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Asymptotic consistency and inconsistency of the chain ladder
Insurance: Mathematics and Economics 51 (2012) 472–479
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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
Asymptotic consistency and inconsistency of the chain ladder Michal Pešta ∗ , Šárka Hudecová Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics, Sokolovská 83, CZ-186 75 Prague 8, Czech Republic
article
info
Article history: Received May 2012 Received in revised form July 2012 Accepted 10 July 2012 JEL classification: C13 C18 G22
abstract The distribution-free chain ladder reserving method belongs to the most frequently used approaches in general insurance. It is well known, see Mack (1993), that the estimators fj of the development factors are unbiased and mutually uncorrelated under some mild conditions on the mean structure and under the assumption of independence of the claims in different accident years. In this article we deal with some asymptotic properties of fj . Necessary and sufficient conditions for asymptotic consistency of the estimators of true development factors fj are provided. A rate of convergence for the consistency is derived. Possible violation of these conditions and its consequences are discussed, and some practical recommendations are given. Numerical simulations and a real data example are provided as well. © 2012 Elsevier B.V. All rights reserved.
MSC: 62F12 62P05 Subject category and insurance branch category: IM10 IM20 Keywords: Claims reserving Chain ladder Asymptotic properties Development factors
1. Introduction Claims reserving is a classical problem in general insurance. A number of various methods have been invented in this field, see England and Verrall (2002) or Wüthrich and Merz (2008) for an overview. Among them, the chain ladder method is probably the most popular and frequently used one for estimating outstanding claims reserves. Besides its simplicity, this approach leads to reasonable estimates of the outstanding loss liabilities under quite mild assumptions on the mean structure and under the assumption of independence of the observations in different accident years. In recent years, many authors investigated the relationship between various stochastic models and the chain ladder technique, see for instance Mack (1994b) or Renshaw and Verrall (1998). A number of different properties of the estimated ultimate claims amount have been studied. The distribution-free approach introduced by Mack (1993) is probably the most famous one.
∗
Corresponding author. Tel.: +420 221 913 400; fax: +420 222 323 316. E-mail addresses: [email protected] (M. Pešta), [email protected] (Š. Hudecová). 0167-6687/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2012.07.004
In this article we deal with some asymptotic properties of the estimators of development factors within this distributionfree framework. Various types of the conditional asymptotic consistency are defined. Necessary and sufficient conditions for being the development factors’ estimates conditionally consistent are proved and discussed. 1.1. Notation We introduce the classical claims reserving notation and terminology. Outstanding loss liabilities are structured in so-called claims development triangles. Let us denote Xi,j all the claim amounts in development year j with accident year i. Therefore, Xi,j stands for the incremental claims in accident year i made in accounting year i+j. The current year is n, which corresponds to the most recent accident year and development period as well. That is, our data history consists of right-angled isosceles triangles Xi,j , where i = 1, . . . , n and j = 1, . . . , n + 1 − i. Suppose that Ci,j are cumulative payments or cumulative claims
j
in origin year i after j development periods, i.e., Ci,j = k=1 Xi,k . Hence, Cij is a random variable of which we have an observation if
M. Pešta, Š. Hudecová / Insurance: Mathematics and Economics 51 (2012) 472–479 Table 1 Run-off triangle for cumulative claims Ci,j .
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for completeness to relevant articles, e.g., Mack (1994a) or Mack (1994b). 1.3. Properties of development factors’ estimators (n)
i + j < n + 1 (run-off triangle, see Table 1). The aim is to estimate the ultimate claims amount Ci,n and the outstanding claims reserve Ri = Ci,n − Ci,n+1−i for all i = 2, . . . , n. 1.2. Distribution-free approach The distribution-free chain ladder reserving technique is still one of the most frequently used approaches in non-life reserving. n,n Suppose that {Ci,j }i,j=1 are random variables on a probability space (Ω , F , P). Assume the following stochastic assumptions: (1) E[Ci,j+1 |Ci,1 , . . . , Ci,j ] = fj Ci,j , 1 ≤ i ≤ n, 1 ≤ j ≤ n − 1; (2) Var[Ci,j+1 |Ci,1 , . . . , Ci,j ] = σj2 Ci,j , 1 ≤ i ≤ n, 1 ≤ j ≤ n − 1; (3) accident years [Ci,1 , . . . , Ci,n ], 1 ≤ i ≤ n are independent vectors. These stochastic assumptions correspond to the distribution-free approach of Mack (1993). The parameters fj are referred to as development factors. If n years of the claims history are available then the estimates of the development factors based on the chain ladder method are given as n−j
Unbiasedness of the development factors estimators fj is often stressed out as an important advantageous property of the chain ladder method. However, unbiasedness of an estimator as such is from the statistical point of view of less importance compared to the consistency. A simple example illustrates why. Suppose that Y1 , . . . , Yn are iid variables sampled from a distribution with finite mean EY . One can use T1 (Y1 , . . . , Yn ) = Y1 as an estimator of the unknown mean EY . This would be, of course, very naive in practice as only the first observation from the sample is used and directly taken as the estimator. However, we take this example because of its simplicity. The estimator T1 is obviously unbiased, but surely inconsistent. On the other hand, the estimator T2 (Y1 , . . . , Yn ) = n 1 1 Y + is biased, but consistent. It is easy to see that i i = 1 n n T2 approaches EY with probability one as n tends to infinity. Alternatively speaking, for n large enough, T2 is very close to the sample average and, hence, provides a reasonable estimate for the mean. The latter simple example illustrates that the unbiasedness of the development factors’ estimators does not guarantee reasonable estimates of the ultimate claims. From an actuarial (n) point of view, the consistency of fj might be a more tempting property of the method. It ensures that a sufficiently large number of observations leads to estimates close to the true quantity. In the claims reserving problem this means that the method provides accurate estimates of the outstanding loss of liabilities. However, the consistency of an estimator is an important (n) property for other reasons as well. Let us present one of them. If fj is an unbiased estimator of fj , then this does not imply (and in the (n)
majority of cases it is not true) that [ fj ]−1 is an unbiased estimate (n)
of fj−1 . In general, [ fj ]−1 can behave quite unpredictably. On the
(n)
other hand, if fj
(n) is a consistent estimator of fj , then [ fj ]−1 is a
fn(n) ≡ 1 (assuming no tail).
consistent estimator of fj−1 . In general, a continuous transformation preserves the property of being a consistent estimator (continuous mapping theorem). This is very useful in many applications. For instance, consider the Bornhuetter–Ferguson method (BF), see Wüthrich and Merz (2008), for reserves estimates. The claims (n) development pattern βj is sometimes estimated using fj as
The upper index in (1) is used in order to emphasize that the estimate of development factor fj depends on n years of history,
βj(n) =
(n) fj =
Ci,j+1
i=1
,
n −j
1 ≤ j ≤ n − 1;
(1)
Ci,j
i =1
(n)
i.e., we prefer fj more than fj from the formal point of view. The ultimate claims amounts Cin are estimated by
Cin =
(n) Ci,n+1−i × fn+1−i
× ···
(n) × fn−1 .
(n)
Mack (1993) proved that the estimators fj are unbiased and mutually uncorrelated under assumptions (1) and (3) together with an additional assumption (4)
n−j i=1
Ci,j > 0,
1 ≤ j ≤ n − 1.
Furthermore, assumption (2) is essential for the calculation of the mean squared error and the standard error of Cin . It has to be remarked that assumption (2) straightforwardly postulates a condition that Ci,j ≥ 0 [P]-a.s. for all i, j ∈ N. If the cumulative claim Ci,n+1−i is equal to zero for some particular accident year i ∈ {1, . . . , n}, then all the consequent predictions Ci,j , where j > n + 1 − i, are zeros as well. But this situation occurs very exceptionally. Independence assumption (3) can sometimes be viewed as slightly unrealistic. In these cases, the chain ladder does not seem to be a suitable choice in reserving. Nevertheless, the assumptions of a distribution-free chain ladder were thoroughly discussed many times and we refer the reader
n−1 1 . (n) k=j f k
(n)
Hence, the consistency of fj
(n)
implies the consistency of βj . On (n)
(n)
the other hand, the unbiasedness of fj does not ‘‘transfer’’ to βj in any sense. Furthermore, an estimate of the mean squared error (MSE) of reserves depends on the estimates of development factors and the dependence is not linear (Mack, 1993, Theorem 3). The same holds for the MSE of prediction. Therefore, the unbiasedness of (n) fj does not preserve the unbiasedness for estimates of the MSE of reserves or prediction. Contrary to unbiasedness, the (n) consistency of development factors’ estimates fj also guarantees the consistency of the reserves’ or prediction’s MSE. Finally, the estimator not only has to stay on target asymptotically but its variability (usually measured by variance) also has to shrink, leading to better accuracy. Since the consistency is only a qualitative property of the estimate, it is needed to characterize the consistency of the development factors’ estimates from a quantitative point of view. Indeed, the variance of the estimates will provide us a rate of convergence of the estimates, as will be pointed out later.
