Individual reserving by detailed conditioning : a parametric approach

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Individual Reserving by Detailed Conditioning —

A parametric approach

Robert Kroon

Graduation Thesis for the Bachelor’s Actuarial Studies University of Amsterdam Faculty Economics & Business Amsterdam School of Economics Author: Robert Kroon Studentnr: 10171428

E-mail: Robertkroon2@hotmail.com Date: July 4, 2014

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Individual Reserving by Detailed Conditioning — Robert Kroon iii

Abstract

Nowadays, many insurers use claim triangles in order to calculate reserves. In this arti-cle we build a parametric alternative for the method used in Godecharle and Antonio (2014). They construct a model in order to calculate reserves using data registered at individual claim level. To do so these authors use so-called claim markers, and condition on these claim characteristics. They then use historical simulation to simulate the IBNR and RBNS reserves. In this article we use the same approach, but use parametric distri-butions to simulate payments. Our results are then compared to results obtained using the method ofGodecharle and Antonio (2014) and to classic bootstrap results. We find that a parametric approach results in similar reserves as the historical approach, and in some cases leads to smaller deviations. We also conclude certain distributions tend to have a bad fit on smaller datasets, which is the main drawback of our approach.

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Preface

In April 2014 the final project of my Bachelor’s Actuarial Studies commenced: the the-sis. For this thesis I studied Individual Claim Reserving, a relatively new subject in the insurance market, especially for me.

First off, I’d like to thank my colleague Yoeri Arnoldus for bringing this interesting subject to my attention. I was immediately intrigued and I learned a lot. Furthermore I’d like to thank all my colleagues for assisting me with this thesis. In particular I want to thank Cees Attema and Yoeri Arnoldus for their technical support where necessary. Finally I want to thank Triple A - Risk Finance for giving me an oppurtunity to write this thesis, and more generally for giving me the chance to develop my actuarial skills. For the assistance at the university I want to express my gratitude to Professor

Katrien Antonio; her comments and assistance were very valuable. I’d also like to thank her and her colleague Els Godecharle for making their data and R-code available to me. Without those, this research would not have been possible.

Robert Kroon

Amsterdam, July 4, 2014.

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Chapter 1

Introduction

In order to calculate the present value of future liabilities, insurers use a variety of methods. Up until a few years ago, the only methods used were based on aggregated data, presented in so-called claim triangles. The methods used on these triangles are in essence variations and extensions of the simple chainladder method. Notable examples are the Mack method, described byMack (1993) and the Bootstrap method, described byEngland and Verrall (1999)among others. The use of these methods can be explained relatively easily, since they are quite straightforward and do not require extensive calcu-lations in order to compute the reserves. These methods were particularly popular when computers did not have a lot of processing power, which is the main reason they were proposed. These methods however face one big drawback: they reduce a large dataset into a small sample design. This means a lot of information is lost by aggregating the data, resulting in higher variances and perhaps even biased estimates1.

That is why, over the last few years, researchers have developed methods that are ca-pable of processing the full dataset. In this paper we will implement ideas suggested by

Drieskens et al. (2012),Rosenlund (2012),Pigeon et al. (2013)and others. They describe ways to calculate reserves using data registered at individual claim level, while working with discrete time intervals. Our starting point is the article written byGodecharle and Antonio (2014) on reserving by conditioning on individual claim markers.

Godecharle and Antonio (2014) expand the method described by Rosenlund (2012)

which he calls the Reserve by Detailed Conditioning (RDC) method. Godecharle and Antonio develop a stochastic version of this method, which allows for conditioning on individual claim characteristics. Then they simulate the claims runoff based on histor-ical data. The main difference between this article and theirs is the way we simulate future payments. Godecharle and Antonio use historical simulation, whereas we will fit distributions to the data, given their characteristics. This might improve the estimates of the reserves and reduce their variance. Thus, the aim of our paper is to formulate a parametric alternative for the approach inGodecharle and Antonio (2014)using empiri-cal distributions for individual claims reserving. Since the same dataset will be used, the results can be directly compared to those of Godecharle and Antonio. As a benchmark, we also include results obtained with the chainladder bootstrap.

The rest of the article is as follows. Chapter 2 describes the theoretical basis of the Bootstrap method and the RDC approach. Chapter 3 describes the data and our ap-proach for simulating.Chapter 4summarizes the results of the parametric RDC method and compares them to similar methods. Chapter 5will conclude.

1_{The findings of}_{Pigeon et al. (2013)}_and_{Godecharle and Antonio (2014)}_{support this claim.}

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Chapter 2

Theoretical Framework

In this chapter we review the most important methods used in this thesis. First we describe the chainladder bootstrap in more detail. Next we propose a way to discretize our data so it is usable for simulating. Lastly, we describe the model set up byGodecharle and Antonio (2014) and outline our adaptations of their model.

2.1 The chainladder bootstrap

The chainladder bootstrap, or bootstrap in short, is a method first described by Eng-land and Verrall (1999). They use a general bootstrap framework described by Efron and Tibshirani (1993). England and Verrall expand this framework in order to calculate reserves using claim triangles. In essence the bootstrap method creates pseudo-data in order to simulate reserves. Resampling from the historical datal allows for calculations of the variance and other risk measures if desired.

