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Modeling and Estimating Modeling and Estimating Parameter UncertaintyParameter Uncertainty
1999 DFA Seminar1999 DFA Seminar
byby
Roger M. Hayne, FCAS, MAAARoger M. Hayne, FCAS, MAAA
Milliman & Robertson, Inc.Milliman & Robertson, Inc.
A Dilemma?A Dilemma?
Year-to-year payments often thought to be Year-to-year payments often thought to be (relatively) stable(relatively) stable
Reserve estimates may have significant Reserve estimates may have significant uncertaintyuncertainty
Dilemma: model liability payouts to reflect both Dilemma: model liability payouts to reflect both of these seemingly contradictory characteristicsof these seemingly contradictory characteristics
Consider Sources of UncertaintyConsider Sources of Uncertainty
ProcessProcess– Everything else knownEverything else known– RandomRandom
ParameterParameter– Models knownModels known– Parameters uncertainParameters uncertain
Specification/Model -- Model itself may be Specification/Model -- Model itself may be uncertainuncertain
Treatment in ModelingTreatment in Modeling
Process - model the processProcess - model the process ParameterParameter
– BayesianBayesian– Can use process model tooCan use process model too
ModelModel– Most difficultMost difficult– May be unquantifiableMay be unquantifiable– Not covered hereNot covered here
Simple ExampleSimple Example
X ~ lognormal(X ~ lognormal(,,²)²) ² is known² is known is uncertain with is uncertain with ~ N(m, ~ N(m,²)²) ² is known² is known Otherwise all random variables independentOtherwise all random variables independent Lognormal only chosen for ease of calculations -- Lognormal only chosen for ease of calculations --
results follow for more general problemsresults follow for more general problems
Simple AlgorithmsSimple Algorithms
IntuitiveIntuitive– Randomly pick Randomly pick from N(m, from N(m,²)²)– Randomly pick X from lognormal (Randomly pick X from lognormal (,,²)²)
““Smarter”Smarter”– Analysis shows X ~ lognormal(m,Analysis shows X ~ lognormal(m,²+ ²+ ²)²)– Randomly pick X from lognormal(m,Randomly pick X from lognormal(m,²+ ²+ ²)²)
Both give same answer Both give same answer
Not-So-Simple ExampleNot-So-Simple Example
Multiple time periodsMultiple time periods– – All independentAll independent– are all knownare all known– where where – b, b, and and ² are known² are known
represents “global” parameter uncertaintyrepresents “global” parameter uncertainty
X i ni i i~ lognormal , , , , , 2 1 2c h
i2
i im ~ ,N b 2c hmi ,
Not-So-Simple AlgorithmsNot-So-Simple Algorithms
IntuitiveIntuitive– Randomly pick Randomly pick from N(b, from N(b,²)²)– Randomly pick from Randomly pick from
““Smarter”Smarter”– Analysis shows Analysis shows – Randomly pick fromRandomly pick from
X i lognormal , mi i2c h
X bm mi i i i~ lognormal , 2 2 2c hlognormal ,bm mi i i 2 2 2c hX i
They Are NOT the Same!They Are NOT the Same!
The two algorithms can give different answers!The two algorithms can give different answers! Example, assume:Example, assume:
m i i
i
b
i
i
0 25 1 2 3
0 1 2 3
12
. , , ,
, , ,
Intuitive AlgorithmIntuitive Algorithm
0
0.5
1
1.5
2
2.5
3
3.5
X(1) X(1) X(3)
““Smarter” AlgorithmSmarter” Algorithm
0
0.5
1
1.5
2
2.5
3
X(1) X(1) X(3)
Why the Difference?Why the Difference?
Structure of the exampleStructure of the example Intuitive approach follows structure, i.e. Intuitive approach follows structure, i.e.
“uniform” uncertainty“uniform” uncertainty ““Smarter” approach loses “uniformity”Smarter” approach loses “uniformity” In the end both will give the same “envelope”In the end both will give the same “envelope” Paths within envelope will differPaths within envelope will differ
Why is This Important?Why is This Important?
