Risk & ReturnParameter Estimation

David Appel, Ph.D.

Milliman & Robertson

Richard A. Derrig, Ph.D.

Senior Vice President

Automobile Insurers Bureau of MA

Casualty Actuarial SocietySeminar on Ratemaking

March 12, 2001Las Vegas, NV

AgendaRisk and Return ModelsTime Value of MoneyCash Flow Patterns & LevelsMeasuring Risk - CAPMIRR ModelsAllocating CapitalCalendar Year Acct ModelsRisk Premium Project (COTOR)Summary

Risk and Return Models

Net Present Value Model: Valuation of Policyholder Flows

Internal Rate of Return Model: Valuation of Shareholder Flows

Calendar Year Accounting Model: Valuation of Company Acct Returns

Risk and Return Models

Net Present Value Model:PV(P) = PV(L) + PV(E) + PV(T)

Internal Rate of Return Model: PV(Shareholder Inv. - Shareholder Dividends) = 0

Calendar Year Accounting Model: Return on Surplus = Return on Investment + Return on Underwriting

Risk and Return Models Family Tree

Profit Models

Historical Prospective

CASH FLOW ACCOUNTING

CYAM ISOSTATE-XNPV

(policyholder perspective)

IRR(shareholderperspective)

Time Value of Money

Premiums and Capital In

Expenses and Claims Out

Risk-Free Interest Rates

Risky Investments

2001 Massachusetts Private PassengerAutomobile Compilation of Treasury

Stirps - Wall Street JournalSummary for 12 months Nov. 1999 - Oct. 2000

Maturity(months)

AverageYield

3 5.6696 5.9869 6.094

12 6.30015 6.38418 6.43221 6.28824 6.43027 6.42830 6.32133 6.44436 6.45539 6.47342 6.45245 6.47748 6.49251 6.49054 6.52357 6.52560 6.540

Cash Flow Patterns & Levels

Premium PaymentsFinance ChargesAcquisition CostsGeneral ExpensePremium TaxesCapital InvestmentInvestment IncomeIncome Taxes

Cash Flow Patterns & Levels

Premium PaymentsFinance ChargesAcquisition CostsGeneral ExpensePremium TaxesCapital InvestmentInvestment IncomeIncome Taxes

Cash Flow Patterns & LevelsIncome Taxes

Federal (35% Marginal Rate)Change in Unearned Premium ReserveUnderwriting: Earned Premium - Formula Discounted Loss

ReservesInvestment: Deductions by Asset Class

• Deduct 70% Stock Dividends• Deduct 85% Tax-Exempt Bond Income• Alternate Minimum Tax• Idea: 20% Minimum Rate on Net Income• Actual: Consolidated Tax Returns

State (Specific Rate)MA: Flat % of Investment Income

Measuring Risk

Cost of CapitalCapital Asset Pricing Model (CAPM)Equity, Asset and Liability BetasMarket Risk PremiumDividend Growth ModelsRisk-Adjusted Discount Rates

Capital Asset Pricing Model(CAPM)

Investors are compensated for non-diversifiable (i.e., “systematic”) risk only

radj = rfree + { x (rmarket - rfree)}

where,

radj = risk adjusted return

rfree = return on risk free investments

rmarket = return on the market

= systematic risk coefficient

Capital Asset Pricing Model(Theoretical Relationship)

Expected Return

Ris

k P

rem

ium

1.0 1.5 = Risk

20.1

R*= 15.4

Rf= 6.0

R* = rf + * (market risk premium)

Expected Return

Capital Asset Pricing Model(Observed Empirical Relationship)

Expected Return

Ris

k P

rem

ium

Beta, Firm Size, Market/Book ratioReturns are higher than predicted for: Low Beta

Small SizeLow M/B

Return

Empirical Observation

CAPM Anomalies

Returns higher for: Low beta firmsSmall firmsLow M/B firms

Insurers tend to be: Average betaRelatively small market capRelatively low market/book

Also, insurers subject to interest rate risk not priced by CAPM

CAPM Issues

•Sample Selection

•Estimation of beta :