Section 2.1.1describes the chainladder method. Insection 2.1.2we introduce the chain-ladder bootstrap.

2.1.1 The chainladder method

This section follows the notations of England and Verrall (2002).

In the chainladder model, we use a claim triangle in order to calculate the ultimate claim and therefore the required reserve. These triangles contain payments of the en-tire portfolio for n periods, or in our case, n years. On the vertical axis we denote the accident year and on the horizontal axis the development year. Since each consecutive accident year yields one year less of observed data, we get the distinct triangular form. In this model, so-called development factors are calculated in order to predict future payoffs. Denote Cij as the incremental value paid in accident year i and development year j. Here, i ranges from 1 to n and j ranges from 1 to n − i + 1. Next, we define the cumulative payments in an accident year as follows:

Dij = j X

k=1

Cik. (2.1)

The chainladder method assumes each consecutive cumulative payment Dij is obtained by taking the cumulative value of the previous (development) year and multiplying it with a development factor denoted by λj (j = 2, ..., n). We estimate these development factors using: ˆ λj = Pn−j+1 i=1 Dij Pn−j+1 i=1 Di,j−1 . (2.2) 2

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Individual Reserving by Detailed Conditioning — Robert Kroon 3

In Figure 2.1 we see an example of a cumulative claim triangle, which we can use to illustrate how the development factors λj are calculated:

Accident Year Development Year

i / j 1 2 3 4 5 6 7 8 1 D1,1 D1,2 D1,3 D1,4 D1,5 D1,6 D1,7 D1,8 2 D2,1 D2,2 D2,3 D2,4 D2,5 D2,6 D2,7 3 D3,1 D3,2 D3,3 D3,4 D3,5 D3,6 4 D4,1 D4,2 D4,3 D4,4 D4,5 5 D5,1 D5,2 D5,3 D5,4 6 D6,1 D6,2 D6,3 7 D7,1 D7,2 To be estimated 8 D8,1

Figure 2.1: Graphical description of a claim triangle

For instance: the first development factor λ2 is calculated by dividing the sum of the second column by the sum of the first column, excluding D8,1. Then, in order to calculate the future payments, labeled with ‘To be estimated’, the following formulas are used:

ˆ

Di,n−i+2= Di,n−i+1λˆn−i+2 (2.3) for the diagonal, and:

ˆ

Di,k = ˆDi,k−1λkˆ k = n − i + 3, n − i + 4, ..., n (2.4) for the lower triangle.

In this manner, the expected ultimate loss can be calculated, and therefore the required reserve. This reserve R is equal to:

R = n X k=1 ˆ Dkn− n X k=1 Dk,n−k+1. (2.5)

2.1.2 The bootstrap method

The bootstrap method has a few advantages over the chainladder method. Whereas the chain ladder method only gives a best estimate of the reserve, the bootstrap method also gives risk measures and a predictive distribution of the reserve. The bootstrap method uses results of the chainladder method in order to simulate the reserve. It utilizes the development factors λj and the diagonal (where i + j = n + 1) in order to create a so-called ’bootstrap triangle’ using:

D_i,j∗ = Di,j, if i + j = n + 1 (2.6)

D_i,j∗ = Di,j+1 λj+1

, if i + j < n + 1. (2.7)

The resulting incremental triangle can be calculated with formula (2.8):

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4 Robert Kroon — Individual Reserving by Detailed Conditioning

The resulting triangle is in general completely different from the actual triangle, with the exception of the last increment of the first row, and the first and only increment of the last row (in our case respectively C1,8 and C8,1). In order to simulate new reserves, an adjusted ’pearson’ residual is defined for each individual increment. This residual is defined as: rp_i,j= s k k − p Ci,j − C_i,j∗ q C_i,j∗ with: (2.9)

p = 2n − 1 The number of parameters (2.10) k = n(n + 1)

2 The number of observations (2.11)

These (adjusted2 and normalised) residuals are then resampled (denoted by r_i,j∗ ) and used to create a new artificial residual triangle. A new artificial triangle results by applying formula (2.12).

C_i,jart=qC_i,j∗ r∗_i,j+ C_i,j∗ (2.12) This new incremental triangle is used to calculate a simulated reserve using the chainlad-der method, as described insection 2.1.1. By doing this a sufficient number of iterations, variances and confidence intervals can be derived for the reserve. For further reading on this method we refer toEngland and Verrall (1999) andEngland and Verrall (2002).

2.2 Reserving by Detailed Conditioning

In the previous section we described the chainladder bootstrap, a method for calculat-ing reserves based on aggregated data displayed in claim triangles. The last few years, researchers have worked on methods that are capable of processing the full dataset of individual records. Notable examples are Norberg (1993), Haastrup and Arjas (1996),

Norberg (1999) andAntonio and Plat (2013). They describe ways to calculate reserves using data registered at individual claim level, while working with continuous time.

Drieskens et al. (2012), Rosenlund (2012), Pigeon et al. (2013) and Godecharle and Antonio (2014) have a similar approach, but instead use discrete time intervals. In this section we describe the method used by Rosenlund (2012) and Godecharle and Antonio (2014)based on reserving by detailed conditioning and propose an alternative. For the sake of consistency, we use the same notation as they do.

