CLAIMS RESERVING USING TWEEDIErsquoSCOMPOUND POISSON MODEL

BY

MARIO V WUumlTHRICH

ABSTRACT

We consider the problem of claims reserving and estimating run-off trianglesWe generalize the gamma cell distributions model which leads to Tweediersquoscompound Poisson model Choosing a suitable parametrization we estimatethe parameters of our model within the framework of generalized linear mod-els (see Joslashrgensen-de Souza [2] and Smyth-Joslashrgensen [8]) We show that thesemethods lead to reasonable estimates of the outstanding loss liabilities

KEYWORDS

Claims Reserving Run-off Triangles IBNR Compound Poisson Model Expo-nential Family GLM MSEP

INTRODUCTION

Claims reserving and IBNR estimates are classical problems in insurance math-ematics Recently Joslashrgensen-de Souza [2] and Smyth-Joslashrgensen [8] have fittedTweediersquos compound Poisson model to insurance claims data for tarificationUsing the connection between tarification and claims reserving analysis (seeMack [3]) we translate the fitting procedure to our run-off problem Our modelshould be viewed within the context of stochastic methods for claims reserv-ing For excellent overviews on this topic we refer to England-Verrall [1] andTaylor [9]

The starting point of this work was the gamma cell distributions modelpresented in Section 75 of Taylor [9] The gamma cell distributions modelassumes that every cell of the run-off triangle consists of rij independent pay-ments which are gamma distributed with mean tij and shape parameter g Theseassumptions enable the calculation of convoluted distributions of incremen-tal payments Unfortunately this model does not allow one to estimate egthe mean square error of prediction (MSEP) since one has not enough infor-mation We assume that the number of payments rij are realisations of randomvariables Rij ie the number of payments Rij and the size of the individual pay-ments X (k)

ij are both modelled stochastically This can be done assuming that

ASTIN BULLETIN Vol 33 No 2 2003 pp 331-346

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Rij is Poisson distributed These assumptions lead to Tweediersquos compound Pois-son model (see eg Joslashrgensen-de Souza [2]) Choosing a clever parametrizationfor Tweediersquos compound Poisson model we see that the model belongs to theexponential dispersion familiy with variance function V(m) = mp p isin (12) anddispersion ƒ It is then straightforward to use generalized linear model (GLM)methods for parameter estimations A significant first step into that directionhas been done by Wright [11]

In this work we study a version of Tweediersquos compound Poisson modelwith constant dispersion ƒ (see Subsection 41) This model should be viewedwithin the context of the over-dispersed Poisson model (see Renshaw-Verrall[6] or England-Verrall [1] Section 23) and the Gamma model (see Mack [3]and England-Verrall [1] Section 33) The over-dispersed Poisson model andthe Gamma model correspond to the two extreme cases p = 1 and p = 2 respOur extension closes continuously the gap between these two models sincep isin (12) To estimate p we additionally use the information rij which is notused in the parameter estimations for p = 1 and p = 2 Though we have oneadditional parameter we obtain in general better estimates since we also usemore information and have more degrees of freedom

Moreover our parametrization is such that the variance parameters p andƒ are orthogonal to the mean parameter This leads to a) efficient parameterestimations (fast convergence) b) good estimates of MSEP

At the end of this article we demonstrate the method using motor insurancedatas Our results are compared to several different classical methods Of coursein practice it would not be wise to trust in just these methods It should bepointed out that the methods presented here are all payment based Usuallyit is also interesting to compare payment based results to results which rely ontotal claims incurred datas (for an overview we refer to Taylor [9] and the ref-erences therein)

In the next section we define the model In Section 3 we recall the defini-tion of Tweediersquos compound Poisson model In Section 4 we apply Tweediersquoscompound Poisson model to our run-off problem In Section 5 we give an esti-mation procedure for the mean square error of prediction (MSEP) Finally inSection 6 we give the examples

2 DEFINITION OF THE MODEL

We use the following (well-known) structure for the run-off patterns the acci-dent years are denoted by i le I and the development periods are denoted byj le J We are interested in the random variables Cij Cij denote the incrementalpayments for claims with origin in accident year i during development periodj Usually one has observations cij of Cij for i + j le I and one tries to complete(estimate) the triangle for i + j gt I The following illustration may be helpful

