Claims/Agency metrics

transcript

Greg Taylor

Taylor Fry Consulting Actuaries

University of Melbourne

University of New South Wales

Casualty Actuarial Society

Special Interest Seminar on Predictive Modeling

Boston, October 4-5 2006

Overview

• Individual claim models• “Paids” models

• “Incurreds” models

• Numerical results

• Adaptive models

Why individual claim models?

Example problem

• Classical workers compensation cost centre allocation problem

• Claim numbers at the leaves of this tree may be small

Total claim cost

Cost centre 1

Cost centre 2

Cost centre m

… … …

Measuring claims performance

• Consider measuring claims performance in a segment of a long tail portfolio

• Likely that adopted metric will require an estimate of the amount of losses incurred but as yet unpaid (loss reserve)• e.g. metric is expected ultimate losses per policy for a

specific underwriting period =

Paid to date + unpaid lossesNumber of policy-years of exposure

= average PTD per policy-year + average unpaid per policy-year

Measuring claims performance in large portfolio segments• Let there be n policy-years of exposure and

ui = i-th amount unpaid

• Consider the ui to be random drawings from some distribution

• Average amount unpaid is

ūi = Σ ui /n = Σ {E[ui] + ui - E[ui]}/n

= E[ui] + Σ {ui - E[ui]}/n

E[ui] as n∞by the large of large numbers

Measuring claims performance in large portfolio segments (cont’d)

ūi E[ui] as n∞• E[ui] = expected size of a randomly drawn claim• This will be the result produced by most

conventional actuarial methods, e.g.• Paid chain ladder• Even incurred chain ladder at early development

• While E[ui] may be a good approximation to ūi for large sample sizes, it may be very poor for small ones• Leading to a highly distorted cost allocation

Measuring claims performance in small portfolio segments

• Effective estimation of small sample average claim cost must somehow take account of the properties of the individual claims

There is a need to change from this…

Data Fitted

Forecast

Model Forecast

Conventional actuarial analysis of loss experience

• Call such models “aggregate models”

…to this

Forecast

Claim 1

Claim 2

Claim 3

Claim n

Claim 1

Claim 2

Claim 3

Claim n

Claim 1

Claim 2

Claim 3

Claim n

Special case of individual claim reserving – statistical case estimation

Individual claim models

Form of such a model

Claim 1

Claim 2

Claim 3

Claim n

Claim 1

Claim 2

Claim 3

Claim n

Claim 1

Claim 2

Claim 3

Claim n

Y=f(β)+ε

Forecast

Form of such a model

Claim 1

Claim 2

Claim 3

Claim n

Claim 1

Claim 2

Claim 3

Claim n

Claim 1

Claim 2

Claim 3

Claim n

Y=f(β)+ε

Yi = f(Xi; β) + εi

Yi = size of i-th completed claim

Xi = vector of attributes (covariates) of i-th claim

β = vector of parameters that apply to all claims

εi = vector of centred stochastic error terms

Forecast

Form of individual claim model

• Convenient practical form is

Yi = h-1(XiT β) + εi [GLM form]

h = link function Error distribution from exponential dispersion family

Linear predictor = linear function of the parameter vector

Form of individual claim model – more specifically• How might one create an individual claim model of the

“paids” type?• Aggregate paids model usually takes the form

Yjk = f(j,k; β) + εjk

j = accident period

k= development period

• Compare with

Not always formulated

Form of “paids” individual claim model

• Possible to mimic aggregate model by defining individual model as just

Yi = h-1(ji,ki; β) + εi

Form of “paids” individual claim model

• Possible to mimic aggregate model by defining individual model as just

Yi = h-1(ji,ki; β) + εi

• But often possible to improve on this, e.g.• Replace development period j with operational

time ti (proportion of accident period’s incurred claims completed) at completion of i-th claim

• Example

Yi = exp [β0+β1ti+β2max(0,ti-0.8)] + εi

Example of “paids” individual claim model

E[Yi] = exp [β0+β1ti+β2max(0,ti-0.8)]Linear predictor of expected claim size as a function of operational

0.16 0.

0.36 0.

0.56 0.

0.76 0.

Operational time

Expected claim size as a function of operational time

$10,000

$20,000

$30,000

$40,000

$50,000

$60,000

$70,000

0.16 0.

0.36 0.

0.56 0.

0.76 0.

Operational time

Example of “paids” individual claim model (cont’d)

• Include superimposed inflation• Let q=j+k=calendar period of claim

completion• Extend model

Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3qi] + εi

• Superimposed inflation at rate exp[β3] per period

• We might wish to model superimposed inflation as beginning at period q=q0

Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3max(0,qi-q0)] + εi

• We might wish to model superimposed inflation as beginning at period q=q0

Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3max(0,qi-q0)] + εi

• …and we might wish to model superimposed inflation with a rate that decreases with increasing operational time

