PREDICTION ERROR OF THE CHAIN LADDER RESERVING METHOD APPLIED TO CORRELATED RUN-OFF TRIANGLES
By M. Merz and M. V. Wu« thrich
abstract
In Buchwalder et al. (2006) we revisited Mack’s (1993) and Murphy’s (1994) estimates for the mean square error of prediction (MSEP) of the chain ladder claims reserving method. This was done using a time series model for the chain ladder method. In this paper we extend the time series model to determine an estimate for the MSEP of a portfolio of N correlated run-off triangles. This estimate differs in the special case N ¼ 2 from the estimate given by Braun (2004). We discuss the differences between the estimates.
keywords
Claims Reserving; Chain Ladder; Mean Square Error of Prediction; Prediction Error;
Correlated Business Lines; Run Off
contact addresses
M. Merz, University of Tu«bingen, Faculty of Economics, D-72074, Germany.
M. V. Wu«thrich, ETH Zu«rich, Department of Mathematics, CH-8092 Zu«rich, Switzerland.
". Motivation
There are several different stochastic models which justify the chain ladder algorithm for claims reserving. The most popular ones are Mack’s (1993) chain ladder model, the overdispersed Poisson model (see Renshaw & Verrall, 1998; Verrall, 2000) and Bayesian models (see, e.g., Gisler, 2006). All these different models give the chain ladder estimate for the claims reserves, however (due to their differences) they lead to different estimators (and values) for prediction errors.
In this paper we consider the claims reserving problem for several correlated claims reserving triangles. We study this multivariate claims reserving problem within the framework of a multivariate time series model for the chain ladder method. This generalises the univariate time series model for the chain ladder model studied in Buchwalder et al. (BBMW) (2006). We have seen in Buchwalder et al. (2006) that there are different approaches for estimating the mean square error of prediction (MSEP) in Mack’s (1993) univariate chain ladder model (unconditional and conditional approaches). Depending on the approach chosen, one A.A.S. 2, I, 25-50 (2007)
obtains different estimators for the MSEP. In BBMW (2006) we applied the conditional approach, which has led to estimators for the MSEP which differ from Mack’s (1993) results. Moreover, we saw that Mack’s (1993) formula is a linear approximation from below to BBMW’s formula (2006). In most practical examples the resulting estimates (Mack (1993) and BBMW (2006)) are very close to each other, which means that it is sufficient to consider the first terms in the Taylor expansions to get the range of possible estimation errors (see, also, Wu«thrich et al., 2006).
The aim of the present article is to extend the MSEP estimate from the univariate chain ladder model to the situation of several correlated run-off triangles. As a result of our extension, we obtain an estimate for the MSEP for aggregated claims reserves of several correlated run-off triangles. Our MSEP formula is compared to Braun’s (2004) MSEP result, which is the bivariate extension of Mack’s formula (see Section 5).
Our studies are motivated by the fact that, in practice, it is quite natural to subdivide a non-life run-off portfolio into several subportfolios, such that each subportfolio satisfies certain homogeneity properties (in our case the chain ladder assumptions). The total reserves are then obtained by the aggregation of the reserves from the single subportfolios. The calculation of the resulting MSEP of the total portfolio is then quite sophisticated if the subportfolios are correlated. In this work we treat such questions precisely. Hence, our work is one step towards the aggregation of different subportfolios. However, we should also remark that, in practice, one often uses different reserving methods for different subportfolios. It remains a challenging open problem to estimate an overall MSEP for aggregated subportfolios (if we use different claims reserving methods in the subportfolios).
An alternative idea for calculating aggregated reserves and their uncertainties is that one only calculates the reserves and their uncertainties on the aggregrated run-off triangle; but one should pay attention to the fact that, if the subportfolios satisfy the chain ladder assumptions, then the aggregated run-off triangle does not necessarily satisfy the chain ladder assumptions (c.f. Anje, 1994). Henceforth, this is not a promising solution to the claims reserving problem on aggregated subportfolios.
Æ. Claims Development Triangles
Assume that we have N 1 run-off triangles. Figure 1 shows the structure of the run-off triangles (claims development data).
For simplicity, we assume that the number of accident years is equal to the number of observed development periods, i.e. I ¼ J. In these triangles the variables:
n, 1 n N, refer to subportfolios (triangles) i, 1 i I, refer to accident years (rows)
j, 1 j J, refer to development years (columns). Usually, at time I, we have observations:
DðnÞI ¼ XðnÞi;j 1 i I and 1 j I ÿ i þ 1 ð2:1Þ
for all N subportfolios. This means that at time I (calendar year) we have observationsDðnÞI (n ¼ 1; . . . ; N), and we need to predict the random variables
in its complement: Dc I¼ [N n¼1 XðnÞi;j 1 i I and I ÿ i þ 1 < j : ð2:2Þ
Here the entries XðnÞi;j denote incremental quantities, and may be interpreted
as: (a) on a paid claims basis; (b) on an occurred claims basis; or (c) number of newly reported claims. Cumulative quantities are denoted by:
Figure 1. The structure of the run-off triangles
Accident Development years j
CðnÞi;j ¼
Xj l¼1
XðnÞi;l: ð2:3Þ
In the following, to simplify our language, XðnÞi;j and C ðnÞ
i;j always refer to
payments.
â. Chain Ladder Estimate and MSEP
The chain ladder method is based on cumulative quantities CðnÞi;j.
Increments XðnÞi;j can easily be derived from X ðnÞ i;j ¼C ðnÞ i;j ÿC ðnÞ i;jÿ1.
3.1 Chain Ladder Algorithm
Often the chain ladder method is understood as a purely mechanical algorithm to estimate claims reserves. For triangle n 2 f1; . . . ; Ng, the algorithmic definition of the chain ladder method at time I (i.e. given the observationsDðnÞI ) reads as follows:
(1) There are constants flðnÞ (l ¼ 1; . . . ; J ÿ 1) so that for all i and
j > I ÿ i þ1: b CðnÞi;j ¼C ðnÞ i;Iÿiþ1f ðnÞ Iÿiþ1f ðnÞ Iÿiþ2. . . f ðnÞ jÿ1 ð3:1Þ
is an appropriate predictor for CðnÞi;j.
(2) The chain ladder factors flðnÞ(age-to-age factors) are estimated by:
bflðnÞ¼ PIÿl i¼1C ðnÞ i;lþ1 SðnÞl ¼X Iÿl i¼1 CðnÞi;l SðnÞl C ðnÞ i;lþ1 CðnÞi;l ð3:2Þ where: SðnÞl ¼ XIÿl i¼1 CðnÞi;l: ð3:3Þ
It is to Mack’s (1993) credit that he first gave a stochastic model which satisfies the chain ladder algorithm, and for which he estimated the MSEP. 3.2 The Chain Ladder Consistent Time Series Model for Several Correlated Run-Off Triangles
Braun (2004) extended Mack’s (1993) method to the multivariate case, in order to estimate the MSEP of the chain ladder method for several subportfolios simultaneously. In this paper we choose a different route, and enlarge our time series framework to N correlated development triangles. Within this framework, we derive an estimate for the MSEP according to the conditional approach described in Buchwalder et al. (2006). The resulting 28 Prediction Error of the Chain Ladder Reserving Method
formula differs from Braun’s formula (2004). The discussion in Section 5 highlights the differences.
