JIA 116 (1989) 559-587

CHAIN LADDER AND INTERACTIVE MODELLING

(CLAIMS RESERVING AND GLIM)

BY A. E. RENSHAW, B.Sc., PH.D.

(of the City University, London)

[Presented at the Seminar, ‘Applications of Mathematics in Insurance, Finance and Actuarial Work’,
sponsored by the Institute of Mathematics and Its Applications, the Institute of Actuaries, and the
Faculty of Actuaries, held at the Institute of Actuaries, 6–7 July 1989.]

1. MOTIVATION

The prediction of outstanding claims amounts in non-life insurance is, by its

very nature, highly speculative. Partially because of this and partially because of
the variety of features suggested by various researchers for possible inclusion in
the structure of the underlying prediction model, the past two decades have seen
a proliferation of methodologies for making such predictions. Specific details of
these developments are contained in a comprehensive and highly detailed survey
conducted by Taylor (1986)(10) in which a taxonomy of methods is established.
One feature common to all of these methods is the utilization of current and past
records of claims amounts—invariably in the form of the familiar so-called run-
off triangle or a variant thereof—to calibrate the proposed prediction model
before use. Prudence dictates that diagnostic checks should then be made to
establish whether or not the data are supportive of the structure imparted to the
prediction model before use, a feature which apart from some notable exceptions
including Zehnwirth (1985)(14) and Taylor (1983),(8) is not always emphasized in
the literature.
Our purpose is not to add to the existing plethora of methodologies but rather
to return to the grass roots of the subject by exploring more fully the statistical
setting for the basic chain-ladder and related techniques. Essentially a determi-
nistic technique, see for example Hossack et al. (1983),(5) it was left to Kremer
(1982)(6) to point out that the mathematical structure underpinning the chain-
ladder technique is identical to that of the linear statistical model involving a log
response variable regressing on two non-interactive convariables. Yet, judging
by the lack of literature, there would not appear to have been a concerted effort to
develop this connection. Perhaps the answer lies partly in the realization, in some
quarters, that the model is heavily parameterized, a phenomenon known to lead
to predictor instability.
The aims therefore are:

(i) To develop more fully the statistical analogue of the original actuarial
chain-ladder technique.
(ii) To investigate the magnitude and nature of predictor instability asso-
ciated with the technique.

559
560 Chain Ladder and Interactive Modelling

(iii) To suggest a method for improving predictor stability.

(iv) To make the methodology readily available to practitioners so that they
may make their own judgements in these matters.

The GLIM software package, because of its user defined macro facility, is an
invaluable tool in achieving these objectives. Indeed we note with interest that
Taylor (1983)(8) and Taylor & Ashe (1983)(9) used the GLIM package to fit Taylor’s
so called ‘invariant see-saw’ model to run-off data.
We identify our philosophical approach to estimating claims whole-heartedly
with the sentiments expressed by Taylor & Ashe (1983)(9) from which we quote
the following passage:
Our view is that claims analysis is a special case of data analysis; that therefore there are few
preconceptions as to what should be done with the data; indeed, anything goes, if it leads to a model
which exhibits acceptable adherence to the data and is plausible in the light of any collateral
information. To us, faced with a problem of multivariate data analysis, regression analysis represents
a most useful exploratory tool.

We would view this application of GLIM to run-off data as the natural extension
of other applications of generalized linear models in actuarial work reported by
Haberman and Renshaw (1988).(4)

2. CLAIMS DATA

Claims run-off data are generated when delay is incurred in settling insurance
claims. Typically the format for such data is that of a triangle (Figure 1.1) in
which the rows (i) denote accident years and the columns (5) delay or
development years. The settlement or payment year is k=i+j–1. The entries in
the body of the triangle are the adjusted (non-cumulative) amounts

Figure 1.1.
 (Claims Reserving and GLIM) 561

Cij=(claims amount) x (inflation factor)

(exposure)

The triangle is augmented each year with the addition of a new diagonal. Two
noteworthy variations of the triangular format are induced by either truncation
after a fixed period of delay or by the removal of data for the early settlement
years.
Additional information in the guise of numbers of claims settled per cell is
required to implement Taylor’s (1983)(8) ‘invariant see-saw’ method.
An obvious first step in any analysis is to plot the adjusted claims against
accident year, against development year and against payment year. One might
even be tempted to use a three-dimensional plot. Such displays can be very
informative about the type of model structure that the data might support.
The remit is essentially to predict likely claim amounts in the incomplete south-
east region bounded by broken lines in Figure 1.1. A two stage modelling/
predicting process is envisaged.

