Claims reserving using copulas : the impact of modeling dependencies

Faculty of Economics and Business

Master’s Thesis - Actuarial Science and Mathematical Finance

Claims reserving using copulas

The impact of modeling dependencies

Kerim Kes

Name: Kerim Kes

Student number: 10542760

E-mail:kerim_kes@hotmail.com Supervisor: dr. Katrien Antonio Date: July 15, 2017

Statement of Originality

This document is written by Student Kerim Kes who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

Abstract

This study examines the implications of using copula regression models to model dependencies for claims reserving with claim triangles. For this purpose, various copula-based dependence models are thoroughly explained and we introduce an own suggestion for a copula regression model. We divide these models into two groups: the pairwise dependence models, which use a cell-by-cell approach and the calendar year dependence models, which link the claims of the same calendar year. Information on the development of claims over time is available in a run-off triangle format. This implies that claims information is aggregated by accident and development periods such that a triangle results. The models under study then capture depen-dencies between cells within a line of business, or across multiple lines of business. Besides a thorough discussion of these models, we replicate the results from Abdallah, Boucher, and Cossette (2016). In order to find the best-fitting models, likelihood-based criteria are used. We compare the three best-fitting dependence models with the model where independence is assumed, by using a backtesting procedure, simulations and a parametric bootstrap. We see that the models which are able to capture a calendar year effect, generally provide the best fits for the dependence structure of the claims data. Furthermore, we show that the choice for a different copula type or a different dependence model might have a great influence on the point estimate of the reserve and its predictive distribution. Finally, we see that a better fitting model does not necessarily perform better according to the backtesting results.

Keywords Claims, Lines of business, Claim triangles, Copula, Dependence, Models, Reserves

Preface

With the completion of this thesis, I end an intensive but beautiful study period. After four years of study together with my classmate Dylan, we both have completed all the courses within the regular time. I am very proud of this achievement and I am grateful to everyone who has supported me. First, I would like to thank Katrien Antonio, my supervisor at the University of Amsterdam, for sharing her knowledge and for taking the time to guide me through my research by providing me useful instructions. I am also grateful to Nico de Boer, my supervisor at Triple A - Risk Finance, for his good advice and support. Thirdly, I am thankful to my parents for supporting me during my entire study period. Finally, I would like to thank my brother Ramon, my girlfriend Rosanna and my friends for clearing my mind in my spare time, which gave me new energy to continue writing the thesis every time.

Contents

1 Introduction 1

2 The copula regression framework for claim triangles 5

2.1 Marginal distribution model . . . 6

2.1.1 Claim severity distributions . . . 7

2.1.2 Maximum Likelihood Estimation (MLE) . . . 8

2.2 Copulas . . . 8

2.2.1 Elliptical copulas . . . 9

2.2.2 Archimedean copulas . . . 10

2.2.3 Hierarchical Archimedean copulas . . . 13

3 Copula dependence structures for claim triangles 15 3.1 Pairwise dependence between cells in claim triangles . . . 15

3.1.1 PWD model . . . 15

3.2 Calendar year dependence between cells in claim triangles . . . 17

3.2.1 ICYD model . . . 18

3.2.2 HCYD model . . . 19

3.2.3 CCYD model . . . 21

4 Model calibration, prediction and validation 24 4.1 Calibration . . . 24 4.2 Reserve prediction . . . 27 4.2.1 Point estimates . . . 27 4.2.2 Simulation . . . 28 4.2.3 Parametric bootstrap . . . 28 4.3 Model validation . . . 29 4.3.1 Likelihood-based criteria . . . 29 4.3.2 Backtesting . . . 30 5 Results 32

5.1 Data . . . 32

5.2 Results for US property-casualty insurer . . . 34

5.2.1 Goodness of fit criteria, parameter estimates and point estimates of the re-serves . . . 34

5.2.2 Backtesting and bootstrap results . . . 39

5.3 Results for Canadian property-casualty insurer . . . 44

5.3.1 Goodness of fit criteria, parameter estimates and point estimates of the re-serves . . . 44

5.3.2 Backtesting and bootstrap results . . . 48

6 Conclusion and outlook 54 6.1 Conclusion . . . 54

6.2 Outlook . . . 55

References 57 A Appendix 58 A.1 The (log-)density of a two-level hierarchical Archimedean copula . . . 58

A.2 Initial parameters and log-likelihood functions inR . . . 59

A.2.1 Log-normal . . . 59

A.2.2 Gamma . . . 59

A.3 Adjustment of thenacLL()function . . . 61

A.4 Parameter estimates of the independence model and the copula regression models . 62 A.4.1 Parameter estimates - US property-casualty insurer . . . 62

A.4.2 Parameter estimates - Canadian property-casualty insurer . . . 67

A.5 Datasets . . . 72

A.5.1 Data US property-casualty insurer . . . 72

1

Introduction

One of the most important tasks in non-life insurance is to determine the outstanding claims re-serves. Insurance companies sell policies that provide financial coverage over a specific time period against arbitrary events that are disadvantageous for the insured. In return for this coverage, the insured pays a premium. The occurrence of such an event, provides the insured the right to claim the amount, which is predetermined in his policy, from the insurer. In the left part of Figure1, we show an example of the development of a single claim for a non-life insurance company.

Figure 1: The development of a single claim and the aggregation of claims

IBNR RBNS Settled

Time Occurrence Reporting Closure

Payments Reporting delay

All claims in portfolio

Compress data Development period A ccident yea r Development period A ccident yea r LoB 1 LoB 2

Usually, there is a time-lag between the occurrence of the event and the reporting of the claim to the insurer. This lag is called the reporting delay (Wüthrich & Merz,2008). Besides that, the time between the reporting and the closure of the claim is called the settlement delay. During this period, the insurer determines which amounts should be paid and these payments are made to the insured. In order to ensure that all the future payments in relation to incurred but not settled claims can be paid out when required, insurance companies keep claims reserves. For accounting purposes these claims reserves are treated as a liability. In determining the total outstanding claims reserve, two main parts are distinguished: the Incurred But Not Reported (IBNR) reserve and the Reported But Not Settled (RBNS) reserve (see Figure1). The IBNR reserve denotes the reserve for claims that have already occurred but have not yet been reported. The RBNS reserve denotes the reserve for the

future payments of claims that have been reported, but have not yet been settled. The outstanding claims reserve is the sum of the IBNR and RBNS reserves.

For the computation of this reserve, different methods are available. More recent studies show methods to evaluate the outstanding claims on an individual claim basis (see for exampleAntonio and Plat(2014)). However, the most commonly used methods are still based on the so-called claim triangles. These claim triangles use the information of all claims belonging to the same line of business (LoB), but aggregate their information per accident and development period such that a triangle results. We provide a visual example of two different sized claim triangles in the right part of Figure 1. When the number of claims is large enough, the past development provides a basis for an estimate of the future claims. Two well-known methods in determining the outstanding re-serves are the Chain Ladder method (seeWüthrich and Merz(2008)) and the Bornhuetter-Ferguson method (Bornhuetter & Ferguson,1972).

One of the downsides of these methods is that they can only be applied to a single claim triangle. This is a disadvantage, because most of the insurance companies have multiple lines of business. Common practice is that the outstanding claims reserves for separate lines of business are added together, assuming that no dependency exists between the risks covered by different lines of busi-ness. However, this assumption may not be valid in practice. Due to new regulatory standards (e.g. Solvency II in Europe) it is not sufficient to only compute a reserve for each line of business, but it is also necessary to estimate the total reserve for all the products of an insurance company. Furthermore, these new standards require that an insurance company needs to better quantify the risks that are associated with its total activities.

