COMPANY NAME

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

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Index-based instruments general concept
Buyer
Seller
Index
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• 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

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

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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)

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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

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An illustration of basis risk
Index-based recovery
Indemnity-based recovery
Reinsureds loss recovery
Reinsureds incurred loss
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• Our tasks
• Quantify/measure basis risk
• Reduce basis risk
• Optimize an index-based hedging program

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• 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

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Measures of basis risk (cont.)
L Incurred loss
LI vs. LR
Basis risk
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Measures of basis risk (cont.)
Comparing LI and LR
Calculate risk measures of L, LI and LR
(denoted yg, yi andyr) Compare the differences
among yg, yi andyr
Define DL LR - LI 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
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• 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)

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• 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

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• 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)

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• 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

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• 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

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

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Example 1 (cont.)
Underlying portfolio Net of benchmark reinsurance Net of ILW
POD (risk measure) yg1.00 yr0.40 yi0.60
Hedging effectiveness hr60.0 hi40.0
Basis risk (a) a33.3
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• 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)

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

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Ways to reduce basis risk (Example 1, cont.)
Cost95M
Cost70M
POD0.2
Cost45M
Limit (M)
POD0.4
Cost20M
POD0.6
POD0.8
technical estimates
Trigger (M)
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Ways to reduce basis risk (Example 1, cont.)
Cost95M
Cost70M
POD0.2
Cost45M
Limit (M)
POD0.4
Cost20M
POD0.6
POD0.8
technical estimates
Trigger (M)
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• 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

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• 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)

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Improvement to a (Example 1, cont.)
Underlying portfolio Net of benchmark reinsurance Net of optimal ILW
POD (risk measure) yg1.00 yr0.40 yi0.40
Hedging effectiveness hr60.0 hi60.0
Basis risk (a) a0 (what about b?)
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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
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• 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

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• 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)

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• Example 2
• Objective
• maximize r expected profit / 99VaR
• Constraints
• 99VaR lt 30M

Inward premium (K) Expected annual loss (K) Expected profit (K) 99VaR (K) r
reinsurer 10,000 2,305 7,695 54,861 14
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• 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)
A 10,000 6 30 The solution space (i.e. to be determined)
B 7,000 10 25 The solution space (i.e. to be determined)
B 20,000 6 50 The solution space (i.e. to be determined)
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• 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 0 17000 24500 29500
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• 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 10,000 5,270 1,312 3,419 14,419 23.7 106,899 15,924
Net of hedging ES 10,000 5,240 1,317 3,443 14,641 23.5 107,093 15,937
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• 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 buyers objectives.
• If the effects of basis risk can be minimized at
  the optimal cost (our topic today).

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• 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

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• 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.