0%
Basis risk (a)
a=33.3%

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 b is measured by the quantile (sq) of the conditional CDF scaled by the limit of the benchmark reinsurance contract (lr)

22. Example 1 (cont.) q b
0.4%
43.4% 1%
41.1% 5%
19.9%
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 ($M)
POD=0.4%
Cost=20M*
POD=0.6%
POD=0.8%
Trigger ($M)
* technical estimates

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

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. Net of benchmark reinsurance
Improvement to a (Example 1, cont.)
Underlying portfolio
Net of benchmark reinsurance
Net of optimal ILW
POD (risk measure)
yg=1.00%
yr=0.40%
yi=0.40%
Hedging effectiveness
hr=60.0%
hi=60.0%
Basis risk (a)
a=0%
(what about b?)

29. Improvement to b (Example 1, cont.)q
b (original)
b (optimal)
0.4%
43.4%
19.3% 1%
41.1%
17.7% 5%
19.9%
1.8%
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. Expected annual loss ($K)
Example 2
Objective:
maximize r = expected profit / 99%VaR
Constraints:
99%VaR < $30M
Inward premium ($K)
Expected annual loss ($K)
Expected profit ($K)
99%VaR ($K) r
reinsurer
10,000
2,305
7,695
54,861
14%

33. Example 2 (cont.) Available ILWs region trigger ($M) rate-on-line
capacity available ($M)
amount to purchase A
3,500
10% 20
The solution space (i.e. to be determined)
10,000 6% 30 B
7,000 25
20,000 50

34. Example 2 (cont.) GA-based vs. exhaustive search (ES) solutions
Amount purchased ($K)
A-3.5b
A-10b
B-7b
B-20b
Genetic algorithm
231
17222
24625
29563
Exhaustive search
17000
24500
29500

35. Example 2 (cont.) Results of optimization Inward premium
Cost of hedging
Expected annual loss
Expected profit
99% VaR r
99% TVaR SD
Underlying portfolio
10,000 -
2,305
7,695
54,861
14.0%
151,513
19,872
Net of hedging – GA
5,270
1,312
3,419
14,419
23.7%
106,899
15,924
Net of hedging – ES
5,240
1,317
3,443
14,641
23.5%
107,093
15,937

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
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),
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,
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,
Zeng, L., 2000: On the basis risk of industry loss warranties, The Journal of Risk Finance, 1(4)