ldf curve-fitting and stochastic reserving: a maximum likelihood approach

LDF Curve-Fitting andLDF Curve-Fitting and Stochastic Reserving: Stochastic Reserving:A Maximum Likelihood A Maximum Likelihood

ApproachApproach

Dave ClarkDave ClarkAmerican Re-InsuranceAmerican Re-Insurance

2003 Casualty Loss Reserve 2003 Casualty Loss Reserve SeminarSeminar

LDF Curve-Fitting and Stochastic LDF Curve-Fitting and Stochastic ReservingReserving

Goals:Goals:1. Describe loss emergence in a 1. Describe loss emergence in a

mathematical model to assist mathematical model to assist in estimating needed reservesin estimating needed reserves

2. Calculate the variability 2. Calculate the variability around the estimated reservesaround the estimated reserves


Growth Curve: Cumulative % of Ultimate

0.0%10.0%20.0%30.0%40.0%50.0%60.0%70.0%80.0%90.0%

100.0%

0 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192Evaluation Age in Months

Perc

ent o

f Ulti

mat

e


Growth Curve = Cumulative % of Growth Curve = Cumulative % of UltimateUltimate

G(tG(t) = 1 / LDF) = 1 / LDFtt

Inverse Power Curve:Inverse Power Curve:

G(t|G(t|,,) = 1 / ) = 1 / [[1+(1+(/t)/t)]]

LDF Curve-Fitting and Stochastic LDF Curve-Fitting and Stochastic ReservingReservingWhy use a continuous curve?Why use a continuous curve?1.1. Smoothing of Development PatternSmoothing of Development Pattern2.2. Interpolation & ExtrapolationInterpolation & Extrapolation

(including tail factor)(including tail factor)3.3. Handle irregular evaluation datesHandle irregular evaluation dates

(e.g., latest diagonal less than 12 months from (e.g., latest diagonal less than 12 months from penultimate diagonal)penultimate diagonal)

4.4. Avoid Over-ParameterizationAvoid Over-Parameterization


Disadvantages of using a Disadvantages of using a continuous curve:continuous curve:

1.1. Need curve-fitting engineNeed curve-fitting engine(answers not in “real time”)(answers not in “real time”)

2.2. May not fit well unless the “right” May not fit well unless the “right” curve form is usedcurve form is used


Basic Model:Basic Model: Convert loss development triangle to an Convert loss development triangle to an

incremental basisincremental basis For each “cell” of the triangle, we haveFor each “cell” of the triangle, we have

cci,ti,t = = actualactual loss for AY loss for AY i,i, between ages between ages tt and and t-1t-1

i,ti,t = = expectedexpected loss for AY loss for AY i,i, between ages between ages tt and and t-t-11

LDF Curve-Fitting and Stochastic LDF Curve-Fitting and Stochastic ReservingReservingTwo Methods for calculating the Two Methods for calculating the

Expected Incremental Loss:Expected Incremental Loss:1. LDF1. LDF

Allows each accident year reserve to beAllows each accident year reserve to be estimated independentlyestimated independently

2. Cape Cod2. Cape CodRequires onlevel premium or other exposureRequires onlevel premium or other exposurebase for each accident yearbase for each accident year


LDF Method:LDF Method:i,ti,t = = UltimateUltimateii x x [[G(G(tt||,,) - G() - G(tt-1 |-1 |,,))]]

n+2 Parameters:n+2 Parameters: UltimateUltimateii expected ultimate loss expected ultimate loss for for accident year accident year ii ““scale” parameter of G(scale” parameter of G(tt)) ““shape” parameter of G(shape” parameter of G(tt))

LDF Curve-Fitting and Stochastic LDF Curve-Fitting and Stochastic ReservingReservingCape Cod Method:Cape Cod Method:

i,ti,t = Premium = Premiumii x x ELRELR x x [[G(G(tt||,,) - G() - G(tt-1 |-1 |,,))]]3 Parameters:3 Parameters: ELRELR expected loss ratio for all yearsexpected loss ratio for all years ““scale” parameter of G(scale” parameter of G(tt)) ““shape” parameter of G(shape” parameter of G(tt))

