Reserve RangesReserve Ranges

Roger M. Hayne, FCAS, MAAARoger M. Hayne, FCAS, MAAAC.K. “Stan” Khury, FCAS, MAAAC.K. “Stan” Khury, FCAS, MAAARobert F. Wolf, FCAS, MAAARobert F. Wolf, FCAS, MAAA2005 CAS Spring Meeting2005 CAS Spring Meeting

Changing SceneChanging Scene

Changes:Changes:– Changes in the 2005 NAIC reporting Changes in the 2005 NAIC reporting

requirements (best estimate, ranges, etc.)requirements (best estimate, ranges, etc.)– SEC pending rule changes about disclosures SEC pending rule changes about disclosures

with respect to items involving uncertainty with respect to items involving uncertainty – Pending changes in the reserving principles Pending changes in the reserving principles – Pending changes in the ASOPPending changes in the ASOP

Unifying theme driving all of these Unifying theme driving all of these changes: changes: – A reserve is really a probability statement A reserve is really a probability statement

consisting of an amount x plus the probability consisting of an amount x plus the probability that the final settlement will not exceed xthat the final settlement will not exceed x

A Range – Gas or A Range – Gas or Electric?Electric? Start simple – a range around what?Start simple – a range around what? Accountants say it is to be a Accountants say it is to be a

“reasonable estimate” of the unpaid “reasonable estimate” of the unpaid claim costsclaim costs

CAS says that “an actuarially sound CAS says that “an actuarially sound loss reserve … is a provision, based on loss reserve … is a provision, based on estimates derived from reasonable estimates derived from reasonable assumptions and appropriate assumptions and appropriate methods…”methods…”

First Question – An First Question – An Estimate of What?Estimate of What? An “estimate” of amount unpaidAn “estimate” of amount unpaid Is it an estimate of the average Is it an estimate of the average

amount to be paid? Noamount to be paid? No Is it an estimate of the most likely Is it an estimate of the most likely

amount to be paid? Noamount to be paid? No It is an estimate of the amount to It is an estimate of the amount to

be paidbe paid

Simple ExampleSimple Example

Reserves as of 12/31/2005Reserves as of 12/31/2005 Claim to be settled 1/1/2006 with Claim to be settled 1/1/2006 with

immediate payment of $1 million times immediate payment of $1 million times roll of fair dieroll of fair die

All results equally likely so some All results equally likely so some accounting guidance says book low accounting guidance says book low end ($1 million), others midpoint ($3.5 end ($1 million), others midpoint ($3.5 million)million)

Mean and median are $3.5 millionMean and median are $3.5 million

An Almost-Simple An Almost-Simple ExampleExample Reserves as of 12/31/2005Reserves as of 12/31/2005 Claim to be settled 1/1/2006 as $1 Claim to be settled 1/1/2006 as $1

million times toss of loaded die:million times toss of loaded die:– Prob(x=1)=Prob(x=6)=1/4Prob(x=1)=Prob(x=6)=1/4– Prob(x=2)=Prob(x=5)=1/6Prob(x=2)=Prob(x=5)=1/6– Prob(x=3)=Prob(x=4)=1/12Prob(x=3)=Prob(x=4)=1/12

What do you book now? What do you book now? Mean and median still $3.5 million Mean and median still $3.5 million ““Most likely” is either $1 million or $6 Most likely” is either $1 million or $6

millionmillion

Traditional ApproachTraditional Approach

Traditional actuarial methods:Traditional actuarial methods:– ““Deestribution? We don’ need no Deestribution? We don’ need no

steenkin’ deestribution.”steenkin’ deestribution.” Traditional methods give “an Traditional methods give “an

estimate”estimate” No assumptions, thus no conclusions No assumptions, thus no conclusions

on distributionson distributions There are stochastic versions of some There are stochastic versions of some

methods (chain ladder, Bornhuetter-methods (chain ladder, Bornhuetter-Ferguson)Ferguson)

Traditional EstimatesTraditional Estimates

Traditional methods give “estimates”Traditional methods give “estimates”– Not estimates of the meanNot estimates of the mean– Not estimates of the medianNot estimates of the median– Not estimates of the modeNot estimates of the mode– Not estimates of a percentileNot estimates of a percentile– Not estimates of any statistic of the Not estimates of any statistic of the

distributiondistribution– Just “estimates”Just “estimates”

Distributions are normally possible Distributions are normally possible onlyonly after added assumptionsafter added assumptions

Range of Reasonable Range of Reasonable ResultsResults Designed for traditional analysisDesigned for traditional analysis Does not address or even talk Does not address or even talk

about distributionsabout distributions Definition is “soft” and talks about Definition is “soft” and talks about

results of “appropriate” methods results of “appropriate” methods under “reasonable” assumptionsunder “reasonable” assumptions

Does Does notnot refer to the distribution of refer to the distribution of potential outcomespotential outcomes

Reasonable?Reasonable?