Model assumptions 3.1 (Multivariate chain ladder time series model) There exist constants flðnÞ;s
ðnÞ
l >0 such that, for all 1 n N, 1 i I
and 2 j J, we have: (T1) CðnÞi;j ¼f ðnÞ jÿ1C ðnÞ i;jÿ1þs ðnÞ jÿ1 ffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1 q eðnÞi;j; with:
(T2) different accident years i are independent, and eðnÞi;j and e ðmÞ k;l are
independent if i 6¼ k or j 6¼ l; (T3) EeðnÞi;j
¼0 and VarÿeðnÞi;j
¼EÿeðnÞi;j 2 ¼1; and (T4) CorÿeðnÞi;j;e ðmÞ i;j ¼EeðnÞi;j e ðmÞ i;j ¼rðn;mÞjÿ1 2 ½ÿ1; 1.
For more technical details we refer to Wu«thrich et al. (2006).
Corollary 3.2. Under model assumptions 3.1, fCðnÞi;jgj1 are Markov chains,
with:
(P1) different accident years of one or of distinct triangles are independent; (P2) ECðnÞi;j C ðnÞ i;jÿ1 ¼fjÿ1ðnÞC ðnÞ i;jÿ1; (P3) VarCðnÞi;j C ðnÞ i;jÿ1 ¼ÿsðnÞjÿ1 2 CðnÞi;jÿ1; and (P4) CovCðnÞi;j; C ðmÞ k;j C ðnÞ i;jÿ1; C ðmÞ k;jÿ1 ¼sðnÞjÿ1s ðmÞ jÿ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q rðn;mÞjÿ1 1fi¼kg.
Observe that (P1) ÿ (P3) are the classical Mack (1993) properties extended to several run-off triangles.
Lemma 3.3. Under model assumptions 3.1, we have that bflðnÞ (given in (3.2))
is (conditionally) unbiased for fðnÞ
l . Moreover, bf ðnÞ l and bf
ðmÞ
k are uncorrelated for
l 6¼ k, and b CðnÞi;J¼C ðnÞ i;Iÿiþ1bf ðnÞ Iÿiþ1bf ðnÞ Iÿiþ2. . . bf ðnÞ Jÿ1 ð3:4Þ
is an (conditionally) unbiased estimator for ECðnÞi;J C ðnÞ i;Iÿiþ1
.
Proof. The proof is similar to the one for Theorems 1 and 2 in Mack (1993).
3.3 Conditional Mean Square Error of Prediction We define the set of total information up to time I by:
DI¼
[N n¼1
DðnÞI : ð3:5Þ
Lemma 3.3 gives unbiased predictors for the ultimate claims. Our goal is to study the conditional mean square error of these predictors. The conditional MSEP is defined as follows (see, e.g., Mack (1993), Section 3):
Definition 3.4 (Conditional mean square error of prediction)
MSEP X I i¼Iþ2ÿJ XN n¼1 b CðnÞi;J ! ¼E X I i¼Iþ2ÿJ XN n¼1 CðnÞi;Jÿ XI i¼Iþ2ÿJ XN n¼1 bCðnÞi;J !2 DI 2 4 3 5: ð3:6Þ
Observe that the predictorsbCðnÞi;J are DI-measurable, hence a constant in the
conditional expectation considered in (3.6). Therefore the conditional MSEP decouples into the conditional process error and the conditional estimation error (see also Mack (1993), p217). Notice that this is different from unconditional MSEP considerations (see, e.g., England & Verrall, 2002). Here we have the following (strict) equality for the conditional MSEP:
MSEP X I i¼Iþ2ÿJ XN n¼1 b CðnÞi;J ! ¼Var X I i¼Iþ2ÿJ XN n¼1 CðnÞi;J DI ! þ X I i¼Iþ2ÿJ XN n¼1 b CðnÞi;JÿE XI i¼Iþ2ÿJ XN n¼1 CðnÞi;J " DI #!2 : ð3:7Þ
The first term on the right-hand side of (3.7) is the conditional process variance and the second term on the right-hand side of (3.7) is the conditional estimation error. In Section 4 we derive estimators for these two terms, which give Result 3.5.
Result 3.5. For N correlated run-off triangles we have the following estimator for the conditional MSEP for the ultimate loss of aggregated subportfolios and accident years:
d MSEP X I i¼Iþ2ÿJ XN n¼1 bCðnÞi;J ! ¼ X I i¼Iþ2ÿJ d MSEP X N n¼1 bCðnÞi;J ! þ2 X Iþ2ÿJi<kI X 1n;mN CðnÞi;Iÿiþ1bC ðmÞ k;Iÿiþ1D ðm;nÞ i;J ð3:8Þ
where the conditional MSEP for the ultimate loss of aggregated subportfolios for a single accident year i 2 fI þ 2 ÿ J; . . . ; Ig is given by:
d MSEP X N n¼1 b CðnÞi;J ! ¼X N n¼1 d MSEPÿbCðnÞi;J þ2 X 1n<mN Gðn;mÞi;J þC ðnÞ i;Iÿiþ1C ðmÞ i;Iÿiþ1D ðn;mÞ i;J ð3:9Þ
and where the conditional MSEP for the ultimate loss for a single accident year i 2 fI þ 2 ÿ J; . . . ; Ig and a single subportfolio n 2 f1; . . . ; Ng is given by: d MSEPÿbCðnÞi;J ¼Gðn;nÞi;J þ ÿ CðnÞi;Iÿiþ1 2 Dðn;nÞi;J ð3:10Þ with: Gðn;mÞi;J ¼bC ðnÞ i;JbC ðmÞ i;J XJÿ1 l¼Iÿiþ1 bsðnÞl bs ðmÞ l br ðn;mÞ l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b CðnÞi;l bC ðmÞ i;l q bflðnÞbf ðmÞ l ð3:11Þ Dðn;mÞi;J ¼ YJÿ1 l¼Iÿiþ1 bflðnÞbf ðmÞ l þbr ðn;mÞ l bsðnÞl SðnÞl bsðmÞl SðmÞl XIÿl i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;lC ðmÞ i;l q ÿ Y Jÿ1 l¼Iÿiþ1 bflðnÞbf ðmÞ l : ð3:12Þ
The estimators bsðnÞl and br ðn;mÞ
l for the variance parameters s ðnÞ
l and the
correlation parameters rðn;mÞl are provided in the next subsection. Formulae
(3.9) and (3.10) are derived in Result 4.2 and formula (3.8) is derived in Result 4.3. Remarks 3.6 ö From (P4) we obtain: rðn;mÞjÿ1 ¼ Cov CðnÞi;j ffiffiffiffiffiffiffiffi CðnÞi;jÿ1 p ; C ðmÞ i;j ffiffiffiffiffiffiffiffi CðmÞi;jÿ1 p CðnÞ i;jÿ1; C ðmÞ i;jÿ1 sðnÞjÿ1s ðmÞ jÿ1 : ð3:13Þ
This means that the correlations between the individual development factors are: Cor CðnÞi;j CðnÞi;jÿ1 ; C ðmÞ i;j CðmÞi;jÿ1 CðnÞ i;jÿ1; C ðmÞ i;jÿ1 ¼rðn;mÞjÿ1 : ð3:14Þ
This differs slightly from Braun’s (2004) covariance notations. Indeed, the rðn;mÞ;Bjÿ1 used in Braun (2004) is not a correlation, but:
rðn;mÞ;Bjÿ1 ¼r ðn;mÞ jÿ1 s ðnÞ jÿ1s ðmÞ jÿ1: ð3:15Þ
Otherwise, the model assumptions in Braun (2004) are (P1) ÿ (P4). ö Our estimate in Result 3.5 differs from the one given in Braun (2004).
