Capital Allocation for Property-Casualty Insurers: A Catastrophe Reinsurance Application

1. 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! • 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 CapitalAmount

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 :

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

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

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

8. Numerical Example

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

10. 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:

11. Point Betas for Some Loss Distributions

12. 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:

13. 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:

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

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

16. 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

17. 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:

18. Example: Annual Aggregate Treaty

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

20. 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

21. 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!