Pricing an absenteeism stop-loss portfolio using extreme value theory

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Pricing an absenteeism stop-loss

portfolio using extreme value

theory

G.E.M. van de Sande-Versendaal

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics

Author: G.E.M. van de Sande-Versendaal Student nr: 10682171

Email: gemvandesande@gmail.com

Date: July 8, 2015

Supervisor: Dr. S.U. Can Second reader: Prof. Dr. R. Kaas

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Abstract

In the Netherlands an employer holds the obligation by law to cover the wage payments for sick employees during the first two years of their illness. An employer has the possibility to protect themselves against this financial risk by entering into a so-called absenteeism insurance contract. Two commonly offered absenteeism products are the absenteeism conventional product and the absenteeism loss product. The focus of this thesis will be the absenteeism stop-loss product. In an absenteeism stop-stop-loss contract there is a retention amount specified. All the wage payments for sick employees below this retention amount are paid by the employer, while the part of the wage payments which are above the retention amount is paid by the insurer. At De Amersfoortse the absenteeism stop-loss product is designed for the larger companies, i.e. companies with 50 employees or more. For large companies the yearly wage payments for sick employees will be relatively stable compared to smaller companies. Therefore, by having an absenteeism stop-loss contract they can protect themselves against the more extreme claim years. Currently, De Amersfoortse uses a pricing scheme that was developed in 2004 by the Verbond van Verzekeraars. More than 10 years later, they wish to update this pricing scheme using their own data. To determine an updated pricing scheme, extreme value theory is applied to study the tail of the distribution underlying the pricing scheme. We determine for which high threshold the tail of the distribution is best described by a Generalized Pareto distribution. Using a threshold model we combine the Generalized Pareto distribution with a distribution below the threshold to obtain the complete distribution. Two approaches are followed: firstly we fit the distributions on the complete dataset and secondly we distinguish in the number of employees on the contract. After we obtain the desired distribution, we determine the updated pricing scheme. The relation between the pricing scheme and the number of employees is determined using a GLM analysis. The range of possible values for the updated pricing scheme is, compared to the current outdated pricing scheme, much larger and therefore more dependent of the characteristics of the contract and company.

Keywords Income protection insurance, Absenteeism, Pricing, Extreme value theory, Threshold model, Generalized Pareto distribution, GLM

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Introduction

The subject of this thesis is the pricing scheme of the absenteeism stop-loss product offered at De Amersfoortse1_{. De Amersfoortse is an insurance company with a focus on income protection}

insurance. The absenteeism type insurance products cover the financial risk of the wage payments of sick employees during the first two years of their illness. De Amersfoortse offers two types of absenteeism products, namely the absenteeism conventional product and the absenteeism stop-loss product. The absenteeism conventional product is designed for small companies, who are willing to cover the wage payments during illness per employee for several days out of their own expenses. If the duration of the illness is larger, De Amersfoortse takes over the wage payments. The absenteeism stop-loss product is designed for larger companies, who are willing to cover the expected yearly claim payments out of their own expenses. However, they can protect themselves against very high yearly claim payments by getting a stop-loss type insurance contract.

De Amersfoortse currently offers the absenteeism stop-loss product using a pricing scheme that was developed by the Verbond van Verzekeraars2_{. This pricing scheme was offered in 2004 to}

the private insurers due to changes in the regulations regarding the coverage of wage payments during absenteeism. Insurers wanted to offer an insurance product, but did not know the risks. The pricing scheme feels a bit like a black box, since it is not clear how all the parameters were estimated and what they really represent. More than 10 years later, De Amersfoortse has collected an enormous amount of data regarding the absenteeism risk and wants to update the current pricing scheme in accordance with the data.

To update the pricing scheme of the absenteeism stop-loss product we will apply extreme value theory to fully describe the tail of the distribution underlying the pricing scheme. Furthermore, we will combine the distribution of the tail with a distribution below the tail, to obtain a complete distribution to determine an updated pricing scheme.

Research objectives & Outline of the report

The objective of this thesis is to determine a distribution (for a quantity to be described below) in order to obtain a pricing scheme for the absenteeism stop-loss product. Using extreme value theory we will properly study the tail of the distribution. However, we are not only interested in the tail of the distribution. Therefore, we will investigate how we can incorporate the results of the extreme value theory to obtain a full distribution. Firstly, the focus will be on the distribution without distinguishing between characteristics of the company. Secondly, we will investigate the effect of the number of employees on the contract on the distribution. Thirdly, after we have obtained the desired distribution, we will compare the results of the current pricing scheme with those of the obtained updated pricing scheme.

1_{Part of a.s.r.}

2_{The Verbond van Verzekeraars is a trade association that represents the interests of private insurance companies}

operating in the Netherlands.

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2 CHAPTER 1. INTRODUCTION

We start in chapter 2, The absenteeism stop-loss product, with a description of the absenteeism stop-loss product and the current pricing scheme. We discuss some shortcomings and improve-ments that are desired for the updated pricing scheme. We conclude this chapter with a discussion of the available data at De Amersfoortse.

Chapter 3, Extreme value theory, gives an introduction to the most important results of extreme value theory. We start with the behavior of maxima and continue with the theory on exceedances over thresholds. We will apply this theory to determine the tail of the distribution of our interest, using the Generalized Pareto distribution. Next, we will discuss some methods which will be used to determine the parameters of the Generalized Pareto distribution. We conclude this chapter with the discussion of a model to combine a distribution below the threshold with the distribution above the threshold.

Chapter 4, Estimation of the distribution of the claims-to-BRA ratio, starts with a summary of the available data and we will discuss some basic statistics. After this, we will implement the extreme value theory to determine the distribution of the tail without distinguishing between characteristics. We repeat the estimation, but now distinguishing between the companies based on the number of employees covered in the contract.

Using the obtained distributions, we will determine a pricing scheme in chapter 5, Pricing scheme for the absenteeism stop-loss product. We will keep the overall structure of the current pricing scheme, to be able to compare both pricing schemes. Using the distributions based on the num-ber of employees, we can determine the relation between the pricing scheme and the numnum-ber of employees using a GLM-analysis. We conclude this chapter with a comparison of the current and the updated pricing schemes.

Finally in chapter 6, Conclusion, we will present the conclusion of this thesis and offer some recommendations for further research.

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Chapter 2

The absenteeism stop-loss product

Before we start with a theoretical background on extreme value theory, we introduce in this chapter the absenteeism stop-loss product that is offered at De Amersfoortse. This will give insight into why extreme value theory might be a possibility to model the claims for the absenteeism stop-loss product.

We start with an introduction of the absenteeism products that are available at De Amersfoortse. Next, we briefly discuss the current pricing scheme of the absenteeism stop-loss product and list some shortcomings of this pricing scheme. Finally, we give a description of the available data at De Amersfoortse to perform the estimations on.

2.1 Product description

In this thesis we consider an insurance product of the type of income protection insurance. The act Wet verlenging loondoorbetaling bij ziekte1 _{(VLZ) obligates an employer to pay a percentage}

of the wage payments to an employee during the first two years of his or her illness. For the first and second years of illness the minimal amount to be paid to the employee is 70 percent of the original wage payment. However, many employers pay especially in the first year of illness a higher percentage than 70 and this percentage is usually determined by the collective labor agreement. The obligation of the employer ends after the initial two years of illness or when the employee recovers, reaches the retirement age or deceases. After the initial two years of illness the income of a disabled employee is regulated by the act Wet werk en inkomen naar arbeidsvermogen2 _(WIA).