Range of reasonable results an Range of reasonable results an attempt to quantify an actuary’s “gut attempt to quantify an actuary’s “gut feel” or “judgment”feel” or “judgment”

Typically you do a lot of methodsTypically you do a lot of methods– If they “bunch up” you feel “good”If they “bunch up” you feel “good”– If they are “spread out” you feel If they are “spread out” you feel

“uncomfortable”“uncomfortable” In the end – estimate is quite In the end – estimate is quite

subjectivesubjective

Model and MethodModel and Method

A method is a general approachA method is a general approach– Chain ladderChain ladder– Bornhuetter-FergusonBornhuetter-Ferguson

A model usually specifies an underlying A model usually specifies an underlying process or distribution and the focus is process or distribution and the focus is on identifying the parameters of the on identifying the parameters of the modelmodel

Most traditional actuarial forecasting Most traditional actuarial forecasting approaches are methods and not modelsapproaches are methods and not models

Stochastic MethodsStochastic Methods

Stochastic methods have assumptions Stochastic methods have assumptions about underlying modelsabout underlying models

Nearly all focus on a single data set Nearly all focus on a single data set (paid loss triangle, incurred loss (paid loss triangle, incurred loss triangle, etc.)triangle, etc.)

Do not directly model multiple sources Do not directly model multiple sources of information (e.g. counts, paid, and of information (e.g. counts, paid, and incurred at the same time)incurred at the same time)

Mack/Quarg method not yet stochasticMack/Quarg method not yet stochastic

Some VocabularySome Vocabulary

Components of uncertainty:Components of uncertainty:– ProcessProcess– ParameterParameter– Model/SpecificationModel/Specification

Any true estimate of the Any true estimate of the distribution of outcomes distribution of outcomes ordinarily would recognize all ordinarily would recognize all threethree

ProcessProcess

Uncertainty that cannot be avoidedUncertainty that cannot be avoided Inherent in the processInherent in the process Example – the throw of a fair dieExample – the throw of a fair die

– You completely know the processYou completely know the process– You cannot predict the result with You cannot predict the result with

certaintycertainty Usually the smallest component of Usually the smallest component of

insurance distributions (law of large insurance distributions (law of large numbers)numbers)

ParameterParameter

Uncertainty about the parameters of Uncertainty about the parameters of models (Note: Some models are not models (Note: Some models are not parametric)parametric)

The underlying process is knownThe underlying process is known Just the position of some “knobs” is Just the position of some “knobs” is

notnot Example – flip of a weighted coinExample – flip of a weighted coin

– Uncertainty regarding the expected Uncertainty regarding the expected proportion of headsproportion of heads

Model/SpecificationModel/Specification

The uncertainty that you have the The uncertainty that you have the right model to begin withright model to begin with

Not just what distributions, but Not just what distributions, but what form the model should takewhat form the model should take

Most difficult to estimateMost difficult to estimate Arguably un-estimable for P&C Arguably un-estimable for P&C

insurance situationsinsurance situations

Distribution of Distribution of OutcomesOutcomes Combines all sources of Combines all sources of

uncertaintyuncertainty Gives potential future payments at Gives potential future payments at

point in time along with an point in time along with an associated likelihoodassociated likelihood

Must be estimatedMust be estimated Estimation is itself subject to Estimation is itself subject to

uncertainty, so we are not away uncertainty, so we are not away from “reasonableness” issuesfrom “reasonableness” issues

What is Reasonable?What is Reasonable?

I use a series of methodsI use a series of methods My “range of reasonable estimates” is My “range of reasonable estimates” is

the range of forecasts from the various the range of forecasts from the various methodsmethods

Is this reasonable?Is this reasonable? What if one or more of the assumptions What if one or more of the assumptions

or methods is really “unreasonable”?or methods is really “unreasonable”? Is something outside this range Is something outside this range

necessarily “unreasonable”?necessarily “unreasonable”?