The differences are analysed in Section 5. 3.4 Estimation of Model Parameters
Our extended time series model specified by model assumptions 3.1 has the parameters: f1ðnÞ;. . . ; f ðnÞ Jÿ1 and s ðnÞ 1 ;. . . ; s ðnÞ Jÿ1 and r ðn;mÞ 1 ;. . . ; r ðn;mÞ Jÿ1 ð3:16Þ
for 1 n m N, which we have to estimate. To estimate the chain ladder factors flðnÞ, we use the standard formulae (3.2) - (3.3). The appropriate
(unbiased) estimator for the varianceÿsðnÞjÿ1
2
, 2 j J, is given by (see Mack (1993), p217): bsðnÞjÿ1 2 ¼ 1 I ÿ j X Iÿjþ1 i¼1 CðnÞi;jÿ1 CðnÞi;j CðnÞi;jÿ1 ÿbfjÿ1ðnÞ !2 : ð3:17Þ
In fact, the proof of the unbiasedness is a straightforward calculation using the model assumptions. There remains to give an estimator for the correlation coefficient rðn;mÞjÿ1 . If s
ðnÞ jÿ1and s
ðmÞ
jÿ1 are known, then the following estimator is an
unbiased estimator for rðn;mÞjÿ1 :
erðn;mÞjÿ1 ¼cj X Iÿjþ1 i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q sðnÞjÿ1s ðmÞ jÿ1 C ðnÞ i;j CðnÞi;jÿ1 ÿbfjÿ1ðnÞ ! C ðmÞ i;j CðmÞi;jÿ1 ÿbfjÿ1ðmÞ ! ð3:18Þ where: cj ¼ 1 I ÿ j ÿ1 þ w2j and w2 j ¼ PIÿjþ1 i¼1 C ðnÞ i;jÿ1C ðmÞ i;jÿ1 1=2 2 SðnÞjÿ1S ðmÞ jÿ1 : ð3:19Þ
For the proof of the unbiasedness we refer to the Appendix.
Hence, a sound estimator for the correlation coefficient rðn;mÞjÿ1 is provided
by (2 j J):
brðn;mÞjÿ1 ¼cj X Iÿjþ1 i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q bsðnÞjÿ1bs ðmÞ jÿ1 C ðnÞ i;j CðnÞi;jÿ1 ÿbfjÿ1ðnÞ ! C ðmÞ i;j CðmÞi;jÿ1 ÿbfjÿ1ðmÞ ! ð3:20Þ
for n 6¼ m andbrðn;nÞjÿ1 ¼1 for n ¼ m.
Remark 3.7. Note that, in general, we are not able to estimate sðnÞJÿ1 and
rðn;mÞJÿ1 from the data and from (3.17) and (3.20). This is due to the lack of data
in the tails. This means that the tail parameters (and tail chain ladder factors) cannot be estimated appropriately from data. There is a whole philosophy about estimating tail parameters. We do not want to enter this discussion here, but we simply extrapolate the last parameters by exponentially decreasing series (see Mack (1993) p217 and formulae (6.1) - (6.2)).
ª. Derivation of the MSEP Estimate
In this section we derive the estimators for the conditional MSEP which are given in Result 3.5.
4.1 MSEP for Single Accident Years
As usual, the conditional MSEP of the sum bCðnÞi;j þbC ðmÞ
i;j can be split
into two parts: (a) the stochastic error (conditional process variance); and (b) the conditional estimation error (c.f. Mack, 1993, Section 3). We assume, in this section, that j þ i > I þ 1, hence we consider:
MSEPÿbCðnÞi;j þbC ðmÞ i;j ¼E b CðnÞi;j þbC ðmÞ i;j ÿ ÿ CðnÞi;j þC ðmÞ i;j 2 DI ¼E CðnÞi;j þC ðmÞ i;j ÿE CðnÞi;j þC ðmÞ i;j DI 2 DI |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼VarÿCðnÞ i;jþC ðmÞ i;j DI þ bCðnÞi;j þbC ðmÞ i;j ÿE C ðnÞ i;j þC ðmÞ i;j DI h i 2 : ð4:1Þ
Note that, because of the DI-measurability of bC ðnÞ i;j þbC
ðmÞ
i;j, we have an exact
equality for the conditional MSEP in (4.1). The conditional process variance VarÿCðnÞi;j þC
ðmÞ i;j DI
originates from the stochastic movement of the processes, whereas the conditional estimation error reflects the uncertainty in the estimation of the expectations (mean values). We derive estimates for both the process variance and the estimation error for N correlated run-off triangles. For a quick reference to the relevant formulae for the different error terms, see Table 1.
4.1.1 Conditional process variance for a single accident year
The derivation of an estimator for the conditional process variance is straightforward, because all terms can be calculated explicitly. For the univariate case we refer to Mack (1993, Theorem 3) or England & Verrall (2002, Appendix 1).
For the conditional process variance in formula (4.1), we obtain: VarCðnÞi;j þC ðmÞ i;j DI ¼Var C ðnÞ i;j DI i þVar CðmÞi;j DI i þ2 Cov CðnÞi;j; C ðmÞ i;j DI i h h h ð4:2Þ for 1 i I, I ÿ i þ 1 < j J and 1 n; m N.
Using (P2) and (P4) we obtain the recursion: CovCðnÞi;j; C ðmÞ i;jjDI ¼CovÿECðnÞi;j C ðnÞ i;jÿ1 ;ECðmÞi;j C ðmÞ i;jÿ1 DI þECovÿCðnÞi;j; C ðmÞ i;j C ðnÞ i;jÿ1; C ðmÞ i;jÿ1 DI ¼fjÿ1ðnÞf ðmÞ jÿ1Cov CðnÞi;jÿ1; C ðmÞ i;jÿ1 DI þsðnÞjÿ1s ðmÞ jÿ1E ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q DI rðn;mÞjÿ1 ð4:3Þ
for the conditional covariance term in formula (4.2). From this we deduce recursively the estimator:
d CovCðnÞi;j; C ðmÞ i;j DI ¼bfjÿ1ðnÞ bf ðmÞ jÿ1 Covd CðnÞi;jÿ1; C ðmÞ i;jÿ1 DI þbsðnÞjÿ1bs ðmÞ jÿ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b CðnÞi;jÿ1bC ðmÞ i;jÿ1 q brðn;mÞjÿ1 ð4:4Þ
for the conditional covariance term CovCðnÞi;j; C ðmÞ i;jjDI
(see also Remarks 4.1). With the definition:
Gðn;mÞi;j ¼Covd CðnÞi;j; C ðmÞ i;j DI ð4:5Þ Table 1. Estimators of conditional process variance, conditional estimation error and conditional MSEP for N correlated run off triangles
Ncorrelated triangles
Single Conditional process variance (4.7)
accident Conditional estimation error (4.22)
years Conditional MSEP (4.23)
Aggregated a.y. Conditional MSEP (4.32)
this can be rewritten in a recursive form: Gðn;mÞi;j ¼bf ðnÞ jÿ1bf ðmÞ jÿ1 G ðn;mÞ i;jÿ1þbs ðnÞ jÿ1bs ðmÞ jÿ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b CðnÞi;jÿ1bC ðmÞ i;jÿ1 q brðn;mÞjÿ1 ð4:6Þ
with Gðn;mÞi;Iÿiþ1¼0, or, in an explicit form, this gives (3.11).
