Capital Allocation for Property-Casualty Insurers: A Catastrophe Reinsurance Application CAS Reinsurance Seminar June 6-8, 1999 Robert P. Butsic Fireman’s Fund Insurance

2 Yes, Capital Can Be Allocated! 4 Outline of Presentation: 4 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 4 Reinsurance (and primary insurance) application: the layer as a policy 4 Semi-realistic catastrophe reinsurance example 4 Results and conclusions

3 Economic Role of Capital in Insurance 4 Affects value of default when insolvency occurs 4 Default = expected policyholder deficit (market value) 4 More capital implies smaller default value (good) 4 But more capital implies higher capital cost (bad) 4 Equilibrium: Capital Cost Solvency Benefit Capital Amount

4 Fair Premium Model 4 For all an insurer’s policies: 4 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 4 Single policy model :

5 Allocation Economics 4 Capital ratios to losses are constant: 4 Premiums are homogeneous: 4 Implies that 4 And marginal shift in line mix doesn’t change default ratio: 4 Solve this equation for

6 Lognormal Model 4 To solve for we need to specify relationship between L, C and D 4 Assume that loss and asset values are lognormal 4 D is determined from Black-Scholes model 4 Final result (modified Myers-Read):

7 Simplifying the Myers-Read Result 4 Assume that loss-asset correlation is small 4 Define Loss Beta: 4 Result: 4 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

8 Numerical Example

9 Negative Capital Example Assumptions: losses are independent no asset risk total losses are lognormal

10 Reinsurance Application 4 For policy/treaty, capital allocation to layer depends on: –covariance of layer with that of unlimited loss –covariance of unlimited loss with other risks 4 Layer Beta is analogous to loss beta 4 Capital ratio for policy/layer within line/policy: 4 Point beta for layer is limit for narrow layer width:

11 Point Betas for Some Loss Distributions

12 Market Values and Risk Loads 4 Layer Betas depend on market values of losses 4 Market values depend on risk loads 4 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: 4 In finance, standard risk process is GBM lognormal –Risk load equals location parameter shift:

13 Reinsurance Risk Loads 4 Risk-neutral valuation insures value additivity of layers 4 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 4 Point risk load –load for infinitesimally small layer –parallel concept to point beta 4 Simple formula:

14 General Layer Risk Load Properties 4 Monotonic increasing with layer 4 Generally unbounded 4 Zero risk load at lowest point layer 4 Lognormal example: location PS

15 PRL and the Generalized PH Transform 4 Location parameter shift may not be “risky” enough 4 Wang’s Proportional Hazard transform 4 More general form: 4 Gives all possible positive point risk loads 4 Fractional transform: –No economic basis –But it works

16 Parameter Estimation 4 Market valuation requires modified statutory data 4 Representative insurer concept necessary for capital requirements –particular insurer could have too much/little capital, risk, line mix, etc. –industry averages can be biased 4 Overall capital ratio 4 CV estimates –losses: reserves and incurred losses, cat losses –assets 4 Catastrophe beta

17 Catastrophe Pricing Application 4 Difficult, since high layers significantly increase estimation error 4 But, made easier because cat losses are virtually independent of other losses 4 Present value pricing model has 3 parts: –PV of expected loss: –PV of risk load: –PV of capital cost:

18 Example: Annual Aggregate Treaty

19 Return on Equity for Treaty 4 Look at point ROE 4 Varies by layer 4 Equals risk-free interest rate at zero loss size

20 Summary 4 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 4 Reinsurance and layer results –Layer betas are monotonic, zero to extremely high –Layer risk loads are monotonic, zero to extremely high 4 ROE pricing method has severe limitations –ROE at fair price will vary by line and layer –capital requirement can be negative

21 Conclusion 4 Capital allocation is essential to an ROE pricing model –capital is the denominator –but this model has severe problems 4 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) 4 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 4 We’ve got a lot of work to do!