Insurance companies offer income protection insurances to take over the financial risks corre-sponding to the wage payments of sick employees. Two commonly offered insurances are the Verzuimverzekering Conventioneel (Absenteeism Conventional) and the Verzuimverzekering Stop-Loss(Absenteeism Stop-Loss). The first is based on a threshold measured in days, i.e. the insurance will cover the wage payments of a sick employee if the duration of the illness is longer than the number of days specified in the contract. The latter is based on a threshold measured in money, i.e. the insurance will cover the wage payments of sick employees if the total amount of wage payments for sick employees exceeds a pre-determined threshold for a certain calendar year. The premium payments for the absenteeism stop-loss product during a financial year should cover the claim amount of the same financial year. This premium principal of the stop-loss insurance differs from many income protection type insurances, where the premium should cover the claim amount of all the claims that originated in the financial year.

De Amersfoortse offers the absenteeism stop-loss product for companies with more than 50 em-ployees. The stop-loss product is more attractive for larger companies, since their yearly costs for

1_{Extension of Wage Payments during Illness Act, installed on 1 January 2004} 2_{Work and Income according to Labor Capacity Act}

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4 CHAPTER 2. THE ABSENTEEISM STOP-LOSS PRODUCT

wage payments for sick employees are much more predictable than for smaller companies. There-fore, they desire to protect themselves against the event when the wage payments for disabled employees exceed a certain threshold, i.e. the retention amount (RA).

The absenteeism stop-loss has the following coverage options:

The retention percentage; The retention amount is expressed as a percentage of the basis

retention amount (BRA), i.e. the retention percentage. The BRA is simply the expected claim amount for the company per calendar year, given its properties and the coverage options. How we determine the BRA is described in section 2.2.1. We offer the retention percentages in the range of 90 percent to 150 percent with a step size of 5 percent, the standard option being 125 percent.

The maximal insured amount; The maximal insured amount above the retention amount,

i.e. the ceiling of the insurance. The standard is an unlimited coverage, i.e. basically there is no ceiling. Other options are two, three or four times the retention amount.

The duration of the coverage; The period which is covered by the insurance. We have the

option of one or two years. A duration of one year covers only the wage payments during the initial first year of illness.

The coverage percentage; The percentage of the wage payments which are paid during the

first two years of illness. The coverage percentage is in the range of 70 percent until 100 percent and flexible to choose for each month. The default setting is 100 percent for the first year of disability and 70 percent for the second year.

The end age of the coverage; The maximal age of an employee which is covered. Usually

this is the legal retirement age, but every age in the range of 55 to 70 years is possible.

Employer costs; Additional to the wage payments an employer has to pay employer costs,

examples are costs associated with social law taxes and pension premiums. An employer can choose whether he or she wishes to include these costs in the insurance.

Figures 2.1 and 2.2 show a hypothetical example of the claim amount for calendar year t. We consider a company with three disabled employees, the first disability originating in calendar year t − 2 (blue), the second in calendar year t − 1 (green) and the third in calendar year t (red). We assume that for the first year of illness the payments correspond to the original wage payment, i.e. a coverage of 100 percent. In the second year of illness the payment is reduced to 70 percent of the original wage payment. Furthermore, we assume that the three disabled employees do not recover in the first two years of their disability. Figure 2.1 shows the evolution of the claim amount per disabled employee. Note that the total claim amount of year t is determined by claims that originated in the calendar years t − 2, t − 1 and t. Figure 2.2 shows the evolution of the claim amount for calendar year t. The claim amount which is lower than the retention amount (grey area) is paid by the company itself. Above the retention amount the claims are paid by the insurer.

2.2 Pricing formulas

In this section we discuss the current pricing scheme of the absenteeism stop-loss product of De Amersfoortse. The pricing scheme is based on the advice published by the Verbond van Verzekeraars (2003). On 1 January 2004 the VLZ was installed and from then on the employer holds the obligation for the wage payments during the first two years of illness of an employee. The VLZ is an extension of the older Dutch act Wet uitbreiding Loondoorbetaling bij Ziekte3_(Wulbz),

which only covered the wage payments during the first year of illness of an employee. With the installment of the VLZ the obligation to cover the wage payments during the second year of the illness introduced a new and unknown risk for insurers. There was no data available to determine a suitable pricing scheme. Therefore, the Verbond van Verzekeraars combined data for the Wulbz

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2.2. PRICING FORMULAS ₅

t − 2 t − 1 t

Figure 2.1: Hypothetical example of the cashflows for year t

t

RA

Figure 2.2: The total claim amount during year t.

and the act Wet op de arbeidsongeschiktheidsverzekering4 _{(WAO), together covering jointly the}

period of the first two years of illness.

To determine the pricing scheme for the absenteeism stop-loss product, as published by the Ver-bond van Verzekeraars, the following assumptions were made:

During the first year of illness the employer holds the obligation to pay 100% of the wage

payment, during the second year of illness the obligation is reduced to 70% of the wage payment.

The premium of the stop-loss product covers the wage payments during illness that are made

during the corresponding year.

The pricing scheme is applicable for companies with 25 employees or more.

The retention amount is at least the basis retention amount and not more than 1.5 times

the basis retention amount.

The probability of an extreme year, where the expected claim amount and its variance is 15

percent higher than in regular years, is 10 percent.

The ratio of the claim amount to the basis retention amount BRA, i.e. the claims-to-BRA

ratio, is normally distributed.

The net actuarial premium for a company for the absenteeism stop-loss product N PSL is given by the following formula;

N PSL = BRA · max(fSL, fminSL) · fcoverage, (2.2.1) where BRA denotes the basis retention amount, fSL the stop-loss factor, fminSL the minimal stop-loss factor and fcoverage a factor to take a coverage larger than the retention amount into account. In the following subsections we describe the different components of formula (2.2.1).

2.2.1 The basis retention amount

BRA

The basis retention amount BRA is the best estimate of the expected payments during a financial year for a company. For the calculation of the BRA De Amersfoortse does not follow the advice of

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the Verbond van Verzekeraars, but developed their own pricing scheme. The BRA is determined by formula (2.2.2);

BRA = frun-in · fexp · RPstat, (2.2.2)

with frun-in the factor whether De Amersfoortse covers the complete claim history (run-in risk) of the company, fexp the experience rating factor given the claim history of the company and RPstat the statistical risk premium depending on the coverage options.

The factor frun-in equals one if the run-in risk is insured, i.e. claims that occurred before the effective date of the insurance contract are included in the claim amount. When these claims are not included in the claim amount, the expected claim amount for the first two calendar years will be smaller than when run-in risk is insured, i.e. frun-in < 1. Companies with an effective date in calendar year t − 3 have their complete run-in risk included for calendar year t.

The experience rating factor fexp describes the ratio of the experienced claim history to the expected claims. If for several years in a row a company realizes a lower claim amount than the in general expected claim amount given the properties of the company, the factor fexp will be smaller than one. Hence, we expect that for the upcoming calendar year the claim amount will be lower than their general expected claim amount. Otherwise, if the realized claim amounts are higher than the general expected claim amounts, the factor will be larger than one. The calculation is based on three years of claim history and the size of the company. For small companies the own realizations are not as valuable as for larger companies. When a company does not provide any claim history, e.g. a start-up company, the experience rating factor is not applied.