This setting motivates a careful investigation of the dependence between multiple lines of busi-ness. Wüthrich and Merz (2008) propose a multivariate Chain Ladder method and a multivariate additive loss reserving method for this purpose. Besides that, Zhang (2010) proposes a general multivariate Chain Ladder model that uses the seemingly unrelated regression technique. These methods mainly focus on the mean squared error of predictions. Another very useful tool for deal-ing with dependencies is a copula. BothCôté, Genest, and Abdallah(2016),Shi and Frees(2011) andAbdallah et al.(2016) focus on the parametric approach to model dependencies with the usage of copulas.Côté et al.(2016) use rank-based methods for selecting, estimating and validating

cop-et al. (2016) propose copula regression models that can be estimated completely with Maximum Likelihood Estimation.

The model ofShi and Frees(2011) captures the dependence structure between lines of business with a copula that links the claims with the same accident and development year in a pairwise man-ner. This model allows for the usage of different parametric severity distributions for the marginal loss distributions. Abdallah et al. (2016) propose a model that uses Archimedean copulas to al-low for dependencies between all the observed claims that have the same calendar year for each line of business. To link the claims of calendar years of different lines of business, they impose another dependence structure which can be constructed using hierarchical Archimedean copulas. Both models (Abdallah et al.,2016;Shi & Frees,2011) can be extended with a parametric bootstrap to determine the entire predictive distribution of the outstanding reserves.

Dependence modeling allows analysts in risk capital analysis to quantify the diversification effect. However, it is of the utmost importance that the correct copula regression model is used. Wrong model specifications or ignoring existing dependence structures might significantly impact the estimates and the distribution of the reserves, which could have major consequences for an insurer. Therefore, we address the following question in this thesis: What are the implications of using copula regression models to model dependencies within claims reserving?

In order to give an answer to this question, we examine the models ofShi and Frees(2011) and Abdallah et al. (2016), together with a newly proposed copula regression model which uses only one copula to capture the dependence structure between all the claims belonging to the same cal-endar year. This paper extends the existing literature on claims reserving with copulas in multiple ways. First we replicate the results ofAbdallah et al.(2016) regarding the estimation of the claims reserve. Thereafter, we complement these results with the corresponding results of the newly pro-posed model. Furthermore, we use these results to find the three best-fitting copula regression models and compare the reserve of these models with the reserve resulting from the model where independence is assumed (the independence model). Finally, we use a backtesting procedure, sim-ulations and a parametric bootstrap to obtain more information regarding the performance of these four models and the corresponding predictive distributions of the claims reserve.

The rest of the paper is structured as follows. Section2describes the theory, which is used as a building block for the copula-based dependence models. We thoroughly describe these models in

Section3. After that, the model calibration, reserve prediction and model validation are described in Section4. Here, we mention which model specifications are assumed and what tools are used to validate the various models. Thereafter, we present and discuss the results in Section5. Finally, in Section6we summarize the results, provide a discussion of research performed and mention some proposals for further research.

2

The copula regression framework for claim triangles

In this section, we sketch the theoretical framework for claims reserving with copula regression models. First of all, we define the basic concepts and introduce the marginal distribution model. After that, we describe the specifications of the claims severity distributions and provide an estima-tion method to obtain the parameter estimates. Finally, we explain the copula concept and describe different copula structures, where we mention some of the most well-known copula families.

The losses in the claim triangles may be represented as incremental payments or cumulative pay-ments, depending on the context. In this thesis, all the models apply to incremental paid claims. We consider an insurance portfolio that consists of N lines of business, each represented by a claim triangle. This implies that the claims information of each single line of business is aggregated by accident and development years such that a triangle results (see Figure 1). Further, let i denote the accident year (i.e., the year in which the accident has occurred) and j denote the development year (i.e., the number of years from the moment of occurrence to the time of payment). Then, X_{i j}(n) is defined as the incremental claims paid in the ith accident year and the jth development year. The superscript n ∈ {1, . . . , N}, refers to the nth line of business. Consequently, the multivariate incremental claims can be expressed by random vector (Shi & Frees,2011)

Xi j = (X (1) i j , . . . , X

(Ni j)

i j ), i∈ {1, . . . , I} and j∈ {1, . . . , J},

where I and J respectively denote the most recent accident year and the latest development year. We assume that all the claims are closed in J years and that typically I = J. Hence the number of accident years is equal to the number of development years in a particular triangle.

Furthermore, we allow for imbalance in the multivariate triangles, in the same way as inShi and Frees(2011). This imbalance (Ni j< N) could be due to the different size of each line of business or due to missing values in the triangles. Here, Ni jis defined as the dimension of the incremental claim vector for accident year i and development year j (Shi & Frees,2011). For example, consider an insurance portfolio that contains two lines of business with a different size for each claim triangle (see Figure1). Let the first claim triangle have I = J = 10 and the second claim triangle I = J = 8. In this case, N = 2 and Ni j = 2 for all i ∈ {1, . . . , 8} and j ∈ {1, . . . , 8}. However, if 8 < i ≤ 10 or

8 < j ≤ 10, then Ni j = 1 and consequently Ni j< N = 2.

Finally, using the earlier mentioned notations, we compute the outstanding claims reserve at time I for the nth line of business by

I

∑

∑

Because we work with copula regression models it is necessary to study the following building blocks: marginal distributions and copulas. The next subsection thoroughly explains the marginal distribution model. After that, we introduce the copula concept.

2.1

Marginal distribution model

For the marginal modeling of incremental payments we use typical claim severity distributions. This is due to the characteristically small sample size of claim triangles. FollowingShi and Frees (2011) the data is standardized so that the volume of each line of business is taken into account. The standardized incremental claims are defined as Y_{i j}(n)= X_{i j}(n)/ω_i(n), where ω_i(n) represents the exposure for the ith accident year in the nth triangle. This exposure variable can, for example, be equal to the earned premiums or the number of policies. The marginal distribution of Y_{i j}(n) is given by:

F_{i j}(n)= Prob(Y_{i j}(n)≤ y(n)_{i j} ) = F(n)(y(n)_{i j} ; η_{i j}(n), γ(n)), n= 1, . . . , N.

Here, F_{i j}(n) is the marginal cumulative distribution function of cell (i, j) in the nth claim triangle. We allow different marginal distribution functions F(n)(·) for the incremental claims from different lines of business. Further, the parameter η_{i j}(n)denotes the systematic component, which determines the location. The idea is to use two parameters α_i(n)(i ∈ {1, 2, . . . , I}) and β(n)_j ( j ∈ {1, 2, . . . , J}) to characterize the effect of the accident year and the development year corresponding to cell (i, j) in triangle n. This causes the systematic component for the nth line of business to be equal to

η_{i j}(n)= ζ(n)+ α_i(n)+ β(n)_j , n= 1, . . . , N, (2)

iden-predictor of the well-known Chain Ladder model. Besides the systematic component, the vector γ(n) summarizes the additional parameters in the distribution of Y_{i j}(n). Except for the location pa-rameters, parameters are equal for all cells within a claim triangle. These specifications are in line with the set-up ofShi and Frees(2011) andAbdallah et al.(2016).

