Risk & Return Parameter Estimation David Appel, Ph.D. Milliman & Robertson Richard A. Derrig, Ph.D. Senior Vice President Automobile Insurers Bureau of MA Casualty Actuarial Society Seminar on Ratemaking March 12, 2001 Las Vegas, NV

Agenda  Risk and Return Models  Time Value of Money  Cash Flow Patterns & Levels  Measuring Risk - CAPM  IRR Models  Allocating Capital  Calendar Year Acct Models  Risk Premium Project (COTOR)  Summary

uRisk 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

uRisk 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 HistoricalProspective CASH FLOWACCOUNTING CYAM ISO STATE-X NPV (policyholder perspective) IRR (shareholder perspective)

uTime Value of Money  Premiums and Capital In  Expenses and Claims Out  Risk-Free Interest Rates  Risky Investments

uCash Flow Patterns & Levels  Premium Payments  Finance Charges  Acquisition Costs  General Expense  Premium Taxes  Capital Investment  Investment Income  Income Taxes

uCash Flow Patterns & Levels  Premium Payments  Finance Charges  Acquisition Costs  General Expense  Premium Taxes  Capital Investment  Investment Income  Income Taxes

uCash Flow Patterns & Levels Income Taxes  Federal (35% Marginal Rate)  Change in Unearned Premium Reserve  Underwriting: Earned Premium - Formula Discounted Loss Reserves  Investment: 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

uMeasuring Risk  Cost of Capital  Capital Asset Pricing Model (CAPM)  Equity, Asset and Liability Betas  Market Risk Premium  Dividend Growth Models  Risk-Adjusted Discount Rates

Capital Asset Pricing Model (CAPM) Investors are compensated for non- diversifiable (i.e., “systematic”) risk only r adj = r free + {  x (r market - r free )} where, r adj = risk adjusted return r free = return on risk free investments r market = return on the market  = systematic risk coefficient

Capital Asset Pricing Model (Theoretical Relationship) Expected Return Risk Premium   = Risk 20.1 R*= 15.4 R f = 6.0 R* = r f +  * (market risk premium) Expected Return

Capital Asset Pricing Model (Observed Empirical Relationship) Expected Return Risk Premium  Beta, Firm Size, Market/Book ratio Returns are higher than predicted for: Low Beta Small Size Low M/B Return Empirical Observation

CAPM Anomalies Returns higher for: Low beta firms Small firms Low M/B firms Insurers tend to be:Average beta Relatively small market cap Relatively 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 premium Ibbotson Associates Greenwich Associates

Capital Asset Pricing Model Beta Coefficients Sources: Value Line Investment Survey, Part 3, The Ratings & Reports, June 2, 2000 and June 30, 2000.

Excess Market Risk Premium  Definition: MRP = RM - RF  RF depends on horizon length  RF = 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?

Simple CAPM is Deficient Add Small Stock Effect

IRR - Relevant Cash Flows  Shareholders commit equity capital to support sale of insurance  In 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 surplus  IRR cash flows conditional on accounting conventions (usually SAP) and tax rules

IRR - Algebra IRR = r = discount rate such that Set price such that IRR = target return (COK) (Cf i ) (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)

FinalNPV of Final TimeCash FlowCash Flow SUM = 0.0 IRR = 16.5% IRR CASH FLOWS IN NOMINAL 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 If D grows at constant annual rate, g, then and P o =

DCF Issues  Sample selection  Estimation of growth rate (g) - Historical data or analysts forecasts - Earnings, dividends or book value  Single growth rate or multi-stage model

Discounted Cash Flow Analysis Estimated Dividend Yield Estimated Dividend Yield 21th Century ACE Limited Allmerican Finan Allstate Corp. Amer Intl Group American Finan Berkley (W.R.) Chubb Cincinnati Fin Fremont General GAINSCO HCC Ins Holdings HSB Group Hartford Finan Mercury General Ohio Casualty Old Republic Progressive RLI Corp Reliance Selective St. Paul Transatlantic Unitrin XL Capital Limited SAFECO Average nil 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 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 Sources: Value Line Investment Survey, Part 3, The Ratings & Reports, June 2, 2000 and June 30, 2000.

uAllocating Capital  Standard Allocations  Premium  Liabilities  Discounted Liabilities  Myers-Cohn Allocation  Myers-Read

Calendar Year Accounting Model - CYAM Total Return = (UW Profit + IY Reserves) + IY Surplus = Return on Operations + Return from Investment of Surplus Return on Operations= Return attributable to undertaking the risk of the insurance transaction

CYAM - Algebra 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

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 ANALYSIS STEADY STATE / GROWTH RATE = 0% AY1AY2AY3AY Calendar Year ROE =( )/75 = 22.3%

CALENDAR/ACCIDENT YEAR ANALYSIS STEADY STATE / GROWTH RATE = 16.5% NPV OF AY1AY2 AY3 AY4 AY Calendar Year ROE =( )/101.8 = 16.5%

CALENDAR/ACCIDENT YEAR ANALYSIS STEADY STATE / GROWTH RATE = 25.0% NPV OF AY1AY2AY3 AY4AY Calendar Year ROE =( )/117.2 = 14.2%

uModel Outputs  NPV: Underwriting Profit Provision  IRR: Expected Return to Capital  CYA: Return to Capital: Actual or Expected  Model Outputs Consistent if Inputs are Consistent

uOther Issues  Guaranty Funds  Residual Markets  Reinsurance  Default Risk  Risk Premium Project  Excess Capital

CAS Risk Premium Project  Committee on Theory of Risk  Discount Rate for Liabilities  Literature Review  Actuarial: Process and Parameter Risk  Financial: Systematic Risk  Academic: Dave Cummins, Rich Phillips  Industry: Bob Butsic, Rich Derrig

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, % (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 Regression  P & C: Equity Beta 12/31/1998 of 0.92; 3/31/2000 of 1.15

Surplus Allocation  Surplus by Company stands behind all lines  Surplus 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

References  Almagro, 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.

References  Cummins, J. David, (1990), Multi-Period Discounted Cash Flow Ratemaking Models in Property-Liability Insurance, Journal of Risk & Insurance, 57:1, , March.  Cummins, J. David, (1988), Risk-Based Premiums for Insurance Guaranty Funds, Journal of Finance, 43, , 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, , 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.

References  Fairley, William B., (1979), Investment Income and Profit Margins in Property-Liability Insurance: Theory and empirical Results, The Bell Journal of Economics, 10, , 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.

uSummary  Models follow policyholder or shareholder perspectives  Cash flows are modelled according to perspective  Pricing models are prospective and by line of business  Capital must be allocated  Model outputs are consistent with consistent parameters