by Louise Francis Louise_francis@msn

Fair Value Accounting and Actuaries in the Post-Enron

World

Sepember, 2002 Casualty Loss Reserve Seminar

by Louise [email protected]

Fair Valuation Task Force

• Much of material is based on work the CAS task force on fair value of liabilities

• White paper presenting the task force’s work is on CAS web site

• Focus is on valuing liabilities.

Fair Value

• For Assets : Fair Value = Market Value

• For Liabilities: Market Value generally not available– Fair Value = PV(Liabilities)@rf + risk

load+other adjustments

Some Alternatives to Fair Value

• Undiscounted expected values

• PV at risk free rate

• PV using industry standard risk adjustment

• Mixture of fair value and alternative

• Entity specific measure

Methods Section

• PV(Expected Liabilities)@rf considered straightforward to estimate using standard actuarial procedures– Use treasury rate for average duration of

liabilities or use a maturity schedule applied to cash flow

• This section focuses on a less familiar area: 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 Kosick, PCAS, 1991– Recent research funded by CAS and AERF has

addressed some of CAPM problems

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 withdrawls equals required rate of return

Method 2: IRR

• Requires a surplus allocation

• Requires an estimate of ROE

• Assumes risk load on reserves lies on a continuum with risk load used in pricing

Method : Risk Adjusted Discount Method

• A pricing based method

• Discount = risk free rate minus a risk adjustment

• Uses relationship between required ROE, expected investment return, income tax rate and ROE

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

Method 5: Loss 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

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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 $.14M 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(Assets)

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

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Power Transform of Lognormal Aggregate

TransformCumulative Distribution

Transform distribution 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

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

Method 10: Other

• Research on this subject is ongoing

• One method recently discussed is based on utility theory– Risk load based on stochastic analysis of

program and surplus used in adverse scenarios. A or charge is applied to surplus useage

Credit Standing and Fair Values

• Adjustment would recognize that a financially weak company would be less likely to satisfy its obligations in full than a financially strong company

• Reduce expected liabilities by expected amount not to be paid because of default

• A number of methods for estimating presented in white paper

Documents

by Louise Francis Louise_francis@msn