Henceforth, putting all estimates together, we obtain from (4.2) and (4.6) the following recursion for the estimator of the process variance of N correlated triangles: d Var X N n¼1 CðnÞi;j DI ¼X N n¼1 d VarCðnÞi;j DI þ2 X 1n<mN d CovCðnÞi;j; C ðmÞ i;j DI ¼X N n¼1 Gðn;nÞi;j þ2 X 1n<mN Gðn;mÞi;j : ð4:7Þ Remarks 4.1
ö Notice that, for one single run-off triangle (univariate case N ¼ 1), we have exactly the well-known estimator for the process variance (see, e.g., England & Verrall, 2002, Section 7.5.4).
ö The estimator for E
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q DI
given in (4.4) is not unbiased. Indeed: E ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q DI ¼Cov ffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1 q ; ffiffiffiffiffiffiffiffiffiffi CðmÞi;jÿ1 q DI þE ffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1 q DI E ffiffiffiffiffiffiffiffiffiffi CðmÞi;jÿ1 q DI : ð4:8Þ We could replace the estimate in (4.4) by an upper bound for the term in (4.8) (using Jensen’s inequality):
E ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q DI E CðnÞi;jÿ1C ðmÞ i;jÿ1 DI h i1=2 ¼ÿCovCðnÞi;jÿ1; C ðmÞ i;jÿ1 DI þECðnÞi;jÿ1 DI ECðmÞi;jÿ1 DI 1=2 : ð4:9Þ Hence, in that case the recursion (4.6) is replaced by:
e Gðn;mÞi;j ¼bf ðnÞ jÿ1bf ðmÞ jÿ1eG ðn;mÞ i;jÿ1 þbs ðnÞ jÿ1bs ðmÞ jÿ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eGðn;mÞi;jÿ1þbC ðnÞ i;jÿ1bC ðmÞ i;jÿ1 q brðn;mÞjÿ1 ð4:10Þ
witheGðn;mÞi;Iÿiþ1¼0, and we obtain an estimation for an upper bound on the
conditional process variance. Observe that this difficulty is not further commented on in Braun (2004).
4.1.2 Conditional estimation error for a single accident year
The conditional estimation error for two aggregated run-off triangles n and m is given by the last term on the right-hand side of formula (4.1). Before we derive an estimator for this term, we discuss the difficulties in the derivation of an estimator for the conditional estimation error (this is similar to Buchwalder et al., 2006). If we have one single run-off triangle, the conditional estimation error for j > I ÿ i þ 1 is given by (see formula (4.11) in Buchwalder et al., 2006): ÿb CðnÞi;j ÿE CðnÞi;j DI 2 ¼ÿCðnÞi;Iÿiþ1 2 ÿbfIÿiþ1ðnÞ . . . bf ðnÞ jÿ1ÿf ðnÞ Iÿiþ1. . . f ðnÞ jÿ1 2 : ð4:11Þ Observe that the realisations bflðnÞ are known at time t ¼ I, but (of course)
the true chain ladder factors fðnÞl are unknown (otherwise we would not need
to estimate them). Hence, the right-hand side of (4.11) cannot be calculated explicitly, and needs to be estimated using an appropriate technique. The chain ladder model allows for different approaches to estimate the right-hand side of (4.11), and most of them try to estimate the possible fluctuations of the estimators bflðnÞ around the true values f
ðnÞ
l . Such methods involve
resampling techniques (e.g. non-parametric bootstrap methods (see England & Verrall, 2002), closed analytical techniques (see, e.g., Mack, 1993), upper and lower bounds (see, e.g., Wu«thrich et al., 2006), etc.). In the present work, we derive a closed analytical estimate which is based on the conditional approach described in Buchwalder et al. (2006). This closed analytical estimate can also be interpreted as a parametric bootstrap method, for which we can calculate the resulting estimators exactly. Henceforth, we use the terminology ‘resampling’, though we are able to calculate all terms in a closed form.
For the estimation error in formula (4.1) we obtain the following decomposition (i þ j > I þ 1): ÿb CðnÞi;j þbC ðmÞ i;j ÿE CðnÞi;j þC ðmÞ i;j DI 2 ¼ÿbCðnÞi;j ÿE CðnÞi;j DI 2 þÿbCðmÞi;j ÿE CðmÞi;j DI 2 þ2 ÿbCðnÞi;j ÿE CðnÞi;j DI ÿbCðmÞi;j ÿE CðmÞi;j DI : ð4:12Þ
Using (P2) from Corollary 3.2 we have the representation: ÿb CðnÞi;j ÿE CðnÞi;j DI ÿbCðmÞi;j ÿE CðmÞi;j DI ¼CðnÞi;Iÿiþ1 ÿb fIÿiþ1ðnÞ . . . bf ðnÞ jÿ1ÿf ðnÞ Iÿiþ1. . . f ðnÞ jÿ1 CðmÞi;Iÿiþ1 ÿb fIÿiþ1ðmÞ . . . bf ðmÞ jÿ1ÿf ðmÞ Iÿiþ1. . . f ðmÞ jÿ1 : ð4:13Þ
As described above, we are not able to calculate the right-hand side of (4.13) explicitly, because the true chain ladder factors flðnÞ are not known.
Therefore, we study the volatility of the estimators bflðnÞaround the true values
fðnÞl to get a reasonable range for the sizes on the right-hand side of (4.13).
We therefore apply the conditional approach technique presented in Buchwalder et al. (2006). This technique proposes to sample new observations eflðnÞ for bf
ðnÞ
l , given the observations S ðnÞ
l (conditional resampling).
This means that the development period l in the time series is resampled at a time, based on the known volume SðnÞl , for l ¼ 2; . . . ; J.
Note that the observed chain ladder factor estimators satisfy (see (T1) and (3.2)): bfjÿ1ðnÞ ¼f ðnÞ jÿ1þ sðnÞjÿ1 SðnÞjÿ1 X Iÿjþ1 i¼1 ffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1 q eðnÞi;j: ð4:14Þ
Now we generate a set of ‘new’ observations for bfjÿ1ðnÞ, as follows. Set:
efjÿ1ðnÞ ¼f ðnÞ jÿ1þ sðnÞjÿ1 SðnÞjÿ1 X Iÿjþ1 i¼1 ffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1 q eeðnÞi;j ð1 n N; 2 j JÞ ð4:15Þ with: ÿ eeðnÞ i;j
n¼1;...;N;i¼1;...;I;j¼2;...;Jindependent copy of
ÿ eðnÞi;j
n¼1;...;N;i¼1;...;I;j¼2;...;J: ð4:16Þ
Observe that we have the following distributional equalities for the conditional distributions (j ¼ 1; . . . ; J ÿ 1 and n ¼ 1; . . . ; N):
efjðnÞjDI ¼ ðdÞ efjðnÞj CðmÞi;l;1 m N; 1 i I; 1 l j ¼ðdÞbfjðnÞj CðmÞi;l;1 m N; 1 i I; 1 l j : ð4:17Þ
This means that, givenDI, the random variable ef ðnÞ
j has the same distribution
as the observed estimator bfjðnÞ, conditioned on all observations C ðmÞ i;l before
time j þ 1. Therefore, we use efjðnÞ for deriving an estimate for the
right-hand side of (4.13).