3. LOG-NORMAL MODELS

Let

Yij= log(Cij )

and consider the class of log-normal models defined by

with

and

Here we have assumed that the normal responses Yij decompose (additively)
into deterministic non-random components (means) mij and independent
homoscedastic normally distributed random error components about a zero
mean. It will be necessary to monitor these assumptions by displaying various
residual plots on fitting specific model structures to the logarithms of the adjusted
claims data.
A number of specific model structures are of interest. These include:

Case (I) (3.1)

with accident and development years treated as non-interactive covariates. This

structure is identical to that used in a two-way analysis of variance (ANOVA), but
based on the incomplete data sketched in Figure 3.1(a). Indeed, our brief is to
estimate the incomplete south-east triangular region. The structure is identical to
that associated with the traditional actuarial chain-ladder technique.

Case (II)
562 Chain Ladder and Interactive Modelling

Figure 3.1. Typical run-off domains and prediction regions.

with development and settlement years treated as non-interactive covariates. The

structure is motivated by the traditional actuarial so-called separation method,
see, for example, Hossack et al. (1983)(5); and was first treated statistically by
Taylor (1979).(7) Depicting the various levels of k along the rows while still
representing the levels ofj as columns distorts the basic data matrix into the form
 (Claims Reserving and GLIM) 563
sketched in Figure 3.1(b). This time our brief is the seemingly difficult one of
predicting values in the lower protruding triangular region.
Case (III) (3.2)
with d=j–1 treated as a continuous regressor variable. A version of this
structure is discussed by Dejong and Zehnwirth (1983)(2)in which parameters are
estimated recursively using the Kalman filter. Practical implementation is
possible using Zehnwirth’s (1985)(14) ICRFS purpose designed software package.
The untransformed model structure is

so that yi<0 ensures claims amounts ultimately decay. Referring to the data
matrix sketched in Figure 3.1(c), prediction beyond the observed limit of d as well
as in the south east triangular region is feasible.

Case (IV) M:

Here we have written d for j when j exceeds some fixed integer q. The model is
clearly a mixture of Case I and Case III applied to separate parts of the data
matrix.
Each of the models discussed above has obvious submodels. We concentrate
on Case I.

4. MODEL FITTING

Consider the two-way ANOVA model structure

with an incomplete experimental design dictated by the pattern of adjusted claim

amounts illustrated in Figure 4.1; obviously, g=0, w=0 for a run-off triangle,
while j= 1, 2, . . ., l; i= 1,2, , . ., r in general. It is well known that whereas this
parametric representation of the model structure involves a total of r+l+1
parameters, it contains only r+l–1 so-called free parameters. Consequently two
contraints must be imposed on the parameters before estimation can proceed.
The GLIM system sets a1= 1= 0 and computes maximum likelihood estimates for
the parameters. As a direct consequence of the normal error structure this is
equivalent to estimation by least squares.
Define indicators ij for all cross-classified factor levels (i,j) according to
ij= 1 if Cij>0, ij= 0 otherwise.
Then
564 Chain Ladder and lnteractive Modelling

Figure 4.1. Typical claims data format.

denote the total number of observations, the number of observations in row iand
the number of observations in column j respectively.
We choose µ, ai, µ, âi,ßj(i,j#
1) SO as to minimize