Definition of the model

1 The number of payments Rij are independent and Poisson distributed withparameter lijwi gt 0 The weight wi gt 0 is an appropriate measure for the volume

332 MARIO V WUTHRICH

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2 The individual payments X (k)ij are independent and gamma distributed with

mean tij gt 0 and shape parameter g gt 0

3 Rij and X (k)mn are independent for all indices We define the incremental pay-

ments paid in cell (i j) as follows

iij ij

ij

C X Y C Wand1 gt( )

ij Rk

k

R

ij01

ij$= =

=

(21)

Remarks

bull There are several different possibilities to choose appropriate weights wi egthe number of policies or the total number of claims incurred etc If onechooses the total number of claims incurred one needs first to estimate thenumber of IBNyR cases (cases incurred but not yet reported)

bull Sometimes it is also convenient to define Rij as the number of claims withorigin in i which have at least one payment in period j

bull Yij denotes the normalized incremental payments in cell (i j)

bull One easily sees that conditionally given Rij the random variable Cij is gammadistributed with mean Rijtij and shape parameter Rijg (for Rij gt 0)

3 TWEEDIErsquoS COMPOUND POISSON MODEL

In this section we formulate our model in a reparametrized version this hasalready been done in the tarification problems of [2] and [8] Therefore wetry to keep this section as short as possible and give the main calculations inAppendix A

CLAIMS RESERVING USING TWEEDIErsquoS COMPOUND POISSON MODEL 333

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For the moment we skip the indices i and j The distribution Y (for givenweight w) is parametrized by the three parameters l t and g We now choosenew parameters m ƒ and p such that the density of Y can be written as y ge 0(see (A2) below and formula (12) in [2])

expz zz

f y w p c y w p w y p pmm m1 2Y

p p1 2

=-

--

- -

^ ^ fh h p 4 (31)

where c(y ƒ w p) is given in Appendix A and

p = (g + 2) (g + 1) isin (1 2) (32)

m = l middot t (33)

ƒ = l1ndash pt 2ndash p (2 ndash p) (34)

If we set q = m1ndash p (1 ndash p) we see that the density of Y can be written as (see also[2] formula (12))

p

p

( )

( )

expz zz

f y w p c y w p w y

p pwith

m q k q

k q q2

11

Y

pp

1

2

-

=-

-

=

-

-

^ ^ `

^^

h h j

h h

( 2

(35)

Hence the distribution of Y belongs to the exponential dispersion family withparameters m ƒ and p isin (12) (see eg McCullagh-Nelder [5] Section 222)We write for p isin (12)

zY wED m( )p` ^ h (36)

For ( )zY wED m( )p` we have (see [2] Section 22)

E [Y ] = kp(q) = m (37)

Var(Y) = ( ) z zw V wm mp$ = (38)

ƒ is the so-called dispersion parameter and V(middot) the variance function withp isin (12) For our claims reserving problem we consider the following situa-tion

Constant dispersion ƒ (see Subsection 41) p isin (12) and Yij are independentwith

ij i ij j zz

Y w E Y Y wED Varandm m m( )pij i i i

pamp` = = ijj` `j j8 B (39)

334 MARIO V WUTHRICH

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Interpretation and Remarks

bull Tweedie [10] seems to be the first one to study the compound Poisson modelwith gamma severeties from the point of view of exponential dispersionmodels For this reason this model is known as Tweediersquos compound Pois-son model in the literature see eg [8]

bull p = (g + 2) (g + 1) is a function of g (shape parameter of the single paymentsdistributions X (k)

ij ) Hence the shape parameter g determines the behaviourof the variance function V(m) = mp Furthermore we have chosen a parame-trization (m ƒ p) such that m is orthogonal to (ƒ p) in the sense that the Fisherinformation matrix is zero in the off-diagonal (see eg [2] page 76 or [8])Ie our parametrization focuses attention to variance parameters (ƒ p) anda mean parameter m which are orthogonal to each other This orthogonalityhas many advantages to alternative parametrizations Eg we have efficientalgorithms for parameter estimations which typically rapidly converge (seeSmyth [7]) Moreover the estimated standard errors of m which are of mostinterest do not require adjustments by the standard errors of the varianceparameters since these are orthogonal