Although example extreme (no process Although example extreme (no process uncertainty) it is close to insurer realityuncertainty) it is close to insurer reality
Meyers & Schenker indicates this is soMeyers & Schenker indicates this is so Usually number of insured claims “large” allowing Usually number of insured claims “large” allowing
“law of large numbers”“law of large numbers” There is still uncertainty that cannot be There is still uncertainty that cannot be
“diversified” by law of large numbers -- e.g. “diversified” by law of large numbers -- e.g. parameter (or model)parameter (or model)
Why is This Important?Why is This Important?
Parameter uncertainty often reflects uncertainty Parameter uncertainty often reflects uncertainty regarding future “state” of the universeregarding future “state” of the universe
In the example, In the example, bb represents various possible represents various possible different “futures”different “futures”
The Intuitive approach shows the paths of various The Intuitive approach shows the paths of various futuresfutures
The “Smarter” approach shows possible outcomes, The “Smarter” approach shows possible outcomes, not necessarily “futures”not necessarily “futures”
Which is Correct?Which is Correct?
Depends on applicationDepends on application If “parameter” uncertainty assumed to reflect If “parameter” uncertainty assumed to reflect
different “futures” then Intuitive Algorithm is different “futures” then Intuitive Algorithm is correctcorrect
For DFA applications (for uncertain cash flows) it For DFA applications (for uncertain cash flows) it is more likely that non-process uncertainty is more likely that non-process uncertainty reflects different futuresreflects different futures
Practical ConsiderationsPractical Considerations
With limited “process” uncertainty but relatively With limited “process” uncertainty but relatively more parameter uncertainty, one would expect more parameter uncertainty, one would expect different “paths” each of which are relatively different “paths” each of which are relatively smooth smooth
““Smarter” Algorithm, though correct at each Smarter” Algorithm, though correct at each point, does not produce these types of pathspoint, does not produce these types of paths
Resulting DFA model with Intuitive Algorithm Resulting DFA model with Intuitive Algorithm may be more “credible” to non-actuariesmay be more “credible” to non-actuaries
More Refined ExampleMore Refined Example
Multiple time periodsMultiple time periods– – for some and all for some and all ii– Here represents the inverse normal, i.e. that Here represents the inverse normal, i.e. that
value such that value such that – are all known are all known
Thus, individually,Thus, individually,
X i ni i i~ lognormal , , , , , 2 1 2c h i i im p 1af
i i im2 2, , and
0 1 p
X m i ni i i i~ lognormal , , , , , 2 2 1 2 c h
1 pafP ~ N ,Z p Z p 1 0 1af afc h
More Refined Example (Cont.)More Refined Example (Cont.)
Assumes each “path” can be determined by the Assumes each “path” can be determined by the probability level of the uncertainty parametersprobability level of the uncertainty parameters
Once a path is chosen only process uncertainty Once a path is chosen only process uncertainty causes variationcauses variation
Convenient way to model liability cash flowsConvenient way to model liability cash flows Quantification of “parameter” can be analytic or Quantification of “parameter” can be analytic or
judgmentaljudgmental
More Refined Example (Cont.)More Refined Example (Cont.)
Model of (current level) aggregate reserve runoff Model of (current level) aggregate reserve runoff for an insurer (for an insurer (hypothetical)hypothetical), for all , for all ii
Assume, N claims, distribution X, aggregate T:Assume, N claims, distribution X, aggregate T:
Year E(X) cv(X) N E(T)1 5,000 3.0 1,000 5,000,000 2 11,000 2.5 300 3,300,000 3 13,000 2.0 150 1,950,000 4 20,000 1.5 50 1,000,000 5 25,000 1.0 20 500,000 6 30,000 0.5 7 210,000 7 40,000 0.3 1 40,000
i 0 5.