2.1.1 Claim severity distributions

Now that the design of the distributional model is clear, we examine which marginal distribution functions F(n)(·) are well-suited for fitting the incremental claims from the different claim triangles. According to Denuit, Dhaene, Goovaerts, and Kaas (2008), suitable marginals for this purpose are: the log-normal, Gamma, inverse Gaussian and Pareto distributions. In this thesis, we only use the log-normal and the Gamma distributions in order to be able to replicate the results of Abdallah et al.(2016). Specifically, we consider the form η_{i j}(n)= µ_{i j}(n)for a log-normal distribution with location parameter µ_{i j}(n) and shape parameter σ(n). For a Gamma distribution with location parameter µ_{i j}(n)and shape parameter φ(n),Shi and Frees(2011) use the canonical inverse link µ_{i j}(n)=

1 η_{i j}(n)φ(n)

. However, in this thesis we use the exponential link µ_{i j}(n)= exp(η

(n) i j )

φ(n) , which is in the same vein asAbdallah et al.(2016). This is due to the fact that the canonical link of the Gamma regression model may lead to undesirable negative values in the claim triangle, so the exponential link is used to ensure positive means of all these cells. Using the above notations, the probability density function of the log-normal distribution for cell (i, j) in triangle n becomes:

f_{i j}(n)(y(n)_{i j} ) = 1 y(n)_{i j} σ(n) √ 2π e− 1 2( log(y(n)_{i j} )−µ_{i j}(n) σ(n) )2 , y(n)_{i j} > 0

with scale parameter σ(n)> 0. Additionally, the probability density function for the Gamma distri-bution is given by:

f_{i j}(n)(y(n)_{i j} ) =   y(n)_{i j} µ_{i j}(n)   φ(n) e −y (n) i j µ_{i j}(n) Γ(φ(n))y(n)_{i j} , y(n)_{i j} > 0

with shape parameter φ(n) > 0 and scale parameter µ_{i j}(n) > 0. See Denuit et al. (2008) for more details considering these marginal distributions.

2.1.2 Maximum Likelihood Estimation (MLE)

In order to obtain the parameter estimates of the marginal distribution models, we use Maximum Likelihood Estimation. We also apply this estimation method to the forthcoming copula regression models. Using the resulting parameter estimates, we are able to estimate the future standardized claims. For the log-normal distribution, the standardized incremental claims can be estimated by

ˆ E[y(n)i j ] = ˆy (n) i j = exp  ˆ µ_{i j}(n)+1₂( ˆσ(n))2 

and for the Gamma distribution, ˆ_E[y(n)_{i j} ] = ˆy(n)_{i j} = ˆµ_{i j}(n)φˆ(n). Here, ˆµ_{i j}(n), ˆσ(n) and ˆφ(n)denote respectively the estimated location and shape parameters. At this point, the structure of the models for marginal lines of business is clear. The next step is to introduce the copula concept, after which the various dependence models are discussed.

2.2

Copulas

According to Sklar’s theorem (see Nelsen(2006)), a copula function is able to uniquely represent the joint distribution of the random variables X1, . . . , Xd as

F(x1, . . . , xd) = C(F1(X1), . . . , Fd(Xd); θ),

where C(·, θ) denotes the copula function from [0, 1]d to [0, 1] with dependence parameter vector θ, and F1, . . . , Fd are the continuous marginal distribution functions of X1, . . . , Xd. This definition implies that a copula itself is a cumulative distribution function, which describes the dependence structure separated from the marginals. Because of this, copulas also have a density function de-fined as c(u) := ∂ d_C(u 1, . . . , ud) ∂ u1. . . ∂ ud (3) if the copula is sufficiently differentiable (Schmidt,2006). The copula density function is required for the Maximum Likelihood Estimation of the various copula regression models that are consid-ered in this thesis. In order to find the right copula regression model for modeling the dependence structure, it is also necessary to look at the case of independence which serves as a basic, benchmark model. For this purpose, the well-known Independence copula is given by (Nelsen,2006):

C(u) = Π(u) = d

It is easy to show that the copula density of this copula is equal to c(u) = 1. Besides the Indepen-dence copula, elliptical copulas and (hierarchical) Archimedean copulas are used in the different dependence models (see Joe (1997) and Nelsen (2006)). These types of copulas are described during the remaining part of this subsection.

2.2.1 Elliptical copulas

Elliptical copulas are copulas that are extracted from elliptical distributions. In order to obtain an elliptical copula, we first define the multivariate elliptical distribution. Consider a d-dimensional random vector X = (X1, X2, . . . , Xd)T which has a multivariate elliptical distribution with location parameter vector µ and dispersion matrix Σ. Random vector X does not generally have a density, but if it exists, the density is defined as follows (Landsman & Valdez,2003):

fX(x) = c_d p|Σ|gd  1 2(x − µ) T_Σ−1_{(x − µ)}  .

Here, g_d(·) is called the density generator function and c_d is a normalizing constant. Besides that, consider the density and marginal distribution functions to be given by f (·) and F(·) respectively. Further, let FX(·) denote the multivariate elliptical distribution function. Using these notations, the d-dimensional elliptical copula can be defined by (Shi & Frees,2011):

C(u1, . . . , ud) = FX(F−1(u1), . . . , F−1(ud)),

which is a function of (u1, . . . , ud) ∈ [0, 1]d. Consequently, the corresponding copula density is given by: c(u₁, . . . , u_d) = f_X(F−1(u₁), . . . , F−1(u_d)) d

∏

This type of copula possesses the property that the dependence structure is easily captured through the dispersion matrix Σ (Shi & Frees,2011).

In the elliptical family of copulas, the multivariate Gaussian copula is the most widely known copula. According to Abdallah et al. (2016), this type of copula is often considered as a bench-mark. Therefore, we also explore dependence models based on multivariate Gaussian copulas. The

multivariate Gaussian copula and its probability density function are given by:

C(u) = ΦΣ(Φ−1(u1), . . . , Φ−1(ud)) and c(u) = |Σ|−1/2exp  −1 2ξ T_(Σ−1_{− I)ξ}_, where ξ = Φ−1(u1), . . . , Φ−1(ud)
T

(see,McNeil, Frey, and Embrechts(2005) andAbdallah et al. (2016)). Here, ΦΣ(·) is the joint distribution function with mean vector 0 and correlation matrix Σ, and Φ−1(·) is the inverse cumulative distribution function of a standard normal variable. For the bivariate case, this copula boils down to (Schmidt,2006):

C(u₁, u₂) = Φ−1(u1) Z −∞ Φ−1(u2) Z −∞ 1 2πp1 − ρ2exp  −s 2 1− 2ρs1s2+ s22 2(1 − ρ2₎  ds₁ds₂,

where ρ denotes the linear correlation between Φ−1(u1) and Φ−1(u2). The structure of a Gaussian copula implies that it offers a lot of flexibility to model dependencies. This can simply be done by changing the correlation matrix (Σ) (Abdallah et al.,2016). A typical drawback of a Gaussian copula and elliptical copulas in general is that their lower and upper tail dependence are equal. In order to allow more kinds of dependencies, (hierarchical) Archimedean copulas are considered. These types of copulas are discussed hereafter.

2.2.2 Archimedean copulas

The main advantage of using Archimedean copulas is that these copulas have quite a simple form, which can explicitly be defined. In contrast, elliptical copulas do not possess this nice prop-erty. These copulas are derived from elliptical multivariate distributions and have no closed form. Archimedean copulas can be defined in terms of a so-called Archimedean copula generator func-tion ψ(·). According to Hofert, Mächler, and McNeil (2012b), this generator is a continuous, decreasing function from [0, ∞] to [0, 1] which satisfies ψ(0) = 1 and ψ(∞) = limt→∞ψ (t) = 0. Furthermore, they state that the generator is strictly decreasing on [0, inf{t : ψ(t) = 0}]. Using this generator ψ(·), the d-dimensional Archimedean copula is defined by (Hofert et al.,2012b):