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M. Pešta, Š. Hudecová / Insurance: Mathematics and Economics 51 (2012) 472–479
(n)
2. Consistency of the chain ladder estimates
and the estimate fj
The consistency of the development factors’ estimators discussed above is of indisputable importance. In spite of that, it has not been investigated or discussed in the literature according to our best knowledge.
(n)
means that fj
(n)
is calculated from (1). The consistency of fj
approaches (in some sense) the true value fj as (n)
n → ∞. Equivalently, the difference fj − fj converges to zero (in some sense) as n → ∞. According to (2), we can express this difference as n−j
n −j
2.1. Conditional convergence In order to formulate and prove the consistency of the chain ladder correctly, we introduce conditional convergence almost surely, in probability and in mean square. For detailed information see, e.g., Belyaev (1995). Suppose that ζ and χ are random ∞ variables, and {ξn }∞ n=1 , {ζn }n=1 are sequences of random variables with a finite mean on a probability space (Ω , F , P). Let us define a conditional probability given some random variable ζ Pζ [·] := EP [I(·)|ζ ],
where I(·) is an indicator function. Definition 2.1 (Convergence in Conditional Probability). To say that ξn converges to χ [Pζ ]-almost surely as n tends to infinity with [Pζ ]-a.s. probability one, i.e., ξn −−−−→ χ , [P]-a.s. means n→∞
P Pζ
lim ξn = χ
n→∞
= 1 = 1.
To say that ξn converges to χ in probability Pζn as n tends to infinity Pζ n
with probability one, i.e., ξn −−−→ χ , [P]-a.s. means n→∞
∀ε > 0 : P lim Pζn {|ξn − χ| ≥ ε} = 0 = 1. n→∞
For p ≥ 1, to say that ξn converges to χ in Lp (Pζn ) as n tends to Lp (Pζn )
infinity with probability one, i.e., ξn −−−→ χ , [P]-a.s. means n→∞
P
lim Eζn |ξn − χ |p = 0 = 1.
n→∞
The conditional convergence in probability and in Lp along some sequence of random variables {ζn }∞ n=1 can be defined, because the concept of these two types of convergence comes from a topology. Despite that, the almost sure convergence does not correspond to a convergence with respect to any topology and, hence, it is not metrizable. Thereafter, the conditional convergence almost surely cannot be defined along a sequence of random variables, but only given one random variable. 2.2. Consistency of chain ladder
(n) fj − fj =
=
n −j
Ci,j
i =1
Var[εi,j+1 |Ci,1 , . . . , Ci,j ] = σ
(n)
Dj = {Ci,k : k ≤ j, i ≤ n − j + 1} and Dj = {Ci,k : k ≤ j, i ∈ N}. For every j ∈ N, the consequent statements are equivalent:
[P]-a.s.;
(ii) P (n) D
j (n) fj −−−→ fj ,
n→∞
[P]-a.s.;
(iii)
L2 P (n)
;
for all 1 ≤ i ≤ n and 1 ≤ j ≤ n − 1. In the following, we investigate the asymptotic properties of (n) the estimated development factors fj . Hence, we assume that the number of accident years n tends to infinity. For each n ∈ N, we observe variables Ci,j , i = 1, 2, . . . , n, j = 1, . . . , n + 1 − i,
Ci,j
Theorem 2.1. Let us consider the chain ladder with assumptions (1)–(3). Suppose that ∞ i=1 Ci,j > 0 [P]-a.s. for all j ∈ N, and denote
n→∞
2 j Ci,j
(3)
The term on the right hand side of the Eq. (3) is a random variable and, therefore, several different kinds of convergence can be considered. We will deal with the almost sure convergence, convergence in probability, and L2 convergence. It is common to work with the conditional probability (and expectation) given the observed data Ci,j , i = 1, . . . , n, j = 1, . . . , n + 1 − i in the claims reserving. Hence, we follow this convention (as is reasonable from the practical point of view) and we investigate the conditional convergences, as defined in the previous section, instead of the unconditional ones. In particular, for each n ∈ N we define (n) Dj = {Ci,k : k ≤ j, i ≤ n − j + 1} the set of cumulative claims Ci,k observed so far such that the development year k is less than or equal to j. Subsequently, we investigate the limiting (n) behavior of (3) conditional on Dj . However, it is mentioned at the end of Section 2.1 that the conditional convergence almost surely cannot be defined along a sequence of random variables, but only given a single random variable. For this reason, the almost sure consistency is studied conditional on the set Dj = {Ci,k : k ≤ j, i ∈ N} of all the past and future variables Ci,k such that k ≤ j. The following theorem provides the sufficient as well as the necessary condition for the weak (in probability) and strong (n) (almost sure) consistency of the development factors’ estimate fj . This condition distinguishes whether the usage of the chain ladder is a consistent approach or not. Furthermore, we show that if this condition holds, then all the mentioned ‘‘kinds of consistency’’ (according to the kind of convergence) are equivalent. In practice, of course, we always deal only with finite data sets. However, the results of the following theorem apply to the finite data as well as ensuring that the number of accident years n is large enough. We discuss the practical consequences later on in the next subsection.
[PD ]-a.s.
If we look at assumption (1) from a ‘‘regression’’ point of view, then Ci,j s play the role of the ‘‘regressors’’, and εi,j are the ‘‘disturbances’’. Considering Ci,j as fixed, the variance of the ‘‘responses’’ Ci,j+1 as well as the variance of the ‘‘errors’’ εi,j equals σj2 Ci,j . More formally, we have
.
n−j i=1
j (n) fj −−−−→ fj ,
(2)
εi,j
i =1
(i)
1 ≤ i ≤ n, 1 ≤ j ≤ n − 1.
E[εi,j+1 |Ci,1 , . . . , Ci,j ] = 0,
i =1
Let us define variables εij as
εi,j := Ci,j+1 − fj Ci,j ,
Ci,j+1 − fj Ci,j
D j
(n) fj −−−−−→ fj , n→∞
[P]-a.s.;
(iv) n −j i=1
Ci,j −−−→ ∞, n→∞
[P]-a.s.
M. Pešta, Š. Hudecová / Insurance: Mathematics and Economics 51 (2012) 472–479
Remark 2.1. Due to the independence of the different accident years (assumption (3)), statements (ii) and (iii) of Theorem 2.1 can (n)
be equivalently replaced by fj fj , [P]-a.s., respectively.
PD
L2 (PD )
n→∞
n→∞
(n) −−−→ −−−→ fj , [P]-a.s. and fj j
j
1 bn
i=n0 ai bi → 0 as n → ∞. To complete the proof, we need to show that there exists a random variable ξ such that n i=n0
Proof. Let j ∈ N be fixed. For the whole proof, we think of all (n) random variables as conditional ones, conditioned on Dj or Dj , respectively. (iii) ⇐⇒ (iv) We calculate the conditional mean square error of the development factors’ estimate taking into account assumptions (1)–(3) and Eq. (3),
εi,j i
[PDj ]-a.s.
−−−−→ ξ , n→∞
[P]-a.s.
Ck,j
k=1
Indeed, this is true, because ∞
ε i,j i
∞ σj2 Ci,j = 2 i i =n
VarPD j
i=n0
Ck,j
0
Ck,j
k=1 n−j
EP
D
2 (n) fj − fj =
(n)
VarP
i =1
n−j
j
(n)
D j
(εi,j )
2
σj2
=
n −j
Ci,j
,
[P]-a.s.
(4)
εi ,j
[PDj ]-a.s.
i=1
−−−−→ 0,
n −j
[P]-a.s.
n→∞
Ci,j
i=1
Hence,
n −j i=1
[PDj ]-a.s.