Yi = exp [β0+β1ti+β2max(0,ti-0.8)+(β3-β4ti) max(0,qi-q0)] + εi

etc etc

Example of “paids” individual claim model (cont’d) • “Paids” estimate of

loss reserve scaled to baseline $1,000M

• Prediction CoV = 5.3%

• Mack (incurreds) estimate is $887M with CoV = 10.5%

• Mack estimate produces negative reserves for the old years of origin

• “Paids” chain ladder fails completely

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Accident Year

Mack -1 s.d

Mack +1 s.d

Time Paids

Time Paids -1 s.d

Time Paids +1 s.d

• This model is very economical

• Contains only 9 parameters to represent many thousands of claims

Name Value Standard Error Standard Error (%)Mean 9.755 0.18697 1.9

l_Qtrly optime(^1) 0.0086 0.00167 19.5

l_Qtrly optime<12(^2) -0.0158 0.00049 3.1

l_Finalisation year<1981(^1) 0.1917 0.02029 10.6

l_Finalisation year1981-87(^1) 0.0795 0.00844 10.6

l_Finalisation year1987-90(^1) 0.0244 0.01281 52.4

l_Finalisation year>1990(^1) 0.0386 0.00379 9.8

l-Qtrly optime<70(^1) 0.0034 0.00204 60.3

l_Finalisation year>1990(^1).l-Qtrly optime<70(^1) 0.0008 0.0001 12.7

Further extension of “paids” individual claim model

Yi = exp [β0+β1ti+β2max(0,ti-0.8)+(β3-β4ti) max(0,qi-q0)] + εi

• May include claim characteristics other than time-related, e.g.• Nature of injury

• Claim severity (MAIS scale)

• Pre-injury earnings

Yi = exp [β0+β1ti+β2max(0,ti-0.8)+(β3-β4ti) max(0,qi-q0) +

more terms] + εi

Example of “incurreds” individual claim model

• Similar to “paids” model

• Basic set-up is still

Yi = h-1(ji,ki,qi,ti,other;β) + εi

• Example

Yi = exp(Ci,ji,ki,qi,ti,other;β) + εi

where Ci = current manual estimate of incurred cost of i-th claim

Example of “incurreds” individual claim model (cont’d)• In fact, the model requires more structure than this because

of claims and estimates for nil cost

• Let (for an individual claim)• U = ultimate incurred (may = 0)

• C = current estimate (may = 0)

• X = other claim characteristics

Model of Prob[U=0|C,X]

Model of U|U>0,C=0,X

Model of U/C|U>0,C>0,X

Prob[U=0]

Prob[U>0]

If C=0

If C>0

Example of “incurreds” individual claim model (cont’d) • “Paids” estimate of

loss reserve = $1,000M

• “Incurreds” estimate of loss reserve = $1,040M

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Accident Year

Time Paids

Time Paids -1 s.d

Time Paids +1 s.d

Incurreds

Incurreds -1 s.d

Incurreds +1 s.d

Adaptive reserving

Static and dynamic models

• Return for a while to models based on aggregate (not individual claim) data

• Model form is still Y=f(β)+ε

• Example• j = accident quarter

• k = development quarter

• E[Yjk] = a kb exp(-ck) = exp [α+βln k - γk]

• (Hoerl curve for each accident period)

Static and dynamic models (cont’d)

• ExampleE[Yjk] = a kb exp(-ck) = exp [α+βln k - γk]• Parameters are fixed• This is a static model

But parameters α, β, γ may vary (evolve) over time, e.g. with accident period

Then• E[Yjk] = exp [α(j)+β(j) ln k - γ(j) k]• This is a dynamic model, or adaptive model

Illustrative example of evolving parameters

Separate curves represent different accident periods

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Development period

Formal statement of dynamic model

• Suppose parameter evolution takes place over accident periods

• Y(j)=f(β(j)) +ε(j) [observation equation]

• β(j) = u(β(j-1)) + ξ(j) [system equation]

• Let (j|s) denote an estimate of β(j) based on only information up to time s

Some function Centred stochastic perturbation

Adaptive reserving

(1|q)(2|q)

q-th diagonal

Forecast at valuation date

Adaptive reserving (cont’d)

• Reserving by means of an adaptive model is adaptive reserving

• Parameter estimates evolve over time• Fitted model evolves over time• The objective here is “robotic reserving” in

which the fitted model changes to match changes in the data• This would replace the famous actuarial

“judgmental selection” of model

Special case of dynamic model: DGLM

• Y(j)=f(β(j)) +ε(j) [observation equation]• β(j) = u(β(j-1)) + ξ(j) [system equation]• Special case:

• f(β(j)) = h-1(X(j) β(j)) for matrix X(j)• ε(j) has a distribution from the exponential dispersion family

• Each observation equation denotes a GLM• Link function h• Design matrix X(j)

• Whole system called a Dynamic Generalised Linear Model (DGLM)

• Individual claim models can also be converted to adaptive form• Just subject parameters to evolutionary model

• We have experimented with this type of model and adaptive reserving• Moderately successful

Adaptive form of individual claim models

Conclusions

• Effective forecast of costs of small samples of claims requires individual claim models

• Such models condition the forecasts on much more information than aggregate models

• Even for large samples, individual claim models may yield considerably more efficient forecasts• Lower coefficient of variation

• This may save real money• Lower uncertainty implies lower capitalisation

• Adaptive forms of individual claim models may further improve the tracking of claims experience over time

Claims/Agency metrics

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