C(u) = ψ(ψ−1(u1) + · · · + ψ−1(ud)) = ψ(t(u)), where t(u) = d

Here, ψ−1(·) denotes the inverse generator function from [0, 1] to [0, ∞], which satisfies ψ−1(0) = [inf{t : ψ(t) = 0}].McNeil and Nešlehová(2009) show that an Archimedean copula is only defined by a generator if ψ(·) is d-monotone. This means that ψ(·) admits derivatives ψ(k)up to order d − 2 that satisfy (−1)kψ(k)(t) ≥ 0 ∀k ∈ {0, . . . , d − 2}, t ∈ (0, ∞), ψ(·) is continuous on [0, ∞), and (−1)d−2ψ(d−2)(t) is convex and decreasing on (0, ∞) (McNeil & Nešlehová,2009). Furthermore, they state that if and only if ψ(d−1)exists and is absolutely continuous on (0, ∞), an Archimedean copula admits a density c(·) which is given by:

c(u) = ψ(d)(t(u)) d

∏

There are a few Archimedean copula families that are well-known and widely used in appli-cations. Three of them are the Frank copula, Clayton copula and Gumbel copula. Just like the Gaussian copula, these copulas are also considered in the dependence models of this thesis. For the bivariate case, these copulas are given respectively by (Nelsen,2006):

CFr_θ (u₁, u₂) = −1 θ ln  1 +(e −θ u1_{− 1) · (e}−θ u2_{− 1)} e−θ− 1  θ ∈ R\{0}, CCl_θ (u1, u2) =  max{u−θ₁ + u−θ₂ − 1, 0}− 1 θ θ ∈ [−1, ∞)\{0}, and CGu θ (u1, u2) = exp  −(− ln u1)θ+ (− ln u2)θ 1_θ θ ∈ [1, ∞). (5)

For some particular values of the dependence parameters, these copulas are equal to the Inde-pendence copula given in Equation 4. For example, the Gumbel copula is equal to the Indepen-dence copula when θ = 1 and the Clayton copula tends to the IndepenIndepen-dence copula when θ → 0 (seeSchmidt(2006) andMcNeil et al. (2005)). Generally, this also applies for more than two di-mensions. In the bivariate form, the copulas have a simple expression and it is straightforward to derive the density functions. However, for the multivariate case where d > 2, this becomes a bit more complicated. We now list these copula densities in order to use them later on in the thesis. Let us start with the dependence parameter range and the generator function for the mentioned copula families. For these families,Hofert et al.(2012b) present explicit formulas which are given in Table

Table 1: Archimedean generators ψ(·) for Frank, Clayton and Gumbel families.

Family Parameter ψ (t)

Frank θ ∈ (0, ∞) − log(1 − (1 − e−θ) exp (−t))/θ Clayton θ ∈ (0, ∞) (1 + t)−1/θ

Gumbel θ ∈ [1, ∞) exp (−t1/θ)

Using the generators ψ(t) for the Frank, Clayton and Gumbel families,Hofert, Mächler, and Mc-Neil(2012a) report the corresponding copula densities, which are given by:

1. Frank copula density,

c_θ(u) = 

θ 1 − e−θ

d−1

Li__(d−1)(hF_θ(u))exp (−θ ∑ d j=1uj) hF θ(u) , where Li_s(z) = ∑∞k=1z k

ks and hF_θ(u) = (1 − e−θ)1−d∏d_j=1(1 − exp (−θ uj)).

2. Clayton copula density,

c_θ(u) = d−1

∏

∏

3. Gumbel copula density,

c_θ(u) = θdexp (−t_θ(u)α₎∏ d j=1(− log uj)θ −1 t_θ(u)d_∏d j=1uj P_d,αG (t_θ(u)α_), where α = 1/θ , P_d,αG (tθ(u)α) = ∑dk=1aGdk(α)x k_, and aG_dk(α) =d!_k!∑kj=1 kj
_{α j} d
(−1) d− j_{, k ∈ {1, . . . , d}.}

In order to be able to model multiple levels of dependence, the concept of Archimedean copulas can be extended to the so-called hierarchical Archimedean copulas. This is described next.

2.2.3 Hierarchical Archimedean copulas

Hierarchical Archimedean copulas were first considered in three- and four-dimensional cases by Joe (1997). Later, McNeil (2008) expanded this concept to the d-dimensional case (see Hofert and Pham (2013)). These copulas are interesting, because they allow for partial symmetry in the class of Archimedean copulas. Hierarchical Archimedean copulas can be used to model different dependencies within and between various groups of random variables. In fact, such a copula is an Archimedean copula with (some) arguments replaced by another (hierarchical) Archimedean copula. In this thesis, we study in particular two-level hierarchical Archimedean copulas to model the dependencies first within and then between different claim triangles. Hofert and Pham(2013) define this copula class by:

C(u) = C0(C1(u1), . . . ,Cd0(ud0)), u = (u1, . . . , ud0)

T_{, (6)}

where each copula Cs∈ {0, . . . , d0} is an Archimedean copula with a completely monotone genera-tor function ψs(·) and d0is the dimension of the so-called parent copula C0. We define the set of all completely monotone generators by Ψ∞. Completely monotone means that ψs(·) : [0, ∞] → [0, 1] is continuous, ψs(∞) = limt→∞ψs(t) = 0, ψs(0) = 1, and (−1)kψ

(k)

s (t) ≥ 0 ∀t ∈ (0, ∞), k ∈ N0 (Hofert & Pham, 2013). Furthermore, Hofert and Pham (2013) state that a sufficient, but not per se necessary, nesting condition for this hierarchical Archimedean copula is that the nodes

˚

ψ0s= ψ₀−1◦ ψs, s ∈ {1, . . . , d0} have first order derivatives which are completely monotone. For the earlier mentioned Frank, Clayton and Gumbel families, this condition is satisfied as long as θ0≤ θs, s ∈ {1, . . . , d0} and all generators belong to the same family (Hofert & Pham,2013). This implies that the degree of dependence is lower for parent copula C0 than for the so-called child copulas C1, . . . ,Cd0.

Now that the structure of the two-level hierarchical Archimedean copula is known, the next step is to define the (log-)density which is necessary for Maximum Likelihood Estimation. However, this becomes quite challenging because the derivative of Equation (6) is not so straightforward anymore. For example, we consider the following hierarchical Archimedean copula:

This copula consists of one Archimedean parent copula C0, with two Archimedean child copulas C1and C2as arguments. C1models the dependence structure between u1and u2and C2models the dependence structure between u3 and u4. On top of that, C0 models the dependence structure be-tween C1(u1, u2) and C2(u3, u4). Using Equation (3), the density of this hierarchical Archimedean copula becomes (Abdallah et al.,2016):

c(u) = ∂ 4 ∂ u1∂ u2∂ u3∂ u4 C0(C1(u1, u2),C2(u3, u4)) = ∂ 3 ∂ u1∂ u2∂ u3 C₀(0,1)(C1(u1, u2),C2(u3, u4))C₂(0,1)(u3, u4) = ∂ 2 ∂ u1∂ u2 h C₀(0,2)(C1(u1, u2),C2(u3, u4))C(1,0)₂ (u3, u4)C₂(0,1)(u3, u4) + C₀(0,1)(C1(u1, u2),C2(u3, u4))C(1,1)₂ (u3, u4) i = ∂ ∂ u1 h C₀(1,2)(C1(u1, u2),C2(u3, u4))C₁(0,1)(u1, u2)C(1,0)₂ (u3, u4)C₂(0,1)(u3, u4) + C₀(1,1)(C1(u1, u2),C2(u3, u4))C(0,1)₁ (u1, u2)C₂(1,1)(u3, u4) i = C₀(2,2)(C₁(u₁, u₂),C₂(u₃, u₄))C(1,0)₁ (u₁, u₂)C₁(0,1)(u₁, u₂)C(1,0)₂ (u₃, u₄)C₂(0,1)(u₃, u₄) +C₀(1,2)(C₁(u₁, u₂),C₂(u₃, u₄))C₁(1,1)(u₁, u₂)C₂(1,0)(u₃, u₄)C(0,1)₂ (u₃, u₄) +C₀(2,1)(C1(u1, u2),C2(u3, u4))C₁(1,0)(u1, u2)C₁(0,1)(u1, u2)C(1,1)₂ (u3, u4) +C₀(1,1)(C1(u1, u2),C2(u3, u4))C₁(1,1)(u1, u2)C₂(1,1)(u3, u4),

where C(i, j)(u, v) = ∂i+ jC(u,v)

∂ ui∂ vj for i, j ∈ {0, 1, 2}. This density is rather complicated due to the appearing inner derivatives as a consequence of the Chain Rule.