2.2.1 Discretizing the data

Following the approach used in Godecharle and Antonio (2014) and Rosenlund (2012)

we discretize the data into years3 and work with the discrete dataset. We convert the occurence, reporting and closing dates into the following variables for each claim k:

• the occurence period for claim k, i(k) (with i(k) ∈ 1, ..., n).

• the reporting period for claim k, W (k). If the claim is reported in the same year as it occured W (k) = 1.

• the closing period for claim k, F (k). If the claim is closed in the same year as it occured F (k) = 1.

2

Note that the adjustment factor asymptotically vanishes, and only exists for n > 2.

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• the sum of all payments in year j, denoted by Y (k, j), with j = 1, ..., F (k). Using this notation, we summarize a claim k as:

{i(k), W (k), F (k), Y (k, 1), ..., Y (k, F (k))} (2.13) For the claims that have already been closed, this information set is complete. For the unclosed claims the missing information has to be simulated, which leads to simulated values of the reserve. In order to illustrate the conversion of the dataset, we have added

Figure 2.2, taken fromGodecharle and Antonio (2014). Here we convert a claim k∗ and its continuous characteristics into the discrete characteristics described in this section.

Event Payment Date Our Notation

Accident 05/17/1997 i(k∗) = 1

Reporting 02/02/1998 W (k∗) = 2 Cash Flow e200 11/24/1998 Y (k∗, 2) = 200

e150 02/08/1999 Y (k∗, 3) = 250 e100 05/11/1999 Y (k∗, 4) = 0 e50 02/23/2001 Y (k∗, 5) = 50 Y (k∗, 6) = 0 Closure 03/12/2002 F (k∗) = 6

Figure 2.2: Example of discretization of a claim, taken from Godecharle and Antonio (2014), page 3

For this fabricated claim k∗, i(k∗) = 1 since the accident occured in the starting year of the datset. Because the accident was reported after a year, we have W (k∗) = 2. Then, after four more years the claim is closed, so F (k∗) = 6. Next, we can see that it is possible for an accident to have multiple or zero payments in a single year. We accumulate payments made in the same year so we do not have to worry about multiple payments in one year. In this thesis we allow payments Y (k, j) to be equal to zero.

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2.2.2 Claim Markers

As mentioned before, we will simulate the future of a claim by conditioning on so-called claim markers. A claim marker is a marker based on the characteristics of a claim. We assume that closing probabilities and payments of claims with the same markers are independent and identically distributed (i.i.d.). Following the approach of Godecharle and Antonio (2014), we use three different markers, which we will describe in this section. These markers are:

• the claim length, from now on denoted by L(k).

• the (quantile of the) last observed cumulative payment, Q_t(k). • the reporting delay W (k).

Note that we use a generic framework. With enriched data, variables used in pricing theory such as gender and age could be added as well. This might be interesting for insurers with the available data.

Claim Length

We specify the length of a claim as:

L(k) = F (k) − W (k) + 1 (2.14)

Using equation (2.14)and W (k) we derive two important characteristics of a claim: • if W (k) ≤ n − i(k) + 1, a claim is reported.

• if L(k) ≤ n − i(k) − W (k) + 2, a claim is closed. This is equivalent with requiring that F (k) ≤ n − i(k) + 1.

With these two characteristics we can define three types of claims. First we define closed claims as claims that are both reported and closed. For these claims both L(k) and W (k) are known. For a claim that is reported but not settled (RBNS), only the first condition must be satisfied. This means both the length and the reporting delay are known, but the length L(k) is censored at the moment of evaluation. Therefore we observe a length that is smaller or equal to the actual length of the claim. For Incurred but not Reported claims (IBNR), neither condition must be satisfied. So for an IBNR claim both the length of the claim and the reporting delay are unknown.

Therefore we define Lmin(k) as the observed length of an unclosed claim. For a re-ported claim we have Lmin(k) = n − i(k) − W (k) + 2 and for an unreported claim Lmin(k) = 0. In chapter 3 we discuss a method to simulate the actual length of both reported and unreported claims.

Note that we assume claims have completely developed 8 years after occurring (which is consistent with the triangle range). It would however be quite simple to add so-called tail development, as Drieskens et al. (2012)among others have done.

Cumulative Payments

As the second marker, we use the last known quantile of the cumulative payments of a claim. FollowingRosenlund (2012), we define the cumulative claims for a claim k up to time t as: H(k, t) = t X h=1 Y (k, h + W (k) − 1) (2.15)

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with t ∈ (0, 1, . . . , n), the period since reporting. If t = 0, no data is available and the claim is not yet reported. Additionally, if t = L(k), it means there is no further devel-opment, so that H(k, L(k)) is the total cost for claim k. The reserve we are interested in then equals H(k, L(k)) − H(k, Lmin(k)), since this expression is equal to all future payments of claim k. For the claim in Figure 2.2we have:

H(k∗, 1) = 200, H(k∗, 2) = 450, H(k∗, 3) = 450, H(k∗, 4) = 500, H(k∗, 5) = 500 We apply quantile binning to structure the information on cumulative payments, i.e. we divide claims into a fixed number (q0) of quantiles, and assume the payments in each quantile have the same distribution. We define Qt(k) as the quantile claim k belongs to at time t. Since there is no claim information at t = 0, we have Q0(k) = 1 for every k. For a more detailed description of quantile binning, we refer toGodecharle and Antonio (2014).