εi,j −−−−→ 0 [P]-a.s., because the denominator conn→∞
verges to a finite variable. Furthermore, we have VarPD
∞ j
εi,j = σj2
i=1
∞
Ci,j < ∞,
[P]-a.s.,
i =1
εij is summable in L2 (PDj ) with ∞ i=1 εi,j ] < ∞ under the j n −j stated assumptions. But then, it cannot be true that i=1 εi,j conditionally converges [PDj ]-a.s. to zero with probability one as n apand, therefore, the series
∞ σj2 Cn0 ,j + ≤ i 2 n0 i=n0 +1
Ck,j
Ci,j
It is clear that the expression (4) converges to zero with probability one if and only if (iv) holds. (iii) =⇒ (ii) The convergence in mean square implies the convergence in probability, and this implication holds with probability one as well. (i) =⇒ (iv) We prove this implication by a contradiction. ∞ Suppose that i=1 Ci,j < ∞ [P]-a.s. and statement (ii) holds. In view of relation (3) statement (ii) means that n −j
k=1
∞
i=1
probability one. Moreover, 0 < VarPD [
proaches infinity. (ii) =⇒ (iv) This implication is proved similarly as the previous one. Additionally, one needs to realize two facts. Firstly, conditional converges in probability PD(n) coincides with conditional converj
gence in probability PDj due to independence assumption (3).
σj2 Ci,j i −1
Ck,j
Ck,j
k=1
k=1
k=1
i =1
i =1
475
n
∞ σj2 Cn0 ,j 1 2 = − σ i j 2 n 0 i=n0 +1
Ck,j
Ck,j
k=1
k=1
σj2 Cn0 ,j σj2 = + n0 2 n0 Ck,j
k=1
−
< ∞,
1 i−1
Ck,j
k=1
[P]-a.s.
Ck,j
k=1
The previous Theorem 2.1 postulates a complete characterization of the conditional convergence of development factors’ estimate, because it gives a necessary and sufficient condition (iv) for the convergence almost surely, in probability, and in mean square. If condition (iv) of Theorem 2.1 holds, we refer to the correspond(n) ing estimator fj as consistent, dropping the term conditionally. As the three types of convergence are equivalent, the type of consistency (strong, weak, L2 ) is left out as well. Note that the assumptions of Theorem 2.1 consist exactly of the same assumptions as the classical distribution-free chain ladder ∞ approach. The condition i=1 Ci,j > 0 is not restrictive at all, because assumption (2) implies Ci,j ≥ 0[P]-a.s. for all i, j ∈ N. It (n)
ensures that the estimators fj are well defined for all j ∈ N and all n ∈ N. The necessary and sufficient condition for the consistency are studied in more detail in Section 2.4. The practical consequences of Theorem 2.1 are discussed in Section 3. Remark 2.2. We have explained above that the conditional consistency is natural to be considered in the claims reserving. However, for the sake of completeness we briefly discuss the unconditional consistency as well. Consider, for instance, the L2 (n) convergence. It follows from the proof of Theorem 2.1 that fj converges to fj in L2 (unconditionally) as n → ∞ if and only if
1 −j n
→ 0,
Secondly, series i=1 εi,j is conditionally summable in probability PDj with probability one if and only if it is conditionally summable [PDj ]-almost surely with probability one. (iv) =⇒ (i) We apply Kronecker’s lemma (Shiryaev, 1996, Lemma IV.3.2). Denote
E
εi,j ai := , i
However, this condition is obviously more complicated than condition (iv) in Theorem 2.1, and it is practically unverifiable. Thus, the conditional convergence is not only a more natural one in this case, but an even more convenient one.
∞
Ck,j
bi :=
i
Ck,j .
k=1
Ci,j
n → ∞.
i =1
k=1
Let n0 be the first integer n such that i=1 Ci,j > 0 [P]-a.s. If ∞ C = ∞ [ P ] -a . s . , then for all i ≥ n holds i , j 0 i =1 0 < bi ≤ bi+1 ↗ ∞ as i → ∞. Kronecker’s lemma states that if ∞ i=n0 ai < ∞, then
n
2.3. Rate of convergence Consistency of an estimator is a very important but only qualitative property. If we want to measure consistency, it is
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M. Pešta, Š. Hudecová / Insurance: Mathematics and Economics 51 (2012) 472–479
necessary to evaluate its rate. This quantitative attribute can be obtained directly from the proof of Theorem 2.1 via the mean square error of the development factors’ estimate. Proposition 2.2. Consider the assumptions of Theorem 2.1 and let j ∈ N be fixed. Denote the conditional mean square error of the estimate of development factor fj as
2 (n) (n) (n) (n) MSE f := E f − E f . D j
j
j
i =1
Proof. A direct consequence of Eq. (4) and unbiasedness of j
(n) f .
Crucial interpretation of previous Proposition 2.2 is as follows: ∞ the slower (faster) divergence of i=1 Ci,j implies the slower (faster) realization of consistency of the development factors’ estimates. 2.4. On the necessary and sufficient condition Let us investigate the necessary and sufficient condition (iv) from Theorem 2.1 Ci,j −−−→ ∞, n→∞
[P]-a.s.
(5)
in more detail. First, recall that variables Ci,j are called summable [P]-a.s. if ni=1 Ci,j converges [P]-a.s. to a finite variable as n → ∞. Assumption (2) of the distribution-free approach implies that Ci,j ≥ 0 [P]-a.s. for all i, j ∈ N and, therefore, it is not restrictive to assume that Ci,j ≥ 0 for all i, j ∈ N. In this case, Ci,j are either summable [P]-a.s. or condition (5) holds. Proposition 2.3. Consider the chain ladder assumptions (1)–(3). with ∞ Suppose that Ci,j ≥ 0 for all i, j ∈ N, and i=1 Ci,j > 0 [P]-a.s. for all j ∈ N. Let j ∈ N be fixed. Then the following conditions are equivalent. (i) The condition (5) holds. (ii) ∞
ECi,1 = ∞.
(6)
i=1
(iii) The condition (5) holds for j0 ∈ N, j ̸= j0 . Proof. Since Ci,j , i = 1, 2, . . . , are independent variables, it follows that the summability [P]-a.s. of Ci,j is equivalent to the condition ∞
ECi,j < ∞.
(7)
i =1
Assumption (1) further implies that ECi,j = E[E(Ci,j |Ci,1 , . . . , Ci,j−1 )]
= fj−1 E(Ci,j−1 ) = · · · = fj−1 × fj−2 × · · · × f1 × ECi,1 . Hence, (7) can be rewritten as ∞ i =1
ECi,j =
j −1 ∞ fk
i =1
k=1
ECi,1 =
∞
(n)
n → ∞.
j
i=1
Corollary 2.4. Consider the assumptions of Proposition 2.3 and let j ∈ N be fixed. (n)
Then, with probability one holds
n −j
∞
(i) The estimator fj is consistent for fj if and only if i=1 ECi,1 = ∞. n (n) (ii) The estimator fj is consistent for fj if and only if i=1 Ci,1 → ∞ [P]-a.s. as n → ∞.
j
−1 n−j ( n ) , MSE f = O Ci,j
This shows that condition (7) is equivalent to i=1 ECi,1 < ∞. Using the same procedure in the opposite direction, we get the equivalence of the summability [P]-a.s. of Ci,j0 , i = 1, 2, . . . , for an arbitrary j0 ∈ N, j ̸= j0 . Then, the statement of Proposition 2.3 follows from the fact that Ci,j ≥ 0.
j −1 k=1
fk
∞ i=1
ECi,1 .
It follows from Corollary 2.4 that either fj is consistent for fj for all j ∈ N, or none of them is consistent. Furthermore, the consistency of fj depends only on the asymptotic behavior of the cumulative sums of ECi,1 (or Ci,1 ) and there is no effect of the true values of fj , j ∈ N on the property of being a consistent estimator or not. Condition (i) of Corollary 2.4 is better understandable compared to (5), because it is formulated in terms of divergence of a series of real numbers instead of divergence of random variables. Condition (ii) of Corollary 2.4 is useful for practice. Here, we observe just one data set and, therefore, condition (6) cannot be verified. On the other hand, we can deduce conclusions about the n behavior of i=1 Ci,1 from the observations so far. We discuss this more deeply in the next section. 3. Practical aspects In practice, one might be interested whether his/her estimator
(n) fj is consistent for fj or not. Therefore, it is important to verify
one of the necessary and sufficient conditions for the estimator’s consistency given above. Since these conditions are formulated in terms of asymptotic properties of some series of random variables or numbers, they cannot be exactly verified in practice. However, we give an informal procedure which might give us a clue whether the consistency holds or not. (n) Recall that Corollary 2.4 claims that the consistency of fj is equivalent to the condition
n
i =1
Ci,1 → ∞ [P]-a.s. as n → ∞.