Refined Example -- BandsRefined Example -- Bands
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
7,000,000
8,000,000
9,000,000
1 2 3 4 5 6 7
Year
More Refined ExampleMore Refined Example
0
2,000,000
4,000,000
6,000,000
8,000,000
10,000,000
12,000,000
1 2 3 4 5 6 7
Year
95%
5%
Refined Example -- Fully RandomRefined Example -- Fully Random
0
2,000,000
4,000,000
6,000,000
8,000,000
10,000,000
12,000,000
1 2 3 4 5 6 7
Year
95%
5%
ConsiderationsConsiderations
Lognormal assumption is for ease onlyLognormal assumption is for ease only– Closed form solutions to distributions at various pointsClosed form solutions to distributions at various points– Calculations relatively easy to doCalculations relatively easy to do
Bayesian approach also for convenienceBayesian approach also for convenience Process variance relatively easy to model using Process variance relatively easy to model using
collective risk, intuitive toocollective risk, intuitive too Analyze total variation to obtain an estimate of the Analyze total variation to obtain an estimate of the
various valuesvarious values i
Collective Risk ModelCollective Risk Model
Heckman & Meyers consider the algorithm:Heckman & Meyers consider the algorithm:– Randomly select number of claims, nRandomly select number of claims, n– Randomly select n independent claims and totalRandomly select n independent claims and total– Randomly select scale parameter Randomly select scale parameter from a distribution from a distribution
with mean 1 and variance bwith mean 1 and variance b– Divide total by Divide total by
Generally collective risk model with “parameter” Generally collective risk model with “parameter” uncertainty built in with uncertainty built in with
Collective RiskCollective Risk
Claim count distribution also “uncertain”Claim count distribution also “uncertain”– Select Select from a distribution with E(from a distribution with E()=1, Var()=1, Var()=)=cc– Select n, the number of claims, randomly from a Select n, the number of claims, randomly from a
Poisson distribution with parameter Poisson distribution with parameter In this caseIn this case
– E(n)=E(n)=, , Var(n)= Var(n)= +c+c²² If If is gamma, is gamma, cc<0 gives binomial, <0 gives binomial, cc=0 gives =0 gives
Poisson, Poisson, cc>0 gives negative binomial>0 gives negative binomial
Collective Risk (Cont.)Collective Risk (Cont.)
With this algorithm the total T hasWith this algorithm the total T has– E(E(TT) = ) = E(E(XX))– Var(Var(TT)= )= E(E(XX²)(1+²)(1+bb)+ )+ ²E²(²E²(XX)()(bb++cc++bcbc))
Other than independence and existence of Other than independence and existence of moments not many assumptions needed for these moments not many assumptions needed for these relationsrelations
Formula holds any time the count distribution hasFormula holds any time the count distribution has– Var(Var(nn)= E()= E(nn) + ) + ccE²(E²(nn))
Collective Risk (Cont.)Collective Risk (Cont.)
Allowing uncertainty in claim counts (i.e. not Allowing uncertainty in claim counts (i.e. not requiring requiring cc=0) but not in the scale (or “mixing”) =0) but not in the scale (or “mixing”) parameter (i.e. parameter (i.e. bb=0) gets:=0) gets:– E(E(T|bT|b=0=0)) = = E(E(XX))– Var(Var(T|bT|b=0)= =0)= E(E(XX²)+ ²)+ cc²E²(²E²(XX))– cv²(cv²(T|bT|b=0)==0)=cc+(1+cv²(+(1+cv²(XX))/))/
Use analysis to get claim size, expected count, Use analysis to get claim size, expected count, and, possibly, variance captured by and, possibly, variance captured by cc
Collective Risk (Cont.)Collective Risk (Cont.)
Assume from other analysis we can estimate total Assume from other analysis we can estimate total variance of reserve amounts, and thus cv(variance of reserve amounts, and thus cv(TT))
We can solve for We can solve for bb to get: to get:
bT T b
T b
T T b
T b T b
cv cv
cv
Var Var
Var E
2 2
2
2
0
0 1
0
0 0
af a fa faf a fa f a f
Collective Risk (Cont.)Collective Risk (Cont.)