Reporting Delay

Finally, we use the reporting delay W (k) as a claim marker. We choose to do so, be-cause we think claims with the same reporting delay tend to have a similar development pattern. Note that the occurrence date of the claim does not matter, since none of our markers depend on i(k).

Extra Parameters

Using these claim markers in order to simulate future claim development may prove to be problematic, since some combinations of these markers may result in very small clusters. This makes them unreliable and susceptible to outliers. In order to assess this, we define a few extra parameters to combine small clusters:

• we combine claims with a long developing period (L(k) ≥ λ₀), adjusting our claim marker gives us the bounded claim length: Lb(k) = Min{L(k), λ0}

• we combine claims with a long reporting delay (W (k) ≥ ω₀), adjusting our claim marker gives us the bounded reporting delay: Wb(k) = Min{W (k), ω0}

• we identify late payments (payments at t ≥ τ₀), we assume payments after t = τ0 have the same conditional distribution as payments made in the year t = τ0 With these adjusted claim markers we set up our stochastic simulation framework. Note that the values of these parameters affect the simulation process. In order to compare our results to those ofGodecharle and Antonio (2014)we use the same parameter values and data as they do. For a sensitivity analysis of the parameters we refer toGodecharle and Antonio (2014).

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Chapter 3

Case Study

Building upon our theoretical framework laid down in the previous section, we now construct a method to simulate the future of individual claims. After analyzing the data, we formulate a way to simulate the actual claim length for unclosed claims. We use the same approach asGodecharle and Antonio (2014), because it is equivalent with fitting a multinominal distribution on the dataset. Next, we need to simulate the number of IBNR claims, since the amount of claims is unknown. Here we will follow the appoach used in Pigeon et al. (2012). Lastly, we will simulate payments for respectively IBNR and RBNS claims, and construct their corresponding reserves. We will do so by fitting distributions to the claims, given their claim markers. This is the main difference of our reseach compared toGodecharle and Antonio (2014), since they use historical simulation for the payments.

3.1 The data

We use the same dataset asAntonio and Plat (2013),Pigeon et al. (2013)andGodecharle and Antonio (2014). This dataset contains payment data and reporting and/or closing dates for general liabilty insurance policies of individuals. It consists of claims related to both Material Damage (MD) and Bodily Injury (BI). If claims have both a MD and a BI part, these will be considered seperately. We have available data for 224,836 MD claims and 4,483 BI claims of which respectively 220,730 and 3,452 are already closed. These claims contain data from January 1997 to August 2009. We will split the data into two subsets, one ranging from January 1997 to December 2004 and the other from January 2005 to August 2009. We will use the first dataset for the actual simulation, the rest of the data will be used as a control group to validate our results.

It should be noted that the payments Y (k, t) in the dataset are adjusted for infla-tion to January 1, 1997. This means the occurence date of the claim does not matter, since we correct for inflation and do not condition on occurence. We also assume that all payments, conditional on the claim markers, are independent and identically distributed.

Figure 3.1 and 3.2 show the incremental claim triangles for both claim types. As can be seen, the development of the claim types is quite different. We see that BI claims tend to have a longer development period than MD claims. Additionally the payments for individual BI claims are higher than the MD payments. Note that we use the upper triangle for the actual simulation, and use the lower triangle for validating our results.

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Individual Reserving by Detailed Conditioning — Robert Kroon 9 Year 1 2 3 4 5 6 7 8 1997 261 614 359 526 546 137 130 339 1998 202 473 307 336 269 56 179 78 1999 238 569 393 270 249 286 132 97 2000 237 557 429 496 406 365 247 275 2001 389 628 529 559 446 375 147 239 2002 260 570 533 444 132 122 332 1,082 2003 236 743 558 237 217 205 171 2004 248 794 401 236 254 98

Figure 3.1: Incremental claim triangle for BI. The values are in thousands

Year 1 2 3 4 5 6 7 8 1997 4,427 992 89 13 39 27 37 11 1998 4,389 984 60 35 76 24 0.6 16 1999 5,280 1,239 76 110 113 12 0.4 0 2000 5,445 1,164 172 16 6 10 0 10 2001 5,612 1,838 156 127 13 3 0.4 3 2002 6,593 1,592 74 71 17 15 9 9 2003 6,603 1,660 150 52 37 18 3 2004 7,195 1,417 109 86 39 15

Figure 3.2: Incremental claim triangle for MD. The values are in thousands The difference in development patterns can be illustrated more clearly if we show the (empirical) distribution of the reporting delay W (k) (Figure 3.3) and the Claim Length L(k) (Figure 3.4) of both claim types.

Based on these figures we can conclude both claim types have a signifcantly differ-ent distribution in terms of claim length and reporting delay. Therefore we have to simulate both claim types seperately.