Denote Sk = i=1 Ci,1 , k = 1, . . . , n the cumulative sums of the cumulative claims Ci,1 in the first development year, i.e., the values in the first column of our cumulative triangle data. This can also be preceded with another development year’s cumulative claims, but we face the issue of a smaller number of observations in the subsequent development years. Now, the question is whether the observed values {Sk }nk=1 so far indicate that the sequence diverges to infinity, or it converges to some finite number as n → ∞. Hence, one may plot the values of Sk and study how they behave when k grows. Alternatively, one can (informally) apply some of the well known criteria for the convergence of real series on the values of Ci,1 . For instance, the ratios Ck+1,1 /Ck,1 or the sequence k Ck,1 can be studied. Let us give an example of this procedure. We consider the well known data of Taylor and Ashe (1983). The sequence Sk of the cumulative sums in the first column of the data is plotted in Fig. 1. One can observe a growing trend without any evidence for a horizontal asymptote. Hence, this gives us confidence that the chain ladder estimates are consistent for this data. Note that the sequence of ratios Ck+1,1 /Ck,1 is inconclusive here and, moreover, the sequence k Ck,1 seems to converge to a number larger than one (corresponding to the divergence of the series), but more observations would be necessary for convincing evidence. In the described procedure, we have looked at the cumulative sums in the first column of our data. One might suggest that according to Proposition 2.3, we could equivalently investigate
k
M. Pešta, Š. Hudecová / Insurance: Mathematics and Economics 51 (2012) 472–479
477
Example 4.1 (Decreasing business). In the first example, we consider a fast decreasing business, i.e., the situation where condition (6) holds. For this case, the random variable Ci,1 was generated from the exponential distribution with the mean i−2 × 106 . The be(n) havior of fj is graphically shown in Fig. 2 for j = 1, 2, 3, 4. Even (n)
though the sample size n increases, the estimates fj do not converge to the true value fj . Their values are close to fj in this setting (exponential distribution with the mean i−2 × 106 ), but the estimates are indeed not consistent. Note that the same simulations were run also for Ci,1 with the uniform distribution. In this setting, (n)
the differences between values of estimates fj fj are more noticeable.
Fig. 1. The sequence of cumulative sums Sk for the Taylor/Ashe data. i=1 Ci,j for some j > 1 as well. However, the largest number of observations is always in the first development year and, therefore, the largest amount of information is obtained from the first column. What kinds of business behavior correspond to the violation of consistency? For instance, condition (5) can be violated if one observes a decreasing trend (decreasing fast enough) in payments across the accident years. So to speak, the corresponding line of business is worsening maybe due to new insurance companies entering the market or decreasing prices of such insurance products. Furthermore, splitting one existing line of business into several others can also cause inconsistency in the estimation of development factors.
k
4. Simulations and examples
and the true values
Example 4.2 (Slowly decreasing business). In this example, we still consider a decreasing business, i.e., a situation where ECi,1 decreases with increasing i, but in a slow manner such that condition (6) does not hold. Hence, we choose ECi,1 = i−1/2 × (n)
106 in the exponential distribution. The behavior of fj for this situation is graphically shown in Fig. 3 for j = 1, 2, 3, 4. Unlike the estimates in Example 4.1, a clear convergence pattern can be (n) observed, confirming that the estimates fj are consistent in this case. Example 4.3 (Growing business). Finally, we consider a situation with a growing business, i.e., the case where ECi,1 increases with increasing i. The parameters of the exponential distribution were √ (n) set such that ECi,1 = i × 106 . The estimates fj for this setting are plotted in Fig. 4 for j = 1, 2, 3, 4. The figure obviously confirms (n) that the estimates are consistent. Moreover, fj converges to the true values fj much faster than in Fig. 3 of Example 4.2. (n)
In this section, some examples, which illustrate the results of Theorem 2.1, are presented. In particular, several data sets according to the distribution-free approach are simulated, i.e., satisfying (n) assumptions (1)–(3), and the consistency of fj was studied.
The data DN = {Ci,j : 1 ≤ i, j ≤ N } for N ∈ N accident years were simulated using the following procedure. First, the variable Ci,1 was generated such that Ci,1 ∈ L2 and Ci,j ≥ 0. Then, for a given sequence {fj }Nj=−11 the variables Ci,2 , . . . , Ci,N were generated successively such that Ci,j satisfies (1) and (2). Hence, for each j the variable Ci,j is drawn from a distribution with mean fj−1 Ci,j−1 and variance σj2 Ci,j−1 for some σj2 ∈ (0, ∞). Since the accident years are assumed to be independent (assumption (3)), the rows of the data sets were generated separately, using the same approach. The variables Ci,j for j > n − i are, of course, not used for the (n)
computation of fj , but they can be used for assessment of the estimation of the claims reserves. (n) Having the data DN , the estimated development factors fj were computed using only a subset of the first n accident years. If (n) the estimator is consistent, then the values of fj should approach fj as n grows. For inconsistent estimators, no convergence pattern could be observed. In the following examples, Ci,1 were drawn from the exponential distribution, and the Ci,j was generated from the Poisson distribution with the parameter fj−1 Ci,j−1 for j = 2, . . . , N. However, different distributions can be considered as well. For instance, a suitable uniform distribution can be considered for Ci,1 as well. With this choice one can also independently model the variance of Ci1 . The development factor fj was set as fj = j/(j + 1) in all the presented examples. Recall that the choice of {fj } does not impact whether
(n) f are consistent or not. j
For the whole analysis, software R by R Development Core Team (2011) was used and the seed of the random number generator was set to 1234.
Comparison. One can observe that a convergence of fj to fj is visible in Figs. 3 and 4. In contrast to this, no convergence pattern is obvious in Fig. 2. The comparison of Figs. 3 and 4 also illustrates that ∞ the rate of convergence depends on the rate of convergence of i=1 Ci,1 , as discussed theoretically in Section 2.3. 5. Conclusions The property of being consistent has indisputable importance for an estimator. From some point of view, consistency is more important than unbiasedness, as is motivated in this paper. Consistency and inconsistency of the development factors’ estimate in the distribution-free chain ladder method is investigated in the whole paper. Firstly, various kinds of conditional asymptotic consistency are defined. Secondly, a necessary and sufficient condition for the conditional consistency of the development factors’ estimates is derived. It is also proved that the weak, strong consistency, and consistency in the mean square are equivalent under the distributionfree chain ladder framework. Consistency of the estimators does go beyond the unbiasedness and emphasizes that their variability deserves equal attention. That is why the consistency of the development factors’ estimates is also investigated from the quantitative point of view, i.e., how fast the estimates converge to the unknown development factors and such a convergence rate is provided. The necessary and sufficient condition can unambiguously distinguish whether the chain ladder approach provides consistent estimates for the development factors or not. Therefore, such a very important and crucial condition is elaborated. Practical recommendations, on how to check this condition, are discussed. Thirdly, a real data example and numerical simulations are shown in order to illustrate the performance of the development factors’ estimates. Possible violation of the necessary and sufficient
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(n)
Fig. 2. Behavior of fj
(n)
Fig. 3. Behavior of fj
as n increases in Example 4.1. The dashed lines correspond to the true values of fj .
as n increases in Example 4.2. The dashed lines correspond to the true values of fj .
M. Pešta, Š. Hudecová / Insurance: Mathematics and Economics 51 (2012) 472–479
( n)
Fig. 4. Behavior of fj
479
as n increases in Example 4.3. The dashed lines correspond to the true values of fj .
condition ensuring the consistency with the consequences is demonstrated. Acknowledgments This paper was written with the support of the Czech Science Foundation project ‘‘DYME Dynamic Models in Economics’’ No. P402/12/G097. The authors thank the anonymous reviewer for suggestions that improved this paper. References Belyaev, Y.K., 1995. Bootstrap, resampling and Mallows metric. In: Lecture Notes, vol. 1. Institute of Mathematical Statistics Umeå University.
England, P.D., Verrall, R.J., 2002. Stochastic claims reserving in general insurance (with discussion). British Actuarial Journal 8 (3), 443–518. Mack, T., 1993. Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin 23 (2), 213–225. Mack, T., 1994a. Measuring the variability of chain ladder reserve estimates. In: Proceedings of the Casualty Actuarial Society Forum, Spring 1994. pp. 101–182. Mack, T., 1994b. Which stochastic model is underlying the chain ladder model? Insurance: Mathematics and Economics 15, 133–138. R Development Core Team, 2011. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, ISBN 3900051-07-0. http://www.R-project.org. Renshaw, A., Verrall, R., 1998. A stochastic model underlying the chain-ladder technique. British Actuarial Journal 4, 903–923. Shiryaev, A.N., 1996. Probability, second ed.. Springer, New York. Taylor, G.C., Ashe, F.R., 1983. Second moments of estimates of outstanding claims. Journal of Econometrics 23, 37–61. Wüthrich, M., Merz, M., 2008. Stochastic Claims Reserving Methods in Insurance. In: Wiley Finance Series, John Wiley & Sons.