GivenDI, the random variable ef ðnÞ
j satisfies:
(1) the random variables ef0ðnÞ;. . . ;ef ðnÞ
Jÿ1are conditionally independent givenDI;
(2) efjðnÞand ef ðmÞ
l (n 6¼ m) are conditionally independent givenDIif j 6¼ l;
(3) Eefjÿ1ðnÞ DI ¼fjÿ1ðnÞ for 1 j J; (4) Eÿefjÿ1ðnÞ 2 DI ¼ÿfjÿ1ðnÞ 2 þ ÿ sðnÞjÿ1 2 SðnÞjÿ1 for 1 j J;
(5) Eefjÿ1ðnÞef ðmÞ lÿ1 DI ¼fjÿ1ðnÞ f ðmÞ lÿ1 for 1 j J if j 6¼ l; and (6) Eefjÿ1ðnÞef ðmÞ jÿ1 DI ¼fjÿ1ðnÞ f ðmÞ jÿ1 þ sðnÞjÿ1s ðmÞ jÿ1 SðnÞjÿ1SðmÞjÿ1 X ÿjþ1 i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q rðn;mÞjÿ1 for 1 j J. In fact (1) and (2) are the crucial steps, which differ from the MSEP derivation of Braun (2004) (Braun uses only the conditional uncorrelatedness of the bfjðnÞ). Using this conditional resampling technique we obtain conditional
independence (see (1) and (2)), which leads to a product structure of the conditional estimation error. This product structure is the same as the one derived by Murphy (1994), however it is based on different arguments. (For a discussion we refer to Buchwalder et al. (2006); Mack et al. (2006); Gisler (2006); and Venter (2006).)
Properties (1) and (2) imply that: EefIÿiþ1ðnÞ ef ðmÞ Iÿiþ1. . . ef ðnÞ jÿ1ef ðmÞ jÿ1 DI ¼ Y jÿ1 l¼Iÿiþ1 EeflðnÞef ðmÞ l DI : ð4:18Þ Using (1) - (3) we have: EhÿefIÿiþ1ðnÞ . . . ef ðnÞ jÿ1ÿf ðnÞ Iÿiþ1. . . f ðnÞ jÿ1 ÿefIÿiþ1ðmÞ . . . ef ðmÞ jÿ1 ÿf ðmÞ Iÿiþ1. . . f ðmÞ jÿ1 DI i ¼ Y jÿ1 l¼Iÿiþ1 EeflðnÞef ðmÞ l DI ÿ Y jÿ1 l¼Iÿiþ1 flðnÞf ðmÞ l : ð4:19Þ
Notice that (4.19) can be calculated for the random variables eflðnÞ
explicitly, i.e. there is no approximation involved here. Using (6) we can calculate the covariance term in (4.19). This leads to the following estimator for the right-hand side of (4.13). We replace the unknown parameters sðnÞl , f
ðnÞ l
and rðn;mÞl by their estimators, and we obtain the following estimators for the
estimation error terms in (4.12): bEÿbCðnÞi;j ÿE CðnÞi;j DI ÿbCðmÞi;j ÿE CðmÞi;j DI DI ¼CðnÞi;Iÿiþ1C ðmÞ i;Iÿiþ1D ðn;mÞ i;j ð4:20Þ with Dðn;mÞi;j given by (3.12). D
ðn;mÞ
i;j can be rewritten in a recursive form:
Dðn;mÞi;j ¼bf ðnÞ jÿ1bf ðmÞ jÿ1 D ðn;mÞ i;jÿ1þ bsðnÞjÿ1 SðnÞjÿ1 bs ðmÞ jÿ1 SðmÞjÿ1 X Iÿjþ1 i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q brðn;mÞjÿ1 Y jÿ2 l¼Iÿiþ1 bflðnÞbf ðmÞ l þ bsðnÞl SðnÞl bs ðmÞ l SðmÞl X Iÿl i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;l C ðmÞ i;l q brðn;mÞl ð4:21Þ with Dðn;mÞi;Iÿiþ1¼0.
Note, that for n ¼ m this is exactly the term given in Buchwalder et al. (2006, formula (4.22)), which corresponds to the estimation error term in Murphy (1994), but is different from the estimation error term in Mack (1993).
Hence, from (4.12) and (4.21) we obtain the following recursion for the estimator of the conditional estimation error of N correlated triangles:
b E X N n¼1 b CðnÞi;j ÿE XN n¼1 CðnÞi;j DI #!2 DI " # ¼X N n¼1 CðnÞi;Iÿiþ1 2 Dðn;nÞi;j þ2 X 1n<mN CðnÞi;Iÿiþ1C ðmÞ i;Iÿiþ1D ðn;mÞ i;j : ð4:22Þ
From (4.7) and (4.22) we obtain a recursive estimator for the MSEP for a single accident year of N correlated run-off triangles.
Result 4.2. For N correlated run off triangles, we have the following estimator for the conditional MSEP for a single accident year:
d MSEP X N n¼1 b CðnÞi;j ! ¼X N n¼1 d MSEPÿbCðnÞi;j þ2 X 1n<mN ÿ Gðn;mÞi;j þC ðnÞ i;Iÿiþ1C ðmÞ i;Iÿiþ1D ðn;mÞ i;j ð4:23Þ where: d MSEPÿbCðnÞi;j ¼Gðn;nÞi;j þ ÿ CðnÞi;Iÿiþ1 2 Dðn;nÞi;j : ð4:24Þ
4.2 Conditional MSEP for Aggregated Accident Years
Consider two different accident years i < k. From our assumptions, we know that the ultimate losses CðnÞi;J and C
ðmÞ
k;J are independent for all
n; m 2 f1; . . . ; Ng. Nevertheless, we have to be careful if we aggregate bCðnÞi;J and
b
CðmÞk;J. If n ¼ m, the estimators are no longer independent, since they use the
same observations for estimating the age-to-age factors fjðnÞ. If n 6¼ m, the
estimators are also no longer independent, since they use the dependent estimators bfjðnÞ and bf
ðmÞ
j for estimating the age-to-age factors f ðnÞ j and f
ðmÞ j ,
respectively.