The statistical risk premium RPstat is the expected statistical claim amount given the detailed information of the employees of the company and the coverage options. For each employee i we determine their corresponding statistical risk premium RPstat,i and to determine the statistical

risk premium of the company we sum over all the employees. The formula for the individual RPstat,i per employee is given by equation (2.2.3);

RPstat,i= fclaim · fsector · focc-group · RPrisk,i, (2.2.3)

with fclaim a factor for when no claim history is supplied, fsector an adjustment factor for the business sector of the company (e.g. a medical company, a financial institution, etc.), focc-group a factor for the type of activities of the employee (e.g. administrative activities or manual labor) and RPrisk,ithe risk premium per employee depending on the age, salary and coverage options. The

risk premium per employee is determined by a cashflow projection model, for further information we refer to Stolwijk (2013).

2.2.2 The stop-loss factor

f

SL

The stop-loss factor fSL in equation (2.2.1) denotes the percentage of the basis retention amount that is needed for the stop-loss premium. The stop-loss premium is equal to;

E [(X− d)+] ,

where d denotes the retention percentage. As described in Kaas et al. (2008), if X is normally distributed with mean µ and variance σ2 _{the stop-loss premium can be written as;}

E [(X− d)+] = σφ d − µ σ − (d − µ) 1 − Φ d − µ σ , (2.2.4)

with φ(·) and Φ(·) denoting the probability density and cumulative distribution functions, respec-tively, of the standard normal distribution. For the absenteeism stop-loss product X equals the claims-to-BRA ratio, and the retention d equals the ratio RA/BRA, i.e. the retention percentage. As described in the assumptions of the pricing scheme, there is a probability of 90% that we ex-perience a normal claim year and a probability of 10% that we exex-perience an extreme claim year.

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2.3. SHORTCOMINGS OF THE CURRENT PRICING SCHEME ₇

For a normal year the expectation µ equals one, i.e. the basis retention amount divided by itself. For an extreme year µ equals 1.15, i.e. in the extreme case we expect 15 percent more claims than the basis retention amount. As described in the assumptions, also the variance is multiplied by 1.15 and not by (1.15)2. Hence, the stop-loss factor is given by;

fSL = 0.9 · E " ˆ X −_BRARA + # + 0.1 · E " ˜ X −_BRARA + # , (2.2.5)

with ˆX ∼ N(1; σ2_{) and ˜}_{X ∼ N(1.15; 1.15 · σ}2_{). The standard deviation σ is prescribed as;}

σ = pµ2+ µ

2

1· min(1, max(0, (N − 100)/900))

13.93 ·√N · vz · µ1

, (2.2.6)

with µ1and µ2constants and N the number of employees. The parameter vz denotes the expected

stop-loss absenteeism percentage, which is determined by dividing BRA by the total insured amount covering the first and second years of absenteeism. As expected, the standard deviation decreases when the number of employees increases. The yearly claim amounts of larger companies will on average be more stable than the yearly claim amounts of small companies. Furthermore, for companies with a high expected stop-loss absenteeism percentage the standard deviation is smaller than for companies with lower stop-loss absenteeism percentages.

Using equations (2.2.4) - (2.2.5) we get the following expression for the factor fSL;

fSL = 0.9 · ( σφ RA BRA− 1 σ ! − _RA BRA− 1 " 1 − Φ RA BRA− 1 σ !#) (2.2.7) + 0.1 ·(√1.15σφ RA BRA− 1.15 σ ! − _RA BRA− 1.15 " 1 − Φ RA BRA_√ − 1.15 1.15σ !#) .

2.2.3 The minimal stop-loss factor

f

minSL

Due to the assumption that the claims-to-BRA ratio is normally distributed, the Verbond van Verzekeraars concludes that the right-tail of the distribution is not fat enough. Especially for high values of the retention amount, the stop-loss factor decreases too rapidly to zero. Therefore, the minimal stop-loss factor is introduced to make a correction for this behavior of the normal distribution. The minimal stop-loss factor is given by;

fminSL = 15 − 7.5 ·

RA BRA

100 . (2.2.8)

2.2.4 The maximal coverage factor

f

coverage

The advice of the Verbond van Verzekeraars holds if the maximal coverage of the product is one time the retention amount, i.e. there is a ceiling in the coverage. Therefore, the coverage factor fcoverage is introduced to correct for a maximal coverage larger than the retention amount. The value of fcoverage depends on the chosen retention percentage and the ceiling of the contract.

2.3 Shortcomings of the current pricing scheme

The current pricing scheme of De Amersfoortse for the absenteeism stop-loss product is primarily based on the advice of the Verbond van Verzekeraars from 2003, only the calculation of the basis retention amount is adjusted. De Amersfoortse now has over ten years of data concerning the first two years of illness. Hence, the development of a pricing scheme based on their own data is desired. From the discussion of the current pricing scheme we list the following shortcomings of the current pricing scheme.

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1. The developers of the pricing scheme needed to introduce a minimal stop-loss factor fminSL to correct for the too thin tails of the normal distribution. This suggests that the normal distribution is not the best available distribution to describe the risks arising from the absen-teeism stop-loss product. A distribution with fatter tails could be a better fit to determine the stop-loss premiums. Also, the normal distribution is a symmetric distribution, and this is not expected for the distribution of the claims-to-BRA ratio. Since the minimal value of the claim amount is zero, the left side of the distribution should be bounded by zero. 2. There is a lack of argumentation for the probability of an extreme year every one in ten

years. This choice seems rather arbitrary.

3. The additional claim amount of 15 percent for an extreme year also lacks argumentation. 4. Formula (2.2.6) to compute the standard deviation for a company is a rather complex

func-tion. In the published advice of the Verbond van Verzekeraars (2003), there are no comments on how the constants µ1, µ2 and 13.93 are determined. This makes it nearly impossible to

change the values of these parameters.

5. The factor fcoverage needs to be applied if the maximal coverage exceeds the value of the retention amount. This needs to be done, since in their estimations they used the retention amount as the maximal coverage. However, it is not clear which parameters are influenced by this choice. This choice is not visible in the stop-loss factor fSL. The relation in equation (2.2.4) holds if the maximal coverage equals infinity. It could be that the standard deviation is influenced by the maximal coverage, but this is not clear from the formula. This suggests that the factor fcoverage is introduced to compensate for the underestimation of the tail of the distribution.

6. The pricing scheme is based on the standard coverage percentages of 100 percent for the first year and 70 percent for the second year. It is not intuitively clear how a different combination of coverage percentages would influence the values of the parameters.

2.4 Data

In this section we describe the available data at De Amersfoortse to perform the estimations on. We distinguish between available data on the absenteeism conventional product and the absenteeism stop-loss product. For the absenteeism conventional product there is very detailed claim information available, for the absenteeism stop-loss product there is not. Therefore, in this thesis we will perform the estimations based on the absenteeism conventional product.

2.4.1 Absenteeism conventional product

The target market for the absenteeism conventional product are the smaller companies with the number of employees up to 100. The available data is very detailed and is divided over three files, i.e. the policy file, the claim file and the payment file. The data is available from calendar year 2004 and further. For the estimations we will focus on the data of calendar years 2007 until 2013. calendar year 2014 is left out, since the claim and payment data may not be fully complete due to delays in the administrative system.