Contents lists available at SciVerse ScienceDirect
Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
Asymptotic consistency and inconsistency of the chain ladder Michal Pešta ∗ , Šárka Hudecová Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics, Sokolovská 83, CZ-186 75 Prague 8, Czech Republic
article
info
Article history: Received May 2012 Received in revised form July 2012 Accepted 10 July 2012 JEL classification: C13 C18 G22
abstract The distribution-free chain ladder reserving method belongs to the most frequently used approaches in general insurance. It is well known, see Mack (1993), that the estimators fj of the development factors are unbiased and mutually uncorrelated under some mild conditions on the mean structure and under the assumption of independence of the claims in different accident years. In this article we deal with some asymptotic properties of fj . Necessary and sufficient conditions for asymptotic consistency of the estimators of true development factors fj are provided. A rate of convergence for the consistency is derived. Possible violation of these conditions and its consequences are discussed, and some practical recommendations are given. Numerical simulations and a real data example are provided as well. © 2012 Elsevier B.V. All rights reserved.
MSC: 62F12 62P05 Subject category and insurance branch category: IM10 IM20 Keywords: Claims reserving Chain ladder Asymptotic properties Development factors
1. Introduction Claims reserving is a classical problem in general insurance. A number of various methods have been invented in this field, see England and Verrall (2002) or Wüthrich and Merz (2008) for an overview. Among them, the chain ladder method is probably the most popular and frequently used one for estimating outstanding claims reserves. Besides its simplicity, this approach leads to reasonable estimates of the outstanding loss liabilities under quite mild assumptions on the mean structure and under the assumption of independence of the observations in different accident years. In recent years, many authors investigated the relationship between various stochastic models and the chain ladder technique, see for instance Mack (1994b) or Renshaw and Verrall (1998). A number of different properties of the estimated ultimate claims amount have been studied. The distribution-free approach introduced by Mack (1993) is probably the most famous one.
∗
Corresponding author. Tel.: +420 221 913 400; fax: +420 222 323 316. E-mail addresses: [email protected] (M. Pešta), [email protected] (Š. Hudecová). 0167-6687/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2012.07.004
In this article we deal with some asymptotic properties of the estimators of development factors within this distributionfree framework. Various types of the conditional asymptotic consistency are defined. Necessary and sufficient conditions for being the development factors’ estimates conditionally consistent are proved and discussed. 1.1. Notation We introduce the classical claims reserving notation and terminology. Outstanding loss liabilities are structured in so-called claims development triangles. Let us denote Xi,j all the claim amounts in development year j with accident year i. Therefore, Xi,j stands for the incremental claims in accident year i made in accounting year i+j. The current year is n, which corresponds to the most recent accident year and development period as well. That is, our data history consists of right-angled isosceles triangles Xi,j , where i = 1, . . . , n and j = 1, . . . , n + 1 − i. Suppose that Ci,j are cumulative payments or cumulative claims
j
in origin year i after j development periods, i.e., Ci,j = k=1 Xi,k . Hence, Cij is a random variable of which we have an observation if
M. Pešta, Š. Hudecová / Insurance: Mathematics and Economics 51 (2012) 472–479 Table 1 Run-off triangle for cumulative claims Ci,j .
473
for completeness to relevant articles, e.g., Mack (1994a) or Mack (1994b). 1.3. Properties of development factors’ estimators (n)
i + j < n + 1 (run-off triangle, see Table 1). The aim is to estimate the ultimate claims amount Ci,n and the outstanding claims reserve Ri = Ci,n − Ci,n+1−i for all i = 2, . . . , n. 1.2. Distribution-free approach The distribution-free chain ladder reserving technique is still one of the most frequently used approaches in non-life reserving. n,n Suppose that {Ci,j }i,j=1 are random variables on a probability space (Ω , F , P). Assume the following stochastic assumptions: (1) E[Ci,j+1 |Ci,1 , . . . , Ci,j ] = fj Ci,j , 1 ≤ i ≤ n, 1 ≤ j ≤ n − 1; (2) Var[Ci,j+1 |Ci,1 , . . . , Ci,j ] = σj2 Ci,j , 1 ≤ i ≤ n, 1 ≤ j ≤ n − 1; (3) accident years [Ci,1 , . . . , Ci,n ], 1 ≤ i ≤ n are independent vectors. These stochastic assumptions correspond to the distribution-free approach of Mack (1993). The parameters fj are referred to as development factors. If n years of the claims history are available then the estimates of the development factors based on the chain ladder method are given as n−j
Unbiasedness of the development factors estimators fj is often stressed out as an important advantageous property of the chain ladder method. However, unbiasedness of an estimator as such is from the statistical point of view of less importance compared to the consistency. A simple example illustrates why. Suppose that Y1 , . . . , Yn are iid variables sampled from a distribution with finite mean EY . One can use T1 (Y1 , . . . , Yn ) = Y1 as an estimator of the unknown mean EY . This would be, of course, very naive in practice as only the first observation from the sample is used and directly taken as the estimator. However, we take this example because of its simplicity. The estimator T1 is obviously unbiased, but surely inconsistent. On the other hand, the estimator T2 (Y1 , . . . , Yn ) = n 1 1 Y + is biased, but consistent. It is easy to see that i i = 1 n n T2 approaches EY with probability one as n tends to infinity. Alternatively speaking, for n large enough, T2 is very close to the sample average and, hence, provides a reasonable estimate for the mean. The latter simple example illustrates that the unbiasedness of the development factors’ estimators does not guarantee reasonable estimates of the ultimate claims. From an actuarial (n) point of view, the consistency of fj might be a more tempting property of the method. It ensures that a sufficiently large number of observations leads to estimates close to the true quantity. In the claims reserving problem this means that the method provides accurate estimates of the outstanding loss of liabilities. However, the consistency of an estimator is an important (n) property for other reasons as well. Let us present one of them. If fj is an unbiased estimator of fj , then this does not imply (and in the (n)
majority of cases it is not true) that [ fj ]−1 is an unbiased estimate (n)
of fj−1 . In general, [ fj ]−1 can behave quite unpredictably. On the
(n)
other hand, if fj
(n) is a consistent estimator of fj , then [ fj ]−1 is a
fn(n) ≡ 1 (assuming no tail).
consistent estimator of fj−1 . In general, a continuous transformation preserves the property of being a consistent estimator (continuous mapping theorem). This is very useful in many applications. For instance, consider the Bornhuetter–Ferguson method (BF), see Wüthrich and Merz (2008), for reserves estimates. The claims (n) development pattern βj is sometimes estimated using fj as
The upper index in (1) is used in order to emphasize that the estimate of development factor fj depends on n years of history,
βj(n) =
(n) fj =
Ci,j+1
i=1
,
n −j
1 ≤ j ≤ n − 1;
(1)
Ci,j
i =1
(n)
i.e., we prefer fj more than fj from the formal point of view. The ultimate claims amounts Cin are estimated by
Cin =
(n) Ci,n+1−i × fn+1−i
× ···
(n) × fn−1 .
(n)
Mack (1993) proved that the estimators fj are unbiased and mutually uncorrelated under assumptions (1) and (3) together with an additional assumption (4)
n−j i=1
Ci,j > 0,
1 ≤ j ≤ n − 1.
Furthermore, assumption (2) is essential for the calculation of the mean squared error and the standard error of Cin . It has to be remarked that assumption (2) straightforwardly postulates a condition that Ci,j ≥ 0 [P]-a.s. for all i, j ∈ N. If the cumulative claim Ci,n+1−i is equal to zero for some particular accident year i ∈ {1, . . . , n}, then all the consequent predictions Ci,j , where j > n + 1 − i, are zeros as well. But this situation occurs very exceptionally. Independence assumption (3) can sometimes be viewed as slightly unrealistic. In these cases, the chain ladder does not seem to be a suitable choice in reserving. Nevertheless, the assumptions of a distribution-free chain ladder were thoroughly discussed many times and we refer the reader
n−1 1 . (n) k=j f k
(n)
Hence, the consistency of fj
(n)
implies the consistency of βj . On (n)
(n)
the other hand, the unbiasedness of fj does not ‘‘transfer’’ to βj in any sense. Furthermore, an estimate of the mean squared error (MSE) of reserves depends on the estimates of development factors and the dependence is not linear (Mack, 1993, Theorem 3). The same holds for the MSE of prediction. Therefore, the unbiasedness of (n) fj does not preserve the unbiasedness for estimates of the MSE of reserves or prediction. Contrary to unbiasedness, the (n) consistency of development factors’ estimates fj also guarantees the consistency of the reserves’ or prediction’s MSE. Finally, the estimator not only has to stay on target asymptotically but its variability (usually measured by variance) also has to shrink, leading to better accuracy. Since the consistency is only a qualitative property of the estimate, it is needed to characterize the consistency of the development factors’ estimates from a quantitative point of view. Indeed, the variance of the estimates will provide us a rate of convergence of the estimates, as will be pointed out later.