Reporting Delay 1 2 3 4 5 6 7 8 Total

Bodily Injury 1,337 1,392 414 173 81 36 15 4 3,452

% 29.8% 31.1% 9.2% 3.9% 1.8% 0.8% 0.3% 0.3% 77.0%

Material Damage 198,426 21,091 757 199 77 123 20 37 220,730

% 88.3% 9.4% 0.3% 0.1% 0.0% 0.1% 0.0% 0.0% 98.2%

Figure 3.3: Claim Length of both claim types

Reporting Delay 1 2 3 4 5 6 7 8 Total

Bodily Injury 4,077 332 46 13 9 6 0 0 4,483

% 90.9% 7.4% 1.0% 0.3% 0.2% 0.1% 0.0% 0.0% 100.0%

Material Damage 216,906 7,764 137 23 9 7 0 0 224,836

% 96.5% 3.5% 0.1% 0.0% 0.0% 0.0% 0.0% 0.0% 100.0%

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Lastly, we report the distribution of the actual payments, since they will be used to fit the distributions of future payments. We can see Payments made for BI claims tend to be more right skewed and have longer development periods than MD paymens. We will refer to these tables in section 4.1 to set parameter values of the parameters described insection 2.2.2.

Period Since Reporting 1 2 3 ≥ 4

Bodily Injury Observed 4,483 2,669 1,071 1,046 Mean 575.5 1,787.6 2,346.4 3,533.1 Median 89.1 281.0 100.0 0.0 95% quantile 2,411.9 8,340.2 10,814.1 18,648.5 Maximum 143,248 112,825 212,167 351,860 % zero payments 39.0% 32.8% 44.4% 53.1% Material Damage Observed 224,836 22,872 1,489 1,408 Mean 212.0 328.4 356.2 405.6 Median 81.2 126.3 0.0 0.0 95% quantile 801.1 1,115.6 1,197.3 271.9 Maximum 153,816 174,387 105,114 87,232 % zero payments 25.2% 29.7% 78.7% 93.0%

Figure 3.5: Characteristics of payments of both claim types, given their reporting delay

3.2 Simulating (actual) claim length

In this section we will drop subscript k for ease of notation. For a claim that is not yet closed, we can specify the one year probability that the claim is closed the following year (or equivalently that L = λ). This probability is defined as:

r(λ; q, w, t) = P (L = λ|L ≥ λ, Qt= q, Wb = w) (3.1) We assume this probabilty is the same for claims with the same claim markers. Using this one year probabilty we can define the distribution of the claim length at period t since reporting as:

p(λ; q, w, t) = P (L = λ|L > t, Qt= q, Wb= w) (3.2) Or equivalently, using (3.1): p(λ; q, w, t) = r(λ; q, w, t) · λ−1 Y m=t+1 [1 − r(m; q, w, t)] (3.3)

In order to estimate the one way probabilities r we will use a binomial maximum-likelihood estimator. We therefore define:

• IF_{(λ; q, w, t) as the number of finalized claims with L = λ, given L > t, Q} t = q and Wb = w, and

• J (λ; q, w, t) as the number of reported claims with L = λ, given L > t, Q_t= q and Wb = w.

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We then estimate r(λ; q, w, t) with the following equations:

ˆ r(λ; q, w, t) = I F_{(λ; q, w, t)} J (λ; q, w, t) (λ < n − w + 1) (3.4) ˆ r(n; q, w, t) = 1 (3.5)

Equation (3.5) ensures claim development is restricted to the triangle. With the esti-mates of r we calculate values of p(λ; q, w, t) using equation(3.3). Using these probaili-ties, we simulate the claim length of unclosed claims, which we will denote by Ls(k) as inGodecharle and Antonio (2014). Note that we use both finalized and open claims to calculate the closing probailities. This greatly reduces the variance of our estimates, as more observations are used.

When simulating we condition on the most recent information available. For RBNS claims this means we use the information of the last known year. For IBNR claims this means we are unable to condition on observed claim markers, since there is no information available for these claims.

3.3 Simulating claim development

After simulating lengths for open claims, we need to simulate payments. For RBNS claims we have t = n − i(k) − W (k) + 2 and simulated claim length Ls(k). To simulate future payments we assume claims with the following characteristics have the same distribution:

• the same bounded simulated claim length Ls

b(k) = min{Ls(k), λ0}. • the same bounded reporting delay W_b(k).

• the same cumulative quantile Q_t(k), with t as above.

We use these claim markers, based on the last observed t, throughout the simulation. This means the claim markers are not dynamic, but stay constant throughout the sim-ulation of the payments. Note that since IBNR claims are not yet reported, we cannot condition on Qt(k), and have to simulate the amount of claims and their reporting delay as well. Because we simulate the claim length and reporting delay for the IBNR claims, we can condition on these two markers.

As mentioned before, our subsets of claims with these markers contain both open and closed claims. We use these observations to fit a distribution to the payments of each cluster, using the Akaike Information Criterion4 (AIC) as a statistical measure of the goodness of fit. To do so we will use the R-package ”GAMLSS”. With this package we can fit distributions to our data, without being restricted to the exponential family. Distributions such as the Pareto and Lognormal distribution can then be fit as well. These distributions are ideal for fitting claim payments, since they are right-skewed and have a large variance. Insection 4.2we discuss what distributions we use in more depth. Using this package we simulate payments of a claim k that is not yet closed, i.e. Y (k, W + t − 1) for t ∈ {Lmin(k) + 1, ..., Ls(k)}. If t ≥ τ0 (defined in section 2.2.2) we cluster all individual payments to prevent clusters from getting too small. With the simulated payments we construct a distribution of respectively the RBNS and the IBNR reserves. We will do so in the next subsections.