bull Our model closes continuously the gap between the over-dispersed PoissonModel (see Renshaw-Verrall [6] or England-Verrall [1] Section 23) where wehave a linear variance function (p = 1)

i j zY wVar mi i$= j` j (310)

and the Gamma model (see Mack [3] and England-Verrall [1] Section 33)where

i j zY wVar mi i2$= j` j (311)

In our case p is estimated from the data using additionally the informationrij (see (46)) The information rij is not used in the boundary cases p = 1 andp = 2

bull Naturally in our model we have p isin (12) since g gt 0 We estimate p fromthe data so theoretically the estimated p could lie outside the interval [12]which would mean that none of our models fits to the problem (eg p = 0implies normality p = 3 implies the inverse Gaussian model) In all ourclaims reserving examples we have observed that the estimated p was lyingstrictly within (12)

4 APPLICATION OF TWEEDIErsquoS MODEL TO CLAIMS RESERVING

41 Constant dispersion parameter model

We assume that all the Yij are independent with Yij simED(p) (mijƒwi) ie Yij belongsto the exponential dispersion family with p isin (12) and

CLAIMS RESERVING USING TWEEDIErsquoS COMPOUND POISSON MODEL 335

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i ij j z

E Y Y wVarandm mi i ip= =j j` j8 B (41)

We use the notation m = (m00hellip mIJ) Given the observations (rij yij) i + j le I

i rj ij gt 0 the log-likelihood function for the parameters (m ƒ p) is given by(see Appendix A and [2] Section 3)

i i

( ) ( )

log logzz

z

L p rp p

w yr r y

wy p p

g

m m

Gm1 2

1 2

i

i ii i ij

i j

iij

p p

g

g g1

1 2

=- -

-

+-

--

+

- -j j

jj

j j^^

``hh

j j

R

T

SSSS

V

X

WWW

4

4

(42)

Formula (42) immediately shows that given p the observations yij = cij wi aresufficient for MLE estimation of mij (one does not need rij) Moreover for con-stant ƒ the dispersion parameter has no influence on the estimation of m

Next we assume a multiplicative model (often called chain-ladder type struc-ture) ie there exist parameters (i) and f ( j) such that for all i le I and j le J

( ) ( ) i f jmi $=j (43)

After suitable normalization can be interpreted as the expected ultimateclaim in accident year i and f is the proportion paid in period j It is nowstraightforward to choose the logarithmic link function

( ) log xj m bi i i= =j j j (44)

where b = (log (0)hellip log(I) log f(0)helliplog f(j)) and X = (x00hellipxIJ) is theappropriate design matrix

Parameter estimation

a) For p known We deal with a generalized linear model (GLM) of the form (41)-(44) Hence we can use standard software packages for the estimation of m

b) For p unknown Usually p and g resp are unknown Henceforth we studythe profile likelihood for g (here we closely follow [2] Section 32) For m andp given the MLE of ƒ is given by (see (42))

ƒp

i i

( )

r

w y p p

g

m m

1

1 2

ii j

i i

p p

i j

1 2

=+

--

--

- -j j

j

j

J

L

KKK

N

P

OOO

(45)

336 MARIO V WUTHRICH

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From this we obtain the profile likelihood for p and g resp i gtr 0j ija k as

Lm(p) = L(m p ƒp) = ( ) logrw

g1 1

ii

i j+ -

pzj f p

(46)

log logr p py

r gG2

11

i

ii

i ji j

g

+- -

-jj

jJ

L

KK f `

N

P

OOp j

Given m the parameter p is estimated maximizing (46)

c) Finally we combine a) and b) The main advantage of our parametrizationis (as already mentioned above) the orthogonality of m and (ƒ p) m can beestimated as if (ƒ p) were known and vice versa Alternating the updatingprocedures for m and (ƒ p) leads to an efficent algorithm Set initial valuep(0) and estimate m (1) via a) Then estimate p(1) from m (1) via (46) and iterate thisprocedure We have seen that typically one obtains very fast convergence of(m (k) p(k)) to some limit (for our examples below we needed only 4 iterations)

