A Comparison of Property-Liability Insurance Financial 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

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

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

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

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

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

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

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

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

DCF Premium Formula

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

Parameters Necessary for Implementing Financial Models

Base Case Parameters Company Equity $ 189,360 Expected Losses$ 193,605 Investment rate of return 8.0% SD of investment returns20.0% Equity beta 1.00 Funds generating coefficient 1.18

Base Case Parameters Economic Risk-free rate 5.0% Market risk premium 8.0% Risk Adj./Risk-free ratio60.0% U/W beta 0.0 SD of market returns22.0% SD of losses 48,401 CPI change 3.0% CPI beta 0.50 Industrial prod. growth 2.0% Industrial prod. beta 0.25

Base Case Parameters Government Policy Tax rate34.0% Investment / total tax rate80.0% Tax discount factor 7.0%

Base Case Results ModelIndicated UPM Target UPM5.0% Internal Rate of Return1.7 Option Pricing0.2 Discounted Cash Flow0.1 Arbitrage Pricing -2.9 Target Total Rate of Return -3.6 Insurance CAPM -4.9

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%

U.S. Treasury Bill Returns (%)

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%

Sensitivity to Risk-Free Rate

Sensitivity to Premium/Equity Ratio

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

“No distinction is introduced here between the market value of equity, V E, 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 V E 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

Definitions of Surplus (Equity) ModelSurplus Definition Target Total Rate of Return 1 Insurance CAPM1 and 2 Discounted Cash Flow 2 Internal Rate of Return1 and 2 Option Pricing1 and 2 Arbitrage Pricing Model1 and 2

Calculation of Adjusted Surplus Statutory Surplus150,958 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 Surplus189,360 Market Value of Company220,399

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

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