In this way, the density becomes computationally very intensive in high dimensions. To give a solution to this problem, Hofert and Pham (2013) derive general formulas for the density and the log-density of a two-level hierarchical Archimedean copula. These are given in AppendixA.1, using the same notations as in this subsection. In the next section, we describe the various copula-based dependence models for reserving with claim triangles.

3

Copula dependence structures for claim triangles

Determining the total reserve by simply summing the reserve for each line of business, is equal to the assumption of independence between the cells in different claim triangles. However, multi-ple lines of business may be affected simultaneously by common economic or social factors, like inflation or jurisprudence (Abdallah et al., 2016). In order to improve the diversification and the determination of the risk capital, we investigate possible dependence structures between the dif-ferent claim triangles of the portfolio. Particularly, we examine two difdif-ferent structures: pairwise dependence and calendar year dependence.

3.1

Pairwise dependence between cells in claim triangles

Shi and Frees (2011) propose a copula regression model that links the claims of different lines of business in a pairwise manner. This model is called the pairwise dependence model (PWD) just as in the paper of Abdallah et al. (2016). Specifically, this model assumes that claims from different triangles with the same accident and development year are dependent. Using the notations in Section 2.1, this means for example that the standardized claims from - say two - different different lines of business Y_{i j}(1) and Y_{i j}(2) are correlated for a given pair (i, j). A visual example of this model is shown in Figure2.

Copula

Figure 2: PWD model for two claim triangles

3.1.1 PWD model

In order to estimate the claims reserves corresponding to a set of claim triangles with the PWD model, it is first required to specify the model in terms of a joint distribution. For this model, the

joint distribution of the standardized incremental claims (Y_{i j}(1),Y_{i j}(2), . . . ,Y(Ni j) i j ) is given by: F_{i j}(y(1)_{i j} , . . . , y(N_{i j}i j)) = Prob(Y_{i j}(1)≤ y(1)_{i j} , . . . ,Y_{i j}(Ni j)≤ y(Ni j) i j ) = C(F (1) i j , . . . , F (Ni j) i j ; θ),

where F_{i j}(1), . . . , F_{i j}(Ni j)are the marginal distributions for cell (i, j) in triangle 1 to Ni j as mentioned in Section2.1. Besides that, the dependence is captured by the parameter vector θ. This specification enables the possibility to model incremental claims from multiple lines of business with different marginal distributions. Furthermore, because of this specification the model can easily be estimated with MLE (see, Section2.1.2).

The likelihood function, which is equivalent to the joint density of (Y_{i j}(1), . . . ,Y_{i j}(Ni j)), can be expressed by: f_{i j}(y(1)_{i j} , . . . , y_{i j}(Ni j)) = c(F_{i j}(1), . . . , F_{i j}(Ni j); θ) Ni j

∏

where c(·) denotes the probability density function corresponding to copula C(·). If the situation occurs where the data is imbalanced (Ni j < N), the density of the sub-copula corresponding to copula C(·) replaces the copula density in Equation (7) (Shi & Frees, 2011). Taking the logarithm of the above equation results into the log-likelihood for (Y_{i j}(1), . . . ,Y(Ni j)

i j ). The total log-likelihood for all cells (i, j) in triangle 1 to Ni j which satisfy i + j − 1 ≤ I is given by:

L= I

∑

∑

∑

∑

∑

Here, it is assumed that the pairwise dependence between cells in the triangles is the same, regard-less of the accident or development year (Shi & Frees,2011). Hence, the same copula density c(·) and the same parameter vector θ is used.

This assumption can be relaxed by taking different copulas for each accident year, development year or calendar year. If for example a different copula is specified for each accident year, the joint density and log-likelihood become respectively (Shi & Frees,2011):

fi j(y(1)_{i j} , . . . , y (Ni j) i j ) = ci(F (1) i j , . . . , F (Ni j) i j ; θi) Ni j

∏

L= I

∑

∑

∑

∑

∑

In this specification, copula densities ci for i ∈ {1, . . . , I} could be from the same copula family with different dependence parameters, or they are based on completely different copula families for each accident year (Shi & Frees, 2011). For example, the Clayton copula is used for the first few accident years, while the Gumbel copula is used for the other accident years.

Maximizing the total log-likelihood of the PWD model results in estimates of the required parameters, which are used to compute the outstanding claims reserve. These parameter estimates are influenced by the fact that the dependency structure is now included in the MLE. This changes the estimates of the location and shape parameters for each claim triangle in comparison with the parameter estimates of the independence model. Eventually, using the resulting location and shape parameters from the PWD model, we are able to predict the future claims with the specifications in Section 2.1.2. The PWD model captures the dependence between the lines of business in a simple pairwise manner. In the next subsection the assumption of independence for cells within each triangle is relaxed, which results in a more general structure of dependence.

3.2

Calendar year dependence between cells in claim triangles

Abdallah et al. (2016) propose a model which captures the dependence structure within and be-tween claim triangles by considering calendar year dependence. This dependence structure as-sumes that all claims with the same calendar year are dependent. Such a dependence may arise by changes on paid claims in a calendar year due to inflationary trends or strategic decisions for example (Abdallah et al.,2016). This is also called a calendar year effect. One easy way to model the calendar year effect is by adding a parameter to the systematic component in Equation (2) (see Abdallah et al.(2016)). This results into:

η_{i j}(n)= ζ(n)+ α_i(n)+ β(n)_j + ϒ(n)_t , n= 1, . . . , N,

where ϒ(n)_t (t = i + j − 1) denotes the calendar year effect. However, asAbdallah et al.(2016) state, this approach has some major disadvantages in comparison with the copula-based approach.

estimated. Due to the small sample size of a claim triangle, this might lead to over-parametrization. Besides that, the covariates have no predictive power, because they are only capable of modeling a calendar year effect for past calendar years. Therefore, they can not be used to estimate the outstanding claims reserve. Furthermore, capturing the dependence by using copulas allows for more flexibility and complexer dependence structures.

3.2.1 ICYD model

The second model in this thesis incorporates the calendar year dependence within each triangle, but supposes independence between the different lines of business. Using again the notation of Abdallah et al. (2016) this model is called the independence calendar year dependence model (ICYD). We show the structure of this model for two claim triangles in Figure3.

Copula Copula

Figure 3: ICYD model for two claim triangles

In a triangle, cells on the same diagonal represent incremental claims paid in the same calendar year. We denote a calendar year with t and cell (i, j) in a triangle corresponds to calendar year t= i + j − 1. In the ICYD model, the dependence relation between these cells is based on a copula. As mentioned earlier, in this model we assume that the claims of different triangles are independent of each other. This results in the following joint distribution of the standardized claims:

F_t(n)(yt− j+1, j, . . . , y1,t) = Prob(Y (n) t− j+1, j≤ y (n) t− j+1, j, . . . ,Y (n) 1,t ≤ y (n) 1,t) = C(F (n) t− j+1, j, . . . , F (n) 1,t ; θ)j=1,...,t, where t ∈ {2, . . . , I}. Note that the values in the first diagonal of each triangle are assumed to be independent of other values. Eventually, the log-likelihood function of the ICYD model for the nth

claim triangle is given by (Abdallah et al.,2016): L(n)= I

∑

∑

∑

where we assume that the dependence structure is equal for all claims in the diagonals t ∈ 2, . . . , I. This implies that the same copula c(n)(·) and parameter vector θ are used for each diagonal in the nth triangle. However, just like the PWD model, this assumption can be relaxed by taking different copulas or using different dependence parameters for each calendar year. In this case, the log-likelihood can be written as:

L(n)= I

∑

∑

∑

Finally, the total log-likelihood for all (N) lines of business can simply be computed by summing over all log-likelihoods L(n). In the next calendar year models, the assumption of independence between lines of business is relaxed.

