Capital Allocation for Property-Casualty Insurers: A Catastrophe Reinsurance

Application

CAS Reinsurance SeminarJune 6-8, 1999

Robert P. ButsicFireman’s Fund Insurance

2

Yes, Capital Can Be Allocated!

Outline of Presentation: General approach: Myers-Read model

– Joint cost allocation is a common economics problem– Another options-pricing application to insurance– Extensions, simplification and practical application of MR method

Reinsurance (and primary insurance) application: the layer as a policy

Semi-realistic catastrophe reinsurance example Results and conclusions

3

Economic Role of Capital in Insurance

Affects value of default when insolvency occurs Default = expected policyholder deficit (market value) More capital implies smaller default value (good) But more capital implies higher capital cost (bad) Equilibrium:

Capital Cost

Solvency Benefit

Capital Amount

4

Fair Premium Model

For all an insurer’s policies:

Important Points:– Shows cost and benefit of capital– All quantities at market values (loss includes risk load)– Loss can be attributed to policy/line– But C and D are joint

Single policy model :

CTDLP

iiii CTDLP

5

Allocation Economics

Capital ratios to losses are constant:

Premiums are homogeneous:

Implies that

And marginal shift in line mix doesn’t change default ratio:

Solve this equation for

iiLcC

iiii LPLP

dLDLDd iii

dLD i

ic

6

Lognormal Model

To solve for we need to specify relationship between L, C and D

Assume that loss and asset values are lognormal D is determined from Black-Scholes model Final result (modified Myers-Read):

ic

)1()1()(

)()1(2

2

AL

ALiA

L

LiLi yN

ynccc

7

Simplifying the Myers-Read Result

Assume that loss-asset correlation is small Define Loss Beta:

Result:

Implications:– Relevant risk measure for capital allocation is loss beta– Capital allocation is exact; no overlap– Allocated capital can be negative– Z value is generic for all lines

2/ LiLi

Zcc ii )1(

8

Numerical Example

Table 1.1

Loss Beta and Capital Allocation for Numerical Example

Liability Loss Loss Capital/

Value CV Beta Liability Capital

Line 1 500 0.2000 0.8463 0.3957 197.87Line 2 400 0.3000 1.3029 0.7055 282.19

Line 3 100 0.5000 0.5568 0.1993 19.93

Total 1000 0.2119 1.0000 0.5000 500.00

9

Negative Capital Example

Original Add Low- AdjustCase Risk Policy Capital

Number of PoliciesPolicy 1 1000 1000 1000Policy 2 0 1 1

Loss CVPolicy 1 10.0 10.0 10.0Policy 2 1.0 1.0Total 0.3162 0.3159 0.3159

Expected Loss 1000.000 1001.000 1001.000Capital 500.000 500.000 499.660Default 16.594 16.586 16.610

Default Ratio 1.6594% 1.6569% 1.6594%

Capital/Loss, Policy 2 -0.340

Assumptions:

• losses are independent

•no asset risk

•total losses are lognormal

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Reinsurance Application

For policy/treaty, capital allocation to layer depends on:– covariance of layer with that of unlimited loss– covariance of unlimited loss with other risks

Layer Beta is analogous to loss beta Capital ratio for policy/layer within line/policy:

Point beta for layer is limit for narrow layer width:

1

)()(1)(

0

12 xEX

xEs

x

Zcc kikkik

iki

1

11

Point Betas for Some Loss Distributions

0.00

5.00

10.00

15.00

20.00

25.00

0 50 100 150 200 250 300 350 400

x

Legend: right hand side (x = 400), top to bottomParetoLognormalExponentialGammaNormal

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Market Values and Risk Loads

Layer Betas depend on market values of losses Market values depend on risk loads Modern financial view of risk loads

– Adjust probability of event so that investor is indifferent to the expected outcome or the actual random outcome

– Risk-neutral valuation– General formula:

In finance, standard risk process is GBM lognormal– Risk load equals location parameter shift:

0 )(ˆ)1(ˆ dxxfxXX

)exp()exp(1

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Reinsurance Risk Loads

Risk-neutral valuation insures value additivity of layers Risk load for a layer

– integrate R-N density instead of actual density, giving pure premium loaded for risk

– risk load is difference from unloaded pure premium

Point risk load– load for infinitesimally small layer– parallel concept to point beta

Simple formula:

.1)()(ˆ

)( xGxGx

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General Layer Risk Load Properties

Monotonic increasing with layer Generally unbounded Zero risk load at lowest point layer Lognormal example:

location PS

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 200 400 600 800 1000

x

15

PRL and the Generalized PH Transform

Location parameter shift may not be “risky” enough Wang’s Proportional Hazard transform

More general form:

Gives all possible positive point risk loads Fractional transform:

– No economic basis– But it works

qxGxG )]([)(ˆ 10 q

)()]([)(ˆ xqxGxG

mxqxxq

)(

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Parameter Estimation

Market valuation requires modified statutory data Representative insurer concept necessary for capital

requirements– particular insurer could have too much/little capital, risk, line mix, etc.– industry averages can be biased

Overall capital ratio CV estimates

– losses: reserves and incurred losses, cat losses– assets

Catastrophe beta

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Catastrophe Pricing Application

Difficult, since high layers significantly increase estimation error But, made easier because cat losses are virtually independent of

other losses Present value pricing model has 3 parts:

– PV of expected loss:

– PV of risk load:

– PV of capital cost:

)1/(),( rbaX

)1/(),(),( rbabaX

)1)(1(),()],(1)[,(

trrtbacbabaX

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Example: Annual Aggregate Treaty

a bExpected

LossCapital

CostRiskLoad

FairPremium

ExpectedLoss Ratio

ImpliedROE

0 100 31.51 0.74 8.78 41.04 0.814 0.272100 200 6.83 0.73 4.73 12.29 0.589 0.213200 300 3.01 0.58 2.64 6.24 0.512 0.180300 400 1.66 0.46 1.67 3.79 0.463 0.161400 500 1.03 0.38 1.14 2.55 0.427 0.149500 600 0.69 0.32 0.82 1.83 0.398 0.140600 700 0.48 0.27 0.62 1.37 0.374 0.133700 800 0.35 0.23 0.47 1.06 0.354 0.127800 900 0.27 0.20 0.37 0.84 0.336 0.123900 1000 0.21 0.18 0.30 0.69 0.321 0.119

1000 Infinity 1.13 2.33 2.03 5.48 0.219 0.094

0 Infinity 47.17 6.42 23.58 77.18 0.648 0.137

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Return on Equity for Treaty

Look at point ROE Varies by layer Equals risk-free interest rate at zero loss size

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0 20 40 60 80 100

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Summary

How to allocate capital to line, policy or layer– Key intuition is to keep a constant default ratio– Relevant risk measure is loss or layer beta– Allocated capital is additive

Reinsurance and layer results– Layer betas are monotonic, zero to extremely high– Layer risk loads are monotonic, zero to extremely high

ROE pricing method has severe limitations– ROE at fair price will vary by line and layer– capital requirement can be negative

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Conclusion

Capital allocation is essential to an ROE pricing model– capital is the denominator– but this model has severe problems

It’s less (but still) important in a present value pricing model

– capital determines the cost of double taxation– this model works pretty well (cat treaty example)

The real action is in understanding the risk load process – knowing the capital requirement doesn’t give the price– because the required ROE is not constant

We’ve got a lot of work to do!