Separates “process” and “parameter” uncertainty Separates “process” and “parameter” uncertainty for modelingfor modeling
Relatively easy to implement, even without Relatively easy to implement, even without “neat” distributions“neat” distributions
Relatively easy to explain and support in Relatively easy to explain and support in discussionsdiscussions
Collective Risk ExampleCollective Risk Example
Use basic distributions from earlier exampleUse basic distributions from earlier example Assume Assume c=c=0 (for simplicity)0 (for simplicity)
Year E(X) cv(X) N E(T|b=0) cv(T|b=0) cv(T) b1 5,000 3.0 1,000 5,000,000 0.100 0.333 0.1002 11,000 2.5 300 3,300,000 0.155 0.390 0.1253 13,000 2.0 150 1,950,000 0.183 0.434 0.1504 20,000 1.5 50 1,000,000 0.255 0.501 0.1755 25,000 1.0 20 500,000 0.316 0.566 0.2006 30,000 0.5 7 210,000 0.423 0.666 0.225
Collective Risk ExampleCollective Risk Example
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
7,000,000
1 2 3 4 5 6 7
Year
Somewhat Realistic ExampleSomewhat Realistic Example
Expected Standard Deviation ImpliedYear Cum. Paid Process Total b Value
1 $213,000 $5,900 $60,700 0.08042 431,000 15,400 114,300 0.06903 668,000 27,500 169,400 0.06254 923,000 41,200 222,800 0.05625 1,197,000 54,800 278,900 0.05216 1,491,000 66,800 337,100 0.04907 1,807,000 79,400 396,700 0.04628 2,144,000 84,700 457,900 0.04409 2,506,000 90,700 515,800 0.0410
10 2,894,000 94,500 574,400 0.0383
Somewhat Realistic ExampleSomewhat Realistic Example
0500,0001,000,0001,500,0002,000,0002,500,0003,000,0003,500,0004,000,0004,500,0005,000,000
1 2 3 4 5 6 7 8 9 10
Year
Fully RandomFully Random
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
3,000,000
3,500,000
1 2 3 4 5 6 7 8 9 10
Year
Second PaperSecond PaperEstimating the ParametersEstimating the Parameters
If you can estimate total uncertainty and If you can estimate total uncertainty and “process” you can “realistically” model liability “process” you can “realistically” model liability cash flowscash flows
Formula for “b” parameter usefulFormula for “b” parameter useful How can we derive the various estimates?How can we derive the various estimates?
Example FrameworkExample Framework
Lifetime medical care cases (workers compensation, auto Lifetime medical care cases (workers compensation, auto no-fault)no-fault)
Annuity model base “process”Annuity model base “process” Uncertainty in future payments due to:Uncertainty in future payments due to:
– Uncertain exit estimatesUncertain exit estimates– Uncertain inflationUncertain inflation– Uncertain cost estimatesUncertain cost estimates– Uncertain frequency estimatesUncertain frequency estimates
Concepts can carry over to other applicationsConcepts can carry over to other applications
Simple Annuity - One ClaimantSimple Annuity - One Claimant
Pay for claimant’s medical care (resulting from Pay for claimant’s medical care (resulting from accident) for lifeaccident) for life
Estimate liability in two steps:Estimate liability in two steps:– Estimate probability the claimant remains in population Estimate probability the claimant remains in population
incurring costs for each future yearincurring costs for each future year– Estimate costs incurred in each future yearEstimate costs incurred in each future year
Total reserves can be expected value (given the Total reserves can be expected value (given the probability distribution inherent in the “survival”)probability distribution inherent in the “survival”)
Similarly for annual payments Similarly for annual payments
One Claimant CalculationOne Claimant Calculation
First assume no uncertainty in any parameters or First assume no uncertainty in any parameters or estimatesestimates
