hedging catastrophe risk using index-based reinsurance instruments lixin zeng
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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. Presentation Highlights Index-based instruments can play a key role in managing catastrophe risk and reducing earnings volatility - PowerPoint PPT PresentationTRANSCRIPT
Hedging Catastrophe Risk Using Index-Based Reinsurance Instruments
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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
Hedging Catastrophe Risk Using Index-Based Reinsurance Instruments
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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
Hedging Catastrophe Risk Using Index-Based Reinsurance Instruments
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Index-based instruments: general concept
BuyerBuyer SellerSellerFixed premiumFixed premium
IndexIndex
Variable payoutVariable payout
Hedging Catastrophe Risk Using Index-Based Reinsurance Instruments
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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
Hedging Catastrophe Risk Using Index-Based Reinsurance Instruments
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Industry loss warranty (ILW)
* Payoff XI might not exceed actual loss, depending on accounting treatment
t
tI iI
iIlX
,0
,
Payoff*Payoff*Industry Industry
lossloss
Industry loss Industry loss triggertrigger
limitlimit
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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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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)
Some advantages of index-based instruments
Hedging Catastrophe Risk Using Index-Based Reinsurance Instruments
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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
Potential drawbacks of index-based instruments
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An illustration of basis risk
Reinsured’s incurred loss
Re
insu
red
’s lo
ss r
eco
very Index-based recovery
Indemnity-based recovery
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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.)
LL = Incurred loss = Incurred loss
XXII = Index-based payoffIndex-based payoff
L*L*II = L -= L - XXII = loss net of index-based payoff
XXRR = Benchmark payoffBenchmark payoff
L*L*RR = L - = L - XXRR = loss net of benchmark payoff
L*L*I I vs. L*L*RR
Basis riskBasis risk
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Measures of basis risk (cont.)
ComparingComparing L*L*I I and L*L*RR
CalculateCalculate risk measures of L, L*L*I I
andand L* L*RR (denoted yygg, yyii andyyrr)
Compare the differences among Compare the differences among yygg, yyii andyyrr
Type-I basis riskType-I basis risk ((
Related to hedging effectivenessRelated to hedging effectiveness
Define Define LL = = L*L*RR - - L*L*I I = = XXII - - XXRR
Analyze the conditional Analyze the conditional probability distribution of probability distribution of LL
Type-II basis riskType-II basis risk ((
Related to payoff shortfallRelated to payoff shortfall
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Type-I basis risk ()
Hedging effectiveness
Basis risk
Related references: Major (1999), Harrington and Niehaus (1999), Cummins, et. al. (2003), and Zeng (2000)
gii
grr
yyh
yyh
/1
/1
ri hh /1
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Type-II basis risk ()
Based on the payoff shortfall L
- L is a problem only when a large loss occurs
- We are primarily concerned about negative L
- Calculate the conditional cumulative distribution function (CDF) of L:
)0(
)0()0|(
R
RR Xprob
XsLprobXsLprob
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Type-II basis risk (, cont.)
Basis risk is measured by
- The quantile (sq) of the conditional CDF
- Scaled by the limit of the benchmark reinsurance contract (lr)
r
q
l
s )0,max(
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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 reinsuranceRetention = 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: ILWIndex = 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 ( and ) for this example
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Type-I basis risk ()
Hedging effectiveness
Basis risk
gii
grr
yyh
yyh
/1
/1
ri hh /1
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Example 1 (cont.)
company loss ($M)
CD
F
1000 2000 3000 4000 5000 6000
0.90
0.92
0.94
0.96
0.98
1.00
Underlying portfolio
Net of benchmark
reinsurance
Net of ILW
POD (risk measure)
yg=1.00% yr=0.40% yi=0.60%
Hedging effectiveness
hr=60.0% hi=40.0%
Basis risk () =33.3%
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Type-II basis risk ()
Based on the payoff shortfall L- L is a problem only when a large loss occurs- We are primarily concerned about negative L- The conditional cumulative distribution function (CDF) of L:
Basis risk is measured by the quantile (sq) of the conditional CDF scaled by the limit of the benchmark reinsurance contract (lr)
)0|( RXsLprob
r
q
l
s )0,max(
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Example 1 (cont.)
payoff differential ($M)
CC
DF
-450 -400 -350 -300 -250 -200 -150
0.0
0.02
0.04
0.06
0.08
(II)
L
cond
ition
al C
DF
q
0.4% 43.4%
1% 41.1%
5% 19.9%
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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, should be the primary measure
- to address the bias toward traditional indemnity-based reinsurance, should be the primary measure
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Trigger ($M)
Lim
it ($
M)
6000 8000 10000 12000 14000
050
010
0015
0020
0025
00
Ways to reduce basis risk (Example 1, cont.)
Trigger ($M)
Lim
it ($
M)
POD=0.6%
POD=0.4%
POD=0.8%
POD=0.2% Cost=45M*
Cost=20M*
Cost=70M*Cost=95M*
* technical estimates
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Trigger ($M)
Lim
it ($
M)
6000 8000 10000 12000 14000
050
010
0015
0020
0025
00
Ways to reduce basis risk (Example 1, cont.)
Trigger ($M)
Lim
it ($
M)
POD=0.6%
POD=0.4%
POD=0.8%
POD=0.2% Cost=45M*
Cost=20M*
Cost=70M*Cost=95M*
* technical estimates
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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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company loss ($M)
CD
F
1000 2000 3000 4000 5000 6000
0.90
0.92
0.94
0.96
0.98
1.00
Improvement to (Example 1, cont.)
Underlying portfolio
Net of benchmark
reinsurance
Net of optimal
ILW
POD (risk measure)
yg=1.00% yr=0.40% yi=0.40%
Hedging effectiveness
hr=60.0% hi=60.0%
Basis risk () =0%(what about ?)
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payoff differential ($M)
CC
DF
-450 -400 -350 -300 -250 -200 -150
0.0
0.02
0.04
0.06
0.08
(IV)
L
cond
ition
al C
DF
q (original)
(optimal)
0.4% 43.4% 19.3%
1% 41.1% 17.7%
5% 19.9% 1.8%
Improvement to (Example 1, cont.)
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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 / 99%VaR
Constraints:99%VaR < $30M
Inward premium
($K)
Expected annual loss
($K)
Expected profit ($K)
99%VaR ($K)
r
reinsurer 10,000 2,305 7,695 54,861 14%
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Example 2 (cont.)
Available ILWs
region trigger ($M) rate-on-line capacity available ($M)
amount to purchase
A 3,500 10% 20 The solution space (i.e. to be determined)
A 10,000 6% 30
B 7,000 10% 25
B 20,000 6% 50
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Example 2 (cont.)
GA-based vs. exhaustive search (ES) solutions
Amount purchased ($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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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
Example 2 (cont.)
Results of optimization
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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 buyer’s 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.