Claims Reserving Using GAMLSS

Claims Reserving using GAMLSS

Gian Paolo Clemente α ∗ Giorgio Spedicato α †

α Catholic University, Milan, Italy

Abstract

The assessment of predictive distribution of outstanding claims is a keyissue for non-life insurers in order to quantify both the economic balancesheet and the solvency capital requirement. To this aim generalized linearmodels (GLM) have been applied by England and Verrall ([10]). Throughan over-dispersed Poisson framework, they obtain an estimation of theprediction error of the point estimate of claims reserve derived through astandard Chain-Ladder. This approach paves the way toward stochasticclaims reserve and it is extensively used in practice. The aim of this paperis to introduce the General Additive Models for Location, Shape and Scale(GAMLSS) within the stochastic loss reserving framework and to revisethe approach proposed by England and Verrall. The GAMLSS regression[21] provides extended capabilities compared to GLM by allowing all theparameters of the dependent variable to be modelled as functions of co-variates. Formulas for loss reserving derived under a GLM framework canbe extended using GAMLSS in order to estimate the variability of unpaidclaim estimate. The dispersion parameter of the dependent variable canbe indeed expressed as function of explanatory variables, as accident, de-velopment or calendar years. Numerical examples found in [10] will bereviewed under the GAMLSS reserving framework as well as prospectiveenhancements and limitations of GAMLSS reserving will be discussed. R[18] software and ChainLadder R package [13] will be used throughoutexamples.

Keywords: Claims Reserving; GAMLSS models; GLM models.

∗corresponding author, Dipartimento di Discipline Matematiche, Finanza Matematica,Econometria. Via Necchi 9, 20123 Milan. [email protected]†PhD, ACAS, spedicato [email protected]

1

1 INTRODUCTION AND EXISTING LITERATURE

1 Introduction and existing literature

The estimation of claims reserve plays a central role in the economics of insur-ance business and many algorithmic methods have been developed from decadesto obtain an estimated value of claims reserve, the so - called ”best estimate”(BE). Recently, stochastic models for outstanding claims evaluation have beendeveloped with the aim to assess not only the BE value, but also the variabilityof the reserve distribution, that may be measured by statistics like the Coeffi-cient of Variation, CV. Furthermore, the claim reserve BE and its variabilityestimate allow a first assessment of the insurers solvency position with respectto Reserve Risk (see [5] and [12] for the European and US solvency approachrespectively).Mack [16] proposed a first approximate approach to quantify the standard errorof the chain-ladder reserve. A distribution-free formula has been derived withthe aim to allow for the uncertainty arising both from the reserve developmentintrinsic variability (Process Error) and from the uncertainty of parameters’estimation of the selected model (Estimation Error). A different way basedon regression has been analysed by Zehnwirth [3] by developing a log-normalregression model to assess the claim reserves. Wright and Renshaw ([28] and[19]) focused on a log-linear approach typical of GLM that has been properlyextended by England and Verrall. Furthermore, Generalized Additive Models(GAM) were used in some regression equations shown by England and Verralpapers ([10] and [27]) to model the effect of either accident or development yearusing a flexible non - parametric term within the link function.

A new class of statistical models, namely GAMLSS, has been introducedby [21] with the aim to provide a flexible regression framework. These modelsallow to describe not only the mean (or location) but also the other parame-ters of the distribution underlying to the dependent variable, as functions ofexplanatory variables using parametric and/or additive nonparametric termswithin the equation. Moreover, a wide choice of marginal distribution for theresponse variable in the GAMLSS framework can be selected (up to 60 as ofcurrent version of the R package) that are not bounded to the exponential fam-ily. GAMLSS framework provides tools to assess the adequacy of the dependentvariable distribution form, the adequacy of the functional relationship expressedin the regression equation and the overall goodness of fit of the model.Few applications of GAMLSS in the actuarial fields and none in stochastic lossreserving have been found at the time this paper is being written. Venter ([26])applied GAMLSS to model mortality trend, whilst claim cost distribution hasbeen described by GAMLSS in [14]. Finally, these models has been used in [24]in order to quantify premium risk capital requirement for a MTPL portfolio.Therefore this paper aims to apply GAMLSS framework in stochastic loss re-serving in order to assess both the BE and the distribution of claims reserve.All calculations underlying the exemplified numerical application will be per-formed by the aid of open - source R statistical software [18] and the ChainLad-der package [13].