Let:Let:– aaxsxs denote payments for claimant denote payments for claimant xx in year in year ss
– ppxsxs denote the probability claimant denote the probability claimant xx incurs a loss in incurs a loss in
year year ss and then exits the claimant population and then exits the claimant population Definition guaranteesDefinition guarantees 1 xs
s
p
One Claimant, One Year DistributionOne Claimant, One Year Distribution
Distribution of payments for claimant Distribution of payments for claimant xx in year in year ss
ProbabilityProbability AmountAmount
xtt s
p
1
xtt s
p
xsa
0
One Claimant, One Year MomentsOne Claimant, One Year Moments
Moments are easy to calculate (for this one):Moments are easy to calculate (for this one):
E
s xs xtt s
X a p
2Var 1
s xs xt xtt s t s
X a p p
One Claimant, Total MomentsOne Claimant, Total Moments
Assuming all years are statistically independent, total Assuming all years are statistically independent, total moments for one claimant not too difficult:moments for one claimant not too difficult:
Assuming claims independent these add for total Assuming claims independent these add for total populationpopulation
1
E
xs xts t s
X a p
1 1 max( , ) min( , )
Var 1
xs xr xt xts r t r s t r s
X a a p p
A Little BreatherA Little Breather
Last formula a bit complicated, but closed formLast formula a bit complicated, but closed form Benefit of annuity -- a known underlying Benefit of annuity -- a known underlying
distribution (the decrement table)distribution (the decrement table) Generally for other casualty applications Generally for other casualty applications
distribution not as well knowndistribution not as well known Collective risk model helps, but not necessarily Collective risk model helps, but not necessarily
on timingon timing
Add Some UncertaintyAdd Some Uncertainty
Decrement table is uncertainDecrement table is uncertain– Mortality for target population may not match available Mortality for target population may not match available
tablestables– Claimants can exit for reasons other than mortality Claimants can exit for reasons other than mortality
(recovery, settlement, etc.)(recovery, settlement, etc.) Future payments affected by economic conditionsFuture payments affected by economic conditions
– InflationInflation– Investment income (linked to inflation?)Investment income (linked to inflation?)
Payment estimates uncertainPayment estimates uncertain
Add Some UncertaintyAdd Some Uncertainty
Can deal with these elements one at a timeCan deal with these elements one at a time Exit probabilities:Exit probabilities:
– Tables usually constructed based on a sample Tables usually constructed based on a sample populationpopulation
– Statistical theory can quantify uncertainty in binomial Statistical theory can quantify uncertainty in binomial parameter estimate given sample sizeparameter estimate given sample size
– Simplifying assumption of uniform uncertainty factor Simplifying assumption of uniform uncertainty factor for all ages makes estimates tractablefor all ages makes estimates tractable
Add Some UncertaintyAdd Some Uncertainty
Assume one-year retention rate is uncertain, i.e.Assume one-year retention rate is uncertain, i.e.
yy (note uniform for all ages and claimants) is assumed (note uniform for all ages and claimants) is assumed binomial with unknown parameter binomial with unknown parameter
If If is estimated using a sample of size is estimated using a sample of size nn with with zz observed claims remaining after a year then observed claims remaining after a year then will have will have a Beta distribution with parameters a Beta distribution with parameters z+1z+1 and and n-z+1n-z+1..
