2000 CLRS - September 18th CAS Fair Value Task Force White Paper Methods of Estimation Louise Francis Francis Analytics and Actuarial Data Mining, Inc.

2000 CLRS - September 18th Methods Section Discusses how fair values are estimated For Assets : Fair Value = Market Value For Liabilities: Market Value generally not available –Fair Value = f + risk load + other adjustments

2000 CLRS - September 18th Methods Section f considered straightforward to estimate using standard actuarial procedures This section focuses on methods of computing risk loads

2000 CLRS - September 18th The Methods 1.CAPM based methods 2.IRR approach 3.Single Period RAD 4.Methods that use historical underwriting data 5.Methods using probability distributions 6.Using reinsurance data 7.Direct Estimation Method 8.Transformed Distributions 9.Rules of thumb 10.Other

2000 CLRS - September 18th 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

2000 CLRS - September 18th Method 1: CAPM Based CAPM for assets: –r A = r f + β A (r M – r f ) CAPM for liabilities –r L = r f + β L (r M – r f ) β A is positive, β L is negative

2000 CLRS - September 18th Method 1: CAPM Based A number of different ways to estimate β L 1.Compute β e and β A for insurance companies. Get β L by subtraction. 2.Regress accounting underwriting profitability data on stock market index 3.Regress accounting underwriting profitability data by line on industry all lines profitability

2000 CLRS - September 18th Method 1: CAPM Method is controversial –Estimates of β L very sensitive to estimates of β A because of leverage –Accounting data biased –CAPM under attack in Finance literature –See Kozik, PCAS, 1994

2000 CLRS - September 18th 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

2000 CLRS - September 18th 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

2000 CLRS - September 18th Method 2: IRR

2000 CLRS - September 18th Method 2: IRR

2000 CLRS - September 18th 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

2000 CLRS - September 18th Method 3: Risk Adjusted Discount Method Example Leverage (S/L).5 ROE.13 E(r I ).07 E(r F ).06 E(t) 0 E(L) $100 Risk Adj = (S/L)*(ROE - E(r I )) +E(r F ) -E(r I ) =.5* ( ) =.02

2000 CLRS - September 18th 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

2000 CLRS - September 18th Method 4: Based on Underwriting Data c = (1+r F ) -u – e(1+r F ) -w – l(1+r A ) -t c is ratio of PV(profit) to premium r F is risk free rate, r A 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+r F )c/V m V m = PV(.5*(1+f)*L)), f is % losses outstanding at end of year

2000 CLRS - September 18th Method 4: Based on Underwriting Data

2000 CLRS - September 18th 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

2000 CLRS - September 18th 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.

2000 CLRS - September 18th Method 5: Distribution Based Methods

2000 CLRS - September 18th 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* A variance based risk load might have a λ of –Risk load = *1.9* =190,000

2000 CLRS - September 18th 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-r f )/(1+ROE)*S –If ROE =.13 and r f =.06 –Risk Margin =( )/1.13*4,340,000=230,442

2000 CLRS - September 18th 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-r f )S t /(1+ROE) t

2000 CLRS - September 18th 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 F to get risk load –Might need to make adjustments for riskiness of layers

2000 CLRS - September 18th 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)

2000 CLRS - September 18th Method 8: Distribution Transform Method Based on transforming aggregate probability distribution –Simple example: x -> kx –Where k>1

2000 CLRS - September 18th 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

2000 CLRS - September 18th Method 8: Distribution Transform Method Applied to Lognormal Aggregate Probability Distribution Transform distribution with p of.75. Mean 10% higher than original mean.

2000 CLRS - September 18th 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 )) (qp-1)

2000 CLRS - September 18th 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

2000 CLRS - September 18th 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

2000 CLRS - September 18th 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

2000 CLRS - September 18th Method 10: Other Intended to account for new methods which are developed and reasonable methods not covered here Risk margin should be positive