The paper will be organized as follows: section 2 will review underlying sta-

2

2 THE USE OF GLM IN STOCHASTIC LOSS RESERVING

tistical and actuarial theory, focusing on the approach proposed by England andVerrall, while section 3 will introduce a different approach based on GAMLSSmodels. In section 4 an applied example will be shown, in order to compareGLM based and GAMLSS based results. Finally section 5 will discuss resultsand draft conclusions. Models logs will be shown in the appendix.

2 The use of GLM in stochastic loss reserving

Several approaches have been proposed by actuarial literature with the aimto obtain the variability or the full distribution of claims reserve. GLM havebeen already introduced in early works (see [15], [20] and [28]). As well known,GLM generalize ordinary least square (OLS) regression allowing distributionsother than gaussian for the dependent variable. The mean and variance of theresponse variable ( yi, i = 1, ...N ) are expressed as functions of covariates xi asEquation 1 shows: {

E [yi] = µi = g−1 (ηi) = f (xi)

var [yi] = φ · V (µi)ωi

(1)

where g−1(·) is the link function, V (µi), the variance function, is a functionof the mean specific for the distribution family, φ is a constant that can beestimated from the data and ωi a prior weight of the i-th observatio, that canbe set equal to one for all cases (see [1] and [17] for details). However, standardGLM framework leads to restrictive modeling for the variance of yi since itdepends on µi as expressed within the variance function.

Focusing now on claims reserve, we can consider a generic loss developmenttriangle with dimension (I, J) where rows (i = 1, . . . , I) represent the claimsaccident years and columns (with j = 1, . . . , J) describe development years forpayments. It needs to be emphasized that the number of columns may differfrom the number of rows, for example due to a tail in payments development.We define now Pij as the incremental paid estimate.Renshaw and Verrall ([20]) casted the chain ladder method into the frameworkof GLM with an over-dispersed Poisson model for incremental payments asEquation 2 shows:

E [Pij ] = mij

var [Pij ] = φmij

mij = xiyjln (mij) = ηij = c+ αi + βj

(2)

This approach is based on an over-dispersed Poisson framework since it assumesthe incremental claims Pij to be distributed as independent over-dispersed Pois-son random variables, with mean and variance defined by previous relations.Here, xi represents the expected ultimate claims and yj represents the propor-tion of ultimate claims to emerge in each development year (with the constraint

3

2 THE USE OF GLM IN STOCHASTIC LOSS RESERVING

∑Jj=1 yj = 1). The over-dispersion is introduced by the parameter φ, which is a-

priori unknown and estimated from the data. The allowance for over-dispersiondoes not affect estimation of the parameters, but it yields to increase their stan-dard errors.

A flexible framework, within which previous model could be regarded asa special case, is reported in Equation 3 (see [10]). The first two items inEquation 3 bundle the claim reserving within the GAM framework. The choiceof dependent variable’s distribution is driven by the ρ parameter, being Normal,Poisson, Gamma and Inverse Gaussian specified by setting ρ equal to either 0,1, 2, or 3, respectively. The predictor is linked to the expected value of theresponse by means of the logarithmic link function. The last item of Equation 3defines the central estimate as a function of accident and development years.This relation extends the last term in Equation 2 by introducing two optionalterms (the offset, uij , and the inflation, δt) and by taking into account the effectof the accident year i and the development year j by means of smoothing splines.It is worth to be noted that the accident and development years are treated asfactors (as Equation 2 shows) when both θj and θi are set equal to zero.

E [Pij ] = mij

var [Pij ] = φmρij

ln (mij) = ηij = uij + δt+ c+ sθi (i) + sθj (j) + sθi (ln (j))(3)

Within this framework, the authors derive the prediction error of the singleincremental payments Pij through the square root of the next formula:

E

[(Pij − Pij

)2]≈ φmρ

ij + m2ijV ar (ηij) . (4)

The estimation variance (second component in Equation 4) is usually com-puted directly by the implementation of the model, letting the prediction errorto be evaluate.