Value Line, S&P,Merrill Lynch, Wilshire

•Market risk premiumIbbotson AssociatesGreenwich Associates

Capital Asset Pricing Model Beta Coefficients

Company Beta Coefficients 21st Century 0.75ACE Limited 1.20Allmercan Finan 1.15Allstate Corp. 1.25Amer. Intl. Group 1.35American Finan 0.95Berkley (W.R.) 0.85Chubb 1.20Cincinnati Fin 0.80Fremont General 0.90GAINSCO 0.65HCC Ins. Hldngs 1.10HSB Group 0.90Hartford Finan 1.20Mercury General 1.20Ohio Casualty 0.80Old Republic 1.00Progressive 1.35RLI Corp. 0.75Reliance 1.10Selective 0.65St Paul 1.05Transatlantic 0.70Unitrin 0.75XL Capital Lmtd. 0.85SAFECO 1.10

Average 0.98

Sources: Value Line Investment Survey, Part 3, The Ratings & Reports, June 2, 2000 and June 30, 2000.

Excess Market Risk Premium

Definition: MRP = RM - RFRF depends on horizon lengthRF = T-Bill, Int. Govt, Long Govt.MRP = RM-Tbill,

RM-Int .Govt.

RM-Long Govt.

Excess Market Risk Premium

Problem 1: How Do I Estimate MRP Value?

Problem 2: Does RF + Beta * MRP Work?

2001 Massachusetts Private Passenger Auto Ibbotson Market Data 1926-99*

Year Common StocksTotal Annual

Returns

U.S. TreasuryBills

Total annualReturns

ArithmeticEquity

Risk Premia

1926 11.62% 3.27% 8.36%

1927 37.49% 3.12% 34.36%

1928 43.61% 3.56% 40.05%

1929 8.42% 4.75% -13.16%

1930 -24.90% 2.41% -27.31%

2001 Massachusetts Private Passenger Auto Backward Moving Average Risk Premium

Year RiskPremium

# Years inAvg.

Yr. X to 1999Average Risk

Premium

1926 8.36% 74 9.44%

1927 34.36% 73 9.46%

1928 40.05% 72 9.11%

1929 -13.16% 71 8.68%

1930 -27.31% 70 8.99%

2001 Massachusetts Private Passenger Auto Forward Moving Average Risk Premium

Year RiskPremium

# Years inAvg.

Average RiskPremium

1926 8.36% 1 8.36%

1927 34.36% 2 21.36%

1928 40.05% 3 27.59%

1929 -13.16% 4 17.40%

1926-99 74 9.44%

Market Risk Premium

-60%

-40%

-20%

0%

20%

40%

60%

1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

Year

Actual MRP Long-term Mean Mean pre-1960 Mean 1960+

Long-term Mean =9.44%

Mean 1926 to 1959 = 11.94%

Mean 1960 to 1999 = 7.32%

Market Risk Premium - Backward Moving Average(Year Y to 1999)

0%

5%

10%

15%

20%

25%

1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

Starting Year (Y)

Long-term average = 9.44%

40 year average = 7.32%

Market Risk Premium - Forward Moving Average(Year 1926 to Y)

-5%

0%

5%

10%

15%

20%

25%

30%

1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

Ending Year (Y)

Long-term average = 9.44%

Simple CAPM is DeficientAdd Small Stock EffectArithmetic Return on Decile

Mean In Excess of SizeDecile Return Risk-Free Rate Beta Beta * MRP Premium

1 12.13% 6.93% 0.90 7.28% -0.35%2 13.55% 8.34% 1.04 8.36% -0.02%3 13.92% 8.71% 1.08 8.76% -0.05%4 14.55% 9.35% 1.12 9.07% 0.28%5 15.28% 10.08% 1.15 9.31% 0.76%6 15.44% 10.24% 1.18 9.49% 0.74%7 15.75% 10.54% 1.23 9.90% 0.64%8 16.80% 11.60% 1.27 10.22% 1.38%9 17.59% 12.38% 1.33 10.77% 1.61%10 20.73% 15.52% 1.43 11.57% 3.95%

*Note: Analysis done using 30-day Treasury bill as riskless rate.