3.2.2 HCYD model

The final model of Abdallah et al. (2016) is called the hierarchical calendar year dependence model(HCYD). This model uses a two-level hierarchical Archimedean copula (see (6)) to model the dependence structure within and between different lines of business. For each claim triangle n, a child copula Cnis used to model the calendar year dependence within the diagonals. This copula is the same for all the calendar years within the nth triangle. On top of that, a parent copula C0 models the dependence structure between the diagonals of different claim triangles. For two lines of business, the structure of the HCYD model is shown in Figure4.

Child Copula Child Copula Parent Copula

Figure 4: HCYD model for two claim triangles

same diagonal t is given by (Abdallah et al.,2016):

Ft(y(1)_{t− j+1, j}, . . . , y(1)_1,t, y_{t− j+1, j}(2) , . . . , y(2)_1,t, . . . , y(N)_{t− j+1, j}, . . . , y(N)_1,t )

= Prob(Y_{t− j+1, j}(1) ≤ y(1)_{t− j+1, j}, . . . ,Y_1,t(N)≤ y(N)_1,t ) = C(F_{t− j+1, j}(1) , . . . , F_1,t(N); θ)j=1,...,t

= C0(C1(F_{t− j+1, j}(1) , . . . , F_1,t(1); θ1),C2(F_{t− j+1, j}(2) , . . . , F_1,t(2); θ2), . . . ,CN(F_{t− j+1, j}(N) , . . . , F_1,t(N); θN); θ0)j=1,...,t,

where t ∈ {2, . . . , I} and θ = {θ0, θ1, θ2, . . . , θN}. Note again that the values in the first diagonal of each triangle are assumed to be independent of other values. This is due to the fact that the other diagonals are modeled with a two-level hierarchical Archimedean copula, which is not possible for the first diagonal. However, one could also consider to model the dependence between the values of all the first diagonals (t = 1) with a Archimedean copula, which uses the same dependence parameter vector θ0 as C0. This causes the dependence relation between different triangles to be the same for all diagonals. Furthermore, just as with the previous two dependence models, the assumption of using the same copula for each calendar year can be relaxed. Finally, this results into the following log-likelihood (seeAbdallah et al.(2016)):

L= N

∑

∑

∑

∑

θt= {θ0t, θ1t, θ2t, . . . , θNt}.

The advantage of the HCYD model is that it models the calendar year dependence within each triangle separate from the dependence structure between multiple lines of business. According to Abdallah et al. (2016), this offers a better and more realistic interpretation of the dependence between lines of business. Modeling the dependence between different lines of business by linking diagonals instead of cells with a copula, has the advantage of being applicable even when some data is missing in one of the claim triangles (Abdallah et al.,2016). This makes the model very flexible. However, due to this flexibility it also has some drawbacks. As mentioned earlier in Section2.2.3, hierarchical Archimedean copulas have to satisfy a few conditions. For the well-known families of Archimedean copulas, this means that the degree of dependence has to decrease with each level of hierarchy. In other words, the dependence parameters of the child copulas have to be larger than the dependence parameter of the parent copula. Nevertheless, this does not have to be true in practice, since the dependence between different lines of business could be larger than the calendar year dependence within each triangle. Moreover, the (log-)density of a hierarchical Archimedean copula is quite complex and hard to compute.

3.2.3 CCYD model

In order to model the dependence within and between claim triangles without relying on the rather complex hierarchical Archimedean copulas, we propose a model which assumes that all claims corresponding to the same same calendar year are dependent, regardless of the line of business. This model is called the complete calendar year dependence model (CCYD). The CCYD model is in fact a simplification of the HCYD model, using only one copula type to capture the entire dependence structure. A visual example of this model is shown below in Figure5.

Although the CCYD model does not possess the nice property of a separate modeling of the dependence within and between claim triangles, the model has a clear motivation. Inflationary trends, strategic decisions or regulatory changes might affect all the values in a specific calendar year simultaneously. In this case, the calendar year effect does not necessarily depend on the line of business, which means that the dependence structure within and between triangles does not have to be modeled separately. Furthermore, less parameters are needed when only one copula is used to model the entire dependence structure. Since claim triangles have rather small sample sizes, this

Copula

Figure 5: CCYD model for two claim triangles

is really an advantage of the CCYD model in comparison with the HCYD model. On top of that, any properly defined copula can be used to model this. Therefore, the model is not restricted by the nesting condition and the complexity of a hierarchical Archimedean copula.

The joint distribution of the CCYD model is given by:

Ft(y(1)_{t− j+1, j}, . . . , y(1)_1,t, y_{t− j+1, j}(2) , . . . , y(2)_1,t, . . . , y_{t− j+1, j}(N) , . . . , y(N)_1,t )

= Prob(Y_{t− j+1, j}(1) ≤ y_{t− j+1, j}(1) , . . . ,Y_1,t(1)≤ y(1)_1,t, . . .Y_{t− j+1, j}(N) ≤ y_{t− j+1, j}(N) , . . . ,Y_1,t(N)≤ y(N)_1,t ) = C(F_{t− j+1, j}(1) , . . . , F_1,t(1), F_{t− j+1, j}(2) , . . . , F_1,t(2), . . . , F_{t− j+1, j}(N) , . . . , F_1,t(N); θ)j=1,...,t,

where t ∈ {1, . . . , I} and C(·) denotes the copula with parameter vector θ. Besides that, the log-likelihood of this model is given by:

L= N

∑

∑

∑

∑

Note that this model also captures the dependence between the values in the first diagonals

copula for each calendar year t can be relaxed. In this case, the log-likelihood becomes: L= N

∑

∑

∑

∑

4

Model calibration, prediction and validation

In this section, we describe which tools and procedures are used to obtain the parameter estimates for the various models. Furthermore, we show how point estimates of the reserves are obtained and we describe simulation and bootstrap techniques that are used to obtain the predictive distribution of the outstanding claims reserve. Finally, we introduce different criteria and backtesting mechanisms, which we use to validate and compare the different models.

4.1

Calibration

In order to find the impact of using the various models on the outstanding claims reserve, we first need to calibrate these models. For this purpose, we use the open source programming language R. The calibration process for the independence model can be described in a number of steps. The first step of the calibration is to convert the claims X_{i j}(n)in each dataset to standardized claims Y_{i j}(n) by dividing the claims that belong to a certain accident year by the corresponding premium for that year, which serves as the exposure. Using these standardized claims, the second step is to estimate the parameters of the linear predictor for each claim triangle together with its additional shape parameter. We use both the log-normal distribution and the Gamma distribution (see Section2.1.1) to fit each claim triangle by maximizing the corresponding log-likelihood functions. First of all, we use functions inRwith regard to (generalized) linear models to obtain parameter estimates that serve as initial input for these log-likelihood optimizations. Specifically, we perform the following commands:

## Fit lognormal to std.claims with Linear Model

fit.lnorm <- lm(log(std.claims[,n])~as.factor(i)+as.factor(j))

## Fit gamma to std.claims with vector Generalized Linear Model fit.gamma <- vglm(std.claims[,n]~as.factor(i)+as.factor(j),

family=gamma2(lmu="loge", lshape="identitylink"))

ing the resulting parameter estimates as initial input, we perform the followingRcode to maximize the corresponding log-likelihoods:

## Fit log-normal to std.claims with Maximum Likelihood MLM.L <- optim(ini.L, loglikL, method=c("L-BFGS-B"),

lower=c(rep(-Inf, (2*LoT-1)), 0), upper=rep(Inf, (2*LoT)))

## Fit Gamma to std.claims with Maximum Likelihood MLM.G <- optim(ini.G, loglikG, method=c("L-BFGS-B"),

lower=c(rep(-Inf, (2*LoT-1)), 0), upper=rep(Inf, (2*LoT))).

