Fundamentación del plan de negocios

We first introduce some notation. LetCi,j denote the cumulative claims of acci-

dent yeari ₌1, . . . ,I and development yearj ₌0, . . . ,JwithJ_≤I₋1. We assume that the ultimate claim for accident yeari is given byCi,J, i.e. there is no further

development after development yearJ. Moreover, denote by

D_I=©_{Ci,j : 1≤i≤I, 0≤j≤J,i+j≤I}ª_,

the trapezoid of observations up to theI-th accounting year, illustrated in Fig- ure 1.

Arbenz, Salzmann 151 Ci , J ul ti m ate c laims 1 j J I i D_I ac cident y ear i 1 2 to be predicted 2 0 observations ofCi,j development yearj . . . . . . 3 .. . .. .

Figure 1: The trapezoid D_{I of cumulative claims known in accounting yearI.}

The lower right triangle is yet unknown and has to be predicted.

Stochastic reserving methods aim to predict the yet unknown lower triangle

Dc

I, i.e.Ci,j ∉DI, along with a measure of uncertainty. In particular we are in-

terested in the prediction of the ultimate claimCi,J for each accident yeari. For

CL and BF, this is done as follows. In brief, CL predictsCi,J by

b

C_iC L_{,J =}Ci,I₋ifbI₋i₊1···fbJ,

where the fbj are estimators of the age-to-age factors fj. These factors indicate

the average relative increase of the cumulative claims in one accident year from one development year to the next development year. On the other hand, BF predictsCi,J by

b

C_iB F_{,J =}Ci,I−i+µi(γbI−i+1+ ··· +γbJ),

where theµi is a prior estimate ofE[Ci,J] and theγbj are estimates ofγj, which

denote the fraction of claims expected in development year j.

The CL predictor has a multiplicative structure, whereas the BF predictor is additive. Moreover the two approaches represent two extreme positions of data reliance and expert opinion. These differences impose restrictions on the applicability of the two approaches. In practice, one often encounters claims reserving triangles in which CL is appropriate for some accident years, whereas BF is suitable for the others. As stated in Neuhaus (1992), this can be caused by data sparsity, e.g., triangles which have missing entries or zero cumulatives for recent accident years. Another reason is illustrated in the following exam- ple. Suppose we are given a short-tailed insurance runoff triangleD_{I. Assume}

the triangle behaves nicely to apply CL, except that in accident yeark, the di- agonal valueCk,I₋k is twice as large asfI₋kCk,I₋k₋1. The latter value represents

152 Paper D

the expectation ofCk,I−k in the CL model, given the information in the previ-

ous accounting year. In the following, we illustrate different possible underly- ing reasons for such an observationCk,I₋k, which imply different appropriate

reserves.

• There was a legal change, allowing each insured of accident yearka dou- bled indemnification. Then the predicted ultimateCbk,J should also be

doubled. In this case, CL is appropriate, asCb_kC L_,J scales proportionally to

Ck,I−k.

• The increaseCk,I−k−fI−kCk,I−k−1is caused by a single event ("outlier"),

which is not systematic and is not expected to happen again. In this case, the predicted ultimate should be increased by this difference, which is re- alised by BF.

• The increase is due to an exceptional early commutation, which causes a claim to be paid earlier than usual. In this case, the predicted ultimate from the previous accounting year should not be changed at all.

The reserving actuary often knows these underlying reasons for seemingly un- usual behaviour in a triangle. However, such information cannot be extracted neither fromD_{I nor from prior estimates of the ultimate claims. Therefore, ac-}

tuarial judgement is necessary and appropriate when choosing the reserving method.

Due to these reasons, it is common practice for reserving actuaries to switch accident year-wise between CL and BF, i.e., decide for each accident year sepa- rately whether the reserves are determined according toCb_iC L_,J orCb_iB F_,J . Such ap- proaches are easily applied, as both CL and BF can be quickly implemented in a spreadsheet. Commercial reserving software such as EMB ResQ™and Milli- man ReservePro®allow to set reserves as a weighted average between different reserving methodologies, with weights different for each accident year.

The crucial assumptions behind CL as introduced in Mack (1993) are that

E[Ci,j|Ci,j−1]= fjCi,j−1and var(Ci,j|Ci,j−1)=σ2_jCi,j−1, where theσ2_j are vari-

ance parameters. BF as introduced in Mack (2008b) assumes increments to be independent, such thatE[Ci,j|Ci,j−1]=Ci,j−1+µiγjand var(Ci,j−Ci,j−1)=σ2_jµi.

As a consequence, we can calculate the conditional covariance of consecutive increments: cov¡Ci,j+2−Ci,j+1,Ci,j+1−Ci,j ¯ ¯Ci,j ¢ = (¡ fj₊2−1 ¢ σ2_j +1Ci,j, for CL, 0, for BF.

This shows that the assumptions behind CL and BF are incompatible. How- ever, these assumptions are applied to the whole triangle when estimating pa- rameters. Thus, when switching accident year-wise between CL and BF, one

Arbenz, Salzmann 153

cannot avoid applying two conflicting sets of assumptions on the triangle D_I.

Due to the different nature of the two methodologies, there is no straight for- ward stochastic representation of an accident year-wise weighting of CL and BF within a distribution-free model.

Reserving models that allow for a combination of CL-style and BF-style re- serves have already been studied in the literature. A first credibility approach is given by the Benktander-Hovinen method, see Benktander (1976). In Neuhaus (1992) and Mack (2000) this credibility mixture of CL and BF is further studied but they do not specify a concrete parametric model and do not deduce a pa- rameter estimation error. Alai (2010) provides a solution within a generalised linear model (GLM) framework. Bayesian approaches are proposed in Verrall (2004) and Section 4.3.2 of Wüthrich and Merz (2008), respectively.

The HCL method as proposed here allows to freely choose a weighting be- tween a multiplicative and an additive behaviour circumventing the problem of conflicting assumptions within a distribution-free framework. The follow- ing three aspects highlight the main differences between HCL and the models referenced above.

• In the existing literature the weight between CL and BF is generally inter- preted as a credibility weight, which is chosen to minimise the prediction uncertainty. This approach neglects special situations often encountered in practice as discussed above. On the contrary, the HCL method sees this weight as a model selection parameter through which the reserving actu- ary can determine whether a multiplicative structure as in CL or an addi- tive structure as in BF is more appropriate to predict the ultimate claim.

• The HCL method provides a stochastic model that allows parameter esti- mation, claims prediction, the calculation of the conditional mean square error of prediction (MSEP), and the calculation of uncertainty in the claims development result (CDR). Furthermore, an easy-to-use implementation is available.

• HCL is based on a distribution-free framework. This enables us to avoid assumptions needed in distribution based methods, such as positivity of incremental claims (in contrast to over-dispersed Poisson and GLM) or distributional assumption (in contrast to Bayesian methods).