Insurance Applications of Insurance Applications of Bivariate DistributionsBivariate Distributions

David L. Homer & David R. David L. Homer & David R. ClarkClark

CAS Annual MeetingCAS Annual Meeting

November 2003November 2003

AGENDA:AGENDA:

Explain the Insurance problem Explain the Insurance problem being addressedbeing addressed

Show the mathematical Show the mathematical “machinery” used to address the “machinery” used to address the problemproblem

Provide a numerical exampleProvide a numerical example

AGENDA:AGENDA:

Explain the Insurance problem Explain the Insurance problem being addressedbeing addressed

Show the mathematical Show the mathematical “machinery” used to address the “machinery” used to address the problemproblem

Provide a numerical exampleProvide a numerical example

The PlayersThe Players::

Insured: Dietrichson DrillingInsured: Dietrichson DrillingA large account with predictable A large account with predictable

annual annual losseslosses

Insurer: Pacific All Risk Insurance Co.Insurer: Pacific All Risk Insurance Co.

Actuary:Actuary: YouYou

The Pricing Problem:The Pricing Problem:

Pacific All-Risk Insurance Company Pacific All-Risk Insurance Company sells a product that provides sells a product that provides coverage on bothcoverage on both

Specific Excess – individual losses above Specific Excess – individual losses above 600,000600,000

Aggregate Excess – above the sum of all Aggregate Excess – above the sum of all retained losses capped at 3,000,000 in retained losses capped at 3,000,000 in the aggregatethe aggregate

3,000,000 8,000,000

1,000,000

600,000

Aggregate Losses

Per-

occ

urr

en

ce

Retained by the insured

Per-Occurrence Layer

Stop Loss Layer

Policy Structure proposed for our insured Dietrichson Drilling

How do we price this product?How do we price this product?

Expected Losses are straight-Expected Losses are straight-forwardforward

Expected Losses for the two Expected Losses for the two coverages are additivecoverages are additive

Separate distributions are straight-Separate distributions are straight-forwardforward

A combined distribution is NOTA combined distribution is NOT

AGENDA:AGENDA:

Explain the Insurance problem Explain the Insurance problem being addressedbeing addressed

Show the mathematical Show the mathematical “machinery” used to address the “machinery” used to address the problemproblem

Provide a numerical exampleProvide a numerical example

How do we estimate a single How do we estimate a single distribution?distribution?

Define frequency and severity Define frequency and severity distributions, then:distributions, then:

Heckman-MeyersHeckman-Meyers Recursive methods (Panjer)Recursive methods (Panjer) SimulationSimulation Fast Fourier Transform (FFT)Fast Fourier Transform (FFT)

Key Elements of FFT Technique:Key Elements of FFT Technique:

Discretized severity vector Discretized severity vector x=(xx=(x00 ,…,x ,…,xn-1n-1)) FFT formulaFFT formula

IFFT formulaIFFT formula

1

0

)/2exp()(~ n

jjkk

nijkxxFFTx

1

0

)/2exp(~1)~(

n

jjkk

nijkxn

xIFFTx

Convolution Theorem:The transform of the sum is equal to the product of the transforms.

To sum up j independent identical variables:

kkk yFFTxFFTyxFFT )()()(

jkk

j xFFTxFFT )()( *

Probability Generating Function:Probability Generating Function:

The PGFPGFN N is a short-cut method for

combining the distributions for each possible number of claims.

It does all of the convolutions for us!

)()Pr()(0

j

NjN tEjNttPGF

Putting it all together we obtain the Putting it all together we obtain the aggregate probability vector aggregate probability vector zz from from the severity probability vector the severity probability vector xx and and the claim count the claim count PGFPGFN N ::

)))((( xFFTPGFIFFTz N

Bivariate case is the same, but using Bivariate case is the same, but using a MATRIX instead of a VECTOR.a MATRIX instead of a VECTOR.

becomesbecomes…

)))((( xFFTPGFIFFTz N

)))((( xNz MFFTPGFIFFTM

AGENDA:AGENDA:

Explain the Insurance problem Explain the Insurance problem being addressedbeing addressed