The policy file

The policy file is a very detailed source of information. On employer level we have the following information available; start date of the policy, end date of the policy, the coverage duration, the business sector, the percentage of employer costs, the number of waiting days and the coverage percentage and scheme. For each employee we have insight into; start date at the firm, end date at the firm, gender, date of birth and yearly salary. The only information missing on employee level is the occupation group of the employee. However, there is a rough estimation available of

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2.4. DATA ₉

the division over the different occupation groups per business sector, and this will be used in the estimations.

The claim file

For the absenteeism conventional product the policyholder has the obligation to report sick em-ployees on the first day of their illness. Hence, using the claim file will give insight into all the claims that are made during a calendar year, even the ones where no payment is made to the em-ployer since the duration of the illness is shorter than the threshold in days. The claim file gives insight into; the first date of illness, the end date of illness and all the mutations of percentage of disability during the period of illness (e.g. 100% if the employee is not able to work at all, 50% if the employee can work on half capacity).

The payment file

The payment file consists of the claims where a payment is made, i.e. the duration of illness is longer than the number of waiting days. The payment amount is based on the disability percentage, the coverage percentage, the percentage of employer costs and the duration of the illness.

Adjustments for the estimations

To use the data of the absenteeism conventional product for the estimations of the absenteeism stop-loss product, we need to make the following adjustments:

The basis retention amount per policy is not available. However, given the characteristics in

the policy file this can easily be determined from equations (2.2.2) and (2.2.3). In this calcu-lation we do not take the factors for experience rating fexp and historic claim information fclaim into account, i.e. we assume that all companies are start-up companies.

We need to adjust the claim payments from the payment file with the number of days

that is for the risk of the company, since for the absenteeism stop-loss product the claim payments from the first day of illness are included in the total claim amount. From the claim file we know the disability percentage during the waiting days, hence the claim amount corresponding to the waiting period can easily be determined.

2.4.2 Absenteeism stop-loss product

The information available for the absenteeism stop-loss product is not as detailed as the informa-tion for the absenteeism conveninforma-tional product. Especially the claim and payment informainforma-tion is very limited. We presume that this information is too limited to perform the estimations on.

The policy file

The policy file consists of the following information on policy holder level; start date of the policy, end date of the policy, coverage duration, business sector, percentage of employer costs, retention percentage and amount, coverage percentage and scheme, number of employees and insured amount. There is no information available on the employee level, such as age, gender and salary. The retention amount is determined at the start of the policy and is not updated thereafter.

The claim and payment file

For the absenteeism stop-loss there is hardly any difference between the claim and the payment file. The policyholder does not have the obligation to report each sick employee to De Amersfoortse. At the end of each calendar year the policyholder has to report the total claim amount and if it is larger than the retention amount a claim and payment is made. Claim amounts smaller than

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the retention amount are not registered by De Amersfoortse, hence there is no insight into the claims smaller than the retention amount. This lack of detail makes it very difficult to use the absenteeism stop-loss data for the estimations.

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Chapter 3

Extreme value theory

In this chapter we give a brief introduction to extreme value theory and discuss the most important results found in literature. Extreme value theory is very widely applied in the field of hydrology. However, in recent years it is increasingly applied to problems originating in the fields of finance and insurance. We start with the behavior of the maxima of a set of random variables. This is not our primary interest, however, the results are needed to describe the behavior of exceedances above a certain high threshold. This will be discussed next. With the help of extreme value theory we want to determine a good formulation of the term max(fSL, fminSL) and fcoverage in equation (2.2.1). For the proofs of the presented theorems we refer to Embrechts et al. (1997).

3.1 Behavior of maxima

Consider a sequence of non-degenerate independent and identically distributed (i.i.d.) random variables X1, X2, . . . with cumulative distribution function F . In extreme value theory we are

interested in the behavior of the sample maxima

Mn= max (X1, X2, . . . , Xn) , n ≥ 2.

As n → ∞ the maxima Mn will be determined by the right tail of the distribution F and will

almost surely converge to the right endpoint xF. The right endpoint xF is defined by

xF = sup {x ∈ R : F (x) < 1} .

This results does not provide a lot of information, hence we are more interested whether we are able to find sequences cn> 0 and dn ∈ R such that

P Mn− dn cn ≤ x

→ H(x), x ∈ R, (3.1.1)

for some non-degenerate distribution function H.

Definition 3.1.1(Maximum domain of attraction). The distribution function F of X belongs to the maximum domain of attraction of the extreme value distribution H if there exist constants cn > 0, dn∈ R such that the limit relation (3.1.1) holds. We use the notation F ∈ MDA(H).

The Fisher-Tippett-Gnedenko theorem (Theorem 3.1.2) forms the basis of the classical extreme value theory and characterizes the possible distribution functions H(x) for which F ∈ MDA(H) can hold.

Theorem 3.1.2 (Fisher-Tippett-Gnedenko). Let (Xn) be a sequence of i.i.d. random variables.

If there exist norming constantscn> 0 and dn∈ R and some non-degenerate distribution function

H such that

Mn− dn

cn → H,

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12 CHAPTER 3. EXTREME VALUE THEORY

thenH must be of the type Hξ, with

Hξ =

exp −(1 + ξx)−1/ξ_, _{ξ 6= 0}

exp (−e−x_{) ,} _{ξ = 0} , 1 + ξx > 0. (3.1.2)

The distribution Hξdefined in equation (3.1.2) is known as the Generalized Extreme Value (GEV)

distribution. Not every distribution function F belongs to a MDA(Hξ), an example is the Poisson

distribution. However, all common continuous distributions are in the MDA(Hξ) for some value

of ξ.

We will discuss the three types of GEV distribution functions corresponding to ξ = 0, ξ < 0 and ξ > 0 briefly. But first we need to formulate the definition of a slowly varying function L. Definition 3.1.3 (Slowly varying function). A positive function L : (0, ∞) → (0, ∞) is slowly varying at ∞ if lim x→∞ L(tx) L(x) = 1, t > 0. Fr´echet distribution

The Fr´echet distribution corresponds to the GEV with ξ > 0. The case of ξ > 0 describes the heavy tailed distributions with infinite higher moments. It holds that the larger ξ the heavier the tails and EXk_{= ∞ for k > 1/ξ. Theorem 3.1.4 gives the relation of the Fr´echet distribution to}

the class of slowly varying functions at ∞.

Theorem 3.1.4. For ξ > 0 , the distribution function F belongs to the maximum domain of attraction ofHξ if and only if ¯F (x) = 1 − F (x) = x−1/ξL(x) for some slowly varying function L

at∞.

In actuarial modeling the case of ξ > 0 is the most observed distribution for the maxima. Gumbel distribution

The Gumbel distribution corresponds to the GEV with ξ = 0. EXk_{< ∞ for all k.}

Weibull distribution

The Weibull distribution corresponds to the GEV with ξ < 0. For the Weibull distribution it is also possible to determine the relation with a slowly varying function at ∞ and is given in theorem 3.1.5.

Theorem 3.1.5. For ξ < 0, the distribution function F belongs to the maximum domain of attraction of Hξ is and only if xF < ∞ and ¯F xF− x−1 = x1/ξL(x) for some slowly varying

function _{L at ∞.}

3.2 Exceedances over thresholds

In the previous section we described the fundamental theory for the behavior of maxima, such as the GEV distribution and the maximum domain of attraction. However, this theory is not applicable to determine a pricing scheme for the absenteeism stop-loss product. Therefore, we continue with the theory concerning exceedances over a certain high threshold u. We start with the introduction of the excess distribution function and the Generalized Pareto distribution (GPD), which both play an important role in the theory of exceendances over high thresholds.

