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

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

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

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

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

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

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)

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

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

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

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

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

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

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)

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:

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)

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

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

Type-I basis risk (a) Hedging effectiveness Basis risk a

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%

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)

Example 1 (cont.) q b 0.4% 43.4% 1% 41.1% 5% 19.9% conditional CDF DL

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

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

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

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

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)

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?)

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

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

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)

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%

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

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

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

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).

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

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.