Partial differentiation with respect to µ, âi, for each i ( # 1) and ßj for each j ( # 1)
leads to the system of linear equations

where
 (Claims Reserving and GLIM) 565
denote the grand total, row totals and column total of the transformed adjusted
claims. The solution of this set of non-singular linear equations yield the required
estimates.
By way of illustration, the artificial data set
j→ 1 2 3 Totals

i1 2 4 6 12
↓2 2 3 4 9
3 4 2 5
4 2

Totals 9 9 10 28

(l = 3, ω = 1, r = 4, g= 0)
gives rise to the system of linear equations

28 3 3
9 2 2 1
9 1 1
3 3 0 0
10 2 0 2 1 0 0
=
9 3 1 1 3 0 0
5 2 1 0 0 2 0
1 0 0 0 0 0
2

which yield the solution

The corresponding fitted and predicted values

are
2·917 3·583 5·500
1·917 2·583 4·500
2·167 2·833 4·50
2·000 2·666 4·583
Scrutiny of these fitted and predicted values reveals the true nature of the
assumed non-interactive model structure which manifests itself in the constant
differences between columns and between rows.
566 Chain Ladder and Interactive Modelling
A noteworthy submodel is that involving development year effects only. The
one-way ANOVA sub-structure is
H: mij=µ+ßj
where, again we define ß1=0 because of overparameterization. This time the
incomplete nature of the data matrix (Figure 4.1) is irrelevant. The parameter
estimates are determined by

The solution is

so that the fitted and predicted values are the column averages. Justification for
using this simplified model is sought by examining the t-statistics associated with
the parameters ai, examination of further residual plots and through a formal
ANOVA F-test based on the statistic

in which RM and RH denote the residual sums of squares or deviance under the
full mode1 M and the submodel H respectively.
Whereas it has been established by Kremer (1982)(6) that the model structure in
use here is identical to that utilized in the standard actuarial chain-ladder
technique as described, for example, in Hossack et al. (1983)(s), the current
treatment of the model differs in two important respects-namely the ways in
which the model parameters are estimated and the predicted values are
constructed.

5. PREDICtED VALUES

The model is fitted on the log-response scale. On this scale

(5.1)
provides a point predictor for the empty (i,j)th cell in the south east triangular
region. Since the mij are linear in the Yijs, they are distributed normally with

(5.2)
and
(5.3)
 (Claims Reserving and GLIM) 567
If, in keeping with common practice, the predictor is augmented by an
independent additive error term, distributed as N(0, 2), then 2 has to be added
to the RHS of (5.3).
Reverting to the original (anti-log) scale, predictors ij are needed where

Since the ijSare normally distributed, the ijSare log-normally distributed with

(5.4)
and

(5.5)
One method of computing predicted values and their standard errors, apparently
favoured by Zehnwirth (1985)(14), is based on (5.4) and (5.5) in which E( ij) and
V(mij) are replaced by their estimated values as dictated, in this instance by (5.2)
and (5.3). It should be stressed, however, that Zehnwirth is working within a
Bayesian framework and would presumably seek to justify the method of
prediction within this framework.

6. PREDICTED TOTALS AND THEIR STANDARD ERRORS

Practitioners have a vested interest in

(i) the predicted row totals

where l and c(i) = l + 1 –i are the upper and lower limits of j;

(ii) the predicted diagonal totals

(iii) the overall predicted total

together with their standard errors.

Consequently, for the predicted row totals, it follows that
568 Chain Ladder and Interactive Modelling
Making use of the Theorem 2.4 of Aitchison and Brown (1969)(1) it can be
shown that

(6.1)
from which (5.5) is retrieved on settingj= k. Further, (5.1) implies that forj#k

(6.2)
This time (5.3) is retrieved on settingj=k. Also note the useful identity

where we have assumed the augmented version of (5.3).

Yet more general versions of (6.1) and (6.2), namely

and

catering for between row dependencies are needed to compute the variances of
the predicted diagonal totals and the overall predicted total. Notice that (6.1) and
(6.2) are retrieved on setting il = i2 = i (together with ji =j, j2 = k).

