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Reserve Ranges. Roger M. Hayne, FCAS, MAAA C.K. “Stan” Khury, FCAS, MAAA Robert F. Wolf, FCAS, MAAA 2005 CAS Spring Meeting. Changing Scene. Changes: Changes in the 2005 NAIC reporting requirements (best estimate, ranges, etc.) - PowerPoint PPT Presentation
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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
BassAndKhury.com
Fre
qu
ency
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
BassAndKhury.com
Fre
qu
ency
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