Decile Largest Company Capitalization

1 General Electric Company $369,722,2142 Ynisys Corp. $10,498,7963 Readers Digest Association $4,221,6014 Sterling Software Inc. $2,203,6715 Steris Corp. $1,304,1316 Unova Inc. $872,2207 Trammell Crow Co. $577,7788 Transaction Network Services Inc. $381,8309 Donna Karen International Inc. $214,64010 Delta Financial Corp. $97,914

IRR - Relevant Cash FlowsShareholders commit equity capital to support sale of

insuranceIn return, receive rights to cash flows from

underwriting and investment activities

- Underwriting cash flow (net of tax)

- Investment income on reserves and surplus

- Flow of surplusIRR cash flows conditional on accounting conventions

(usually SAP) and tax rules

IRR - Algebra

iiiiiii STSRtiUWCFCashFlow 1

IRR = r = discount rate such that

Set price such that IRR = target return (COK)

(Cfi) (1 + r)-i = 0

IRR - Key Inputs/Assumptions

Cash Flow Patterns (especially premium + loss)

Investment Yield Rate (usually current yield)

Leverage (reserves/premiums/other)

Surplus Runoff (flow/block)

INTERNAL RATE OF RETURN ANALYSISALL LINES COMBINED (Policy Year)

TABLE III: PORTFOLIO YIELD AND TAX RATE - CURRENT YIELD(1) (2) (3) (4) (5)

InvestableAsset

Percent ofAssets

Expected Pre-Tax Return

Tax Rate Expected Post-Tax Return

Bonds U.S. Govt 14.6 6.37% 35.00% 4.14% States & territories 21.0 5.49% 5.25% 5.20% Specialrevenue

13.8 4.98% 5.25% 4.72%

PublicUtilities

1.2 7.45% 35.00% 4.84%

Industrial 19.0 7.25% 35.00% 4.71%

Preferredstock

4.5 6.89% 14.18% 5.91%

Commonstock

18.2 14.77% 28.89% 10.50%

MortgageLoans

1.0 8.30% 35.00% 5.40%

Real estate 1.0 9.65% 35.00% 6.27%Cash & short-term invs. 5.7 5.92% 35.00% 3.85%Rate of Return Pre-Inv Expand Prior to MA Inv Tax 100.0 7.75% 24.98% 5.82%MA Inv Tax 0.260% 5.80%Investment Expenses 0.47% 35.00% 0.31%Portfolio Rate of Return 7.28% 24.61% 5.49%Sources:

Value Line Investment Survey, Part II, Selection and Opinion. Federal Reserve Statistical Release, H.15(519). Moody's Credit Perspectives.

Ibbotson Associates, Stocks, Bonds, and Inflation: 1999 Yearbook. Ibbotson and Siegel, AREUEA Journal, 1984.

OneSource 7/31/99 CD, Asset Page, Part 1, Part 1A and Schedule D, Top 30 MA Groups, 12/98. Return on other bonds from Major Lehman Indices, Asset Class U.S. Aggregate, @3/31/99.

Return on other assets from TOP302.WK4.

Final NPV of FinalTime Cash Flow Cash Flow

0 -75.0 -75.01 68.0 58.42 15.6 11.53 8.1 5.1

SUM = 0.0IRR = 16.5%

IRR CASH FLOWS INNOMINAL AND PRESENT VALUE

Setting Target Returns(a.k.a. Estimating Cost of Capital)

Two Important Methods

Dividend valuation (DCF) model

CAPM

Also, comparable earnings

DCF Model

Price of stock equals present value of future cash flows

i

i

rDi

1

)1/(

If D grows at constant annual rate, g, then

g

P

gDr

r

gD

r

gDP

o

o

ooo

1

......)1(

1

1

12

2

and

P o =

DCF Issues

Sample selectionEstimation of growth rate (g)