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3.3.1 IBNR claims

For IBNR claims we have to simulate both the number of claims and the severity of the claims, since it is unknown how many claims will be reported in future years. Due to this, we can not condition on observed claim markers either, simply because no infor-mation is available.

Number of (IBNR) claims

We follow the approach fromPigeon et al. (2013) and Godecharle and Antonio (2014)

to simulate the number of IBNR claims per occurence year i and reporting delay W5. We assume the number of claims in period i is Poisson distributed with mean θ · w(i), with w(i) an exposure measure for occurence year i. We can then estimate W using a geometric distribution (with a degenerate component). Since we only observe reported claims, we should thin the Poisson distribution to:

θw(i)F1(t − 1; ν) (3.6)

With F1(t − 1; ν) the (discrete) distribution of the reporting delay, observed t periods after occurence. For further details, we refer to Pigeon et al. (2013). Note that they use development factors instead of payments to estimate the reserve. Their approach of estimating the amount of IBNR claims, however, remains applicable to our framework.

IBNR claim development

For IBNR claims the individual reserve for claim k equals:

IRk_{IBN R}= H(k, L(k)) − H(k, 0) = H(k, L(k)). (3.7) If we denote KIN BR as the number of IBNR claims in the dataset we can deduce the following formula for the total IBNR reserve:

RIBN R= KIBN R X k=1 IRk_{IBN R}= KIBN R X k=1 H(k(L(k)) (3.8)

After simulating the amount of claims and their length, we estimate distributions for the payments, given Ls(k) and Ws(k). For each cluster we will seperately fit a distribution and use the one that fits best. By approaching the fitting of distributions seperately for each cluster, we retain the possibility to fit a different distribution for each cluster, depending on which one fits best.

Conditional on our values for ω0, λ0 and τ0 we have to fit around 30 (parametric) dis-tributions. Then we use these distributions to simulate future payments for t = Lmin(k) to Ls(k), so we get a simulated reserve for each individual claim. After repeating this process a sufficient number of times for all claims, we can construct the distribution of the IBNR reserve.

3.3.2 RBNS claims

The individual reserve for RBNS claim k is equal to:

IRk_{RBN S} = H(k, L(k)) − H(k, Lmin(k)) = L(k) X h=n−i(k)−W (k)+3 Y (k, h + W (k) − 1) (3.9) 5_{Note that W ≤ n − i + 2.}

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If we denote KRBN S as the number of RBNS claims in the dataset we can deduce the following formula for the total RBNS reserve:

RRBN S = KRBN S

X

k=1

IRk_{RBN S} (3.10)

Similarly to IBNR claims, in order to simulate the payments, we have to simulate from parametric distributions. For RBNS claims we have to estimate around 150 distributions instead of 30. We have to fit additional distributions because we only condition on Ws(k) and Ls(k) for IBNR claims, whereas for RBNS claims we condition on Qt(k) and t as well. These extra parameters result in extra clustering and therefore additional distributions have to be fitted. After fitting these extra distributions, the reserve is constructed in the same way as the IBNR reserve.

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Chapter 4

Results

In this chapter we report the results of our parametric simulation. We use the methods and the data described in previous sections. First we report the parameters we use, next we illustrate the distributions we use to simulate the payments. Finally we report and analyze the results. As a benchmark, results obtained with the method used in

Godecharle and Antonio (2014) and the results from the chainladder bootstrap are added as well.

4.1 Parameters

Before simulating claim payments, we need to determine the values of our parameters λ0, q0, ω0 and τ0. To compare our results to those ofGodecharle and Antonio (2014), we will use the same parameters as they do.

Based on the descriptive statistics of the data, described in section 3.1, we use the following parameters:

• Based onFigure 3.3, we conclude BI claims have a long claim development if L ≥ 4 and MD claims if L ≥ 3. Therefore we set λ0 = 3 for BI claims and λ0 = 4 for MD claims.

• We distinguish three types of claims based on cumulative payments H(·, t): small, intermediate and large claims. Accordingly, we set q0= 3 for both claim types. • FromFigure 3.4, we conclude both BI and MD claims have a long reporting delay

if W ≥ 3, so for both BI and MD claims we set ω0 = 3.

• First, we require that τ0 ≤ λ0, so the final payments of long developing claims are late. Based on Figure 3.5 we conclude that payments are late if they are made after t = 3, hence we set τ0= 3 for both claim types.

These parameters give us a maximum of q0 · λ0 · ω0 · τ02 = 324 possible distributions for RBNS claims6 and λ0· ω0· τ0 = 36 for IBNR claims (since we do not condition on Qt). Note that many combinations will result in empty clusters, so the amount of fitted distributions will always be smaller than this. In practice we have to fit approximately 150 to 160 distributions with these parameters. For a sensitivity analysis on the param-eters, we refer toGodecharle and Antonio (2014). It may also be interesting to look into the optimal values of the parameters. Higher parameter values means more clustering and therefore more accuracy, but will also result in extra distributions that have to be fit. Similar to the AIC, an information criterion might be constructed that favors the optimal parameter set. We will however not do so in this thesis.