Firstly, we consider the following conditional MSEP:
MSEP X N n¼1 b CðnÞi;Jþ XN n¼1 b CðnÞk;J ! ¼Var X N n¼1 CðnÞi;JþC ðnÞ k;J DI " # þ X N n¼1 b CðnÞi;JþbC ðnÞ k;J ÿE X N n¼1 CðnÞi;JþC ðnÞ k;J DI " #!2 : ð4:25Þ
Using the independence of the different accident years, the first term on the right-hand side of (4.25) can easily be decoupled:
Var X N n¼1 CðnÞi;JþC ðnÞ k;J DI " # ¼Var X N n¼1 CðnÞi;J DI " # þVar X N n¼1 CðnÞk;J DI " # : ð4:26Þ
However, for the second term (the conditional estimation error in (4.25) we have to be more careful. We have the following decomposition:
XN n¼1 b CðnÞi;JþbC ðnÞ k;J ÿE X N n¼1 CðnÞi;JþC ðnÞ k;J DI " #!2 ¼ X N n¼1 b CðnÞi;JÿE XN n¼1 CðnÞi;J DI " #!2 þ X N n¼1 b CðnÞk;JÿE XN n¼1 CðnÞk;J DI " #!2 þ2 X N n¼1 b CðnÞi;JÿE XN n¼1 CðnÞi;J DI " #! X N n¼1 bCðnÞk;JÿE XN n¼1 CðnÞk;J DI " #! : ð4:27Þ
For the cross-product in formula (4.27) we obtain:
XN n¼1 b CðnÞi;JÿE XN n¼1 CðnÞi;J DI " #! X N n¼1 b CðnÞk;JÿE XN n¼1 CðnÞk;J DI " #! ¼ X 1n;mN b CðnÞi;JÿE C ðnÞ i;J DI h i bCðmÞk;JÿE C ðmÞ k;J DI h i : ð4:28Þ
Hence, we have the following decomposition for the conditional MSEP of the aggregated ultimate loss of two run-off triangles:
MSEP X N n¼1 bCðnÞi;Jþ XN n¼1 b CðnÞk;J ! ¼MSEP X N n¼1 b CðnÞi;J ! þMSEP X N n¼1 b CðnÞk;J ! þ2 X N n¼1 b CðnÞi;JÿE C ðnÞ i;J DI h i bCðnÞk;JÿE C ðnÞ k;J DI h i þ4 X 1n<mN b CðnÞi;JÿE C ðnÞ i;J DI h i bCðmÞk;JÿE C ðmÞ k;J DI h i : ð4:29Þ
This means that, in addition to the conditional MSEP of single accident years, we need to find estimates for the last two terms in (4.29). For this we proceed as in Section 4.1.2. We obtain:
b CðnÞi;JÿE C ðnÞ i;J DI h i bCðmÞk;JÿE C ðmÞ k;J DI h i ¼CðnÞi;Iÿiþ1 bf ðnÞ Iÿiþ1. . . bf ðnÞ Jÿ1ÿf ðnÞ Iÿiþ1. . . f ðnÞ Jÿ1 CðmÞk;Iÿkþ1 bf ðmÞ Iÿkþ1. . . bf ðmÞ Jÿ1ÿf ðmÞ Iÿkþ1. . . f ðmÞ Jÿ1 : ð4:30Þ
Completely analogously, as in Section 4.1.2, the right-hand side of (4.30) is now estimated by the Pð jDIÞ average when replacing bf
ðnÞ
j by the resampled
values efjðnÞ. Hence, (4.30) is estimated by (i < k):
bE bCðnÞi;JÿE CðnÞi;J DI bCðmÞk;JÿE CðnÞk;J DI DI h i ¼CðnÞi;Iÿiþ1C ðmÞ k;Iÿkþ1f ðmÞ Iÿkþ1. . . f ðmÞ Iÿi Y Jÿ1 l¼Iÿiþ1 EeflðnÞef ðmÞ l DI ÿ Y Jÿ1 l¼Iÿiþ1 flðnÞf ðmÞ l ! : ð4:31Þ
If we plug in the estimators similarly to (4.20), we obtain Result 4.3.
Result 4.3. For N correlated run-off triangles, we have the following estimator for the conditional MSEP of the ultimate loss of aggregated accident years:
d MSEP X I i¼Iþ2ÿJ XN n¼1 b CðnÞi;J ! ¼ X I i¼Iþ2ÿJ d MSEP X N n¼1 b CðnÞi;J ! þ2 X Iþ2ÿJi<kI X 1n;mN CðnÞi;Iÿiþ1bC ðmÞ k;Iÿiþ1D ðn;mÞ i;J : ð4:32Þ
ä. Comparison with the Braun Formula
If we compare Braun’s (2004) bivariate formulae for the MSEP with our results (4.23) and (4.32) for the MSEP in the special case N ¼ 2, we see the following differences:
(1) There is a small distinction between our result (4.4) of the process variance and Braun’s (2004) formula. It comes from the fact that rðn;mÞ;Bjÿ1 ,
considered in Braun (2004), is not a correlation (see (3.15)).
(2) There is a second difference between our formulae and Braun’s result; namely, we observe a difference in the estimation of the estimation error. The difference is of the same nature as the one in the univariate case N ¼1 (compare Buchwalder et al., 2006; and Mack, 1993).
The main difference comes from the fact that in Braun (2004), the recursive formulae for Dðn;mÞi;j uses a linear approximation which involves the
following terms: bDðn;mÞi;j ¼bf ðnÞ jÿ1bf ðmÞ jÿ1bD ðn;mÞ i;jÿ1 þbs ðnÞ jÿ1 SðnÞjÿ1 bs ðmÞ jÿ1 SðmÞjÿ1 X Iÿjþ1 i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q brðn;mÞjÿ1 Yjÿ2 l¼Iÿiþ1 bflðnÞbf ðmÞ l : ð5:1Þ
Our conditional resampling approach leads to the terms (4.21):
Dðn;mÞi;j ¼bf ðnÞ jÿ1bf ðmÞ jÿ1D ðn;mÞ i;jÿ1 þ bsðnÞjÿ1 SðnÞjÿ1 bs ðmÞ jÿ1 SðmÞjÿ1 X Iÿjþ1 i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q brðn;mÞjÿ1 Y jÿ2 l¼Iÿiþ1 bflðnÞbf ðmÞ l þ bsðnÞl SðnÞl bs ðmÞ l SðmÞl X Iÿl i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;l C ðmÞ i;l q brðn;mÞl ! : ð5:2Þ
Hence, our approach gives a slightly larger estimate compared to Braun’s estimate. In fact, Braun’s result is a linear approximation from below to our result (see also Buchwalder et al., 2006, for a similar fact in the univariate case).
Recall that we have used the conditional independence assumptions (1) and (2) in Section 4.1.2. These conditional independence assumptions have allowed 42 Prediction Error of the Chain Ladder Reserving Method
for an explicit calculation of the estimator for the conditional estimation error. Braun (2004) does not have this independence assumption, and, therefore, cannot calculate an estimator for the estimation error explicitly. Indeed, at some point in Braun (2004) one replaces:
E bflðnÞ 2 bfðnÞ r 2 DI by fðnÞl f ðnÞ r ÿ 2 ð5:3Þ
whereas, in our model we calculate this term explicitly:
E eflðnÞ 2 efðnÞ r 2 DI ¼ fðnÞl ÿ 2 þ ÿ sðnÞl 2 SðnÞl ! fðnÞr ÿ 2 þ s ðnÞ r ÿ 2 SðnÞ r ! ð5:4Þ
see (4.18) ff. This is exactly the difference between our estimator and the estimator in Braun (2004). In the practical examples at which we have looked, the numerical differences between the two formulae are rather small (or even negligible).
å. Example
We apply our methods to the data considered by Braun (2004) for CðnÞi;j
(c.f. Tables 1 and 2 in Braun, 2004). The index n ¼ 1 denotes motor third party liability (MTPL) insurance data and n ¼ 2 is general liability (GL) insurance data. The data for MTPL are given in Table 2 and the data for GL are given in Table 3.
Now we calculate the estimated claims reserves bCðnÞi;J, the MSEP formulae
of Result 3.5 and the corresponding parameter estimators. Note that all these calculations can be done easily and explicitly without any simulation behind them. We have done our calculations in a spreadsheet.