7. PREDICTOR INSTABII.ITY

First the comment that the adjusted claim amounts are generally characterized
by significant differences between development years but only small differences
across accident years.
The extent of any instability exhibited by each predicted value depends directly
on the number of parameters used to make the prediction, in this case just three
which is not excessive, and more importantly on the extent to which the estimates
of these parameters are sensitive to fluctuations in the data. Not surprisingly in
view of the nature of the model structure and data format, simulation exercises
confirm that predictions are sufficiently robust to data fluctuations in the heart of
and in the north-west corner of the run-off triangle; and that stability deteriorates
as data points further into the other two corners of the run-off triangle are varied.
However, the instability in the north-east corner is generally not a serious
problem since claims amounts in this region are relatively low in comparison with
the remainder of the data triangle. The position is further improved if truncation
has occurred.
 (Claims Reserving and GLIM) 569

Consequently, it is essential to improve predictor stability for the more recent

accident years. There are a number of possibilities such as the estimation of the
ais by empirical Bayes, see Verrall (1988)(12) or by Kahnan filtering as proposed
by Dejong and Zehnwirth (1983)(2) and applied to Case III (discussed in Section
3). We note with particular interest in passing that were one to attempt to
generate the ais as a first order autoregressive process within GLIM, the facility to
handle non-diagonal weight matrices recently proposed by Green (1988)(3) is
needed.
Another possibility which we have been pursuing is a reduction in the total
number of row parameters based on the multiple comparison t-criteria

The objective is to partition the set of ais by varying the limit h. This would seem
to work well, is objective, intuitively appealing, and induces the required degree
of stability provided no new parameters are allocated to the more recent accident
year.

8. IMPLEMENTATION

This is by user defined macros within GLIM. Essentially four primary macros
are required:
(i) to create related vectors, scalars and to output data plots;
(ii) to do the model fitting and output graphical checks;
(iii) to conduct the multiple comparison t-tests;
(iv) to output further graphical checks; to compute and output the predicted
claims amounts, their totals and standard errors.
It is suggested that these macros could form the basis of a more extensive suite of
macros to be offered to practitioners. It is noted with interest that one such
practitioner, Taylor ( 1988),(11)strongly recommends the use of such regression
methods.

9. AN APPLICATION

Consider the non-cumulative run-off triangle with exposures (Table 9.1)

computed from the data given in Taylor and Ashe (1983)(9) and used by them to
illustrate their ‘invariant see-saw’ method. Inflation effects are not discussed so
we ignore these. The plot of adjusted claims against delay (Figure 9.1) is
informative, hinting that a model of the type defined by (3.2) as well as that
defined by (3.1) might well be appropriate. We concentrate on the latter because
of its historical interest. The remaining adjusted claims plots are relatively
uninformative and are consequently not reproduced here.
570 Chain Ladder and interactive Modelling

Table 9.1 Run-off claims data and exposures

development 1
2 3 4 5 6 7 8 9 10
year j

accident 1 357848 766940 610542 482940 527326 574398 146342 139950 227229 61948

year 2 352118 884021 933894 1183289 445745 320996 527804 266172 425046

(i) 3 290507 1001799 926219 1016654 750816 146923 495992 260405

4 310608 1108250 776189 1562400 272482 352053 206286

5 443160 693190 991983 769488 504851 470639

6 396132 937085 847498 805037 705960

7 440832 847631 1131398 1063269

8 359480 1061648 1443370

9 376686 986608

10 344014

EXPOSURES

610 721 697 621 600 552 543 503 525 420

Figure 9. I.
 (Claims Reserving and GLIM) 571
Residual plots for the two-way ANOVAModel defined by (3.1) (Figures 9.2(a)–
(e)) are reasonably supportive of the model although the histogram is slightly
skewed. Estimates for the model parameters and their standard errors are given
in standard GLIM format (Table 9.2). Here the model parameters of (3.1) have
been recoded according to 1 for µ, the general mean; DY _(j) for βj, the
development year parameters and AY _(i) for αi, the accident year parameters.
The system automatically sets 1α = β 1 = 0, a feature utilized in the development of
Section 4.

