2000 CLRS - September 18th CAS Fair Value Task Force White Paper Methods of Estimation Louise...

2000 CLRS - September 18th

CAS Fair Value Task Force White Paper

Methods of Estimation

Louise FrancisFrancis Analytics and Actuarial Data Mining, Inc.

[email protected]


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


Methods Section

• PV(Liabilities)@rf considered straightforward to estimate using standard actuarial procedures

• This section focuses on methods of computing risk loads


The Methods1. CAPM based methods2. IRR approach3. Single Period RAD4. Methods that use historical underwriting data5. Methods using probability distributions6. Using reinsurance data7. Direct Estimation Method8. Transformed Distributions9. Rules of thumb10. Other


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


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


Method 1: CAPM Based

• A number of different ways to estimate βL

1. Compute βe and βA for insurance companies. Get βL 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


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


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


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


Method 2: IRRFixed Inputs Loss & LAE cash flow patterns

1 Risk-free rate 0.0602 Expected investment return0.080 Proportion3 Income tax rate0.350 Time of Total4 Equity beta 0.800 0 0.0005 market risk premium0.090 1 0.5006 Capital/reserve 0.500 2 0.3007 Loss & LAE 1000.00 3 0.2008 Total 1.0009 Calculated values10 Required ROE0.132011 Risk-adjusted yield0.034612 After-tax risk-free rate0.039013 Premium 968.751415 Iterative Input16 Risk adjustment0.02541718 Balance sheet, at fair value19 0 1 2 320 Assets21 Investments, before dividend960.14 1018.12 469.03 110.5522 Investments, after dividend1432.21 725.53 292.97 0.002324 Liabilities

Time


Method 2: IRRBalance sheet, at fair value

0 1 2 3AssetsInvestments, before dividend960.14 1018.12 469.03 110.55Investments, after dividend1432.21 725.53 292.97 0.00

LiabilitiesLoss & LAE 944.15 476.81 193.31 0.00Income tax liability24.60 10.31 3.01 0.00Capital, before dividend0.00 531.00 272.71 110.55Capital after div (required amount)472.07 238.41 96.66 0.00

IncomeUnderwriting income24.60 -32.67 -16.50 -6.69Investment income 114.58 58.04 23.44Net income, pretax24.60 81.91 41.54 16.75Inv income, capital (risk-adjusted)28.32 14.30 5.80

Insurance Cash FlowsPremium 968.75 0.00 0.00 0.00Loss & LAE 0.00 -500.00 -300.00 -200.00Income tax -8.61 -28.67 -14.54 -5.86

Income tax, capital (risk-adjusted)9.91 5.01 2.03

Time


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


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


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


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


Method 4: Based on Underwriting DataInterest Rate Rf 0.0972Fraction of losses OS after 1 year 0.591Initial Risk Adjustment 0.044

Variable Nominal ValueDuration Discounted Value1 Loss&LAE 0.767 2.300 0.6812 Premium 1.000 0.250 0.9773 UW Expense 0.268 0.250 0.2624 Pol Dividends 0.016 2.250 0.0135 Average Liabilities 0.610 1.800 0.556

Calculation6 Premium-Expenses Discounted

(2) - (3) - (4) 0.7027 Premiums-Expenses-Losses Disc 0.021

(6)-(1)8 C*(1+I) 0.024

(7)*(1+I)9 Z=C*(1+I)/Vm 0.042

(8)/(5)


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


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.


Method 5: Distribution Based Methods

$2,000,000.00 $4,000,000.00 $6,000,000.00 $8,000,000.00 $10,000,000.00

050

100

150

Liability Value

Aggregate Probability Distribution for Liabilities in Line X



• 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



• 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



• 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


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


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)


Method 8: Distribution Transform Method

• Based on transforming aggregate probability distribution– Simple example: x -> kx– Where k>1



• 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


Method 8: Distribution Transform Method Applied to Lognormal Aggregate Probability Distribution

$1,000,000.00$3,000,000.00

$5,000,000.00$7,000,000.00

$9,000,000.00$11,000,000.00LIability

0.0

0.2

0.4

0.6

0.8

1.0

Power Transform of Lognormal Aggregate

TransformCumulative Distribution

Transform distribution with p of .75. Mean 10% higher than original mean.



• 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)



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


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


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


Method 10: Other

• Intended to account for new methods which are developed and reasonable methods not covered here

• Risk margin should be positive

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2000 CLRS - September 18th CAS Fair Value Task Force White Paper Methods of Estimation Louise...

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