6_{We multiply with τ}2

0 because Q also depends on t, so we have to account for this. 14

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4.2 Distributions

For fitting claim payments, there are generally three distributions that are used7. These are the Gamma distribution, the Lognormal distribution and the Pareto distribution. Since none of these distributions can fit zero payments, we exclude these from each (sub)dataset. We then correct for this using a mixed distribution with a positive prob-ability of a zero payment. As we can see inFigure 4.1 andFigure 4.2, the distributions fit relatively well on both the MD and BI datasets. Note that we have used the entire non-zero dataset, and have not partitioned it based on claim markers yet. The densities of the distributions using µ and σ are as follows:

f (y|µ, σ) = y

(1/σ2₋₁₎

exp[−y/(σ2µ)] (σ2_µ)(1/σ2₎

Γ(1/σ2₎ (4.1)

for the Gamma distribution,

f (y|µ, σ) = 1

y√2πσexp[− 1

2σ2(log(y) − µ)

2_] _(4.2)

for the Lognormal distribution,

f (y|µ, σ) = 1 σµ 1 σ(y + µ)− 1 σ+1 _(4.3)

for the Pareto distribution.

Figure 4.1: Fit of distributions on complete non-zero BI dataset with AIC, µ and σ

Figure 4.2: Fit of distributions on complete non-zero MD dataset with AIC, µ and σ

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Based on the AIC it appears the Lognormal distribution fits best for both datasets, and the Gamma distribution has the worst fit. To determine which distribution we use for each subdataset, we will fit every distribution on each cluster and determine the AIC for every distribution8_{. We will then simulate payments for each cluster using the}

distribu-tion with the lowest AIC. This way we are not restricted to fitting a single distribudistribu-tion on the entire dataset.

We have to note that due to the high variance of both the Lognormal and the Pareto dis-tribution, and based onFigure 3.5we decide to cap payments at 400,000, meaning that all simulated payments above this threshold will become equal to 400,000. We expect this slightly reduces (and stabilizes) the mean of the Lognormal and Pareto distribution, while not greatly affecting their standard deviation. We believe this is justified, because the maximum payment in our dataset is 351,860 and the truncation does not affect many simulated payments. If we decided not to truncate our payments, our simulated reserves were heavily disturbed, often caused by a single payment. This truncation is only relevant for BI claims, since MD claims have a lower mean and variance.

4.3 Simulation Results

Our main results are summarized in Figure 4.3 and Figure 4.4. These tables show the IBNR and RBNS reserves calculated with parametric simulation, where we choose the best fitting distribution for each cluster. As a benchmark we have added the results obtained with the method used inGodecharle and Antonio (2014)and the classic boot-strap. Due to the computing time of the parametric bootstrap we only have respectively 1000 and 100 simulations for our BI and MD datasets. Although this is a small amount when simulating, we believe we can still come to some solid conclusions regarding the results of our parametric simulation.

Comparing our results to the classic bootstrap and Godecharle and Antonio (2014), we conclude our reserves are higher than both methods for our BI dataset, which is somewhat troubling. In the case of Material Damage our parametrical reserve is similar to the reserve obtained with the approach of Godecharle and Antonio. This implies our method might not work well on small datasets, since we only have around 4,000 observations for BI against 200,000 for MD. In terms of standard deviance, our method performs slightly worse than the historical simulation of Godecharle and Antonio (2014), but this might also be due to our limited number of simulations.

Method

(BI) Minimum 25% Mean 75% Maximum S. Dev.

IBNR 254,249 585,332 797,725 969,041 1,941,415 286,381 RBNS 6,669,272 8,281,110 8,921,539 9,509,672 12,337,192 912,254 Total 7,449,489 9,058,182 9,719,265 10,320,453 13,563,825 947,942 Historical Simulation 5,931,841 7,695,383 8,194,603 8,651,351 11,576,616 717,868 Bootstrap 4,679,411 7,565,338 8,869,373 10,005,328 15,712,559 1,165,498 Observed 255,221 (IBNR) + 7,256,374 (RBNS) = 7,551,595

Figure 4.3: Results from our BI simulations, compared toGodecharle and Antonio (2014) and the chain ladder bootstrap

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Method

(MD) Minimum 25% Mean 75% Maximum S. Dev.

IBNR 13,162 29,752 42,972 48,908 250,529 26,676 RBNS 1,517,587 1,781,651 1,942,534 2,077,412 2,697,519 219,751 Total 1,571,702 1,827,466 1,985,506 2,130,241 2,736,745 222,116 Historical Simulation 1,498,703 1,895,789 2,043,762 2,169,430 2,862,788 201,424 Bootstrap 1,739,151 2,734,892 3,046,488 3,330,532 4,727,862 341,530 Observed 348,847 (IBNR) + 1,751,449 (RBNS) = 2,100,296

Figure 4.4: Results from our MD simulations, compared toGodecharle and Antonio (2014) and the chain ladder bootstrap

Upon further analyzing our results we conclude that neither the Lognormal nor the Pareto distribution fit well on small datasets, which we show with an example in Ap-pendix A.

In Figure 4.5 we see the same results as in Figure 4.3, but split on distribution (i.e. we only fit a one type of distribution for each run). These results confirm our assump-tion, since the results obtained using the Gamma simulation are fairly close to the historical bootstrap, and those of the Lognormal and Pareto distribution have a sig-nificantly higher mean. This might be explained by their high standard deviation and the small amount of payments that have to be simulated from each distribution, which makes the simulation process more susceptible to outliers.