Figure 9.2(a).

Figure 9.2(b).
572 Chain Ladder and Interactive Modelling

Figure 9.2(c).

Figure 9.2(d).
(Claims Reserving and GLIM) 573

Figure 9.2(e)

Table 9.2
[o]
[o]
[o] estimate s.e. parameter
[o] 1 6.106 0.1646 1
[o] 2 0.9112 0.1607 DY_(2)
[o] 3 0.9387 0.1681 DY_(3)
[o] 4 0.9650 0.1761 DY_(4)
[o] 5 0.3832 0.1857 DY_(5)
[o] 6 -0.0004909 0.1978 DY_(6)
[o] 7 -0.1181 0.2142 DY_(7)
[o] 8 -0.4393 0.2387 DY_(8)
[o] 9 -0.05351 0.2806 DY_(9)
[o] 10 -1.393 0.3786 DY_(10)
[o] 11 0.1938 0.1607 AY_(2)
[o] 12 0.1489 0.1681 AY_(3)
[o] 13 0.1533 0.1761 AY_(4)
[o] 14 0.2988 0.1857 AY_(5)
[o] 15 0.4117 0.1978 AY_(6)
[o] 16 0.5084 0.2142 AY_(7)
[o] 17 0.6731 0.2387 AY_(8)
[o] 18 0.4952 0.2806 AY_(9)
[o] 19 0.6018 0.3786 AY_(10)
[o] scale parameter taken as 0.1162
[o]
574 Chain Ladder and Interactive Modelling
Attempted model simplification by excluding accident year effects leads to an
F-statistic value of 1.481 on 9,36 degrees of freedom with an observed
significance level of approximately 20%. Whereas this is supportive of the
simplification, two of the residual plots (Figures 9.3(a) and (b)) under the
simplified one-way development year effects model become unacceptably
distorted. The explanation for this is possibly to be found in the values of the
parameter estimates (Table 9.2) under the full two-way ANOVAmodel. The t-
statistics (obtained by dividing the estimates by their standard errors) indicate
that the accident year parameters from year six onwards are all in fact significant;
a feature which would appear to synchronize with the residual plots (Figures
9.3a–b). Consequently, we retain the two-way ANOVAmodel for the time being.
We also have a vested interest in investigating the extent of predictor instability
for this model. The run-off claims data, their expected (fitted) values under this
model, the predicted claims values and their standard errors are presented in
Table 9.3 together with the predicted totals and their standard errors.
We are involved in a two stage process in which the data are first utilized to
calibrate/validate the proposed model before moving to the predictive second
stage. Model validation is done through scrutiny of response and residual plots
coupled with attempted model simplifications where appropriate. Given a
satisfactory model, both the magnitude of the standard errors of the predicted
values and the degree of stability exhibited by predicted values to fluctuations in
the data are important aspects of performance with which to assess the
effectiveness of this process. Clearly, if relatively minor fluctuations in the data
induce excessive changes in the predicted values there is cause for concern, a
phenomenon which is well known in the context of predictive regression
modelling.
The extent of any instability exhibited by each predicted value depends directly
in the number of parameters used to make each prediction, in this case just three
(and not directly on the total number of model parameters), together with the
extent to which the estimates of these parameters are sensitive to fluctuations in
the data. We concentrate on the latter source of possible instability since the
number of parameters involved in making each prediction is low. Indeed an
identical number of parameters (three) is involved in each prediction based on
the model defined by (3.2) in which a much more rigid structure is imputed to
development year effects.
Suppose first that g=0, w=0 so that the data are triangular in shape. Not
surprisingly in view of the nature of the model structure, simulation exercise
reveals that predictor stability deteriorates as data points further into the apices
of the run-off triangle are varied. This is illustrated by Figure 9.4(a) in which the
arrows indicate the directions of decreasing predictor stability. However, the
magnitude of predictor instability induced by changes in the data would not
appear to be excessive in our experience except for changes in the last few data
rows and columns. This is hardly surprising as so little data are yet available to
stabilize the estimates of the corresponding row and column parameters.
(Claims Reserving and GLIM) 575

Figure 9.3(a).

