Methods for Estimating Fair Value in Actuarial Science

1. CAS Fair Value Task Force White Paper Methods of Estimation Louise Francis Francis Analytics and Actuarial Data Mining, Inc. louise_francis@msn.com

2. Methods Section • Discusses how fair values are estimated • For Assets : Fair Value = Market Value • For Liabilities: Market Value generally not available • Fair Value = PV(Liabilities)@rf • + risk load • + other adjustments

3. Methods Section • PV(Liabilities)@rf considered straightforward to estimate using standard actuarial procedures • This section focuses on methods of computing risk loads

4. The Methods • CAPM based methods • IRR approach • Single Period RAD • Methods that use historical underwriting data • Methods using probability distributions • Using reinsurance data • Direct Estimation Method • Transformed Distributions • Rules of thumb • Other

5. Two Major Paradigms • Finance Perspective • Only non diversifiable risk included in risk load • Non diversifiable risk used in risk load is systematic risk • Actuarial Perspective • Diversifiable risk matters • Non diversifiable risk used in risk load is parameter risk

6. Method 1: CAPM Based • CAPM for assets: • rA = rf + βA (rM – rf) • CAPM for liabilities • rL = rf + βL (rM – rf) • βA is positive, βL is negative

7. Method 1: CAPM Based • A number of different ways to estimate βL • Compute βe and βA for insurance companies. Get βL by subtraction. • Regress accounting underwriting profitability data on stock market index • Regress accounting underwriting profitability data by line on industry all lines profitability

8. Method 1: CAPM • Method is controversial • Estimates of βL very sensitive to estimates of βA because of leverage • Accounting data biased • CAPM under attack in Finance literature • See Kozik, PCAS, 1994

9. Method 2: IRR • A pricing based method • Uses the IRR pricing method to back into a risk adjusted discount rate • Internal rate of return on • capital contributions and withdrawals equals required rate of return

10. Method 2: IRR • Requires a surplus allocation • Requires an estimate of (target) ROE • Assumes risk load on reserves lies on a continuum with risk load used in pricing

11. Method 2: IRR

12. Method 2: IRR

13. Method : Risk Adjusted Discount Method • A pricing based method • Uses relationship between required ROE, expected investment return, income tax rate and capital to find risk adjusted discount rate

14. Method 3: Risk Adjusted Discount Method Example • Leverage (S/L) .5 • ROE .13 • E(rI) .07 • E(rF) .06 • E(t) 0 • E(L) $100 • Risk Adj = (S/L)*(ROE - E(rI)) +E(rF) -E(rI) = .5* (.13 - .07) + .06 - .07 = .02

15. Method 4: Based on Underwriting Data • Bases risk adjustment on long term averages of profitability observed in underwriting data. • Method first published by Butsic (1988) to compute risk adjusted discount rates • Uses industry wide data, possibly for all lines • Unless data for very long periods is used, results could be unstable

16. Method 4: Based on Underwriting Data • c = (1+rF)-u – e(1+rF)-w – l(1+rA)-t • c is ratio of PV(profit) to premium • rF is risk free rate, rA is risk adjusted rate • e is expense ratio • l is loss and LAE ratio • u is duration of premium, w is duration of expenses, t is duration of liabilities • Ratio c to average discounted losses to get risk adjustment: RA = (1+rF)c/Vm • Vm = PV(.5*(1+f)*L)), f is % losses outstanding at end of year

17. Method 4: Based on Underwriting Data

18. Method 5: Distribution Based Risk Loads • Three classical actuarial risk load formulas • Risk load = λ (sd Loss) • Risk load = λ (var Loss) • U(Equity) = E[U(Equity + Premium - Loss)] • A recent actuarial risk load formula • Risk Load =  Surplus Requirement • Surplus requirement from Expected Policyholder Deficit calculation

19. Method 5: Distribution Based Risk Loads • All four formulas require a probability distribution for aggregate losses • Simulation and Heckman-Meyers are common methods for deriving probability distribution • Probability distribution includes process and parameter risk • Risk load may not be value additive • Typically gives a risk load that is applied to PV(liabilities), not an adjustment to discount rate.

20. Method 5: Distribution Based Methods

21. Method 5: Distribution Based Methods • The aggregate losses displayed in the graph have a mean of $4.7M, and sd of $1.4M and a variance of 1.9*1012. • A variance based risk load might have a λ of 10-7 • Risk load = 10-7*1.9*10-12=190,000

22. Method 5: Distribution Based Methods • Standard deviation based risk loads often use the sd to derive a theoretical surplus: • Surplus (S) = z.999*sd = 3.1* 1.4M = 4,340,000 • Philbrick’s method for converting this into a risk load: • Risk Margin=(ROE-rf)/(1+ROE)*S • If ROE = .13 and rf =.06 • Risk Margin =(.13-.06)/1.13*4,340,000=230,442

23. Method 5: Distribution Based Methods • This result is about 5% of liabilities. • The risk margin might be 5% of liabilities discounted at the risk free rate • A more complicated formula for liabilities paying out over several years • RM=Σ(ROE-rf)St/(1+ROE)t

24. Method 6: Using the Reinsurance Market • Reinsurance surveys • Conceptually similar to PCS Cat options • Extrapolate from companies’ own reinsurance program • Compare price charged by reinsurers to PV(liabilities)@rF to get risk load • Might need to make adjustments for riskiness of layers

25. Method 7: Direct Estimation • Directly uses market values of companies’ equity and assets to derive market value of liabilities • MV(Liabilities) = MV(Assets) – MV(Equity) • Ronn-Verma method used to compute MV(Equity)

26. Method 8: Distribution Transform Method • Based on transforming aggregate probability distribution • Simple example: x -> kx • Where k>1

27. Method 8: Distribution Transform Method • Power transform • S*(x)->S(x)p • S(x) is survival distribution of x ,(1 – F(x)) • p is between 0 and 1 • The tail probabilities increase • Mean also increases • Choice of p depends on riskiness of business

28. Method 8: Distribution Transform Method Applied to Lognormal Aggregate Probability Distribution Transform distribution with p of .75. Mean 10% higher than original mean.

29. Method 8: Distribution Transform Method • Let F(x)=1-(b/(b+x))q, S(x)=b/(b+x)q • S*(x) = (b/(b+x))qp • E(x) =b/(q-1) • E*(x)=b/(qp-1) • ILF*(L)=1-(b/(b+L))qp-1/(1-b/(b+100000))(qp-1)

30. Method 8: Distribution Transform Method b=$5,000 q=1.6 p=.95 • E(x)=5,000/.6=8,333 • E*(x) = 5,000/.52 = 9,615, about 15% higher than E(x) • ILF(1M) =1.142 • ILF*(1M)=1.179

31. Method 9: Rules of Thumb • In some situations there may not be adequate data or other resources to develop risk loads from scratch • Rules of thumb may provide a quick and dirty by adequate approach • Might require an industry committee to develop the rules

32. Method 9: Rules of Thumb • Examples • Compute the risk adjusted discount rate by subtracting 3% from the risk free rate • The risk load should be 10% of the present value of liabilities in the General Liability line and 5% of liabilities in the Homeowners line

33. Method 10: Other • Intended to account for new methods which are developed and reasonable methods not covered here • Risk margin should be positive