- Historical data or analysts forecasts

- Earnings, dividends or book valueSingle growth rate or multi-stage model

Discounted Cash Flow Analysis Estimated Dividend Yield

Estimated Dividend Yield

21th CenturyACE LimitedAllmerican FinanAllstate Corp.Amer Intl GroupAmerican FinanBerkley (W.R.)ChubbCincinnati FinFremont GeneralGAINSCOHCC Ins HoldingsHSB GroupHartford FinanMercury GeneralOhio CasualtyOld RepublicProgressiveRLI CorpRelianceSelectiveSt. PaulTransatlanticUnitrinXL Capital LimitedSAFECOAverage

3.01.90.52.80.23.72.62.12.27.61.41.15.71.63.84.43.30.41.8nil3.23.10.67.0.3.63.52.84

Sources: Value Line Investment Survey, Part 3, The Ratings & Reports, June 2, 2000 and June 30, 2000

Discounted Cash Flow Analysis Earnings Per Share Experience

Annual Rate of ChangeCompany 1990-1999 1995-1999 2000-2004 21st Century 9.5 N.A. 5.5ACE Limited N.A. 27.5 13.5Allmercan Finan N.A. N.A> 5.5Allstate Corp. N.A. 24.5 11.0Amer. Intl. Group 14.5 13.5 13.5Berkley (W.R.) N,.A. 3.5 NilChubb 12.0 12.0 7.5Cincinnati Fin 9.0 7.5 6.0Fremont General 13.0 12.0 10.5GAINSCO N.A. 17.5 5.5HCC Ins. Hldngs N.A. N.A. 10.0HSB Group 7.0 3.0 3.0Hartford Finan N.A. N.A. 10.0Mercury General 22.5 18.5 9.0Ohio Casualty 7.0 4.5 6.0Old Republic 11.5 16.0 11.0Progressive 7.5 4.5 3.0RLI Corp. 8.5 8.0 7.0Reliance 2.5 N.A. NilSelective 4.0 0.5 4.0St Paul 7.0 7.0 4.0Transatlantic 7.5 17.0 12.0Unitrin 19.0 20.0 19.5XL Capital Lmtd. N.A. 26.0 9.0SAFECO 10.0 9.5 4.5

Average 10.08 12.38 8.15Average Dividend Growth Rate 10.20

Sources: Value Line Investment Survey, Part 3, The Ratings & Reports, June 2, 2000 and June 30, 2000.

Discounted Cash Flow Analysis Earnings Per Share Experience

Annual Rate of ChangeCompany 1990-1999 1995-1999 2000-2004 21st Century 0.50 N.A. NilACE Limited N.A. 27.00 7.00Allmercan Finan N.A. N.A> 11.50Allstate Corp. N.A. 49.50 1.00Amer. Intl. Group 13.5 15.0 13.0American Finan N,.A. 9.5 14.0Berkley (W.R.) -5.5 -5.5 15.5Chubb 7.0 8.0 10.0Cincinnati Fin 8.5 6.5 11.5Fremont General 10.0 -1.0 20.0GAINSCO 6.0 -30.0 40.0HCC Ins. Hldngs N.A. 24.0 10.5HSB Group 1.0 10.0 4.5Hartford Finan N.A. 37.0 3.5Mercury General 19.0 13.5 6.0Ohio Casualty 4.0 3.0 5.0Old Republic 12.0 10.5 6.0Progressive 17.5 13.0 17.5RLI Corp. 11.0 27.5 8.0Reliance N.A. N.A. 6.0Selective 4.0 12.0 7.0St Paul 7.0 11.0 5.0Transatlantic 14.5 19.0 9.0Unitrin 27.5 20.0 19.5XL Capital Lmtd. N.A. 20.0 -1.5SAFECO 4.0 -1.0 4.0

Average 8.318 12.98 10.14Average Dividend Growth Rate 10.47

Sources: Value Line Investment Survey, Part 3, The Ratings & Reports, June 2, 2000 and June 30, 2000.

