A Comparison ofProperty-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

UPMS

Pr E r r

IA IR

Sf e m f

[ ( [ ] )

( )]

UPM krt

tE r r

S

Pr

t

tf

i

uu m f f

i

u

1

1 1 ( [ ] )

DCF Premium Formula

P a

(1+ R ) = L

b

(1+ R ) + E

c

(1+ R )

+

(P - E c

(1+ R ))t

1+ R - Lt

b

(1+ R )

1+ R +

R b

(1+ R )

(1+ R )

+ R t

i=0

Ni

fi

i=0

Ni

Li

i=-M

Ni

fi

i=-M

Ni

fi

f

i=1

Ni

Ti-1

L j=2

Ni= j

NT i

Ti- j+1

Lj

fj=1

Ni= j

N

ii=0

j-1

i

fj

S b b

R )

+ P - E - L

(1+

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 forImplementing Financial Models

C om p an yV ariab les

E con om icV ariab les

G overn m en t P o lic y(Tax) V ariab les

V ariab les w ithIm p ac t on

P ric in g

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

Base Case ParametersEconomic

• 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

Base Case Parameters

Government Policy

• Tax rate34.0%

• Investment / total tax rate 80.0%

• Tax discount factor 7.0%

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

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 (%)

0

2

4

6

8

10

12

14

16

1925 1935 1945 1955 1965 1975 1985 1995

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

-15

-10

-5

0

5

10

15

0 5 10 15

Risk-Free Rate (%)

UPM(%)

TUPM

TTRR

ICAPM

DCF

IRR

OPM

APM

Sensitivity to Premium/Equity Ratio

-10

-5

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5

Premium/Equity Ratio

UPM(%)

TUPM

TTRR

ICAPM

DCF

IRR

OPM

APM

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, 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

Definitions of Surplus (Equity)

Model Surplus 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

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

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