42 Dispersion modelling

So far we have always assumed that ƒ is constant over all cells (i j) If we con-sider the definitions (33) and (34) we see that every factor which increases lincreases the mean m and decreases the dispersion ƒ because p isin (12) Increas-ing the average payment size t increases both the mean and the dispersionChanging l and t such that l1ndashp t2ndashp remains constant has only an effect on themean m Hence it is necessary to model both the mean and the dispersion inorder to get a fine structure ie model mij and ƒij for each cell (ij) individuallyand estimate p Such a model has been studied in the context of tarificationby Smyth-Joslashrgensen [8]

We do not further follow these ideas here since we have seen that in our sit-uation such models are over-parametrized Modelling the dispersion parame-ters while also trying to optimize the power of the variance function allowstoo many degrees of freedom eg if we apply the dispersion modelling modelto the data given in Example 61 one sees that p is blown up when allowingthe dispersion parameters to be modelled too It is even possible that there is nounique solution when modelling ƒij and p at the same time (in all our exampleswe have observed rather slow convergence even when choosing ldquomeaningfulrsquorsquoinitial values which indicates this problematic)

5 MEAN SQUARE ERROR OF PREDICTION

To estimate the mean square error of prediction (MSEP) we proceed as inEngland-Verrall [1] Assume that the incremental payments Cij are independent

CLAIMS RESERVING USING TWEEDIErsquoS COMPOUND POISSON MODEL 337

ˆ

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and Cij are unbiased estimators depending only on the past (and hence are

independent from Cij) Assume jij is the GLM estimate for jij = log mij then (seeeg [1] (76)-(77))

jz

C E C C C C

w w

MSEP Var Var

Varm m

C i i i i i

i ip

i i i

2

2

i

$

= - = +

+j

j j j j j j

j j

a a ` a

` `

k k j k

j j

lt F

(51)

The last term is usually available from standard statistical software packagesall the other parameters have been estimated before The first term in (51) denotesthe process error the last term the estimation error

The estimation of the MSEP for several cells (i j) is more complicated sincewe obtain correlations from the estimates We define D to be the unknown tri-angle in our run-off pattern Define the total outstanding payments

C C Cand C( ) ( )

ii j

ii jD D

= =

j j (52)

Then

2( )

j

j j

zE C w w

w w

MSEP Var

Cov

C C m m

m m

( ) ( )

( ) ( )( )( )

C i i ip

i ji i i

i j

i

i j i ji j i j

i j i i j i j i j

D D

D

2

1

1 1 2 2

1 1 2 2

1 1 2 2 2 1 1 2 2

$= - +

+

j j j j

_ ` `

`

i j j

j

9 C

The evaluation of the last term needs some care Usually one obtains a covari-ance matrix for the estimated GLM parameters log (i) and log f ( j) Thiscovariance matrix needs to be transformed into a covariance matrix for j withthe help of the design matrices

6 EXAMPLE

Example 61

We consider Swiss Motor Insurance datas We consider 9 accident years overa time horizon of 11 years Since we want to analyze the different methodsrather mechanically this small part of the truth is already sufficient for drawingconclusions

338 MARIO V WUTHRICH

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Remark As weights wi we take the number of reported claims (the number ofIBNyR claims with reporting delay of more than two years is almost zero forthis kind of business)

a) Tweediersquos compound Poisson model with constant dispersion

We assume that Yij are independent with Yij sim ED(p) (mi j ƒwi) (see (41)) Definethe total outstanding payments C as in (52) If we start with initial valuep(0) = 15 isin (12) and then proceed the estimation iteration as in Subsection 41we observe that already after 4 iterations we have sufficiently converged toequilibrium (for the choice of p one should also have a look at Figure 1)