Definition 3.2.1 (Excess distribution function). Let X be a random variable with distribution function F and right endpoint xF. For a fixedu < xF,

Fu(x) = P(X − u ≤ x|X > u) = F (x + u) − F (u)

1 − F (u) , x ≥ 0 (3.2.1)

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3.3. ESTIMATION OF PARAMETERS ₁₃

The Generalized Pareto distribution is given by Gξ,β(x) =

1 − (1 + ξx/β)−1/ξ_, _{ξ 6= 0,}

1 − exp(−x/β), ξ = 0, (3.2.2)

with ξ the shape parameter and β the scale parameter. As with the Generalized Extreme Value distribution, we can also define three types of the Generalized Pareto distribution.

For ξ ≥ 0 it holds that x ≥ 0. This case corresponds to the ordinary Pareto distribution

with α = 1/ξ and κ = β/ξ.

For ξ = 0 it is the exponential distribution.

For ξ < 0 it holds that 0 ≤ x ≤ −β/ξ and corresponds to a short-tailed Pareto type II

distribution.

An important result and link between the excess distribution function and the Generalized Pareto distribution is given by Pickands (1975). The result is shown in Theorem 3.2.2, which states that if the distribution function F is in the maximum domain of attraction of Hξ, the excess distribution

function is uniformly close to the Generalized Pareto distribution function for a sufficiently high threshold u.

Theorem 3.2.2 (Pickands-Balkema-De Haan). There exists a positive function β(u) such that lim u↑xF sup 0≤x<xF−u Fu(x) − Gξ,β(u)(x) = 0, (3.2.3)

if and only if _{F ∈ MDA (H}ξ) with ξ ∈ R.

3.2.1 Mean excess function

The mean excess function of a random variable X with a finite mean is given by the following definition;

Definition 3.2.3(Mean excess function). The function

e(u) = E [X − u|X > u] , (3.2.4)

is called the mean excess function of X.

The mean excess function is a convenient tool to test whether a distribution F is in the maximum domain of attraction of Hξ for a certain high threshold u. This is done by comparing the mean

excess function of F with the mean excess function of the Generalized Pareto distribution. The mean excess function of the Generalized Pareto distribution is given by;

e(u) = β + ξu

1 − ξ . (3.2.5)

For the derivation of equation (3.2.5) we refer to Appendix A. Equation (3.2.5) shows that the mean excess function of a GPD type distribution is a linear function with respect to the threshold u. Hence, if the distribution F ∈ MDA (Hξ) then the mean excess function of F becomes linear

for a certain high threshold u.

3.3 Estimation of parameters

In this section we discuss some methods to determine the parameters u, ξ and β. First we will discuss a method to determine a suitable high threshold u beyond which the excess distribution function can be well-approximated by a GPD. Then we will discuss four methods to determine the parameters of the excess distribution function. The first three methods are the maximum

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likelihood estimator, the method of moments and the probability weighted moments and are all based on the Generalized Pareto distribution. The fourth method is the Hill estimator, which is based on the result of theorem 3.1.4.

The first three methods are summarized and compared by Hosking and Wallis (1987). They conclude that for sufficiently large datasets the maximum likelihood estimator is the most efficient estimator. The method of moments and the probability weighted moments estimators perform better than the maximum likelihood estimator for smaller datasets. However, their performance depends on the value of the parameter ξ, if ξ > 0.2 the method of moments performs better, if 0 < ξ < 0.2 the probability weighted moments estimator is preferred.

3.3.1 Threshold parameter

u

There is plenty of literature that discusses the estimation of the parameters of the Generalized Pareto distribution. But the literature on the estimation of threshold u is not that elaborate. One often used method to estimate the threshold u is to make an empirical mean-excess-plot for several values of u. As presented by equation (3.2.5), the mean excess function of a Generalized Pareto distribution is a linear function with respect to the threshold u. Hence, one would expect that for a sufficiently high threshold u the mean-excess-plot will also become linear. With the choice of threshold u there is the variance-bias trade-off. A higher choice of the threshold u results in less datapoints that are available to estimate the parameters of the excess distribution function, and hence a large variance in the estimates. But one would also expect that for a higher threshold it is more likely that the data will have a Generalized Pareto distribution, and hence the estimates will have a low bias. A lower choice of the threshold u will result in more datapoints, but the data might not closely follow a Generalized Pareto distribution.

3.3.2 Maximum likelihood estimator

With the likelihood function of the Generalized Pareto distribution function one can determine the values of ξ and β for which the likelihood function is maximized given the observed exceedances above the threshold u. The likelihood function of the Generalized Pareto distribution function is given by; L(ξ, β|x1, . . . , xn) = n Y i=1 1 β 1 + ξ βxi −1/ξ−1 , (3.3.1)

where x1, . . . , xn are the observed exceedances above the threshold u. The maximum of equation

(3.3.1) with respect to the parameters ξ and β cannot be determined analytically. Numerical methods, such as the Newton-Raphson method, are used to determine the solution. For more details on the estimation of the parameters ξ and β based on the likelihood function we refer to appendix B.1.

3.3.3 Method of moments

The Method of Moments uses the assumption that the moments of the datapoints, more precisely the mean and the variance, should equal the theoretical values of the moments of the Generalized Pareto distribution. The mean and variance of a random variable X with a Generalized Pareto distribution are given by;

E[X] = β

1 − ξ and var(X) =

β2

(1 − ξ)2_{(1 − 2ξ)}. (3.3.2)

Setting the expression in equation (3.3.2) equal to the observed mean ¯x and variance ¯s2 _{of the}

sample, we get the following expressions for ξ and β;

β = 1 2x¯ 1 +x¯ 2 ¯ s2 and ξ = 1 2 1 − x¯ 2 ¯ s2 . (3.3.3)

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3.4. THE THRESHOLD MODEL ₁₅

For the derivation of the expressions in equation (3.3.3) we refer to appendix B.2.

3.3.4 Probability weighted moments

The probability weighted moments of a continuous random variable X with cumulative distribution function F are given by;

Mp,r,s= E[XpF (X)r(1 − F (X))s].

Hosking and Wallis (1987) suggest to work with the the moments;

αs= M1,0,s= E[X(1 − F (X))s] =

β2

(s + 1)(s + 1 − ξ),

which exist if ξ < 1. We can determine the parameters ξ and β using α0 and α1, which gives;

β = 2α0α1

α0− 2α1 and ξ = 2 −

α0

α0− 2α1. (3.3.4)

We refer to appendix B.3 for a detailed derivation of the expressions in equation (3.3.4).

The solutions of β and ξ in equation (3.3.4) depend on the values of α0 and α1. Hosking and

Wallis (1987) propose the estimator ˜αr for an observed sample size n of exceedances above the

threshold u; ˜ αr= 1 n n−1 X j=0 1 − j + γ n + δ r x_n−j, (3.3.5)

where xn ≤ . . . ≤ x1 denotes the ordered sample of exceedances above the threshold u, and γ and

δ are suitable constants. Hosking and Wallis (1987) propose the values of γ = −0.35 and δ = 0.

