A Comparison of Property-Liability Insurance Financial Pricing Models

1. A Comparison ofProperty-Liability InsuranceFinancial Pricing Models Stephen P. D’Arcy, FCAS, MAAA, Ph.D. Richard W. Gorvett, FCAS, MAAA, Ph.D. Department of Finance University of Illinois at Urbana-Champaign Presented to the Casualty Actuarial Society Spring Meeting May, 1998

2. Comparison of Loss Reserving and Ratemaking Techniques Loss Reserving • Recognizes that predicting the future is uncertain • Apply a number of different approaches • Attempt to explain outliers • Actuarial judgment to select final value • Expect variation from selected value

3. Comparison of Loss Reserving and Ratemaking Techniques Ratemaking • Often a single model applied • Process relatively mechanical • “Correct” result is expected

4. Comparison of Loss Reserving and Ratemaking Techniques Recommendation • Ratemaking process should be similar to the loss reserving process • Use a number of different methods • Expect model error • Apply actuarial judgment

5. Actuarial Focus on supply / demand in insurance markets Satisfy exogenous constraints Financial Include capital market considerations Consider behavior of insurance company claimholders Actuarial vs. Financial Models

6. Objectives of Paper • Demonstrate the application of financial pricing models to a realistic ratemaking situation • Compare results from different models • Examine how changes in parameter values affect results • Discuss strengths and weaknesses of each model • Focus on most important parameters

7. Methodology • Identify financial pricing models • Determine representative company financial statements • Apply financial pricing models to company to determine indicated UPMs • Test UPM sensitivity to changes in the model parameters • Identify implications and need for additional research

8. Financial Pricing Models • Target total rate of return • Insurance capital asset pricing model • Discounted cash flow • Internal rate of return • Option pricing model • Arbitrage pricing model

9. Target Total RoR and Insurance CAPM Premium Formulas Target Total Rate of Return Insurance CAPM

10. DCF Premium Formula

11. Characteristics of Company • Operates in a single state • Writes one line of business: Private Passenger Auto (These assumptions avoid the need to allocate surplus)

12. Parameters Necessary forImplementing Financial Models

13. Base Case Parameters Company • Equity $ 189,360 • Expected Losses $ 193,605 • Investment rate of return 8.0% • SD of investment returns 20.0% • Equity beta 1.00 • Funds generating coefficient 1.18

14. Base Case Parameters Economic • Risk-free rate 5.0% • Market risk premium 8.0% • Risk Adj./Risk-free ratio 60.0% • U/W beta 0.0 • SD of market returns 22.0% • SD of losses 48,401 • CPI change 3.0% • CPI beta 0.50 • Industrial prod. growth 2.0% • Industrial prod. beta 0.25

15. Base Case Parameters Government Policy • Tax rate 34.0% • Investment / total tax rate 80.0% • Tax discount factor 7.0%

16. Base Case Results Model Indicated UPM Target UPM 5.0% Internal Rate of Return 1.7 Option Pricing 0.2 Discounted Cash Flow 0.1 Arbitrage Pricing - 2.9 Target Total Rate of Return - 3.6 Insurance CAPM - 4.9

17. Reality Check:Target Total Rate of Return Model Target: 13% State Farm Target: 15% (Per 1994 KY Auto Filing) Model UPM Indication: -3.6% State Farm Indication: 0%

18. U.S. Treasury Bill Returns (%)

19. Sensitivity to Risk-Free Rate Indicated UPM • Target Total RoR - 5.2% to 3.9% • Insurance CAPM -14.5% to - 2.9% • Discounted Cash Flow - 0.7% to 0.1% • Internal RoR - 1.2% to 11.4% • Option Pricing - 8.8% to 2.2% • Arbitrage Pricing -12.5% to - 0.9%

20. Sensitivity to Risk-Free Rate

21. Sensitivity to Premium/Equity Ratio

22. Different Concepts of Surplus (Equity) (1) Capital Attraction/Retention Standard • Recognizes that assets can be redeployed to alternative investments • Provides competitive return on this amount of capital (2) Amount of Equity Capital Generating Investment Income for Tax Calculation • Calculates tax impact of investment income on this initial equity • Reflects this taxation in premium level

23. “No distinction is introduced here between the market value of equity, VE, and the various accounting or book values of equity. The two may of course diverge over time, but in competitive markets the expected book and market values of new equity capital put into the insurance business should be the same. Since the Hope standard is a capital-attraction standard, it is appropriate in the analysis of returns and of target returns to treat VE as if it were new equity.”Fairley, 1979, “Investment Income and Profit Margins in Property-Liability Insurance: Theory and Empirical Results,” Bell Journal of Economics

24. Definitions of Surplus (Equity) ModelSurplus Definition Target Total Rate of Return 1 Insurance CAPM 1 and 2 Discounted Cash Flow 2 Internal Rate of Return 1 and 2 Option Pricing 1 and 2 Arbitrage Pricing Model 1 and 2

25. Calculation of Adjusted Surplus Statutory Surplus 150,958 Equity in the UEP Reserve 20,412 Nominal - Discounted Loss Reserves 8,289 Market - Book Value of Bonds 13,928 Non Admitted Assets 946 Tax Liability on Unrealized Capital Gains (5,173) Adjusted Statutory Surplus 189,360 Market Value of Company 220,399

26. Relative Sensitivities of Variables Models Are Generally More Sensitive To: • Level of Equity • Equity Beta • Risk-Free Rate • Underwriting Beta Models Are Generally Less Sensitive To: • Tax Parameters

27. Conclusions • Wide variation in indicated UPMs depending upon model and corporate / economic environment • Implications for insurers and regulators • Use several models • Be aware of operating environment • Note advantages and shortcomings of each model • Insurance is a complex financial transaction