CLAIMS RESERVING USING TWEEDIErsquoS COMPOUND POISSON MODEL 339

TABLE 62

OBSERVATIONS FOR THE NORMALIZED INCREMENTAL PAYMENTS Yij = Cij wi

yij Development period j

AY i 0 1 2 3 4 5 6 7 8 9 10

0 15795 6589 793 361 183 055 014 022 001 014 0001 17686 6031 853 141 063 034 049 101 038 0232 18967 6003 1044 265 154 066 054 009 0193 18915 5771 777 303 143 095 027 0614 18453 5844 696 291 346 112 1175 18562 5659 573 245 105 0936 18103 6235 554 243 3667 17996 5536 599 2748 18801 5586 546

TABLE 63

NUMBER OF PAYMENTS Rij AND VOLUME wi

rij Development period j

AY i 0 1 2 3 4 5 6 7 8 9 10 wi

0 6rsquo229 3rsquo500 425 134 51 24 13 12 6 4 1 112rsquo9531 6rsquo395 3rsquo342 402 108 31 14 12 5 6 5 110rsquo3642 6rsquo406 2rsquo940 401 98 42 18 5 3 3 105rsquo4003 6rsquo148 2rsquo898 301 92 41 23 12 10 102rsquo0674 5rsquo952 2rsquo699 304 94 49 22 7 99rsquo1245 5rsquo924 2rsquo692 300 91 32 23 101rsquo4606 5rsquo545 2rsquo754 292 77 35 94rsquo7537 5rsquo520 2rsquo459 267 81 92rsquo3268 5rsquo390 2rsquo224 223 89rsquo545

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Figure 1 Profile likelihood function Lm(p) (see (46))

For p = 11741 the GLM output is as follows Dispersion ƒ = 29rsquo281 and para-meter estimates

340 MARIO V WUTHRICH

TABLE 64

ESTIMATION OF p

Iteration k 0 1 2 3 4

p(k) 15000 11743 11741 11741 11741Outstanding payments C (k) 1rsquo431rsquo266 1rsquo451rsquo288 1rsquo451rsquo300 1rsquo451rsquo299

TABLE 65

PARAMETERS AND f FOR p = 11741

j 0 1 2 3 4 5 6 7 8 9 10

( )log j ndash5862 ndash5825 ndash5762 ndash5782 ndash5777 ndash5819 ndash5792 ndash5837 ndash5809

( )log f j 1101 990 779 678 645 551 513 508 417 416 000

Altogether this leads to the following result

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The results in Table 66 show that there is considerable uncertainty in thereserve estimates especially in the old years where the outstanding paymentsare small This comes from the fact that we have only little information to esti-mate f (j) for large j and it turns out that the parameter estimation error liveson the same scale as the process error For young accident years we have onthe one hand a lot of information to estimate f (j) for small j and on the otherhand f(j) for j large has a rather small influence on the overall outstanding pay-ments estimate for young accident years in our example Therefore the relativeprediction error is smaller for young accident years

b) Over-dispersed Poisson and Gamma Model

We first compare our result to the two boundary cases p = 1 and p = 2 Thesemodels are described in Renshaw-Verrall [6] or England-Verrall [1] Section 23(over-dispersed Poisson model) and Mack [3] or England-Verrall [1] Section 33(Gamma model) We also refer to (310)-(311) We obtain the following results

CLAIMS RESERVING USING TWEEDIErsquoS COMPOUND POISSON MODEL 341

TABLE 66

ESTIMATED OUTSTANDING PAYMENTS FROM TWEEDIErsquoS COMPOUND POISSON MODEL

Tweedie constant z = 29rsquo281 and p = 11741

AY i Outst payments MSEP12 in Estimation error Process error

1 326 2rsquo636 8089 1rsquo867 1rsquo8602 21rsquo565 26rsquo773 1242 15rsquo584 21rsquo7703 40rsquo716 35rsquo515 872 19rsquo122 29rsquo9274 89rsquo278 53rsquo227 596 25rsquo940 46rsquo4795 138rsquo338 65rsquo977 477 30rsquo529 58rsquo4896 204rsquo269 80rsquo815 396 35rsquo191 72rsquo7517 360rsquo117 111rsquo797 310 45rsquo584 102rsquo0828 596rsquo690 149rsquo775 251 61rsquo212 136rsquo695