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(n)
2. Consistency of the chain ladder estimates
and the estimate fj
The consistency of the development factors’ estimators discussed above is of indisputable importance. In spite of that, it has not been investigated or discussed in the literature according to our best knowledge.
(n)
means that fj
(n)
is calculated from (1). The consistency of fj
approaches (in some sense) the true value fj as (n)
n → ∞. Equivalently, the difference fj − fj converges to zero (in some sense) as n → ∞. According to (2), we can express this difference as n−j
n −j
2.1. Conditional convergence In order to formulate and prove the consistency of the chain ladder correctly, we introduce conditional convergence almost surely, in probability and in mean square. For detailed information see, e.g., Belyaev (1995). Suppose that ζ and χ are random ∞ variables, and {ξn }∞ n=1 , {ζn }n=1 are sequences of random variables with a finite mean on a probability space (Ω , F , P). Let us define a conditional probability given some random variable ζ Pζ [·] := EP [I(·)|ζ ],
where I(·) is an indicator function. Definition 2.1 (Convergence in Conditional Probability). To say that ξn converges to χ [Pζ ]-almost surely as n tends to infinity with [Pζ ]-a.s. probability one, i.e., ξn −−−−→ χ , [P]-a.s. means n→∞
P Pζ
lim ξn = χ
n→∞
= 1 = 1.
To say that ξn converges to χ in probability Pζn as n tends to infinity Pζ n
with probability one, i.e., ξn −−−→ χ , [P]-a.s. means n→∞
∀ε > 0 : P lim Pζn {|ξn − χ| ≥ ε} = 0 = 1. n→∞
For p ≥ 1, to say that ξn converges to χ in Lp (Pζn ) as n tends to Lp (Pζn )
infinity with probability one, i.e., ξn −−−→ χ , [P]-a.s. means n→∞
P
lim Eζn |ξn − χ |p = 0 = 1.
n→∞
The conditional convergence in probability and in Lp along some sequence of random variables {ζn }∞ n=1 can be defined, because the concept of these two types of convergence comes from a topology. Despite that, the almost sure convergence does not correspond to a convergence with respect to any topology and, hence, it is not metrizable. Thereafter, the conditional convergence almost surely cannot be defined along a sequence of random variables, but only given one random variable. 2.2. Consistency of chain ladder
(n) fj − fj =
=
n −j
Ci,j
i =1
Var[εi,j+1 |Ci,1 , . . . , Ci,j ] = σ
(n)
Dj = {Ci,k : k ≤ j, i ≤ n − j + 1} and Dj = {Ci,k : k ≤ j, i ∈ N}. For every j ∈ N, the consequent statements are equivalent:
[P]-a.s.;
(ii) P (n) D
j (n) fj −−−→ fj ,
n→∞
[P]-a.s.;
(iii)
L2 P (n)
;
for all 1 ≤ i ≤ n and 1 ≤ j ≤ n − 1. In the following, we investigate the asymptotic properties of (n) the estimated development factors fj . Hence, we assume that the number of accident years n tends to infinity. For each n ∈ N, we observe variables Ci,j , i = 1, 2, . . . , n, j = 1, . . . , n + 1 − i,
Ci,j
Theorem 2.1. Let us consider the chain ladder with assumptions (1)–(3). Suppose that ∞ i=1 Ci,j > 0 [P]-a.s. for all j ∈ N, and denote
n→∞
2 j Ci,j
(3)
The term on the right hand side of the Eq. (3) is a random variable and, therefore, several different kinds of convergence can be considered. We will deal with the almost sure convergence, convergence in probability, and L2 convergence. It is common to work with the conditional probability (and expectation) given the observed data Ci,j , i = 1, . . . , n, j = 1, . . . , n + 1 − i in the claims reserving. Hence, we follow this convention (as is reasonable from the practical point of view) and we investigate the conditional convergences, as defined in the previous section, instead of the unconditional ones. In particular, for each n ∈ N we define (n) Dj = {Ci,k : k ≤ j, i ≤ n − j + 1} the set of cumulative claims Ci,k observed so far such that the development year k is less than or equal to j. Subsequently, we investigate the limiting (n) behavior of (3) conditional on Dj . However, it is mentioned at the end of Section 2.1 that the conditional convergence almost surely cannot be defined along a sequence of random variables, but only given a single random variable. For this reason, the almost sure consistency is studied conditional on the set Dj = {Ci,k : k ≤ j, i ∈ N} of all the past and future variables Ci,k such that k ≤ j. The following theorem provides the sufficient as well as the necessary condition for the weak (in probability) and strong (n) (almost sure) consistency of the development factors’ estimate fj . This condition distinguishes whether the usage of the chain ladder is a consistent approach or not. Furthermore, we show that if this condition holds, then all the mentioned ‘‘kinds of consistency’’ (according to the kind of convergence) are equivalent. In practice, of course, we always deal only with finite data sets. However, the results of the following theorem apply to the finite data as well as ensuring that the number of accident years n is large enough. We discuss the practical consequences later on in the next subsection.
[PD ]-a.s.
If we look at assumption (1) from a ‘‘regression’’ point of view, then Ci,j s play the role of the ‘‘regressors’’, and εi,j are the ‘‘disturbances’’. Considering Ci,j as fixed, the variance of the ‘‘responses’’ Ci,j+1 as well as the variance of the ‘‘errors’’ εi,j equals σj2 Ci,j . More formally, we have
.
n−j i=1
j (n) fj −−−−→ fj ,
(2)
εi,j
i =1
(i)
1 ≤ i ≤ n, 1 ≤ j ≤ n − 1.
E[εi,j+1 |Ci,1 , . . . , Ci,j ] = 0,
i =1
Let us define variables εij as
εi,j := Ci,j+1 − fj Ci,j ,
Ci,j+1 − fj Ci,j
D j
(n) fj −−−−−→ fj , n→∞
[P]-a.s.;
(iv) n −j i=1
Ci,j −−−→ ∞, n→∞
[P]-a.s.
M. Pešta, Š. Hudecová / Insurance: Mathematics and Economics 51 (2012) 472–479
Remark 2.1. Due to the independence of the different accident years (assumption (3)), statements (ii) and (iii) of Theorem 2.1 can (n)
be equivalently replaced by fj fj , [P]-a.s., respectively.
PD
L2 (PD )
n→∞
n→∞
(n) −−−→ −−−→ fj , [P]-a.s. and fj j
j
1 bn
i=n0 ai bi → 0 as n → ∞. To complete the proof, we need to show that there exists a random variable ξ such that n i=n0
Proof. Let j ∈ N be fixed. For the whole proof, we think of all (n) random variables as conditional ones, conditioned on Dj or Dj , respectively. (iii) ⇐⇒ (iv) We calculate the conditional mean square error of the development factors’ estimate taking into account assumptions (1)–(3) and Eq. (3),
εi,j i
[PDj ]-a.s.
−−−−→ ξ , n→∞
[P]-a.s.
Ck,j
k=1
Indeed, this is true, because ∞
ε i,j i
∞ σj2 Ci,j = 2 i i =n
VarPD j
i=n0
Ck,j
0
Ck,j
k=1 n−j
EP
D
2 (n) fj − fj =
(n)
VarP
i =1
n−j
j
(n)
D j
(εi,j )
2
σj2
=
n −j
Ci,j
,
[P]-a.s.
(4)
εi ,j
[PDj ]-a.s.
i=1
−−−−→ 0,
n −j
[P]-a.s.
n→∞
Ci,j
i=1
Hence,
n −j i=1
[PDj ]-a.s.