Figure 9.3(b).
576 Chain Ladder and Interactive Modelling
 (Claims Reserving and GLIM) 577

Figure 9.4. Arrows indicating direction of decreasing predictor stability.

Comparison of Tables 9.4a–b with Table 9.2 and Tables 9.5a–b with Table 9.3
give an indication of the degree of instability involved. In the construction of
Tables 9.4(a) and 9.5(a) the original claims amount C’32 is changed approxima-
tely 10% from 1001799 to 901799 while Tables 9.4(b) and 9.5(b) are based on a
substantial adjustment to the original claims amount C'28 from 266172 to
166172. We leave the reader to assess for his or herself the magnitude and pattern
of changes induced in the predicted values by these two representative changes in
the claims data by comparing Tables 9.5a–b with Table 9.3. As a further guide
changes to the penultimate row or column of the run-off triangle induce some
changes up to the same order of magnitude in the corresponding row or column

Table 9.4(a)
[o] The parameter estimates are
[o]
[o] estimate s.e. parameter
[o] 1 6.106 0.1644 1
[o] 2 0.8995 0.1604 DY_(2)
[o] 3 0.9395 0.1678 DY_(3)
[o] 4 0.9663 0.1758 DY_(4)
[o] 5 0.3852 0.1854 DY_(5)
[o] 6 -0.002226 0.1975 DY_(6)
[o] 7 -0.1145 0.2139 DY_(7)
[o] 8 -0.4345 0.2383 DY_(8)
[o] 9 -0.05308 0.2802 DY_(9)
[o] 10 -1.393 0.3780 DY_(10)
[o] 11 0.1938 0.1604 AY_(2)
[o] 12 0.1358 0.1678 AY_(3)
[o] 13 0.1539 0.1758 AY_(4)
[o] 14 0.3000 0.1854 AY_(5)
[o] 15 0.4136 0.1975 AY_(6)
[o] 16 0.5112 0.2139 AY_(7)
[o] 17 0.6772 0.2383 AY_(8)
[o] 18 0.5015 0.2802 AY_(9)
[o] 19 0.6022 0.3780 AY_(10)
[o] scale parameter taken as 0.1158
578 Chain Ladder and Interactive Modelling
Table 9.4(b)
[o] The parameter estimates are
[o]
[o] estimate s.e. parameter
[o] 1 6.123 0.1663 1
[o] 2 0.9112 0.1623 DY_(2)
[o] 3 0.9387 0.1697 DY_(3)
[o] 4 0.9650 0.1779 DY_(4)
[o] 5 0.3832 0.1875 DY_(5)
[o] 6 -0.004909 0.1998 DY_(6)
[o] 7 -0.1181 0.2164 DY_(7)
[o] 8 -0.5963 0.2411 DY_(8)
[o] 9 -0.04369 0.2834 DY_(9)
[o] 10 -1.410 0.3823 DY_(10)
[o] 11 0.1415 0.1623 AY_(2)
[o] 12 0.1522 0.1697 AY_(3)
[o] 13 0.1370 0.1779 AY_(4)
[o] 14 0.2824 0.1875 AY_(5)
[o] 15 0.3953 0.1998 AY_(6)
[o] 16 0.4920 0.2164 AY_(7)
[o] 17 0.6568 0.2411 AY_(8)
[o] 18 0.4789 0.2834 AY_(9)
[o] 19 0.5854 0.3823 AY_(10)
[o] scale parameter taken as 0.1185

of predicted values, with changes of a much lower order of magnitude elsewhere

in the predicted values. Changes in the final row or column induce changes of a
greater order of magnitude in that row or column of predicted values while
leaving the remaining predicted values unchanged. We would strongly recom-
mend that any practitioners should conduct their own simulation exercises to

Figure 9.5. Partition of row parameters.