Allocating Capital

Standard AllocationsPremiumLiabilitiesDiscounted LiabilitiesMyers-Cohn AllocationMyers-Read

Calendar Year Accounting Model - CYAM

Total Return = (UW Profit + IY Reserves) + IY Surplus

= Return on Operations +

Return from Investment of SurplusReturn on Operations = Return attributable to undertaking the risk

of the insurance transaction

CYAM - Algebra

S

P.

.1

S

P1

premium

LLAEInc

LLAEInc

reservetitUWROE iu

it1i

Set Total Return = Target Return (COK) and Solve for UW

CYAM - Key Inputs/Assumptions

•Investment Yield Rate = i

•Investible Balance =

•Leverage =

i is usually embedded yield

is usually estimated using recent historical data

is usually normative value; rarely varies by line

Premium

Inc.LLAE

Inc.LLAE

Reserve

Inc.LLAE

Reserve

S

P

S

P

CYAM - Likely Problems

Embedded yield not necessarily good proxy for expected earnings rate

Investible balance may be distorted due to variations in historical growth or loss experience

Leverage is typically insensitive to risk

and,

TIMING OF CASH FLOWS IS IGNORED

CALENDAR/ACCIDENT YEAR ANALYSISSTEADY STATE / GROWTH RATE = 0%

AY1 AY2 AY3 AY40 -75.01 68.0 -75.02 15.6 68.0 -75.03 8.1 15.6 68.0 -75.04 8.1 15.6 68.05 8.1 15.66 8.1

Calendar Year ROE = (68+15.6+8.1)/75 = 22.3%

CALENDAR/ACCIDENT YEAR ANALYSISSTEADY STATE / GROWTH RATE = 16.5%

NPV OFAY1 AY2 AY3 AY4 AY4

0 -75.01 68.0 -87.42 15.6 79.2 -101.83 8.1 18.2 92.3 -118.5 -118.54 9.4 21.2 107.5 92.35 11.0 24.7 18.26 12.8 8.1

Calendar Year ROE =(92.3+18.2+8.1)/101.8 = 16.5%

CALENDAR/ACCIDENT YEAR ANALYSISSTEADY STATE / GROWTH RATE = 25.0%

NPV OFAY1 AY2 AY3 AY4 AY4

0 -75.01 68.0 -93.82 15.6 85.0 -117.23 8.1 19.5 106.3 -146.5 -146.54 10.1 24.4 132.8 114.05 12.7 30.5 22.56 15.8 10.0

Calendar Year ROE = (106.3+19.5+8.1)/117.2 = 14.2%

Model Outputs

NPV: Underwriting Profit ProvisionIRR: Expected Return to CapitalCYA: Return to Capital: Actual or Expected Model Outputs Consistent if Inputs are Consistent

Other Issues

Guaranty FundsResidual MarketsReinsuranceDefault RiskRisk Premium ProjectExcess Capital

CAS Risk Premium ProjectCommittee on Theory of RiskDiscount Rate for LiabilitiesLiterature ReviewActuarial: Process and Parameter RiskFinancial: Systematic RiskAcademic: Dave Cummins, Rich PhillipsIndustry: Bob Butsic, Rich Derrig

http://casact.org/cotor/rpp.htm

Small Stock Effect/Sum Beta

Small Stock Effect: Smaller Decile (MKT CAP) Returns Exceed CAPM Expected

Theory: Non-Systematic Risk Based on Information Flow and Liquidity

Practice: Deciles 5 to 10, 1926-1998 0.87% (5) to 3.75% (10) Excess of CAPM

Example: MA Companies 1.3% Ibbotson, Kaplan & Peterson (1997): Cross-

Autocorrelations in Returns; “Sum Beta” adds One Lag; Sum = + -1

Sum Beta “Explains” Some of Small Stock Effect

Full Information Beta

Problem: Public Firms not all “Pure Play”Solution: Industry Equity Beta via

Sales Weighted Full

Market RegressionP & C: Equity Beta 12/31/1998 of 0.92;

3/31/2000 of 1.15

Surplus AllocationSurplus by Company stands behind all linesSurplus by Line needed to allocate taxes and other by line