Total 1rsquo451rsquo299 271rsquo503 187 179rsquo890 203rsquo355

TABLE 67

ESTIMATED OUTSTANDING PAYMENTS FROM THE OVER-DISPERSED POISSON MODEL

Over-dispersed Poisson model with z = 36rsquo642 and p = 1

AY i Outst payments MSEP12 in Estimation error Process error

1 330 4rsquo947 15007 3rsquo520 3rsquo4752 21rsquo663 34rsquo776 1605 20rsquo386 28rsquo1743 41rsquo007 46rsquo070 1123 24rsquo896 38rsquo7634 88rsquo546 65rsquo229 737 31rsquo786 56rsquo9615 140rsquo150 80rsquo795 576 37rsquo316 71rsquo6626 204rsquo157 95rsquo755 469 41rsquo089 86rsquo4917 362rsquo724 125rsquo433 346 49rsquo421 115rsquo2868 602rsquo784 161rsquo023 267 61rsquo978 148rsquo618

Total 1rsquo461rsquo360 371rsquo208 217 216rsquo965 231rsquo403

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Conclusions It is not very surprising that the over-dispersed Poisson modelgives a better fit than the Gamma model (especially for young accident yearswe have a huge estimation error term in the Gamma model see Table 68)Tweediersquos compound Poisson model converges to the over-dispersed Poissonmodel for p rarr 1 and to the Gamma model for p rarr 2 For our data set p = 11741is close to 1 hence we expect that Tweediersquos compound Poisson results are closeto the over-dispersed Poisson results Indeed this is the case (see Tables 66 and67) Moreover we observe that the estimation error term is essentially smallerin Tweediersquos model than in the over-dispersed Poisson model Two main reasonsfor this fact are 1) For the parameter estimations in Table 66 we additionallyuse the information coming from the number of payments rij (which is used forthe estimation of p) 2) In our model the variance parameters (ƒ p) are orthog-onal to m hence their uncertainties have no influence on the parameter errorterm coming from Var(m )

c) Mackrsquos model and log-normal model

A classical non-parametric model is the so-called chain-ladder method wherewe apply Mackrsquos formulas (see Mack [4]) for the MSEP estimation We applythe chain-ladder method to the cumulative payments

i D C w Yi i i kk

j

k

j

00

= ===

j k (61)

We choose the chain-ladder factors and the estimated standard errors as fol-lows (for the definition of f (j) and s2

j = 2j we refer to Mack [4] formulas (3)

and (5)) Of course there is unsufficient information for the estimation of s10Since it is not our intention to give good strategies for estimating ultimates(this would go beyond the scope of this paper) we have just chosen a valuewhich looks meaningful

342 MARIO V WUTHRICH

TABLE 68

ESTIMATED OUTSTANDING PAYMENTS FROM THE GAMMA MODEL

Gamma model with z = 29rsquo956 and p = 2

AY i Outst payments MSEP12 in Estimation error Process error

1 447 346 773 255 2332 20rsquo248 13rsquo602 672 8rsquo527 10rsquo5973 40rsquo073 20rsquo127 502 13rsquo178 15rsquo2134 122rsquo899 56rsquo984 464 37rsquo465 42rsquo9365 121rsquo740 50rsquo091 411 35rsquo106 35rsquo7306 221rsquo524 91rsquo174 412 66rsquo731 62rsquo1267 331rsquo115 147rsquo730 446 107rsquo386 101rsquo4518 527rsquo988 250rsquo816 475 194rsquo155 158rsquo784

Total 1rsquo386rsquo034 336rsquo842 243 265rsquo771 206rsquo950

use available at httpswwwcambridgeorgcoreterms httpsdoiorg101017S0515036100013490Downloaded from httpswwwcambridgeorgcore University of Basel Library on 11 Jul 2017 at 140836 subject to the Cambridge Core terms of