An onlevel PremiumAn onlevel Premiumii entry for each accident year must entry for each accident year must be supplied by the userbe supplied by the user


Why We Prefer the Cape Cod Method:Why We Prefer the Cape Cod Method:

Provides the model with more information Provides the model with more information (an exposure base in addition to the triangle)(an exposure base in addition to the triangle)

Requires estimation of fewer parametersRequires estimation of fewer parameters More stable estimate of immature year(s)More stable estimate of immature year(s)


And now for the Stochastic And now for the Stochastic part…part…


Assumptions:Assumptions: The expected development in each The expected development in each

cell, cell, i,ti,t is treated as the mean of a is treated as the mean of a distribution.distribution.

Each cell has a different mean, but Each cell has a different mean, but assumed to have the same ratio of assumed to have the same ratio of Variance/Mean, Variance/Mean, 22..


Assumptions:Assumptions: The distribution for each cell follows The distribution for each cell follows

an an Over-dispersed PoissonOver-dispersed Poisson with a with a constant Variance/Mean ratio.constant Variance/Mean ratio.

The model parameters are The model parameters are estimated using estimated using MMaximum aximum LLikelihood ikelihood EEstimation (MLE).stimation (MLE).


What the heck is an What the heck is an Over-Over-dispersed Poissondispersed Poisson distribution? distribution?

A discretized version of the A discretized version of the aggregate loss amount, with the aggregate loss amount, with the same shape as a standard Poisson - same shape as a standard Poisson - commonly used in Generalized commonly used in Generalized Linear Models.Linear Models.


Poisson Distribution

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5 6 7 8 9 10

Over-Dispersed Poisson

0

0.05

0.1

0.15

0.2

0.25

0 150 300 450 600 750 900 1050 1200 1350 1500


Advantages of the Advantages of the Over-dispersed Over-dispersed PoissonPoisson distribution: distribution:

1.1. Sum of ODP is also ODPSum of ODP is also ODP2.2. Can always match mean & varianceCan always match mean & variance3.3. Reflects mass point at zeroReflects mass point at zero4.4. Very convenient mathematicsVery convenient mathematics


Maximizing the Likelihood means Maximizing the Likelihood means solving for solving for andandthat that maximize the expression:maximize the expression:

ti

tititic,

,,, ˆ)ˆln(


The Variance/Mean Ratio, The Variance/Mean Ratio, 22, is , is estimated by:estimated by:

ti ti

titicpn , ,

2,,2

ˆˆ1


Why use Why use MMaximum aximum LLikelihood ikelihood EEstimation (MLE)?stimation (MLE)?

1.1. Familiar methods (LDF and Cape Familiar methods (LDF and Cape Cod) are exact MLE resultsCod) are exact MLE results

2.2. MLE provides estimate of the MLE provides estimate of the uncertainty in the parametersuncertainty in the parameters(“delta method” in (“delta method” in Loss ModelsLoss Models))


Comments & LimitationsComments & Limitations

Independent draws from Identical Independent draws from Identical Distributions (the old “iid”)Distributions (the old “iid”)

Sources of Variance not Sources of Variance not included…included…


Comments & LimitationsComments & LimitationsSources of Variance:Sources of Variance:1.1. ProcessProcess2.2. Parameter or estimation errorParameter or estimation error

MeanMean VarianceVariance

3.3. Model or specification errorModel or specification error4.4. ““State of the World” riskState of the World” risk

These we can do!


Let’s look at an example…Let’s look at an example…

ldf curve-fitting and stochastic reserving: a maximum likelihood approach

Documents

stochastic reservingchart100

stochastic reservinggoals

loss emergence

mathematical model

needed reserves2