Here, we denote the initial parameter vectors byini.Landini.Gand the log-likelihood functions by loglikL and loglikG. The R code that we use for these initial parameter vectors and log-likelihood functions is explicitly shown in AppendixA.2. Besides that,LoTdenotes the ‘Length of Triangle’ which is equal to I in our notation (Section2).

Next to the parameter estimates of the independence model, we have to obtain the parame-ter estimates of the various copula-based dependence models. Therefore, an optimization of the log-likelihood functions corresponding to these models is required (see Section 3 for these log-likelihood functions). We use the parameter estimates of the best-fitting marginal distributions in the independence setting as initial input for these optimizations. Specifically, we consider the dis-tribution with the lowest AIC value as the best-fitting fitting marginal disdis-tribution for a particular claim triangle (see Section4.3). Moreover, we use theR functionnlminb() to optimize the log-likelihood functions corresponding to the various dependence models. In order to perform these optimizations, the (log-)densities of the required copulas in these models have to be computed. The R packagecopula(Hofert, Kojadinovic, Mächler, & Yan,2017) contains several functions which are useful for this purpose. In the models of this thesis, we use copulas from the Gaussian, Frank, Clayton and Gumbel families. The functions from the copulapackage that are used to construct these copulas are respectively given by: normalCopula(), frankCopula(), claytonCopula() andgumbelCopula(). Furthermore, the function that we use to obtain the log-density for a specific copula type is given by

where cop denotes the copula and U denotes the data vector for which the log-density has to be computed.

Not only the densities of the just mentioned Gaussian and Archimedean copulas are required. As we show in Section3.2.2, the HCYD model uses a two-level hierarchical Archimedean copula. Just as with the other copulas, we require the (log-)density of this type of copula for the estimation of the parameters. For this purpose, again theRpackagecopulais used (seeHofert et al.(2017)). This package contains the functionnacLL() (nested archimedean copula Log-Likelihood), which is able to compute the log-likelihood for a particular two-level hierarchical Archimedean copula, given a certain data matrix. Specifically, this function is given by

nacLL(cop, U)

where cop denotes a two-level hierarchical Archimedean copula and U denotes the data matrix. Each row of this data matrix consists of a vector with values that are assumed to be linked by the given copula.

Although thenacLL() function is very useful, it has one minor disadvantage. This function can only be applied to a data matrix which has more than one row. This is a problem, since we use this function to optimize the log-likelihood of the HCYD model, which links cells from diagonals that all differ in size. This means that we have a different sized vector for each calendar year, which consists of values that are all assumed to be linked by the given copula. So in our case, we can not construct one data matrix U with multiple rows of the same length. Instead, we have to apply thenacLL() function multiple times to a single row vector which contains all the values from the diagonals that belong to the same calendar year (see the log-likelihood in Section3.2.2). We solve this issue by adapting thenacLL()function, so that it also works for a single row vector instead of a matrix. We show in AppendixA.3how this function is adjusted. Another drawback of thenacLL() function is that only the Clayton and Gumbel families are supported at the moment. Therefore, we only use these copula families in the HCYD model. Finally, in order to obtain the nacLL()function from thecopulapackage, we perform the followingRcommand:

4.2

Reserve prediction

Now that the calibration of the models is clear, we describe how point estimates of the outstanding claims reserves are obtained. Using these reserve estimates, we are able to compare the different copula regression models and measure the impact of using these models on the claims reserve. In order to expand this analysis, we describe a simulation method and a parametric bootstrap which are used to obtain the entire predictive distribution of the reserves.

4.2.1 Point estimates

We use the parameter estimates resulting from the calibration in the previous section to get point estimates of the outstanding claims reserve. Obviously, we calibrate our models on claim triangles that consist of data regarding past claims. In our notation, these are all the claims X_{i j}(n) that satisfy i+ j − 1 ≤ I. Such a claim triangle is also called the upper left claim triangle because of its shape. Just like the data on past claims, the future claims that have to be estimated also result into a triangle. We refer to this triangle as the lower right claim triangle. The claims X_{i j}(n) in this triangle all satisfy i + j − 1 > I. These future claims form the basis of the outstanding claims reserve.

We obtain a point estimate of the outstanding claims reserve by using the following steps. First of all, we convert the parameter estimates resulting from the calibration to the corresponding loca-tion parameters of each claim triangle by using Equaloca-tion2and the specifications in Section2.1.1. We then use these location parameters together with the estimated shape parameters to compute the expected value of the future standardized claims

 ˆ E[y(n)i j ]



in the lower right claim triangles (see Section2.1.2). These expected values are the point estimates of the future standardized claims ( ˆy(n)_{i j} ). After that, we multiply these point estimates by their exposure (ω_i(n)) in order to obtain estimates of the future claims. We use these estimated future claims as input for Equation (1) to obtain a point estimate of the outstanding claims reserve for each claim triangle. Finally, the to-tal outstanding claims reserve for the entire insurance portfolio is simply equal to the sum of the reserves for all lines of business.

4.2.2 Simulation

The simulation method in this section is equivalent to the ones described in Shi and Frees(2011) andCôté et al.(2016). Thanks to the parametric structure of the copula regression models, such a simulation can easily be performed for these models. We describe the simulation procedure by the following steps:

1. Simulate realizations u(n)_{i j} from the corresponding copula regression model for all accident years i and development years j that satisfy i + j − 1 > I.

2. Transform these realizations to predictions of the standardized claims for the lower right claim triangles by y(n)_{i j} = F(n)(−1)(u(n)_{i j} , ˆη_{i j}(n), ˆγ(n)) for n = 1, . . . , N. Here, ˆη_{i j}(n) and ˆγ(n) re-spectively denote the estimates of the linear predictor and the additional parameters as defined in Section2.1.

3. Obtain a prediction of the outstanding claims reserve for the entire insurance portfolio by: N

∑

∑

∑

where ω_i(n) denotes the exposure for accident year i in the nth claim triangle.

The above procedure is repeated for the total number of simulations to obtain a predictive distribu-tion of the reserves.

4.2.3 Parametric bootstrap

Shi and Frees(2011) show that the above simulation method can be expanded to include parameter uncertainty in the predictive distribution by using a bootstrapping approach. Using this approach, we create for each simulation a new upper left claim triangle and fit the corresponding copula regression model to this new triangle. In this way, we obtain a different parameter set for each simulation instead of always using the same parameters. Consequently, the predictive distribution resulting from these simulations now includes parameter uncertainty. We describe the parametric bootstrap by the following procedure (Shi & Frees,2011):

1. Simulate realizations u(n)_{i j} from the corresponding copula regression model for all accident years i and development years j that satisfy i + j − 1 ≤ I.

2. Transform these realizations to estimates of the standardized claims for the upper left claim triangles by y∗(n)_{i j} = F(n)(−1)(u(n)_{i j} , ˆη_{i j}(n), ˆγ(n)) for n = 1, . . . , N. Here, the y∗(n)_{i j} values denote the so-called pseudo standardized claims.