Let D = {Pij , i+ j > I + 1, 1 < j < J} denote the missing part of the tri-angle, the mean square error of prediction (see [19]) of the claims reserve R =∑Ii=1Ri can be derived as:

E

[(R− R

)2]≈∑i,jεD

φmρij +

∑i,jεD

m2ijV ar (ηij)

+ 2∑

i1, j1εDi2, j2εDi1, j1 6= i2, j2

mi1j1mi2j2Cov(ηi1j1 , ηi2j2). (5)

An alternative approach to estimate the prediction error is based on the useof bootstrap, where the scaled Pearson residuals are commonly used (for furtherdetails see [8] and [11]). However, bootstrap analysis accounts only for the esti-mation variance. The process variance contribution is then computed through

4

3 GAMLSS FOR STOCHASTIC LOSS RESERVING

a closed formula [8] or an additional step based on a simulation of paymentsby the process distribution [7]. According to the ODP framework, the processerror can be obtained by simulating the forecasted value from an overdispersedPoisson with mean Pij and variance Pij˙φ, where Pij is the incremental value

in the lower part of the triangle derived by the bootstrap procedure and φ isobtained as the ratio between the sum of squared pearson residuals and thedegrees of freedom. England [7] makes several suggestions to achieve this goal.

Finally, recent approaches are based on a non-constant scale parameter φj (see[11]), the use of generalized linear mixed models (see [2]) and a reparameterizedversion of original GLM (see [4]).

3 GAMLSS for stochastic loss reserving

GAMLSS is a general class of univariate regression models where the exponen-tial family assumption is relaxed and replaced by a general distribution family.The systematic part of the model allows all the parameters of the conditionaldistribution of the response variable yi to be modelled as parametric or non-parametric functions of explanatory variables within the framework.Let θT = (θ1, θ2, ..., θp) the p parameters of a probability density function

f (yi|θ) modelled using an additive model. θTi = (θi,1, θi,2, ..., θi,p) is a vector ofp parameters related to explanatory variables, where the first two parametersθi,1 and θi,2 are usually characterized as location µi and scale σi. The remainingparameters, if any, are characterized as shape parameters. For many families ofpopulation distributions, a maximum of two shape parameters νi and τi suffices(i.e. p = 4).

Under this condition, we can derive the following model:g1(µ) = η1 = X1β1

g2(σ) = η2 = X2β2

g3(ν) = η3 = X3β3

g4(τ ) = η4 = X4β4

(6)

where we consider only the parametric part (see [21] for an extension of the

model), and where βTk =(β1,k, β2,k, ..., βJ′

k,k

)is a parameter vector of length

J ′k and Xk is a known design matrix of size n× J ′k.In particular, Equation 6 implies that the moments of response variable in

each cell can be directly expressed as a function of covariates after a convenientparametrization.Considering now the claims reserve framework, we can identify the incrementalpayments Pi,j as response variables and derive the following structure:{

E [Pi,j ] = g−11 (η1,i,j)var [Pi,j ] = g−12 (η2,i,j)

(7)

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4 NUMERICAL RESULTS

At this regard, GAMLSS R package [23] supports more than 60 distribu-tions, non-linear and non-parametric relationships (e.g. cubic splines and nonparametric smoothers), random effect modeling, etc. The [22] paper widelydeals with GAMLSS models estimation and selection. The selection processconsists of comparing many different competing models that vary by differentcombinations of either distribution of the response variable, set of link functionfor distribution parameters and set of predictor terms. Within the GAMLSSpackage, a full set of diagnostic tools is available for checking model’s assump-tions. In particular, the Generalized Akaike Information Criterion (GAIC) canbe used to compare alternative models and the normalized randomized quantileresiduals (see Dunn and Smyth, [6]) can be used to check the adequacy of themodel, for example regarding the distribution of the response variable. Theseresiduals are given by ri,j = Φ−1(ui,j) where Φ−1 is the inverse cumulative dis-

tribution function of a standard normal distribution and ui,j = F (Pi,j |θi,j) isderived by the assumed cumulative distribution for the cell (i, j).

The procedure to assess the stochastic distribution of loss reserve proposedby the literature for GLM models can be therefore adapted to GAMLSS asfollows:

1. define the GAMLSS model underlying the claims development triangle,M;

2. compute the residuals ri,j = Φ−1[F (Pi,j |θi,j ];

3. generate N upper triangles of residuals rki,j with k = 1, . . . , N by samplewith replacement;

4. derive N upper triangles of pseudo-incremental payments from the gamlssmodel by the inverse relation: P ki,j = F−1[Φ(rki,j)|θi,j ];

5. refit the gamlss model, M;

6. for each cell of the lower part of the triangle simulate from the processdistribution whose parameters are fitted from M;

7. sum the simulated payments in the lower triangle by origin year and overallto compute the origin year and total reserve estimates, respectively.

This approach leads to the full distribution of claims reserve definition andto the assessment of both the process and the estimation error.Finally, normality of residuals needs to be verified in order to apply the method-ology. In the following numerical results the Shapiro-Wilk test has been used.