Show the mathematical Show the mathematical “machinery” used to address the “machinery” used to address the problemproblem

Provide a numerical exampleProvide a numerical example

Pacific All-Risk: Severity DistributionPacific All-Risk: Severity Distribution

Bivariate Severity Mx

0 0.00%

200,000 37.80%

400,000 23.50%

600,000 14.60%

800,000 9.10%

1,000,000 15.00%

Primary

Excess Marginal

0 200,000 400,000 600,000

0 0.00% 0.00% 0.00% 0.00% 0.00%

200,000 37.80% 0.00% 0.00% 0.00% 37.80%

400,000 23.50% 0.00% 0.00% 0.00% 23.50%

600,000 14.60% 9.10% 15.00% 0.00% 38.70%

800,000 0.00% 0.00% 0.00% 0.00% 0.00%

1,000,000 0.00% 0.00% 0.00% 0.00% 0.00%

Excess Marginal 75.90% 9.10% 15.00% 0.00% 100.00%

Pri

mary

Single Claim Severity x

Negative Binomial PGF for claim Negative Binomial PGF for claim countscounts

with Mean=5 and Variance=6:with Mean=5 and Variance=6:

25)56/(5 )2.2.1())15/6(5/6()(2 tttPGF

Bivariate Aggregate Matrix Mz

0 200,000 400,000 600,000 800,000 1,000,0001,200,000

0 1.05% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

200,000 1.65% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

400,000 2.38% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

600,000 3.09% 0.40% 0.66% 0.00% 0.00% 0.00% 0.00%

800,000 3.34% 0.65% 1.07% 0.00% 0.00% 0.00% 0.00%

1,000,000 3.39% 0.96% 1.58% 0.00% 0.00% 0.00% 0.00%

1,200,000 3.22% 1.27% 2.16% 0.26% 0.21% 0.00% 0.00%

1,400,000 2.86% 1.40% 2.44% 0.44% 0.36% 0.00% 0.00%

1,600,000 2.43% 1.45% 2.59% 0.66% 0.54% 0.00% 0.00%

1,800,000 1.97% 1.40% 2.57% 0.90% 0.78% 0.09% 0.05%

2,000,000 1.54% 1.26% 2.38% 1.02% 0.92% 0.15% 0.08%

2,200,000 1.17% 1.09% 2.12% 1.08% 1.01% 0.24% 0.13%

2,400,000 0.86% 0.90% 1.80% 1.08% 1.05% 0.33% 0.20%

2,600,000 0.62% 0.72% 1.47% 1.00% 1.01% 0.38% 0.24%

2,800,000 0.43% 0.55% 1.16% 0.88% 0.93% 0.42% 0.27%

3,000,000 0.29% 0.41% 0.89% 0.75% 0.82% 0.43% 0.29%

Pri

mary

Excess

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%0

200,

000

400,

000

600,

000

800,

000

1,00

0,00

0

1,20

0,00

0

1,40

0,00

0

1,60

0,00

0

1,80

0,00

0

2,00

0,00

0

2,20

0,00

0

2,40

0,00

0

2,60

0,00

0

2,80

0,00

0

3,00

0,00

0

3,20

0,00

0

3,40

0,00

0

3,60

0,00

0

3,80

0,00

0

4,00

0,00

0

Unconditional Distribution;Mean = 391,000

Conditional on Stop-Loss being Hit;Mean = 830,334

Probability Distribution for Per-Occurrence Excess Losses

Expected Per-Occurrence Loss Expected Per-Occurrence Loss

== 391,000391,000 overalloverall

== 830,334830,334 in scenarios where in scenarios where stop loss is hitstop loss is hit

Both coverages go bad at the same Both coverages go bad at the same time!time!

Other Applications:Other Applications:

Generation of Large & Small losses Generation of Large & Small losses for DFAfor DFA

Loss and ALAE with separate limitsLoss and ALAE with separate limits

Any other bivariate phenomenonAny other bivariate phenomenon(e.g., WC medical and indemnity)(e.g., WC medical and indemnity)

Questions or Comments?Questions or Comments?