A Range IdeaA Range Idea

Take largest and smallest forecast by Take largest and smallest forecast by accident yearaccident year

Add these togetherAdd these together Is this a “reasonable range”Is this a “reasonable range” Example:Example:

– Roll of single fair die, 2/3 confidence interval Roll of single fair die, 2/3 confidence interval is between 2 an 5 inclusiveis between 2 an 5 inclusive

– Roll of a pair of fair dice, 2/3 confidence Roll of a pair of fair dice, 2/3 confidence interval is between 5 and 9 inclusive, not 4 to interval is between 5 and 9 inclusive, not 4 to 10 (5/6).10 (5/6).

You Missed You Missed Again!!Again!!

Your best estimate is $xYour best estimate is $x Actual future payments is $y (>$x)Actual future payments is $y (>$x) Conclusion – you were “wrong”Conclusion – you were “wrong” Why? The myth that the estimate Why? The myth that the estimate

actually will happenactually will happen Problem – a reserve is a Problem – a reserve is a

distributiondistribution, not just a single point, , not just a single point, any other treatment is doomed to any other treatment is doomed to failurefailure

Why Can’t the Why Can’t the Actuaries Get it Right?Actuaries Get it Right? Actually, why can’t the accountants get it Actually, why can’t the accountants get it

right?right? The accountants need to deal with the The accountants need to deal with the

fact rather than the myth that the actual fact rather than the myth that the actual payments will equal the reserve estimatepayments will equal the reserve estimate

Need to Need to – Be able to book a distributionBe able to book a distribution– Recognize the entire distributionRecognize the entire distribution– Recognize context (company environment)Recognize context (company environment)– Realize that future payments = reserves is an Realize that future payments = reserves is an

accident with a nearly 0% chance of accident with a nearly 0% chance of happeninghappening

An Economically An Economically Rational ReserveRational Reserve Why not set reserves so that the loss in Why not set reserves so that the loss in

company value when actual payments company value when actual payments turn out different is the least expectedturn out different is the least expected

Note expectation taken over all Note expectation taken over all possible reserve outcomes (along with possible reserve outcomes (along with their probabilities)their probabilities)

Economically rational – focuses on the Economically rational – focuses on the impact of the final settlement on a impact of the final settlement on a company’s net worthcompany’s net worth

Least PainLeast Pain

Since any single number will be “wrong” Since any single number will be “wrong” let me submit a reasonable estimate of let me submit a reasonable estimate of reserves (compliments of Rodney Kreps)reserves (compliments of Rodney Kreps)

Suppose Suppose – (a really BIG suppose) we know the (a really BIG suppose) we know the

probability density function of future claim probability density function of future claim payments and expenses is f(x) payments and expenses is f(x)

– For simplicity assume a one year time For simplicity assume a one year time horizonhorizon

– g(x,μ) denotes the decrease in shareholder g(x,μ) denotes the decrease in shareholder (policyholder) value of the company if (policyholder) value of the company if reserves are booked at μ but payments are reserves are booked at μ but payments are actually x.actually x.

Least Pain (Cont.)Least Pain (Cont.)

A rational reserve (i.e. “estimate of A rational reserve (i.e. “estimate of future payments”) is that value of future payments”) is that value of μμ that minimizesthat minimizes

i.e. the expected penalty for setting i.e. the expected penalty for setting reserves at reserves at μμ over all reserve over all reserve outcomesoutcomes

0

P g fx, x dx

A Reasonable gA Reasonable g

Likely not symmetricLikely not symmetric Likely flat in a region “near” μLikely flat in a region “near” μ Increases faster when x is above Increases faster when x is above

μ than when x is belowμ than when x is below Likely increases at an increasing Likely increases at an increasing

rate when x is above μrate when x is above μ Such a function generally gives Such a function generally gives

an estimate above the meanan estimate above the mean

Example Distribution IExample Distribution I

Distribution of IBNR Outcomes Using Chain Ladder Method - UnWeighted Analysis

0.00%

0.05%

0.10%

0.15%

0.20%

0.25%

0.30%

0.35%

0.40%

0.45%

0.50%

0.55%

IBNR Amounts

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Example Distribution IIExample Distribution II

Distribution of IBNR Outcomes Using Chain Ladder Method - Incurred - UnWeighted Analysis

0.00%

0.05%

0.10%

0.15%

0.20%

0.25%

0.30%

0.35%

0.40%

0.45%

0.50%

0.55%

0.60%

0.65%

-3.1 -1.7 -0.3 1.2 2.6 4.0 5.5 6.9 8.4 9.8 11.2 12.7 14.1 15.6 17.0 18.4 19.9 21.3 22.7 24.2 25.6 27.1 28.5 29.9 31.4

IBNR Amounts

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