4 Numerical results

As done in [9] example, the Taylor-Ashe triangle [25], TA, available in theChainLadder package (see GenIns triangle in [13]), has been used. TA triangle,a 10x10 incremental payments triangle, has been used in order to assess both

6

4 NUMERICAL RESULTS

the BE and the distribution of claims reserve in order to compare the proposedGAMLSS approach to the classic ODP methodology.

Table 1 shows key figures derived by the use of two classical approachesbased on Chain-Ladder method. In particular, the BE and the CV derivedby using Mack formula and GLM ODP model are compared. Furthermore thecomparison is extended to a GLM based on a Gamma distribution. Finally, the99.5 quantile is obtained under a LogNormal assumption for the Mack formulaand by applying a classical bootstrapping methodology (10.000 simulations) forthe GLM approaches. GLM has been applied here by following England andVerral [11] approach without any judgement in claims reserve evaluation.

model BE CV QuantMack 18680856 0.13 25919050ODP GLM 18680856 0.16 27912402Gamma GLM 18085805 0.15 27710864

Table 1: Classical stochastic reserving methods results on Taylor-Ashe triangle

BE and CV are here reported only for comparison with GAMLSS resultsand values are equivalent to that derived in [8] and [11].

Then, several GAMLSS models were fit on the same triangle and comparedby GAIC index. A wide range of conditional distributions, much more beyondthe classical exponential family, were tested. As expected the GAIC derivedby using a GAMLSS with a Gamma distribution is almost equal to the AIC(roughly 1500) derived by GLM based on the same distribution.Table 2 shows the GAIC fit index and the relating degrees of freedom of variousGAMLSS models applied on triangle. Table 3 confirms results in terms of BE

models df GAICWeibull 20 1495.04NegativeBinomial TypeII 20 1495.25NegativeBinomial 20 1500.77Gamma 20 1500.77Gumbel 20 1515.18InverseGaussian 20 1515.69Exponential 19 1599.88

Table 2: GAMLSS regression fits on Taylor-Ashe triangle

assessment using GAMLSS to be similar to the ones that would be derivedfollowing a straight GLM approach (see Table 1).

The greatest advantage of GAMLSS in reserving application is that morethan one distribution parameter can be explicitely modeled as function of co-variates.For the TA triangle, the dispersion parameter of incremental payments has beenassumed as a function of either origin or development year in order to assure

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4 NUMERICAL RESULTS

distribution BEWEI 19939326GAMMA 18085822IG 17364127NB 18085841GU 23467287NB2 18995459EXP 18085822

Table 3: Best estimates using GAMLSS models

a better fitting on data. The latter one could be described by the followingEquation when the Gamma distribution is considered:{

E [Pi,j ] = exp(c+ αi + βj)var [Pi,j ] = exp (d+ ej)

(8)

The analysis showed that the best fitting model with varying dispersionparameter to be when incremental payments follow a Gamma distribution. Inparticular, Table 4 shows GAIC values determined by assuming the dispersionparameter to vary by development year or by accident year. Figures show thatassuming dispersion to vary by development year significantly enhances theGAIC fit, also with respect to models shown in Table 3.Figure 1 displays the diagnostics plot as given by GAMLSS R package for themodel where dispersion varies by development year. The well behaviour ofresiduals can be shown since no sistematic trend with respect to fitted value orposition appears as well as the shape of normalized quantile residuals can bewell approximated by a Normal distribution as plots in lower section show.

model GAIC BEorigin, factor 1380.79 20387778development, factor 1239.41 20277356

Table 4: GAIC and Best estimates using GAMLSS models with different modelsfor dispersion parameters

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4 NUMERICAL RESULTS

*******************************************************************

Summary of the Quantile Residuals

mean = -0.0004790768

variance = 0.8516852

coef. of skewness = 0.1062681

coef. of kurtosis = 2.586916

Filliben correlation coefficient = 0.9900941

*******************************************************************

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Figure 1: GAMLSS model diagnostic plot for final model

model BEMean 20250642.78CV 0.09Skewness 0.3699.5 Quantile 25807705.98

Table 5: Main characteristics of Claims Reserve using GAMLSS models

Table 5 shows key figures of Claims Reserve distribution derived by theGAMLSS model where the conditional distribution follows a Gamma and the

9

4 NUMERICAL RESULTS

dispersion parameter varies as a function of development year. Lower variabilityand greater skewness than classical models come forth. Furthermore, it couldbe noted that the variability coefficient is roughly equal to the value derived onthe same triangle by England and Verrall [11] where an Over Dispersed PoissonModel with a non-constant scale parameters was used. Finally Figure 2 and3 show the simulated distributions (10,000 simulations) obtained by combiningbootstrap to an additional step to incorporate the process error for classicalGamma model and GAMLSS Gamma model with varying by development yeardispersion parameter respectively.