3.3.5 The Hill method

The Hill method is derived by Hill (1975) and it is not based on the assumption that the excess distribution function Fu(x) converges to the Generalized Pareto distribution, but that it has

a Fr´echet distribution. Hence by Theorem 3.1.4, the excess distribution function is given by ¯

Fu(x) = x−1/ξL(x) with L a slowly varying function and ξ > 0. Hill (1975) considers the special

case with ¯Fu(x) = Cx−α with α = 1/ξ.

Suppose that the dataset consists of N observations, where n observations Xn:N ≤ . . . ≤ X1:N are

above the threshold u. Now the Hill-estimator for C and α are given by;

ˆ α(H)_n,N =   1 n n X j=1 ln Xj:N − ln Xn:N   −1 , ˆ Cn,N = n NX ˆ α(H)_n,N n:N . (3.3.6)

For a derivation of these estimators we refer to appendix B.4.

3.4 The Threshold model

The parameter estimation methods described in section 3.3 determine the parameters of the Gen-eralized Pareto distribution, i.e. the distribution of excesses above a certain threshold u. For the estimation procedures only the observations above the threshold are used. Hence, in the estima-tion of the parameters many observaestima-tions are not taken into account and for a sufficiently high threshold u there could not be enough observations available to perform the estimations on. There-fore, Wong and Li (2010) propose the Threshold model which does take all the observations into

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account. They base the Threshold method on the following partition of the distribution function P (X ≤ x);

P (X ≤ x) = P (X ≤ x ∩ X ≤ u) + P (X > u)P (X ≤ x|X > u). (3.4.1) As u → ∞ the conditional probability P (X ≤ x|X > u) can be approximated by the Generalized Pareto distribution presented by equation (3.2.2). The left-hand side of the sample space (below the threshold u) is modeled by a truncated distribution function L with parameter θ ∈ Rp_{. This}

results in the Threshold model;

F (x; θ, β, ξ) =

L(x|θ), x ≤ u,

L(u|θ) + (1 − L(u|θ))Gξ,β(x − u), x > u. (3.4.2)

The truncated distribution function L(x|θ) should be chosen such that it fits the observations below the threshold best. The parameters u, θ, β and ξ of the Threshold model are determined such that the likelihood of equation (3.4.2) is maximal. A benefit of this procedure is that also the choice of the threshold u is incorporated into the determination of the optimal set of parameters. This is done by the following procedure:

1. Choose a threshold u.

2. Suppose we have the ordered observations XN :N ≤ . . . ≤ X1:N and n observations are above

the threshold u. Then the log-likelihood function of the Threshold model is given by;

ℓ(θ, β, ξ) = N −n X i=0 logl(X_{(N −i):N}|θ) + N X i=N −n+1 log {1 − L(u|θ)} + N X i=N −n+1

loggξ,β(X_{(N −i):N} − u|ξ, β) .

=

N −n

X

i=0

logl(X_{(N −i):N}|θ) + n log {1 − L(u|θ)}

−n log(β) + (−1/ξ − 1) N X i=N −n+1 log 1 + ξ β(X(N −i):N− u) . (3.4.3)

The log-likelihood function should be maximized with respect to the parameters θ, ξ and β. Note that the contribution of the parameters ξ and β in equation (3.4.3) is the same as in equation (3.3.1), the likelihood of only the Generalized Pareto distribution, and can be estimated separately from θ. The expression for θ depends completely on the choice of the distribution function L(x|θ). At this moment we do not know the distribution below the threshold, therefore, cannot determine an expression for θ.

3. Repeat steps 1 and 2 for several values of the threshold. The combination of u, θ, β and ξ for which the expression (3.4.3) approaches its maximum should be chosen.

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Chapter 4

Estimation of the distribution of

the claims-to-BRA ratio

In this chapter we will implement the theory presented in chapter 3. The goal of this chapter is to find a distribution for the claims-to-BRA ratio. Using extreme value theory we will determine the parameters of the Generalized Pareto distribution for observations above a certain high threshold u. By implementing the Threshold model as described in section 3.4, we can determine the distribution for observations below the threshold and combine this with the Generalized Pareto distribution into one distribution for the claims-to-BRA ratio.

Firstly, we will discuss the available data at De Amersfoortse and give some summarizing statistics. Secondly, we present the mean excess plot and the Hill plot. Using the mean excess plot we gain insight for which threshold u the distribution of the observations above the threshold behaves like a Generalized Pareto distribution. The Hill plot can be used to determine whether there is a range of thresholds for which the estimated parameters are expected to be stable. Thirdly, we estimate the parameters of the Generalized Pareto distribution by using the maximum likelihood method, the method of probability weighted moments and the method of moments and discuss the results. Fourthly, we will focus on the Threshold model to obtain a distribution for the whole range of values of the claims-to-BRA ratio. Lastly, we will repeat the parameter estimations where we distinguish between the number of employees per contract.

4.1 Available data and summarizing statistics

As discussed in section 2.4, the estimation of the parameters is performed on the absenteeism conventional product over the years 2007 up to 2013. Table 4.1 gives an overview of the number of contracts (# contr.) and the number of contracts with a reported claim (# claim), where we distinguish between the number of employees on a contract. A contract is included in the count of a calendar year if it has been fully active during this calendar year. With this choice we did not have to manipulate the experienced claim amount or the basis retention amount. For the smaller companies, with up to 10 employees, the percentage of contracts with a reported claim is approximately 40 percent. Especially for the really small companies, with one or two employees, this low percentage could be explained by the relatively large consequences of absenteeism for the continuity of the business. The percentage of contracts with a reported claim increases and approaches, as expected, 100 percent for the larger companies.

In the data the effective date of each contract is available. Hence we are able to determine the effect of the number of active years of the contract on the claims-to-BRA ratio. Remembering figure 2.1, contracts with an effective date before year t − 2 have their complete claim history taken into account in the claim amount of calendar year t. For contracts with an effective date in year t − 2 and especially an effective date in year t − 1 the claim amount in calendar year t will

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18CHAPTER 4. ESTIMATION OF THE DISTRIBUTION OF THE CLAIMS-TO-BRA RATIO

calendar year

2007 2008 2009 2010 2011 2012 2013

# employ. # contr. # contr. # contr. # contr. # contr. # contr. # contr. (# claim) (# claim) (# claim) (# claim) (# claim) (# claim) (# claim)

0 - 10 22,376 23,395 23,950 22,624 22,282 22,451 22,489 (9,024) (9,607) (10,138) (9,230) (9,158) (9,080) (8,722) 10 - 20 5,526 5,895 5,716 5,096 4,925 4,907 4,746 (4,311) (4,681) (4,584) (4,160) (4,028) (3,882) (3,754) 20 - 30 1,643 1,785 1,714 1,513 1,496 1,498 1,423 (1,466) (1,586) (1,531) (1,384) (1,348) (1,334) (1,261) 30 - 40 658 700 673 565 562 546 574 (609) (654) (628) (538) (531) (507) (535) 40 - 50 319 362 342 267 261 253 223 (302) (341) (321) (251) (249) (234) (203) 50 - 60 216 224 206 163 140 134 146 (204) (214) (197) (155) (136) (127) (136) 60 - 70 128 134 118 81 80 78 62 (125) (128) (113) (79) (77) (76) (61) 70 - 80 58 79 84 47 46 43 47 (57) (77) (82) (46) (44) (40) (44) 80 - 90 47 58 40 30 24 28 25 (46) (55) (40) (30) (24) (27) (23) 90 - 100 39 32 35 18 19 20 16 (39) (31) (34) (17) (19) (20) (16) 100+ 84 115 101 71 68 70 59 (83) (115) (101) (70) (68) (68) (58)

Table 4.1: The number of contracts (# contr.) and the number of contracts with a claim in brackets (# claim) per calendar year.

not be based on the full claim history. In equation (2.2.2) this behavior is expressed by the factor frun-in. By determining the average claims-to-BRA ratio per years active, we can easily determine the frun-in parameter. The results are shown in table 4.2 and based on contracts with a reported claim and at least 25 employees. Later on, we will discuss why we only include these contracts into the statistic. A contract with an effective date in year t − 1 is one year active, a contract with an effective date in year t − 2 is two years active and the contracts with an effective date before year t − 2 are three years active. Note that the average claims-to-BRA ratio of contracts that are at least three years active is slightly above 1, which indicates a larger claim amount than expected by the BRA. This suggests that the BRA might not be estimated correctly or that some extreme years has occurred.