The estimators bfjðnÞ for the chain ladder factors f ðnÞ
j , j 13, are calculated
with formula (3.2). The estimatorsbsðnÞj andbr ðn;mÞ
j , for j 12, are calculated
with the help of formulae (3.17) and (3.20), respectively. Note that the estimation of the parameters sðnÞ13 (n ¼ 1; 2) and r
ð1;2Þ
13 cannot be done with the
help of (3.17) and (3.20), respectively. This is due to the lack of data in the tails. We therefore simply apply the formula proposed by Mack (1993):
bsðnÞ13 ¼ def : min bsðnÞ12 2 =bsðnÞ11;bs ðnÞ 11;bs ðnÞ 12 ð6:1Þ and
T a bl e 2 . M T PL ru n-o ff tri a ng le (c um ul a ti ve pa y me n ts C ð1 Þ i; j ) Mo to r th ir d pa rt y li abi li ty ru n-o ff tr ia ng le AY /D Y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 11 4, 42 3 24 7, 96 1 31 2, 98 2 34 4, 34 0 37 1, 47 9 37 1, 10 2 38 0, 99 1 38 5, 46 8 38 5, 15 2 39 2, 26 0 39 1, 22 5 39 1, 32 8 39 1, 53 7 39 1, 42 8 2 15 2, 29 6 30 5, 17 5 37 6, 61 3 41 8, 29 9 44 0, 30 8 46 5, 62 3 47 3, 58 4 47 8, 42 7 47 8, 31 4 47 9, 90 7 48 0, 75 5 48 5, 13 8 48 3, 97 4 3 14 4, 32 5 30 7, 24 4 41 3, 60 9 46 4, 04 1 51 9, 26 5 52 7, 21 6 53 5, 45 0 53 6, 85 9 53 8, 92 0 53 9, 58 9 53 9, 76 5 54 0, 74 2 4 14 5, 90 4 30 7, 63 6 38 7, 09 4 43 3, 73 6 46 3, 12 0 47 8, 93 1 48 2, 52 9 48 8, 05 6 48 5, 57 2 48 6, 03 4 48 5, 01 6 5 17 0, 33 3 34 1, 50 1 43 4, 10 2 47 0, 32 9 48 2, 20 1 50 0, 96 1 50 4, 14 1 50 7, 67 9 50 8, 62 7 50 7, 75 2 6 18 9, 64 3 36 1, 12 3 44 6, 85 7 50 8, 08 3 52 6, 56 2 54 0, 11 8 54 7, 64 1 54 9, 60 5 54 9, 69 3 7 17 9, 02 2 39 6, 22 4 49 7, 30 4 55 3, 48 7 58 1, 84 9 61 1, 64 0 62 2, 88 4 63 5, 45 2 8 20 5, 90 8 41 6, 04 7 52 0, 44 4 56 5, 72 1 60 0, 60 9 63 0, 80 2 64 8, 36 5 9 21 0, 95 1 42 6, 42 9 52 5, 04 7 58 7, 89 3 64 0, 32 8 66 3, 15 2 10 21 3, 42 6 50 9, 22 2 64 9, 43 3 73 1, 69 2 79 0, 90 1 11 24 9, 50 8 58 0, 01 0 72 2, 13 6 84 4, 15 9 12 25 8, 42 5 68 6, 01 2 91 5, 10 9 13 36 8, 76 2 90 9, 06 6 14 39 4, 99 7 So urc e: Br au n (2 00 4)
Ta bl e 3 . GL run -of f tr ia ng le (c um u la tiv e pa y m ent s C ð2 Þ i; j ) Ge ne ra l lia bi li ty run -of f tr ia ng le AY /D Y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 59 ,9 66 16 3, 15 2 25 4, 51 2 34 9, 52 4 43 3, 26 5 47 5, 77 8 51 3, 66 0 52 0, 30 9 52 7, 97 8 53 9, 03 9 53 7, 30 1 54 0, 87 3 54 7, 69 6 54 9, 58 9 2 49 ,6 85 15 3, 34 4 27 2, 93 6 38 3, 34 9 45 8, 79 1 50 3, 35 8 53 2, 61 5 55 1, 43 7 55 5, 79 2 55 6, 67 1 56 0, 84 4 56 3, 57 1 56 2, 79 5 3 51 ,9 14 17 0, 04 8 31 9, 20 4 42 5, 02 9 50 3, 99 9 54 4, 76 9 55 9, 47 5 57 7, 42 5 58 8, 34 2 59 0, 98 5 60 1, 29 6 60 2, 71 0 4 84 ,9 37 27 3, 18 3 40 7, 31 8 54 7, 28 8 62 1, 73 8 68 7, 13 9 73 6, 30 4 75 7, 44 0 75 8, 03 6 78 2, 08 4 78 4, 63 2 5 98 ,9 21 27 8, 32 9 44 8, 53 0 56 1, 69 1 64 1, 33 2 72 1, 69 6 74 2, 11 0 75 2, 43 4 76 8, 63 8 76 8, 37 3 6 71 ,7 08 24 5, 58 7 41 6, 88 2 56 0, 95 8 65 4, 65 2 72 6, 81 3 76 8, 35 8 79 3, 60 3 81 1, 10 0 7 92 ,3 50 28 5, 50 7 46 6, 21 4 62 0, 03 0 74 1, 22 6 82 7, 97 9 87 3, 52 6 89 6, 72 8 8 95 ,7 31 31 3, 14 4 55 3, 70 2 75 5, 97 8 85 7, 85 9 96 2, 82 5 1, 02 2, 24 1 9 97 ,5 18 34 3, 21 8 57 5, 44 1 76 9, 01 7 93 4, 10 3 1, 01 9, 30 3 10 17 3, 68 6 45 9, 41 6 72 2, 33 6 95 5, 33 5 1, 14 1, 75 0 11 13 9, 82 1 43 6, 95 8 80 9, 92 6 1, 17 4, 19 6 12 15 4, 96 5 52 8, 08 0 1, 03 2, 68 4 13 19 6, 12 4 77 2, 97 1 14 20 4, 32 5 So urc e: Br au n (2 00 4)
brð1;2Þ13 ¼ def : min ÿ brð1;2Þ12 bs ð1Þ 12 bs ð2Þ 12 2 brð1;2Þ 11 bs ð1Þ 11bs ð2Þ 11 ; brð1;2Þ 12 bs ð1Þ 12bs ð2Þ 12 ; brð1;2Þ 11 bs ð1Þ 11 bs ð2Þ 11 ( ) bsð1Þ13bs ð2Þ 13 : ð6:2Þ
For our purposes, i.e. to demonstrate the methods, these estimators are sufficient. However, in practice it is a serious issue choosing appropriate tail parameters, which needs a deep discussion which we do not want to enter here. Using our parameter estimators, we obtain the numerical values shown in Table 4.
Using the estimators bfjðnÞ, bs ðnÞ j andbr
ð1;2Þ
j (n ¼ 1; 2), we find the predictors
for the ultimate claims bCðnÞi;J (see Lemma 3.3) and the corresponding claims
reserves defined by:
bR ¼X 14 i¼2 bRi ¼ X14 i¼2 ÿ bCð1Þi;14ÿC ð1Þ i;14ÿiþ1 þÿbCð2Þi;14ÿC ð2Þ i;14ÿiþ1 ð6:3Þ
as well as the estimators for the conditional MSEP given in Result 3.5 (see Table 5).
