Hedging Catastrophe Risk Using Index-Based Reinsurance Instruments Lixin Zeng

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


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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

BuyerBuyer SellerSellerFixed premiumFixed premium

IndexIndex

Variable payoutVariable payout


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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

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


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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 ()


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%


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.)


Net of benchmark

reinsurance

Net of optimal

ILW

POD (risk measure)

yg=1.00% yr=0.40% yi=0.40%


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


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.

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Hedging Catastrophe Risk Using Index-Based Reinsurance Instruments Lixin Zeng