εi,j −−−−→ 0 [P]-a.s., because the denominator conn→∞
verges to a finite variable. Furthermore, we have VarPD
∞ j
εi,j = σj2
i=1
∞
Ci,j < ∞,
[P]-a.s.,
i =1
εij is summable in L2 (PDj ) with ∞ i=1 εi,j ] < ∞ under the j n −j stated assumptions. But then, it cannot be true that i=1 εi,j conditionally converges [PDj ]-a.s. to zero with probability one as n apand, therefore, the series
∞ σj2 Cn0 ,j + ≤ i 2 n0 i=n0 +1
Ck,j
Ci,j
It is clear that the expression (4) converges to zero with probability one if and only if (iv) holds. (iii) =⇒ (ii) The convergence in mean square implies the convergence in probability, and this implication holds with probability one as well. (i) =⇒ (iv) We prove this implication by a contradiction. ∞ Suppose that i=1 Ci,j < ∞ [P]-a.s. and statement (ii) holds. In view of relation (3) statement (ii) means that n −j
k=1
∞
i=1
probability one. Moreover, 0 < VarPD [
proaches infinity. (ii) =⇒ (iv) This implication is proved similarly as the previous one. Additionally, one needs to realize two facts. Firstly, conditional converges in probability PD(n) coincides with conditional converj
gence in probability PDj due to independence assumption (3).
σj2 Ci,j i −1
Ck,j
Ck,j
k=1
k=1
k=1
i =1
i =1
475
n
∞ σj2 Cn0 ,j 1 2 = − σ i j 2 n 0 i=n0 +1
Ck,j
Ck,j
k=1
k=1
σj2 Cn0 ,j σj2 = + n0 2 n0 Ck,j
k=1
−
< ∞,
1 i−1
Ck,j
k=1
[P]-a.s.
Ck,j
k=1
The previous Theorem 2.1 postulates a complete characterization of the conditional convergence of development factors’ estimate, because it gives a necessary and sufficient condition (iv) for the convergence almost surely, in probability, and in mean square. If condition (iv) of Theorem 2.1 holds, we refer to the correspond(n) ing estimator fj as consistent, dropping the term conditionally. As the three types of convergence are equivalent, the type of consistency (strong, weak, L2 ) is left out as well. Note that the assumptions of Theorem 2.1 consist exactly of the same assumptions as the classical distribution-free chain ladder ∞ approach. The condition i=1 Ci,j > 0 is not restrictive at all, because assumption (2) implies Ci,j ≥ 0[P]-a.s. for all i, j ∈ N. It (n)
ensures that the estimators fj are well defined for all j ∈ N and all n ∈ N. The necessary and sufficient condition for the consistency are studied in more detail in Section 2.4. The practical consequences of Theorem 2.1 are discussed in Section 3. Remark 2.2. We have explained above that the conditional consistency is natural to be considered in the claims reserving. However, for the sake of completeness we briefly discuss the unconditional consistency as well. Consider, for instance, the L2 (n) convergence. It follows from the proof of Theorem 2.1 that fj converges to fj in L2 (unconditionally) as n → ∞ if and only if
1 −j n
→ 0,
Secondly, series i=1 εi,j is conditionally summable in probability PDj with probability one if and only if it is conditionally summable [PDj ]-almost surely with probability one. (iv) =⇒ (i) We apply Kronecker’s lemma (Shiryaev, 1996, Lemma IV.3.2). Denote
E
εi,j ai := , i
However, this condition is obviously more complicated than condition (iv) in Theorem 2.1, and it is practically unverifiable. Thus, the conditional convergence is not only a more natural one in this case, but an even more convenient one.
∞
Ck,j
bi :=
i
Ck,j .
k=1
Ci,j
n → ∞.
i =1
k=1
Let n0 be the first integer n such that i=1 Ci,j > 0 [P]-a.s. If ∞ C = ∞ [ P ] -a . s . , then for all i ≥ n holds i , j 0 i =1 0 < bi ≤ bi+1 ↗ ∞ as i → ∞. Kronecker’s lemma states that if ∞ i=n0 ai < ∞, then
n
2.3. Rate of convergence Consistency of an estimator is a very important but only qualitative property. If we want to measure consistency, it is
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necessary to evaluate its rate. This quantitative attribute can be obtained directly from the proof of Theorem 2.1 via the mean square error of the development factors’ estimate. Proposition 2.2. Consider the assumptions of Theorem 2.1 and let j ∈ N be fixed. Denote the conditional mean square error of the estimate of development factor fj as
2 (n) (n) (n) (n) MSE f := E f − E f . D j
j
j
i =1
Proof. A direct consequence of Eq. (4) and unbiasedness of j
(n) f .
Crucial interpretation of previous Proposition 2.2 is as follows: ∞ the slower (faster) divergence of i=1 Ci,j implies the slower (faster) realization of consistency of the development factors’ estimates. 2.4. On the necessary and sufficient condition Let us investigate the necessary and sufficient condition (iv) from Theorem 2.1 Ci,j −−−→ ∞, n→∞
[P]-a.s.
(5)
in more detail. First, recall that variables Ci,j are called summable [P]-a.s. if ni=1 Ci,j converges [P]-a.s. to a finite variable as n → ∞. Assumption (2) of the distribution-free approach implies that Ci,j ≥ 0 [P]-a.s. for all i, j ∈ N and, therefore, it is not restrictive to assume that Ci,j ≥ 0 for all i, j ∈ N. In this case, Ci,j are either summable [P]-a.s. or condition (5) holds. Proposition 2.3. Consider the chain ladder assumptions (1)–(3). with ∞ Suppose that Ci,j ≥ 0 for all i, j ∈ N, and i=1 Ci,j > 0 [P]-a.s. for all j ∈ N. Let j ∈ N be fixed. Then the following conditions are equivalent. (i) The condition (5) holds. (ii) ∞
ECi,1 = ∞.
(6)
i=1
(iii) The condition (5) holds for j0 ∈ N, j ̸= j0 . Proof. Since Ci,j , i = 1, 2, . . . , are independent variables, it follows that the summability [P]-a.s. of Ci,j is equivalent to the condition ∞
ECi,j < ∞.
(7)
i =1
Assumption (1) further implies that ECi,j = E[E(Ci,j |Ci,1 , . . . , Ci,j−1 )]
= fj−1 E(Ci,j−1 ) = · · · = fj−1 × fj−2 × · · · × f1 × ECi,1 . Hence, (7) can be rewritten as ∞ i =1
ECi,j =
j −1 ∞ fk
i =1
k=1
ECi,1 =
∞
(n)
n → ∞.
j
i=1
Corollary 2.4. Consider the assumptions of Proposition 2.3 and let j ∈ N be fixed. (n)
Then, with probability one holds
n −j
∞
(i) The estimator fj is consistent for fj if and only if i=1 ECi,1 = ∞. n (n) (ii) The estimator fj is consistent for fj if and only if i=1 Ci,1 → ∞ [P]-a.s. as n → ∞.
j
−1 n−j ( n ) , MSE f = O Ci,j
This shows that condition (7) is equivalent to i=1 ECi,1 < ∞. Using the same procedure in the opposite direction, we get the equivalence of the summability [P]-a.s. of Ci,j0 , i = 1, 2, . . . , for an arbitrary j0 ∈ N, j ̸= j0 . Then, the statement of Proposition 2.3 follows from the fact that Ci,j ≥ 0.
j −1 k=1
fk
∞ i=1
ECi,1 .
It follows from Corollary 2.4 that either fj is consistent for fj for all j ∈ N, or none of them is consistent. Furthermore, the consistency of fj depends only on the asymptotic behavior of the cumulative sums of ECi,1 (or Ci,1 ) and there is no effect of the true values of fj , j ∈ N on the property of being a consistent estimator or not. Condition (i) of Corollary 2.4 is better understandable compared to (5), because it is formulated in terms of divergence of a series of real numbers instead of divergence of random variables. Condition (ii) of Corollary 2.4 is useful for practice. Here, we observe just one data set and, therefore, condition (6) cannot be verified. On the other hand, we can deduce conclusions about the n behavior of i=1 Ci,1 from the observations so far. We discuss this more deeply in the next section. 3. Practical aspects In practice, one might be interested whether his/her estimator
(n) fj is consistent for fj or not. Therefore, it is important to verify
one of the necessary and sufficient conditions for the estimator’s consistency given above. Since these conditions are formulated in terms of asymptotic properties of some series of random variables or numbers, they cannot be exactly verified in practice. However, we give an informal procedure which might give us a clue whether the consistency holds or not. (n) Recall that Corollary 2.4 claims that the consistency of fj is equivalent to the condition
n
i =1
Ci,1 → ∞ [P]-a.s. as n → ∞.