Method

(BI) Minimum 25% Mean 75% Maximum S. Dev.

Gamma 6,865,362 7,826,000 8,168,627 8,506,449 10,232,973 518,652 Lognormal 7,661,542 9,973,047 10,721,066 11,434,542 14,686,607 1,080,193

Pareto 7,389,496 9,910,934 10,779,922 11,569,508 14,757,226 1,223,915 Historical

Simulation 5,931,841 7,695,383 8,194,603 8,651,351 11,576,616 717,868 Figure 4.5: Total Reserves BI simulations for each distribution, compared to the

method used inGodecharle and Antonio (2014)

Furthermore, the results we found by using only the Gamma distribution have a smaller deviation than the results we get using the method of Godecharle and Antonio. We must note that this decrease in variance might well be due to the choice of distribution, as the Gamma distribution has a smaller variance than the historical variance, which is also shown in Appendix A. To check whether this difference is due to the fit of the distributions or due to the smaller dataset, we do the same for our larger MD dataset, which results inFigure 4.6.

This figure shows our method does seem to work on bigger datasets, as the means of all four methods tend to be close together for our MD dataset. For the three distribu-tions the standard deviation is quite different though, showing the reduction in variance is mainly caused by the choice of distribution.

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Method

(MD) Minimum 25% Mean 75% Maximum S. Dev.

Gamma 1,718,142 1,980,517 2,062,307 2,141,395 2,461,847 145,532 Lognormal 1,678,846 1,861,520 2,019,210 2,158,685 2,509,903 201,627 Pareto 1,631,771 2,028,274 2,319,007 2,496,500 3,771,277 413,677 Historical

Simulation 1,498,703 1,895,789 2,043,762 2,169,430 2,862,788 201,424 Figure 4.6: Total Reserves MD simulations for each distribution, compared to the

method used inGodecharle and Antonio (2014)

Based on the results in this section we conclude that our parametrical bootstrap gives similar results as a historical bootstrap, given we use a big enough dataset. Because of the extra computing time due to the fitting of extra distributions it might not be practical for actual applications, since the reduction in variance does not outweigh the extra distributions that have to be fit. Furthermore, the reduction in variance might be caused by the choice of distribution which means our approach may not always lead to smaller variances.

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Chapter 5

Conclusion

In this thesis we have constructed an alternative method for insurance companies to calculate their reserves, based on individual claim data. We formulated a parametrical alternative to the model used in Godecharle and Antonio (2014). The main difference between the models is that instead of using historical simulation for constructing pay-ments, we use parametrical distributions. Although we did not use many simulations, we think that we can still come to some solid conclusions based on our results. Our main finding is that this new method generally works equally well as its historical equivalent, but fails on smaller datasets due to the large impact of outliers and the bad fits of some distributions. Although our parametrical approach does reduce variance on some occasions (with the Gamma distribution for example), the extra computing time for each simulation does not always outweigh using more simulations to increase accuracy. We believe that, in terms of future research, it would be worthwhile to look into ad-ditional methods to compare the fit of right-skewed distributions on small datasets. Furthermore, a sensitivity analysis on the parameters, accounting for the amount of (extra) distributions that have to be fit might be interesting to look into. Additional ways to improve our model might be intruiging to look into as well, such as adding vari-ables used in pricing theory like age and gender. This would make the framework even more accurate for insurers who have the available data. Finally it might be interesting to compare our parametric RDC to a (parametric) variant where we only condition on historical data or where we fit a single distribution using covariates. For now we have proposed a parametrical variant of the historical RDC, and have reported its advantages and disadvantages over existing methods.

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Appendix A: Fit on small

datasets

Although the Lognormal and Pareto distributions fit fairly well on the entire dataset, results fromsection 4.3show that this is not the case for some small dataclusters, which we will show here with an example. Suppose we have the following non-zero claim pay-ments in a cluster, which is an actual subset of our BI dataset:

290.46; 1,507.51; 90.93; 9,236.97; 2,603.24; 49,032.67; 1,329.29; 129.94; 697.95 If we fit the distributions and compare simulated results to the actual data, we come to a somewhat troubling conclusion. In Figure A.1 we can see that although the Log-normal and Pareto distributions have a smaller AIC, and therefore statistically have a better fit, their means and standard deviatons do not correspond with the actual data. This is probably due to the large tails of these distributions, hence the large standard deviation. This large variance also has an upwards effect on the mean of the Pareto dis-tribution. Truncating the simulations to 400,000 decreases the mean of the Lognormal distribution, but does not have such an effect on the means of the Gamma and Pareto distribution.

Distribution AIC µ σ Mean S. Dev.

Actual - - - 7,213 15,940

Gamma 172.75 8.88 0.49 7,216 11,820

Lognormal 168.90 7.10 0.64 6,925 24,385

Pareto 169.13 6.44 0.33 11,638 48,004

Figure A.1: Fit of the distributions on the example cluster. 10,000,000 simulations are made for each distribution

Based on the mean and the standard deviation of the data and simulations, we conclude that the Gamma distribution is actually the distribution that fits best on these small datasets. That’s why we have also split the results in section 4.3on distribution, where we only fit one type of distribution for the simulation. These results prove the allegations made in this section.

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Individual reserving by detailed conditioning : a parametric approach