Table 4.3 gives the summarizing statistics of the claims-to-BRA ratio per category of number of employees. The claims-to-BRA ratio is corrected with the frun-in parameter from table 4.2 for companies with effective dates in the years t − 1 and t − 2. Furthermore, we only base it on contracts with a reported claim. For the categories 0-10 employees and 10-20 employees the average claims-to-BRA ratio is above one, since there is also a large part of contracts that does not report a claim. If we would take this effect into account, the average claims-to-BRA ratio of all the contracts (with and without reported claims) is approximately one. For the larger companies the

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4.1. AVAILABLE DATA AND SUMMARIZING STATISTICS ₁₉

# years active # obs Av. ratio _frun-in

1 1,330 0.8884 0.8643

2 1,638 0.9658 0.9397

3 9,983 1.0278 1

Table 4.2: _{The average claims-to-BRA ratio per number of years active and the corresponding frun-in}

factor.

average claims-to-BRA ratio is close to one, since it is expected that all the companies will report a claim. There are some larger companies that did not report a claim. We cannot be completely sure that all the companies report their sick employees on the first day of illness. Instead, it might be that they only report a claim if the duration of the illness is longer than a certain number of days. The absenteeism stop-loss product is designed for the larger companies with at least 50 employees. For larger companies it is very unlikely that none of the employees experiences a day off due to illness. From the numbers in table 4.1 with at least 50 employees we observe that approximately 97 percent of the contracts have a reported claim. Therefore, for the estimation of the parameters we will use only the data of contracts with a reported claim.

Table 4.3 shows that the standard deviation of the claims-to-BRA ratio decreases as the number of employees increases. This behavior of the standard deviation is expected. By the law of large numbers, as the company size increases, it becomes more certain that the average claim amount will be close the the expected value, i.e. the basis retention amount. This will decrease the standard deviation of the claims-to-BRA ratio as the company has more employees. This effect is also visible in the maximal value of the claims-to-BRA ratio per category of number of employees. However, the maximal value stabilizes for companies with more than 70 employees, while the standard deviation still seems to decrease.

# employ. # obs Average Std. dev. Minimum Maximum

0 - 10 64,957 2.1905 4.3554 0 654.0930 10 - 20 29,400 1.1327 1.1539 0 12.2093 20 - 25 6,154 1.0582 0.9501 0 9.8510 25 - 30 3,756 1.0314 0.8626 0 11.2511 30 - 40 4,002 1.0266 0.7816 0.0020 6.0747 40 - 50 1,901 1.0153 0.7377 0.0012 5.0978 50 - 60 1,169 1.0568 0.7169 0.0030 4.7950 60 - 70 659 0.9861 0.6403 0.0094 4.1541 70 - 80 390 1.0616 0.6572 0.0040 3.3100 80 - 90 245 1.0644 0.6597 0 2.9741 90 - 100 176 0.9768 0.6121 0.0386 3.2814 100+ 563 1.0201 0.6169 0.0013 4.4960 Total 113,372 1.7228 3.4096 0 654.0930

Table 4.3: Statistics of the claims-to-BRA ratio per category of number of employees.

For the smaller companies the minimum value of the claims-to-BRA ratio equals zero, as there are some contracts where one employee has been sick for one day and the salary of this employee is not known in the administrative system of De Amersfoortse. When no claim payment is made for this employee, the salary will not be registered for this calendar year. There is one contract in the category 80 to 90 employees, where this situation also occurs.

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20CHAPTER 4. ESTIMATION OF THE DISTRIBUTION OF THE CLAIMS-TO-BRA RATIO

For the estimation of the parameters we will only use contracts with at least 25 employees. We made this choice for the following two reasons. Firstly, the average of the claims-to-BRA ratio in these companies is close to one and the standard deviation is significantly lower than for smaller companies. Secondly, it is consistent with the current pricing scheme which is applicable for companies with 25 or more employees. Hence using these contracts to estimate the parameters, makes it possible to compare the results of the current and the proposed pricing scheme.

Table 4.4 gives the summarizing statistics per year of the number of contracts and the claims-to-BRA ratio. These statistics are based on contracts with at least 25 employees and a reported claim. Note that the number of active contracts is decreasing over the years. The average and the standard deviation of the claims-to-BRA ratio fluctuates over the years, but there is no trend visible which should be included in the estimation of the parameters. The maximal value of the claims-to-BRA ratio also fluctuates over the years, where in 2012 there is an outlier in data for a contract in the category 25 to 30 employees.

Year # obs Average Std. dev. Minimum Maximum

2007 1,999 0.9772 0.7590 0.0040 5.0978 2008 2,243 0.9695 0.7048 0 5.6985 2009 2,127 1.0635 0.7923 0.0020 7.1806 2010 1,702 1.0951 0.7824 0.0005 6.0747 2011 1,637 1.0994 0.8208 0.0026 6.1381 2012 1,612 1.0466 0.8316 0.0061 11.2511 2013 1,541 0.9590 0.7221 0 6.2915 Total 12,861 1.0278 0.7738 0 11.2511

Table 4.4: Statistics of the claims-to-BRA ratio per calendar year.

Figure 4.1 shows the distribution of the claims-to-BRA ratio for contracts with reported claims and 25 or more employees. The majority of the contracts has a claims-to-BRA ratio between zero and one, i.e. the experienced claim amount is lower than the expected basis retention amount. The largest part has a claim-to-BRA ratio between a half and one. It is clearly a right skewed distribution. This indicates that the normal distribution is very unsuitable to model the claim-to-BRA ratio, as mentioned in section 2.3.

4.2 Mean excess plot

Figure 4.2 shows the number of observations above the threshold u in the left figure. The right figure shows the empirical mean excess plot for the claims-to-BRA ratio. The mean excess func-tion e(u) is given by equafunc-tion (3.2.4), the empirical mean excess funcfunc-tion is computed by taking the average of the exceedances above the threshold u, for several values of the threshold u. The empirical mean excess function is decreasing for u < 2. For u > 2 the empirical mean excess func-tion is increasing and approximately linear. This indicates the heavy tailedness of the distribufunc-tion of the claims-to-BRA ratio. Therefore, we can conclude that for a threshold u larger than two the distribution of the excesses can be approximated well by the Generalized Pareto distribution. The mean excess function of the Generalized Pareto distribution, given by equation (3.2.5), is also a linear function with respect to the threshold u. Since we observe a linear increasing empirical mean excess function it must hold that ξ is larger than zero.