3. Obtain estimates of the linear predictorsηˆ_{i j}∗(n) 

, the additional marginal parametersγˆ∗(n) and the dependence parameters θˆ∗
by maximizing the log-likelihood of the corresponding copula regression model for the pseudo standardized claims y∗(n)_{i j} .

4. Using these parameter estimates, simulate the outstanding claims reserve for the entire insur-ance portfolio by performing the steps of the simulation procedure in Section4.2.2.

We repeat the above procedure for the total number of simulations to obtain a predictive distribution of the reserves.

4.3

Model validation

In the previous subsection, we describe how the reserves for the different copula-based dependence models are computed. In this subsection, we propose a set of criteria and backtesting mechanisms which are used to compare and validate those models.

4.3.1 Likelihood-based criteria

A well-known and widely used method to compare likelihood-based models is the Akaike Informa-tion Criterion(AIC). The AIC is defined as follows (Akaike,1974):

AIC = −2` + 2k,

where ` denotes the value of the log-likelihood of the model considered and k denotes the corre-sponding number of parameters. Because the datasets for the models in this thesis are relatively small, models using too many parameters would overfit the data. Therefore, we also consider an-other well-known information criterion, which imposes a larger penalty for including too many

parameters in the model. This criterion is called the Bayesian Information Criterion (BIC). The BIC is given by:

BIC = −2` + k log(n),

where n is the number of observations (Schwarz,1978). Both criteria are used to rank the various copula regression models investigated in this thesis. The model with the lowest AIC or BIC value is considered to be the best model.

4.3.2 Backtesting

Besides looking at the likelihood-based criteria, backtesting is another way to validate the dif-ferent models. In contradiction to the likelihood-based criteria, which give in-sample evaluations, backtesting allows for out-of-sample evaluations. We describe the backtesting procedure by the fol-lowing steps. First, we remove the last t diagonals from the original claim triangles. Here, t denotes the number of diagonals. After that, we re-estimate the values belonging to these diagonals, using the calibration and claim estimation procedures as described in Section 4.1and4.2.1. Finally, we compare the observed claims from the original diagonals to the corresponding re-estimated claims by using two different measurements.

Step 1: Remove the last t diagonals of the original data Step 2: Re-estimate the values of the removed diagonals

Step 3: Compare the observed values with the estimated values

We show the backtesting procedure above in Figure 6. Note that we are not able to re-estimate all the cells of the removed diagonals, due to the structure of the copula regression models. For example, consider the backtesting procedure in Figure6where I = J = 8. Removing one diagonal, causes the remaining triangle to have I = J = 7. Consequently, we are only able to re-estimate six values belonging to this removed diagonal.

The two measurements that we use to compute the prediction error are the so-called cell error and total error (Agbeko, Martínez-Miranda, Nielsen, & Verrall, 2014). The cell error is defined as the square root of the sum of squared errors divided by the sum of squared observations. The total error is defined as the absolute value of the sum of errors divided by the sum of observations. Specifically, these measurements are given by

v u u u u t ∑ni j   X_{i j}(n)− ˆX_{i j}(n)2  ∑ni j h X_{i j}(n)2 i and       ∑ni j  X_{i j}(n)− ˆX_{i j}(n)  ∑ni jX (n) i j       (8)

respectively. Both measurements are computed for the entire insurance portfolio. Furthermore, we also compute these measurements for each claim triangle separately, by choosing a fixed value for n.

In this thesis, we specifically perform the backtesting by cutting off one diagonal, two diagonals or three diagonals and then we compute the measurements in Equation (8) using the observed and re-estimated values corresponding to these removed diagonals. Each time backtesting is used, we follow the calibration strategy as outlined in Section 4.1. Hence we first estimate the marginal distributions and then use the parameter estimates of the best-fitting distribution (log-normal or Gamma) as starting values in the calibration of the various dependence models.

5

Results

In this section, we present and discuss the results of applying the various dependence models to two different datasets. In doing so, we set a few objectives. We attempt to replicate the results of Abdal-lah et al.(2016) by using our own code to model the corresponding dependence models. Moreover, we expand our analysis by introducing another copula regression model, the CCYD model. Using all these models, we attempt to find the best model specification for modeling the claims reserves, considering the dependence structure within and between different lines of business. Finally, we show the impact of using those dependence models on the claims reserve, in comparison with the independence model.

First of all, we describe the two datasets that we use in our analysis. Thereafter, we present and discuss the results for each dataset in the subsequent subsections. Specifically, we start in each of these subsections with the goodness of fit criteria of the marginal distributions to determine the best-fitting distribution for each claim triangle. After that, we show the goodness of fit criteria and point estimates of the reserves for the various copula regression models and compare these with the corresponding results for the independence model. Furthermore, we determine which models provide the best fits according to the likelihood-based criteria. For these models and the independence model, we also analyze the results of the backtesting procedure, the simulations and the parametric bootstrap.

5.1

Data

As mentioned above, we use two datasets for the comparison of the different dependence models. Both datasets have two claim triangles with incremental paid claims as entries. The first dataset, which consists of data from a major US property-casualty insurer, is the same as the one that Shi and Frees (2011) and Abdallah et al. (2016) use. The claim triangles in this dataset correspond to the paid claims of an insurance portfolio which consists of two lines of business: personal auto and commercial auto. The claim triangles in the second dataset come from a Canadian property-casualty insurer and have also been used byCôté et al.(2016) andAbdallah et al.(2016). The first triangle in this dataset consists of paid claims of the Accident Benefits coverage from Ontario and

et al.(2016)). Both triangles in each dataset have ten accident years and ten development years.

Figure 7: Standardized claims of US property-casualty insurer

1 1 1 1 1 1 ₁ 1 1 1 2 4 6 8 10 0.00 0.10 0.20 0.30

Incremental claims LoB 1

development year v alue 2 2 2 2 2 2 _{2 2 2} 3 3 3 3 3 3 3 3 4 4 4 4 4 ₄ 4 5 5 5 5 5 ₅ 6 6 6 6 6 7 7 7 7 8 8 8 9 9 0 1 1 1 1 1 1 1 1 1 1 2 4 6 8 10 0.3 0.5 0.7

Cumulative claims LoB 1

development year v alue 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 8 8 8 9 9 0 1 1 1 1 1 1 1 ₁ 1 1 2 4 6 8 10 0.00 0.10 0.20 0.30

Incremental claims LoB 2

development year v alue 2 2 2 2 2 2 2 2 ₂ 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 8 8 8 9 9 0 1 1 1 1 1 1 1 1 1 1 2 4 6 8 10 0.1 0.3 0.5 0.7

Cumulative claims LoB 2

development year v alue 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 8 8 8 9 9 0

Figure 8: Standardized claims of Canadian property-casualty insurer

1 1 1 1 1 1 1 1 1 1 2 4 6 8 10 0.00 0.01 0.02 0.03

Incremental claims LoB 1

development year v alue 2 2 2 2 2 2 2 2 ₂ 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 8 8 8 9 9 0 1 1 1 1 1 1 1 1 1 1 2 4 6 8 10 0.02 0.06 0.10

Cumulative claims LoB 1

development year v alue 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 8 8 8 9 9 0 1 1 1 1 1 1 1 1 ₁ 1 2 4 6 8 10 0.02 0.06 0.10

Incremental claims LoB 2

development year v alue 2 2 2 2 2 2 2 2 ₂ 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 7 7 ₇ 7 8 8 8 9 9 0 1 1 1 1 1 1 1 1 1 1 2 4 6 8 10 0.1 0.2 0.3 0.4

Cumulative claims LoB 2

development year v alue 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 8 8 8 9 9 0