Denote Sk = i=1 Ci,1 , k = 1, . . . , n the cumulative sums of the cumulative claims Ci,1 in the first development year, i.e., the values in the first column of our cumulative triangle data. This can also be preceded with another development year’s cumulative claims, but we face the issue of a smaller number of observations in the subsequent development years. Now, the question is whether the observed values {Sk }nk=1 so far indicate that the sequence diverges to infinity, or it converges to some finite number as n → ∞. Hence, one may plot the values of Sk and study how they behave when k grows. Alternatively, one can (informally) apply some of the well known criteria for the convergence of real series on the values of Ci,1 . For instance, the ratios Ck+1,1 /Ck,1 or the sequence k Ck,1 can be studied. Let us give an example of this procedure. We consider the well known data of Taylor and Ashe (1983). The sequence Sk of the cumulative sums in the first column of the data is plotted in Fig. 1. One can observe a growing trend without any evidence for a horizontal asymptote. Hence, this gives us confidence that the chain ladder estimates are consistent for this data. Note that the sequence of ratios Ck+1,1 /Ck,1 is inconclusive here and, moreover, the sequence k Ck,1 seems to converge to a number larger than one (corresponding to the divergence of the series), but more observations would be necessary for convincing evidence. In the described procedure, we have looked at the cumulative sums in the first column of our data. One might suggest that according to Proposition 2.3, we could equivalently investigate
k
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Example 4.1 (Decreasing business). In the first example, we consider a fast decreasing business, i.e., the situation where condition (6) holds. For this case, the random variable Ci,1 was generated from the exponential distribution with the mean i−2 × 106 . The be(n) havior of fj is graphically shown in Fig. 2 for j = 1, 2, 3, 4. Even (n)
though the sample size n increases, the estimates fj do not converge to the true value fj . Their values are close to fj in this setting (exponential distribution with the mean i−2 × 106 ), but the estimates are indeed not consistent. Note that the same simulations were run also for Ci,1 with the uniform distribution. In this setting, (n)
the differences between values of estimates fj fj are more noticeable.
Fig. 1. The sequence of cumulative sums Sk for the Taylor/Ashe data. i=1 Ci,j for some j > 1 as well. However, the largest number of observations is always in the first development year and, therefore, the largest amount of information is obtained from the first column. What kinds of business behavior correspond to the violation of consistency? For instance, condition (5) can be violated if one observes a decreasing trend (decreasing fast enough) in payments across the accident years. So to speak, the corresponding line of business is worsening maybe due to new insurance companies entering the market or decreasing prices of such insurance products. Furthermore, splitting one existing line of business into several others can also cause inconsistency in the estimation of development factors.
k
4. Simulations and examples
and the true values
Example 4.2 (Slowly decreasing business). In this example, we still consider a decreasing business, i.e., a situation where ECi,1 decreases with increasing i, but in a slow manner such that condition (6) does not hold. Hence, we choose ECi,1 = i−1/2 × (n)
106 in the exponential distribution. The behavior of fj for this situation is graphically shown in Fig. 3 for j = 1, 2, 3, 4. Unlike the estimates in Example 4.1, a clear convergence pattern can be (n) observed, confirming that the estimates fj are consistent in this case. Example 4.3 (Growing business). Finally, we consider a situation with a growing business, i.e., the case where ECi,1 increases with increasing i. The parameters of the exponential distribution were √ (n) set such that ECi,1 = i × 106 . The estimates fj for this setting are plotted in Fig. 4 for j = 1, 2, 3, 4. The figure obviously confirms (n) that the estimates are consistent. Moreover, fj converges to the true values fj much faster than in Fig. 3 of Example 4.2. (n)
In this section, some examples, which illustrate the results of Theorem 2.1, are presented. In particular, several data sets according to the distribution-free approach are simulated, i.e., satisfying (n) assumptions (1)–(3), and the consistency of fj was studied.
The data DN = {Ci,j : 1 ≤ i, j ≤ N } for N ∈ N accident years were simulated using the following procedure. First, the variable Ci,1 was generated such that Ci,1 ∈ L2 and Ci,j ≥ 0. Then, for a given sequence {fj }Nj=−11 the variables Ci,2 , . . . , Ci,N were generated successively such that Ci,j satisfies (1) and (2). Hence, for each j the variable Ci,j is drawn from a distribution with mean fj−1 Ci,j−1 and variance σj2 Ci,j−1 for some σj2 ∈ (0, ∞). Since the accident years are assumed to be independent (assumption (3)), the rows of the data sets were generated separately, using the same approach. The variables Ci,j for j > n − i are, of course, not used for the (n)
computation of fj , but they can be used for assessment of the estimation of the claims reserves. (n) Having the data DN , the estimated development factors fj were computed using only a subset of the first n accident years. If (n) the estimator is consistent, then the values of fj should approach fj as n grows. For inconsistent estimators, no convergence pattern could be observed. In the following examples, Ci,1 were drawn from the exponential distribution, and the Ci,j was generated from the Poisson distribution with the parameter fj−1 Ci,j−1 for j = 2, . . . , N. However, different distributions can be considered as well. For instance, a suitable uniform distribution can be considered for Ci,1 as well. With this choice one can also independently model the variance of Ci1 . The development factor fj was set as fj = j/(j + 1) in all the presented examples. Recall that the choice of {fj } does not impact whether
(n) f are consistent or not. j
For the whole analysis, software R by R Development Core Team (2011) was used and the seed of the random number generator was set to 1234.
Comparison. One can observe that a convergence of fj to fj is visible in Figs. 3 and 4. In contrast to this, no convergence pattern is obvious in Fig. 2. The comparison of Figs. 3 and 4 also illustrates that ∞ the rate of convergence depends on the rate of convergence of i=1 Ci,1 , as discussed theoretically in Section 2.3. 5. Conclusions The property of being consistent has indisputable importance for an estimator. From some point of view, consistency is more important than unbiasedness, as is motivated in this paper. Consistency and inconsistency of the development factors’ estimate in the distribution-free chain ladder method is investigated in the whole paper. Firstly, various kinds of conditional asymptotic consistency are defined. Secondly, a necessary and sufficient condition for the conditional consistency of the development factors’ estimates is derived. It is also proved that the weak, strong consistency, and consistency in the mean square are equivalent under the distributionfree chain ladder framework. Consistency of the estimators does go beyond the unbiasedness and emphasizes that their variability deserves equal attention. That is why the consistency of the development factors’ estimates is also investigated from the quantitative point of view, i.e., how fast the estimates converge to the unknown development factors and such a convergence rate is provided. The necessary and sufficient condition can unambiguously distinguish whether the chain ladder approach provides consistent estimates for the development factors or not. Therefore, such a very important and crucial condition is elaborated. Practical recommendations, on how to check this condition, are discussed. Thirdly, a real data example and numerical simulations are shown in order to illustrate the performance of the development factors’ estimates. Possible violation of the necessary and sufficient
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(n)
Fig. 2. Behavior of fj
(n)
Fig. 3. Behavior of fj
as n increases in Example 4.1. The dashed lines correspond to the true values of fj .
as n increases in Example 4.2. The dashed lines correspond to the true values of fj .
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( n)
Fig. 4. Behavior of fj
479
as n increases in Example 4.3. The dashed lines correspond to the true values of fj .
condition ensuring the consistency with the consequences is demonstrated. Acknowledgments This paper was written with the support of the Czech Science Foundation project ‘‘DYME Dynamic Models in Economics’’ No. P402/12/G097. The authors thank the anonymous reviewer for suggestions that improved this paper. References Belyaev, Y.K., 1995. Bootstrap, resampling and Mallows metric. In: Lecture Notes, vol. 1. Institute of Mathematical Statistics Umeå University.
England, P.D., Verrall, R.J., 2002. Stochastic claims reserving in general insurance (with discussion). British Actuarial Journal 8 (3), 443–518. Mack, T., 1993. Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin 23 (2), 213–225. Mack, T., 1994a. Measuring the variability of chain ladder reserve estimates. In: Proceedings of the Casualty Actuarial Society Forum, Spring 1994. pp. 101–182. Mack, T., 1994b. Which stochastic model is underlying the chain ladder model? Insurance: Mathematics and Economics 15, 133–138. R Development Core Team, 2011. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, ISBN 3900051-07-0. http://www.R-project.org. Renshaw, A., Verrall, R., 1998. A stochastic model underlying the chain-ladder technique. British Actuarial Journal 4, 903–923. Shiryaev, A.N., 1996. Probability, second ed.. Springer, New York. Taylor, G.C., Ashe, F.R., 1983. Second moments of estimates of outstanding claims. Journal of Econometrics 23, 37–61. Wüthrich, M., Merz, M., 2008. Stochastic Claims Reserving Methods in Insurance. In: Wiley Finance Series, John Wiley & Sons.