This leads to the following result

CLAIMS RESERVING USING TWEEDIErsquoS COMPOUND POISSON MODEL 343

TABLE 610

ESTIMATED OUTSTANDING PAYMENTS FROM MACKrsquoS MODEL

Chain-ladder estimates

AY i Outst payments MSEP12 in Estimation error Process error

1 330 3rsquo740 11346 2rsquo661 2rsquo6272 21rsquo663 19rsquo903 919 11rsquo704 16rsquo0993 41rsquo007 30rsquo090 734 15rsquo954 25rsquo5124 88rsquo546 57rsquo012 644 26rsquo295 50rsquo5855 140rsquo150 71rsquo511 510 31rsquo476 64rsquo2126 204rsquo157 75rsquo522 370 31rsquo746 68rsquo5267 362rsquo724 138rsquo915 383 49rsquo300 129rsquo8728 602rsquo784 156rsquo413 259 54rsquo293 146rsquo688

Total 1rsquo461rsquo360 286rsquo752 196 177rsquo616 225rsquo120

TABLE 69

CHAIN-LADDER PARAMETERS IN MACKrsquoS MODEL

j 1 2 3 4 5 6 7 8 9 10

f (j) 13277 10301 10107 10076 10030 10020 10019 10008 10008 10000sj 15728 3416 1417 2331 570 778 867 389 300 050

A look at the results shows that Tweediersquos compound Poisson model is close tothe chain-ladder estimates For the outstanding payments this is not surpris-ing since for p = 11741 we expect that Tweediersquos estimate for the outstandingpayments is close to the Poisson estimate (which is identical with the chain-ladderestimate) For the error terms it is more surprising that they are so similarThe reason for this similarity is not so clear because we have estimated a dif-ferent number of parameters with a different number of observations Further-more MSEP is obtained in completely different ways (see also discussion in [1]Section 76)

An other well-known model is the so-called parametric chain-ladder methodwhich is based on the log-normal distribution (see Taylor [9] Section 73) Weassume that

j j log D D z sN i j i j12++` `j j and are independent (62)

This model is different from the one usually used in claims reserving whichwould apply to incremental data (see eg [1] Section 32) We have chosen themodel from Taylor [9] because it is very easy to handle

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Living in a ldquonormalrsquorsquo world we estimate the parameters as in Taylor [9] for-mulas (711)-(713) ie since we assume that the parameters only depend onthe development period we take the unweighted averages to estimate zj andthe canonical variance estimate for s2

j This implies

344 MARIO V WUTHRICH

TABLE 611

PARAMETER ESTIMATES IN THE LOG-NORMAL MODEL

j 1 2 3 4 5 6 7 8 9 10

z(j) 02832 00293 00106 00077 00030 00020 00019 00008 00008 00000sj 00274 00067 00027 00046 00011 00015 00016 00007 00004 00001

The prediction errors are estimated according to Taylor [9] formulas (729)-(7-35) This leads to the following result

TABLE 612

ESTIMATED OUTSTANDING PAYMENTS FROM THE LOG-NORMAL MODEL

Log-normal model

AY i Outst payments MSEP12 in Estimation error Process error

1 330 3rsquo905 11837 2rsquo761 2rsquo7612 21rsquo603 14rsquo297 662 8rsquo412 11rsquo5613 40rsquo814 26rsquo680 654 13rsquo991 22rsquo7174 88rsquo535 53rsquo940 609 25rsquo130 47rsquo7285 140rsquo739 69rsquo027 490 30rsquo676 61rsquo8366 205rsquo396 71rsquo506 348 31rsquo043 64rsquo4167 367rsquo545 131rsquo216 357 49rsquo386 121rsquo5688 608rsquo277 147rsquo156 242 54rsquo163 136rsquo826

Total 1rsquo473rsquo238 271rsquo252 184 170rsquo789 210rsquo733

The log-normal model gives estimates for the outstanding payments which areclose to the chain-ladder estimates and hence are close to Tweediersquos estimatesWe have very often observed this similarity One remarkable difference betweenTweediersquos MSEP estimates and log-normal MSEP estimates is that the Tweediemodel gives more weight to the uncertainties for high development periodswhere one has only a few observations This may come from the fact that forthe chain-ladder model we consider cumulative data This cumulation hasalready some smoothing effect