Table 5 shows the estimated conditional process standard deviation (conditional estimation error)1=2, the estimator for the conditional MSEP and the conditional standard error of prediction for the aggregated ultimate loss over all the different accident years.
We see that the results from Braun’s (2004) formula and our formula presented in Result 3.5 are nearly the same. However, the fact that Braun’s formula uses a linear approximation for the estimation error (see (5.3)), and that our method considers higher order terms according to (5.4), leads to slightly lower results in Braun’s method. This is also confirmed by the findings in Buchwalder et al. (2006).
In the two last columns of Table 5, the results are given for: (1) the independent aggregation of the estimates for the two subportfolios; and (2) their aggregation, assuming perfect correlation between the two subportfolios. We see that these (rough) calculations lead to a prediction standard error which is about 52,000 lower and 81,000 higher, respectively, than the one taking the estimated correlation between the two subportfolios into account. Furthermore, note that using (4.10) leads to an estimate of 397,065 for the upper bound on the process standard deviation (c.f. Remarks 4.1, second remark). This result is only slightly higher than the estimate of 397,054 in Table 5.
T ab le 4 . Pa ra me te r est im a tes fo r the tw o run -of f tr ia ng le s Dev el op men t ye ar s 1 2 3 4 5 6 7 8 9 10 11 12 13 b f ð1 Þ j 3. 23 5 1. 72 0 1. 35 4 1. 17 9 1. 10 6 1. 05 5 1. 02 6 1. 01 4 1. 01 2 1. 00 6 1. 00 5 1.0 05 1. 00 3 b f ð2 Þ j 2. 22 6 1. 26 9 1. 12 0 1. 06 7 1. 03 5 1. 01 7 1. 01 0 1. 00 0 1. 00 4 0. 99 9 1. 00 4 0.9 99 1. 00 0 ÿ bs ð1 Þ j 2 17 ,6 42 .5 3 7, 02 7. 84 1, 43 2. 51 68 5. 21 14 4. 32 20 9. 99 50 .8 1 52 .0 3 13 6. 96 43 .4 5 2. 66 54 .0 3 2. 66 ÿ bs ð2 Þ j 2 11 ,1 04 .3 8 60 7. 07 32 1. 80 36 3. 48 15 6. 37 30 .8 1 20 .4 1 4. 52 26 .4 5 1. 95 10 .3 1 1. 86 0. 34 br ð1 ;2 Þ j 0. 24 5 0. 49 5 0. 68 2 0. 44 6 0. 48 7 0. 45 1 ÿ 0. 17 2 0. 80 2 0. 33 7 0. 68 7 ÿ 0. 00 4 1.0 01 0. 02 1 Ta bl e 5 . R esu lt s fo r th e wh o le p or tf ol io con si sti ng of the M T PL a nd G L sub p or tf ol ios for a g g re g a te d ac ci de nt y ea rs Br au n (2 00 4) Re su lt 3. 5 Re su lt 3. 5 ö Br au n (2 00 4) Co rr el at io n ¼ 0 Co rr el at io n ¼ 1 Es ti ma ted re se rv es b R 8, 21 8, 87 4 8, 21 8, 87 4 0 8, 21 8, 87 4 8, 21 8, 87 4 Pr oc es s st an dar d de vi ati on 39 7, 05 4 39 7, 05 4 0 35 6, 87 2 46 5, 16 1 ffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffi Es ti ma ti on er ro r p 31 8, 80 7 31 8, 84 1 34 28 5, 94 6 36 2, 47 7 Con dit ion al dMSEP 25 9, 28 9, 67 1, 26 4 25 9, 31 1, 48 6, 65 3 21 ,81 5, 38 9 20 9, 12 2, 99 9, 13 2 34 8, 31 8, 93 8, 70 9 Pr ed ic ti on sta nd ar d er ro r 50 9, 20 5 50 9, 22 6 21 45 7, 30 0 59 0, 18 6
References
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APPENDIX
In this appendix we prove the unbiasedness of erðn;mÞjÿ1 defined in formula
(3.18).
Proof of the unbiasedness oferðn;mÞjÿ1 .
We defineBðnÞj ¼ CðnÞ1;j;. . . ; C ðnÞ Iÿjþ1;j
. Note that, for i I ÿ j þ 1:
fjÿ1ðnÞ ¼E CðnÞi;j CðnÞi;jÿ1 B ðnÞ jÿ1 " # ¼Ebfjÿ1ðnÞ BðnÞ jÿ1 : ðA:1Þ
This implies that:
Eerðn;mÞjÿ1 BðnÞ jÿ1;B ðmÞ jÿ1 ¼cj X Iÿjþ1 i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðnÞi;jÿ1C ðmÞ i;jÿ1 q sðnÞjÿ1s ðmÞ jÿ1 Cov C ðnÞ i;j CðnÞi;jÿ1 ÿbfjÿ1ðnÞ; CðmÞi;j CðmÞi;jÿ1 ÿbfjÿ1ðmÞ BðnÞjÿ1;B ðmÞ jÿ1 ! : ðA:2Þ
For the covariance we have:
Cov C ðnÞ i;j CðnÞi;jÿ1 ÿbfjÿ1ðnÞ; CðmÞi;j CðmÞi;jÿ1 ÿbfjÿ1ðmÞ B ðnÞ jÿ1;B ðmÞ jÿ1 ! ¼Cov C ðnÞ i;j CðnÞi;jÿ1 ; C ðmÞ i;j CðmÞi;jÿ1 BðnÞjÿ1;B ðmÞ jÿ1 ! ÿCov bfjÿ1ðnÞ; CðmÞi;j CðmÞi;jÿ1 BðnÞjÿ1;B ðmÞ jÿ1 ! ÿCov C ðnÞ i;j CðnÞi;jÿ1 ;bfjÿ1ðmÞ B ðnÞ jÿ1;B ðmÞ jÿ1 ! þCovÿbfjÿ1ðnÞ;bf ðmÞ jÿ1 B ðnÞ jÿ1;B ðmÞ jÿ1 ¼rðn;mÞjÿ1 s ðnÞ jÿ1s ðmÞ jÿ1 ÿ CðnÞi;jÿ1C ðmÞ i;jÿ1 ÿ1=2 1 ÿC ðnÞ i;jÿ1 SðnÞjÿ1 ÿC ðmÞ i;jÿ1 SðmÞjÿ1 ! þrðn;mÞjÿ1 s ðnÞ jÿ1s ðmÞ jÿ1 PIÿjþ1 i¼1 ÿ CðnÞi;jÿ1ðmÞi;jÿ1 1=2 SðnÞjÿ1S ðmÞ jÿ1 : ðA:3Þ Henceforth:
Eerðn;mÞjÿ1 B ðnÞ jÿ1;B ðmÞ jÿ1 ¼cj X Iÿjþ1 i¼1 rðn;mÞjÿ1 1 ÿ CðnÞi;jÿ1 SðnÞjÿ1 ÿC ðmÞ i;jÿ1 SðmÞjÿ1 ! þcjr ðn;mÞ jÿ1 ÿ PIÿjþ1 i¼1 ÿ CðnÞi;jÿ1C ðmÞ i;jÿ1 1=22 SðnÞjÿ1S ðmÞ jÿ1 ¼cjr ðn;mÞ jÿ1 I ÿ j þ1 ÿ 2 þ w 2 j ÿ ¼rðn;mÞjÿ1 : ðA:4Þ
This completes the proof.