For very large values of the threshold u, the linearity of the empirical mean excess function disap-pears. Above the claims-to-BRA ratio of five there are only 18 of the total of 12,861 observation left. For these outliers it is difficult to maintain the linearity of the mean excess function.

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4.3. HILL PLOT ₂₁ 0 1 2 3 4 5 6 7 8 9 10 11 0 0.31 Claims-to-BRA ratio P ro b a b ilit y

Figure 4.1: Distribution of the claims-to-BRA ratio for companies with at least 25 employees.

0 2 4 6 8 0 5,000 10,000 Threshold u N u m b er o f o b se rv a tio n s a b ov e u 0 2 4 6 8 1 2 Threshold u M ea n ex ce ss e( u )

Figure 4.2: The left figure shows the number of observations per threshold u. The right figure shows the empirical mean excess plot of the claims-to-BRA ratio.

4.3 Hill plot

Based on the Hill estimator for α, see expression (3.3.6), we are looking for a region where the estimates of α are stable. Therefore, we compute a Hill plot for several values of the threshold u and the corresponding number of upper order statistics n above the threshold u, see figure 4.3. The horizontal axis shows the value of the threshold u (top) and the number of upper order statistics n (bottom). The vertical axis shows the computed value of ˆα(H)n,N (black) and the corresponding

value of the 95% confidence interval (blue). The right figure is an enlargement of the left figure for the higher values of the threshold u. For a threshold u between 2.35 and 1.99 the Hill estimator is quite stable around ˆα(H)_n,N = 4, which indicates a value of ξ = 0.25 for the Generalized Pareto distribution. The region of stability of the Hill estimator corresponds with the findings from the empirical mean excess plot. Above the threshold of 2 the mean excess plot becomes linear and upward, which corresponds to the stability of the Hill estimator above the threshold of 2.

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22CHAPTER 4. ESTIMATION OF THE DISTRIBUTION OF THE CLAIMS-TO-BRA RATIO 18 619 1220 1820 2421 3022 3622 4223 4824 5424 2 4 6 8 4.99 2.51 2.07 1.79 1.60 1.44 1.31 1.19 1.09 1.00 18 170 321 472 623 775 926 1077 1228 1379 4 5 6 7 8 9 4.99 3.35 2.97 2.69 2.50 2.35 2.23 2.14 2.06 1.99

Figure 4.3: Hill plot of the Hill estimator ˆα(H)_n,N. The figure on the right is an enlargement of the left figure.

4.4 Parameter estimation of the Generalized Pareto

distri-bution

The Hill plot shows that the estimator for ˆα(H)_n,N is stable for thresholds in the range of 1.99 up to 2.35. Hence, we expect that the estimated values of the shape parameter ξ of the Generalized Pareto distribution are stable for the same values of the threshold u. We determine the parameters ξ and β of the Generalized Pareto distribution for several values of the threshold u and based on the three methods as described in section 3.3, i.e. the maximum likelihood method, the method of probability weighted moments and the method of moments. Table 4.5 shows the estimated values of the parameters of the Generalized Pareto distribution for several values of the threshold u and computed by the three different methods, for more detailed results we refer to table C.1.

MLE PWM MOM u # obs ξˆ βˆ ξˆ βˆ ξˆ βˆ 1.75 1,939 0.0417 0.6495 0.0187 0.6652 0.0535 0.6417 2.00 1,358 0.0757 0.6172 0.0756 0.6175 0.0870 0.6099 2.25 900 0.0574 0.6589 0.0302 0.6783 0.0768 0.6457 2.50 627 0.0794 0.6450 0.0608 0.6587 0.1020 0.6298 2.75 434 0.0849 0.6533 0.0453 0.6827 0.1154 0.6326 3.00 310 0.1553 0.5911 0.1588 0.5894 0.1660 0.5843

Table 4.5: Estimated values of the shape ξ and scale β parameters of the GPD computed by the three proposed methods. The column # obs indicates the number of observations above the threshold u.

The results in table 4.5 show that the estimated value of the parameters depend on the method which is used to determine the parameter. Especially, the estimated value of ˆξ is different for the three methods, the estimated value of ˆβ is better comparable. Compared to the results of the maximum likelihood estimator, the method of probability weighted moments underestimates the value of ˆξ while the method of moments overestimates the value of ˆξ. The estimated value of the shape parameter does not correspond to the results from the Hill estimator. Based on the Hill

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4.4. PARAMETER ESTIMATION OF THE GENERALIZED PARETO DISTRIBUTION ₂₃

estimator we would expect a value around 0.25, but the results in table 4.5 indicate a much lower value for ˆξ.

Hosking and Wallis (1987) compared the three methods and their performance, and they concluded that the method of moments is preferred if ξ > 0.2. For our dataset, based on the estimated values for ˆξ this is clearly not the case. The values for ˆξ computed by the method of moments differ clearly from the estimated values based on the maximum likelihood method. Therefore, we will not determine the parameters of the Generalized Pareto distribution using the method of mo-ments. Hosking and Wallis (1987) found that the method of probability weighted moments is preferred above the maximum likelihood method if 0 < ξ < 0.2 and the number of observations above the threshold u is limited, since the maximum likelihood estimator does not always con-verge for smaller datasets. In the performed estimations, since there are a sufficient number of observations above the threshold u, the maximum likelihood estimator always converged. Table C.1 shows that, when the stepsize between the subsequent values of the threshold is refined, the estimated parameters based on the method of probability weighted moments are not as stable as the parameters determined by the maximum likelihood estimator. Therefore, we will base the values of the parameters on the results of the maximum likelihood estimator.

Similar to the Hill plot, figure 4.4 shows the estimated values of ˆξ and ˆβ for several values of the threshold u (solid black line) and the corresponding 95% confidence interval (dashed blue line) based on the results of the maximum likelihood estimator. The shape parameter ξ is quite stable for the thresholds in the range of 2 to 2.6. The estimated values for the scale ˆβ are not as stable as the values for ˆξ, but for the stable region of the shape parameter ˆξ the value of ˆβ is also rather stable. 1.8 2 2.2 2.4 2.6 2.8 3 −0.1 0 0.1 0.2 0.3 Threshold u S h a p e (ξ ) w it h p = 0 .9 5 1.8 2 2.2 2.4 2.6 2.8 3 0.4 0.5 0.6 0.7 0.8 Threshold u S ca le (β ) w it h p = 0 .9 5

Figure 4.4: Maximum likelihood estimates for ξ and β. The left figure shows the estimated value ˆξ (black line) and the corresponding 95% confidence interval (dashed blue line). The right figure shows the estimated value ˆβ(black line) and the corresponding 95% confidence interval (dashed blue line).

Figure 4.5 shows, for three values of the threshold u, the fitted Generalized Pareto distribution and the empirical distribution of the excesses. The fit for the thresholds u = 1.75 and u = 2.25 is very good and there is no real difference visible. The values of the parameters ˆξ and ˆβ are also very comparable for the these values of the threshold. For the threshold u = 3 the fit is not as good, the Generalized Pareto distribution slightly overestimates the tail of the excess distribution. This corresponds to the higher estimated value of the shape parameter ˆξ. The choice which value of the threshold u gives the best fit, will be made in section 4.5.

Pricing an absenteeism stop-loss portfolio using extreme value theory