1 1 xt xtq q y
Add Some UncertaintyAdd Some Uncertainty
Assumptions allow us to calculate moments, Assumptions allow us to calculate moments, using the facts thatusing the facts that
and properties of the beta distribution to estimate and properties of the beta distribution to estimate moments for moments for
1
0
1
m
xt x it m i
p q
E ry
Decrement ConsiderationsDecrement Considerations
Standard mortality tables may not be appropriate Standard mortality tables may not be appropriate for catastrophically injured claimantsfor catastrophically injured claimants
May not be many (any) injury-specific tablesMay not be many (any) injury-specific tables Spinal cord injury tables more commonSpinal cord injury tables more common Head injury tables much less commonHead injury tables much less common Also should consider decrements for reasons Also should consider decrements for reasons
other than deathother than death
Economic UncertaintyEconomic Uncertainty
Economic conditions can affectEconomic conditions can affect– Inflation (cost increase)Inflation (cost increase)– Investment returnInvestment return
In DFA models this can be “perfect” linkage In DFA models this can be “perfect” linkage between asset and liability, so do these analyses between asset and liability, so do these analyses on a current-dollar basis and separately on a current-dollar basis and separately incorporate economic assumptionsincorporate economic assumptions
Can build uncertainty for these factors if desiredCan build uncertainty for these factors if desired
Cost Estimate UncertaintyCost Estimate Uncertainty
aaxsxs estimated, so uncertain estimated, so uncertain
Simplifying assumption all estimates all relative Simplifying assumption all estimates all relative errors are uniform across claims/yearserrors are uniform across claims/years
Quantify uncertaintyQuantify uncertainty– Compare actual vs. expectedCompare actual vs. expected– Consider “incurred” developmentConsider “incurred” development
Actual/ExpectedActual/Expected
Payment Forecast Annual PaymentYear Year Actual Expected ln(A/E)
1 0 50,000$ 45,000$ 0.10542 0 40,000 35,000 0.13352 1 40,000 45,000 -0.11783 0 30,000 25,000 0.18233 1 30,000 35,000 -0.15423 2 30,000 30,000 0.0000
Average 0.0249
Incurred DevelopmentIncurred Development
Months of DevelopmentReserve 12 24 36
Year Current Prior Current Prior Current
1995 $100,000 $110,000 $105,000 $107,100 $109,5001996 125,000 143,750 137,5001997 175,000
Development Factors24/12 36/24
1995 1.10 1.02
Moving Beyond Case ReservesMoving Beyond Case Reserves
Calculations so far only address known claims Calculations so far only address known claims with case reserveswith case reserves
There are othersThere are others– Known without case reservesKnown without case reserves– IBNR claimsIBNR claims
Uncertainty estimates for these categories depend Uncertainty estimates for these categories depend on estimation methodologyon estimation methodology
More General AlgorithmMore General Algorithm
Suppose aggregate loss is given bySuppose aggregate loss is given by
Claims with individual reserveClaims with individual reserve
Known with average reserveKnown with average reserve
IBNR ClaimsIBNR Claims
1 11
R B R B
R B
R
R
N N
ii N
N N N
ii N N
N
ii
XT X X
More General AlgorithmMore General Algorithm
Assumes all claims are “generally” drawn from same Assumes all claims are “generally” drawn from same distribution and “global” uncertainty distribution and “global” uncertainty
NNRR number of claims with reserves number of claims with reserves
Modifies Modifies NNBB “formula” claims by a single random “formula” claims by a single random
variable variable Modifies IBNR claims by a single random variable Modifies IBNR claims by a single random variable Uncertain number of IBNR claims Uncertain number of IBNR claims NN
More General AlgorithmMore General Algorithm
Can get estimates for the parameters for Can get estimates for the parameters for and and by reviewing relationships between reserved and by reviewing relationships between reserved and other claimsother claims
Analyze claim emergence and forecasts to Analyze claim emergence and forecasts to estimate uncertainty in the IBNR claim count estimate uncertainty in the IBNR claim count estimatesestimates
Can also be used to model payouts for future Can also be used to model payouts for future exposuresexposures
For a Copy of Either Paper For a Copy of Either Paper Discussed in This SessionDiscussed in This Session
Please contact:Please contact:Roger M. HayneRoger M. Hayne
Milliman & Robertson, Inc.Milliman & Robertson, Inc.70 South Lake Avenue, 11th Floor70 South Lake Avenue, 11th Floor
Pasadena, CA 91101-2601Pasadena, CA 91101-2601Phone: 626-577-1144Phone: 626-577-1144Fax: 626-793-2808Fax: 626-793-2808
e-Mail: [email protected]: [email protected]