(Claims Reserving and GLIM) 579
580 Chain Ladder and Interactive Modelling
 (Claims Reserving and GLIM) 581
familiarize themselves with the nature and magnitude of such instability that
exists.
Predictor instability with increasing development year ceases to be an issue
(see Figure 9.4(b)) either if w>0 or when using a model of the type defined by
(3.2). In addition, as already stated in Section 7, instability in the north-east
corner of the run-off triangle is generally not a serious problem since claims
amounts in this region are relatively low in comparison with the remainder of the
data matrix. One further noteworthy feature of the two-way ANOVAmodel when
w > 0 is the invariance of predicted values to row permutations between the early
accident years i, for which i< w+ 1 in the data matrix.
One potent way of diminishing the degree of instability to satisfactory levels in
the important south-west corner of the run-off triangle is by allocating the same

Table 9.6
[o]
[o] estimate s.e. parameter
[o] 1 6.119 0.1520 1
[o] 2 0.9024 0.1476 DY_(2)
[o] 3 0.9324 01528 DY_(3)
[o] 4 0.9363 0.1598 DY_(4)
[o] 5 0.3522 0.1696 DY_(5)
[o] 6 -0.01988 0.1838 DY_(6)
[o] 7 -0.1330 0.1995 DY_(7)
[o] 8 -0.4500 0.2202 DY_(8)
[o] 9 -0.05353 0.2580 DY_(9)
[o] 10 -1.406 0.3551 DY_(10)
[o] 11 0.1682 0.1267 MAY_(2)
[o] 12 0.3009 0.1746 MAY_(3)
[o] 13 0.5102 0.1467 MAY_(4)
[o] scale parameter taken as 0.1030
[o]
[o]

Figure 9.6(a).
582 Chain Ladder and Interactive Modelling

Figure 9.6(b).

Figure 9.6(c).
(Claims Reserving and GLIM) 583

Figure 9.6(d).

Figure 9.6(e).
584 Chain Ladder and Interactive Modelling
 (Claims Reserving and GLIM) 585

Figure 9.7.

a i parameters to more than one accident year where appropriate. Indeed, this is
vital if acceptable levels of stability are to be induced for the most recent accident
years for which relatively little data are, as yet, available. We stress that this
defect is also present in the traditional actuarial deterministic chain-ladder
technique, giving rise to much concern about the apparant continuing esteem
afforded to the technique.
A way forward is to examine all contrasts

between row parameters, Such contrasts are invariant of the somewhat arbitrary
choice of the two parameter constraints α 1( = β 1 = 0) needed to estimate the ais.
Application of the multicomparison t-criterion

for h=·5, induces the partition in row parameters displayed in Figure 9.5 in
which accident years are represented by numbered nodes; two nodes being linked
if and only if the inequality is satisfied.
This allocates separate row parameters to years 1 and 5 while linking years, 2,3
586 Chain Ladder and Interactive Modelling
and 4 together as well as linking years 6 to 10 inclusive; making a total of just four
row parameters. For sufficiently large h, all nodes are interlinked, while linkages
are shed as h is reduced.
The residual plots (Figures 9.6(a)–(e)), the parameter estimates (Table 9.6) and
predicted values (Table 9.7) are presented for scrutiny.
Verrall (1989)(13) has conducted a comparative study of estimates for the αis
based on a variety of estimation methods for these data. A graphical comparison
of least squares, empirical Bayes, Kalman filter and multi comparison estimators
is presented in Figure 9.7.

10. POSTSCRIPT
Possible future developments for incorporating within GLIM include:

(i) alternative methods of mapping back from the logarithmic modelling space;
(ii) use of the other model structures discussed in Section 3 (partially
developed);
(iii) use of methods other than the multicomparison tests to induce predictor
stability.

We would like to acknowledge the financial support received from the

Commercial Union Insurance Company together with the encouraging ongoing
discussions held with Stavros Christofides and Peter Crane from that company.
We are most interested to hear from any further practitioners interested in these
highly practical developments.

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