1. Allocating the Cost of Capital CAS Spring Meeting May 19-22, 2002 Robert P. Butsic Fireman’s Fund Insurance

2. 2 Why is Capital Necessary? The answer is not obvious: –We can’t have enough capital to eliminate insurance insolvency –So why not have minimal capital and let guaranty funds protect policyholders? Answer is: frictional insolvency costs –Additional system costs due to insolvency: –Legal fees, market disruption, extra claims handling costs

3. 3 Optimal Capital Level Frictional Insolvency Costs Frictional Capital Costs Capital Amount

4. 4 What is the Cost of Capital? Investor supplies capital and expects return commensurate with risk to which the capital is exposed This return is the cost of capital –Traditional view of insurance management –But, look from the modern finance perspective

5. 5 Base Cost of Capital A: Investor invests capital in a levered fund –borrow cash and invest all assets –identical to the insurer’s assets Investor’s expected return in A is called Base Cost of Capital B: Insurer has same balance sheet –But insurer has higher COC

6. 6 Frictional Costs of Capital The insurance mechanism will introduce extra costs –Government, regulation and organization –Illiquid nature of insurance liability –Information asymmetry (opaqueness) These are frictional costs of capital –key one is double taxation –Most easily quantified

7. 7 Double Taxation Example Investor can directly invest in security with 10% return, but invests in ABC Insurance, who puts money in same security ABC gets 10% return, pays 35% tax and gives 6.5% net back to investor A losing deal unless PH can make up the difference

8. 8 Other Frictional Costs Regulatory costs –Capital can’t be easily moved, so investment is illiquid Agency costs –Misalignment of owners’ and managers’ interests (Enron a classic example) Financial distress costs –Legal fees –Distressed sale of assets

9. 9 Financial Pricing Model Fair premium = total present value of –loss & LAE (including risk margin) –UW expenses –Frictional capital costs Note that traditional (base) cost of capital is embedded in risk margin

10. 10 Risk Margin and COC Example Assumptions –Fair premium is $1000, paid up front, $1040 loss paid in one year –Risk-free rate of 6%, $500 of capital required –No frictional COC, taxes or expenses Calculation –Initial assets of $1500 grow to $1590, leaving $550 for a 10% return (COC) –Risk margin is $18.87 = 1000 - 1040/1.06

11. 11 RM and COC Example, Cont. Which comes first, the RM or the COC? –Each implies the other In determining a fair premium, it must be the risk margin: –Products have different levels of risk –What COC should a riskless coverage have? Thus, the COC is not fixed for an insurer -- it varies by product

12. 12 Allocating Capital Costs For pricing or performance measurement, must allocate capital costs to product If we know the RM, then we need to allocate the frictional COC If we don’t know the RM, and use a COC pricing model, then we allocate both frictional and base COC

13. 13 Capital Allocation In order to assess capital costs by product, we first need to allocate capital to product There are many methods –Lots of ad hoc models –Very few economically sound models One of them is the general Myers-Read method

14. 14 Myers-Read Method Uses the expected default (PH deficit) as a solvency measure –Others, such as default (ruin) probability will also work (and may be better) Major assumptions –predetermined capital ratios exist: –A marginal change in the line mix keeps the default measure at a constant rate:

15. 15 More on M-R Model Other inputs –Probability distribution of loss and asset values –Means, correlations and volatilities We solve for capital ratio Result: –Beta is covariance/variance –Z is distribution-dependent

16. 16 Loss Beta Relevant risk measure for capital allocation is loss beta –Volatility, correlation with portfolio and weight determine loss beta –Strong parallel with asset pricing, CAPM, portfolio optimization Capital allocation is exact; no overlap –No order dependency

17. 17 Numerical Example

18. 18 Application to Coverage Layers For policy/treaty, capital allocation to layer depends on covariance of layer with that of unlimited loss Layer covariance depends only on loss size distribution Layer beta and capital/loss increase with limits

19. 19 General Layer Beta Properties Monotonic increasing with layer, with zero layer beta at lowest point layer Generally unbounded Legend: RHS top to bottom Pareto Lognormal Exponential Gamma Normal

20. 20 Practical Applications Best measure of capital is economic (fair) value As an approximation, capital is proportional to the loss/layer beta For allocating a company’s capital, the relevant time horizon is one year –Allocation base is reserves plus next year’s AY incurred losses

21. 21 Summary Importance of frictional costs in theory of solvency and capital allocation Myers-Read method is economically sound, with friendly (to user) results We’ve still got a long road ahead before common agreement on capital allocation methodology

22. 22 Further Reading John Hancock, Paul Huber, Pablo Koch, 2001 The economics of insurance: How insurers create value for shareholders, Swiss Re Publishing http://www.swissre.com/ Myers, Read, 2001, Capital Allocation for Insurance Companies, Journal of Risk and Insurance, 68:4, 545- 580 Butsic, 1999, Capital Allocation for Property Liability Insurers: A Catastrophe Reinsurance Application. Casualty Actuarial Society Forum, Fall 1999 http://www.casact.org/pubs/forum/99spforum/99spftoc.h tm

23. 23 Further Reading II Butsic, Cummins, Derrig, Phillips, 2000, The Risk Premium Project, Phase I and II Report, CAS Website, http://casact.org/cotor/rppreport.pdf