Costs.Myers-Read (1999): Theory Allows Unique Additive

Allocation of Capital by “Fairness” to Guaranty Fund Criteria and Options Pricing Methods

Properties: Higher Line Covariance with Liab (Asset) Portfolio Implies Higher (Lower) Surplus

Key Equation: Default Option = F (Liabilities, Assets, A/L)

Loss Distribution Betas

CAPM Loss Beta (Fairley, 1979) has

= F(A,L,T,S, More (?)), No Default Problem: All Liability Dollars Have Same Risk Butsic (1999): Unique Surplus Allocation if Price Homogeneity

(Same Marginal Default Option). Surplus Allocation Across Coverage Layers (Loss Distribution) Layer Beta and Surplus Increasing by Limit Risk Loads by Layer Example: Catastrophe Risk, Layer Betas 0.18 to 8.29 Stay Tuned for More Developments

ReferencesAlmagro, Manuel and Thomas L. Ghezzi, (1988), Federal Taxes Provisions Affecting Property-Casualty Insurers, Proceedings of the Casualty Actuarial Society, LXXV.

Brealey, Richard A. and Stewart C. Myers, (2000), Principles of Corporate Finance, Sixth Edition, McGraw-Hill Higher Education.

Butsic, Robert P., (1991), Loss Reserve Valuation Using A Risk-Adjusted Discounting Interest Rate, Managing the Insolvency Risk of Insurance Companies, J. David Cummins and R. A. Derrig (Eds), Kluwer Academic Publishers

Butsic, Robert P., (1999), Capital Allocation for Property-Liability Insurers: A Catastrophe Reinsurance Application, Casualty Actuarial Society Forum, Spring.

Cummins, J. David, (1990), Asset Pricing Models and Insurance Ratemaking, ASTIN Bulletin, 20:2.

ReferencesCummins, J. David, (1990), Multi-Period Discounted Cash Flow Ratemaking Models in Property-Liability Insurance, Journal of Risk & Insurance, 57:1, 79-109, March.

Cummins, J. David , (1988), Risk-Based Premiums for Insurance Guaranty Funds, Journal of Finance, 43, 823-839, September.

Derrig, Richard A., (1994), Theoretical Considerations of the Effect of Federal Income Taxes on Investment Income in Property-Liability Ratemaking,Journal of Risk and Insurance, 61:4, 691-709, December

Derrig, Richard A., (1989), Solvency Levels and Risk Loadings Appropriate for Fully Guaranteed Property-Liability Insurance Contracts: A Financial View, Financial Models of Insurance Solvency, J. David Cummins and R. A. Derrig (Eds), Kluwer Academic Publishers

Doherty, Neil A. and James R. Garven, (1991), Capacity and the Cyclicality of Insurance markets, Third International Conference on Insurance, Finance and Solvlency, Rotterdam, The Netherlands, May.

ReferencesFairley, William B., (1979), Investment Income and Profit Margins in Property-Liability Insurance: Theory and empirical Results, The Bell Journal of Economics, 10, 192-210, Spring.

Kaplan, Paul D. and James D. Peterson, (1998), Full-Information Industry Betas, Financial Management, Summer.

Ibbotson, Roger G, Paul D. Kaplan and James D. Peterson, (1997), Estimates of Small Stock Betas are Much Too Low, Journal of Portfolio Management, Summer.

Mahler, Howard C., (1985), An Introduction to Underwriting Profit Models, Proceedings of the Casualty Actuarial Society, Volume LXXII.

Myers, Stewart C. and Richard A. Cohn, (1987), A Discounted Cash Flow Approach to Property-Liability Insurance Rate Regulations, Fair Rate of Return in Property-Liability Insurance, J. David Cummins and Scott E. Harrington (Eds).

Myers, Stewart C. and James A. Read, Jr., (2000), Capital Allocation for Insurance Companies, AIB Working Paper, Nov.

SummaryModels follow policyholder or shareholder perspectivesCash flows are modelled according to perspectivePricing models are prospective and by line of businessCapital must be allocatedModel outputs are consistent with consistent parameters