CONCLUSIONS

Of course we start the actuarial analysis of our claims reserving problem bythe chain-ladder method The chain-ladder reserve can very easily be calculated

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But we believe that it is also worth to perform Tweediersquos compound Pois-son method Using the additional information rij one obtains an estimate forthe variance function V(m) = m p If p is close to 1 Tweediersquos compound Pois-son method supports that the chain-ladder estimate Whereas for p differentfrom 1 it is questionable to believe in the chain-ladder reserve since Tweediersquosmodel tells us that we should rather consider a different model (eg the Gammamodel for p close to 2)

A REPARAMETRIZATION

We closely follow [2] We skip the indices i j The joint density of (Y R) is

R

r

g g

lt lt

( ) ( )

( ) ( )

exp exp

exp

exp

zz

z

zz

f r y dy P y Y y dy R r P R r

ryw

yw rw

w wdy

y w r r yw y dy

p pw y

r r yw y p p dy

l t g

gg t

tg l

l

l g t g tg

l

gm m

G

G

G

1

1 21

1 2

Y

r r

r

r p p

g g

g

g

g g

1

1

1 1 2

$

$

= + = =

= - -

= - -

=- - -

--

-

+

+ - -

^

^ ]

^^

d

^

^f

h

h g

hh

n

h

hp

6 6

(

$ (

2 +

2

4 4

(A1)

Hence the density of Y can be obtained summing over all possible values of R

R

p

( )

( ) ( )

( ) ( )

exp

exp

z

zz

zz

f y w p f r y

p pw y

r r yw y p p

c y w p w y

m l t g

gm m

q k q

G1 21

1 2

Y Yr

r

r

p p

g

g g1 1 2

$ $

=

=- - -

--

= -

+ - -

J

L

KKK

^

^

^f

`

N

P

OOO

h

h

hp

j

4 4

1

(A2)

This proves that Y belongs to the exponential dispersion family ED(p)(m ƒ w)

ACKNOWLEDGEMENTS

The author thanks the anonymous referees for their remarks which have sub-stantially improved this manuscript especially concerning Subsection 42 andthe examples section

CLAIMS RESERVING USING TWEEDIErsquoS COMPOUND POISSON MODEL 345

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REFERENCES

[1] ENGLAND PD and VERRALL RJ (2002) Stochastic claims reserving in general insuranceInstitute of Actuaries and Faculty of Actuaries httpwwwactuariesorgukfilespdfsessionalsm0201pdf

[2] JOslashRGENSEN B and DE SOUZA MCP (1994) Fitting Tweediersquos compound Poisson modelto insurance claims data Scand Actuarial J 69-93

[3] MACK T (1991) A simple parametric model for rating automobile insurance or estimatingIBNR claims reserves Astin Bulletin 21 93-109

[4] MACK T (1997) Measuring the variability of chain ladder reserve estimates Claims ReservingManual 2 D61-D665

[5] MCCULLAGH P and NELDER JA (1989) Generalized linear models 2nd edition Chap-man and Hall

[6] RENSHAW AE and VERRALL RJ (1998) A stochastic model underlying the chain-laddertechnique BAJ 4 903-923

[7] SMYTH GK (1996) Partitioned algorithms for maximum likelihood and other nonlinearestimation Statistics and Computing 6 201-216

[8] SMYTH GK and JOslashRGENSEN B (2002) Fitting Tweediersquos compound Poisson model toinsurance claims data dispersion modelling Astin Bulletin 32 143-157

[9] TAYLOR G (2000) Loss reserving an actuarial perspective Kluwer Academic Publishers[10] TWEEDIE MCK (1984) An index which distinguishes between some important exponential

families In Statistics Applications and new directions Proceeding of the Indian StatisticalGolden Jubilee International Conference Eds JK Ghosh and J Roy 579-604 Indian Sta-tistical Institute Calcutta

[11] WRIGHT TS (1990) A stochastic method for claims reserving in general insurance JIA117 677-731

MARIO V WUumlTHRICH

Winterthur InsuranceRoumlmerstrasse 17 PO Box 357CH-8401 WinterthurSwitzerlandE-mail mariowuethrichwinterthurch

346 MARIO V WUTHRICH

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