Claims Reserve (Gamma GLM)

Reserve

Freq

uenc

y

1.0e+07 1.5e+07 2.0e+07 2.5e+07 3.0e+07

020

040

060

080

0

Figure 2: Claim reserve distribution obtained by a Gamma GLM

10

5 CONCLUSIONS

Claims Reserve (Gamma GAMLSS)

Reserve

Freq

uenc

y

1.0e+07 1.5e+07 2.0e+07 2.5e+07 3.0e+07

020

040

060

080

0

Figure 3: Claim reserve distribution obtained by a GAMLSS model with aGamma distribution

5 Conclusions

This paper aims to propose a methodology to assess the claims reserve distri-bution based on a gamlss framework. The approach appears more flexible thanclassical GLM since it tries to describe the variance effect as a function of avail-able covariates, either accident or development year. Furthermore GAMLSSmethodology allows a wide range of response variable distribution to be used,not bounded to exponential family. Similarly it permits flexibility in specifythe regression relationship for example allowing the use of non - parametricsmoothers.A numerical exemplification was performed by testing the well-known Taylor-Ashe triangle. The best fitting model for such triangle is assuming Gammadistribution for incremental payment where dispersion parameter varies as afunction of development year.While it is difficult to make any final conclusions from the single triangle whichhas been analyzed in this paper, it is interesting to note the improvement in theGAIC fit when the variance is explicitly described. Further areas of researchlie in extending to gamlss the analytical expression of the estimation varianceas derived for GLM in [20]. Finally it is noteworthy that despite the flexibilityof GAMLSS, this methodology could result overparameterized when few data(small triangles) are considered.

11

A MODEL LOGS

Appendix

A Model logs

The selected GAMLSS model used to compute reserve distribution is reportedin R log below shown.

*******************************************************************

Family: c("GA", "Gamma")

Call:

gamlss(formula = value ~ factor(origin) + factor(dev), sigma.formula = ~factor(dev),

family = GA, data = GenInsDF, control = con)

Fitting method: RS()

-------------------------------------------------------------------

Mu link function: log

Mu Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 12.412282 7.331e-02 1.693e+02 7.802e-54

factor(origin)2 0.425016 1.588e-07 2.677e+06 5.507e-205









factor(dev)2 0.913387 1.053e-01 8.671e+00 2.426e-10

factor(dev)3 0.909820 7.331e-02 1.241e+01 1.442e-14

factor(dev)4 1.022382 1.409e-01 7.258e+00 1.510e-08

factor(dev)5 0.464730 1.530e-01 3.037e+00 4.424e-03

factor(dev)6 0.163759 2.541e-01 6.445e-01 5.234e-01

factor(dev)7 -0.002649 2.046e-01 -1.295e-02 9.897e-01

factor(dev)8 -0.390936 1.000e-01 -3.908e+00 3.937e-04

factor(dev)9 0.027096 1.021e-01 2.655e-01 7.922e-01

factor(dev)10 -1.285784 7.331e-02 -1.754e+01 3.220e-19

-------------------------------------------------------------------

Sigma link function: log

Sigma Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -1.54747 0.2219 -6.9725 1.119e-08

factor(dev)2 0.09349 0.3222 0.2902 7.730e-01

12

A MODEL LOGS

factor(dev)3 -14.45481 0.3319 -43.5454 1.966e-38

factor(dev)4 0.40252 0.3440 1.1700 2.482e-01

factor(dev)5 0.43574 0.3601 1.2099 2.326e-01

factor(dev)6 0.93870 0.3747 2.5051 1.593e-02

factor(dev)7 0.58500 0.4105 1.4252 1.610e-01

factor(dev)8 -0.59069 0.4638 -1.2735 2.094e-01

factor(dev)9 -0.75070 0.5463 -1.3742 1.762e-01

factor(dev)10 -14.68125 0.7337 -20.0091 4.969e-24

-------------------------------------------------------------------

No. of observations in the fit: 55

Degrees of Freedom for the fit: 29

Residual Deg. of Freedom: 26

at cycle: 10

Global Deviance: 1181.415

AIC: 1239.415

SBC: 1297.628

*******************************************************************

13

REFERENCES REFERENCES

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Documents

Claims Reserving Using GAMLSS