Profitability-testing of a Bonus-Malus System on a portfolio of automobile insurance

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Profitability-testing of a

Bonus-Malus System on a portfolio

of automobile insurance

Ngonidzashe Fungura

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: Ngonidzashe Fungura Student nr: 11085673

Email: ngonifungura@gmail.com

Date: July 24, 2016 Supervisor: Dr. S.U. Can Second reader: Dr. K. Antonio

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Statement of Originality

This document is written by Ngonidzashe Fungura who declares to take full responsibility for the contents of this document.

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

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

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“No great discovery was ever made without a bold guess.”

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Abstract

This thesis provides an analysis of how profits on an automobile insurance portfolio of an insurer react to the implementation of a Bonus-Malus System (BMS) in the pricing process of automobile insurance contracts. Over the last few decades most insurers have implemented this system either through the act of law, such as in France, or to beat the competitive environment. The motive behind a BMS is to ensure that each policyholder in an automobile insurance portfolio pays a premium which correctly reflects the expected value of his claims and therefore not be subsidised by other policyholders or him subsidising other policyholders. This paper then analyses an automobile insurance portfolio where in the first 10 years the premiums are only determined by the risk profile of the policyholders and in the next 10 years the premiums are determined by both the risk profile and the claims history. Claim amounts are known to be a stochastic liability which differs from one period to another and after implementing a BMS, claim amounts will decline due to the “hunger for bonus” though they still remain stochastic. However, before implementing a BMS premium income to be collected from policyholders is deterministic which means it is known in advance. The premium income to be collected from policyholders becomes a stochastic element for some years after implementing a BMS as policyholders will be moving to different premium discount levels but when the BMS achieves an equilibrium the premium income becomes approximately deterministic again. These changes have effects on the profitability of the insurer and this paper analyses the impact of these changes.

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Acknowledgements

First and foremost, I thank the Lord God for being my strength and guide throughout the writing of my thesis. I am deeply humbled that He has taken me this far.

I also take this opportunity to express my invaluable gratitude to my thesis advisor, Dr. Sami Umut Can, for his patience, support and guidance throughout the writing of this thesis. Each time I visited his office with burning issues on this thesis, I felt welcome and with his great insight of the subject he would always lead me to the solution. He also allowed my thesis to be my own work but he would manoeuvre me in the right direction when necessary. I will always remember him for the rest of my career and life and I will always consult him after my Masters Degree.

My special thanks also goes to the following; my brother, Dr. Martin Fungura, and his wife, Felistas Mbirizah Fungura, my sister, Dr. Catherine Fungura Gohori, and her husband, Kundai Walter Gohori for paying my tuition and to ensure that I had enough money throughout the year of my study. If it was not for their generosity, I would have not afforded such a life changing opportunity. I really appreciate your love. Moreover, I extend my sincere thanks to the unconditional love that my parents, Kudzayi and Augustine Fungura, and my sisters, Simbisai Fungura Kaseke and Irene Neganje showed me throughout my study period.

To Tendayi Dhura, thank you so much for always being there for me, even though you were 8,500 miles away. The journey would not have been easy without your unconditional support and love.

Finally, I thank my colleagues who assisted me in different ways especially Barry Coleman and James Mills.

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List of Figures

4.1 Loss Ratios for the original BMS (black line) . . . 29

4.2 Loimaranta for the original BMS (black line) . . . 31

4.3 Loss Ratios with reduced number of Bonus classes (red line) . . . 32

4.4 Loimaranta Efficiency with reduced number of Bonus classes (red line) . . 33

4.5 Loss Ratios with reduced BM factor (blue line) . . . 34

4.6 Loimaranta Efficiency with reduced BM factor (blue line) . . . 35

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List of Tables

1.1 Canonical Link Functions . . . 3

1.2 Deviances . . . 5

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Chapter 1

Introduction

According to the European Motor Insurance Market Journal (2015)[2], “motor insurance is the most widely purchased non-life insurance product in Europe, accounting for 27.4% of non-life business”. The same trend prevails in other continents because it is usually a regulatory requirement by most governments for automobile owners to have third-party liability insurance. This ensures that at least the cost of bodily harm to a third party and/or the cost of physical damage to the property of a third party are covered during the period of ownership of the automobile by an individual. However, most owners of automobiles pay more in premiums than the minimum required for third-party liability insurance in order to cover for the cost of damage to their own automobiles, which is known as Comprehensive Insurance Cover.

1.1 Premium Determination

Defining X as a risk and π[X] as the premium associated with X, Kaas et al (2008)[3] state that a premium without a positive loading will lead to ruin with certainty because it does not take into account the variability of the risk, and the maximal loss premium is a boundary case because the premium can not exceed the maximum value of the risk. The first property of premium principle is known as the non-negative loading which is represented as;

π_{[X] ≥ E[X]} (1.1)

and the second property is known as no rip-off represented as;

π[X] ≤ max[X] (1.2)

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Chapter 1 Introduction 2 Premiums of an automobile portfolio can be determined using, for example, Generalized Linear Models (GLMs) or Credibility Theory.

1.1.1 Generalized Linear Models (GLMs)

Ohlsson and Johansson (2010)[10] state that for each policy, the premium is determined by the values of a number of covariates and to deduce the relationship between claim amounts and the covariates, a statistical model is employed. The covariates acknowledged fall in the following categories; properties of the policyholder (e.g. age and gender), properties of the insured objects (e.g. model of a car) and properties of the geographic region (e.g. population density of the policyholder’s residential area).

The statistical model most widely employed to determine these premiums is called the GLM which the duo define as a rich class of statistical methods. GLMs are used to analyse how a dependent variable responds to some covariates in a multiple linear regression. 1.1.1.1 Components of a GLM

Kaas et al (2008)[3] note that GLMs have three components: • Stochastic component

The stochastic component of the model states that the observations are independent random variables Yi, i = 1, . . . , nwith a probability density function in the exponential

dispersion family with a constant scale parameter. • Systematic component

The systematic component of the model assigns to every observation, a linear predictor ηi = xi1β1+ xi2β2+ . . . + xipβp, linear in the parameters β1, β2, . . . , βp.

The xi1, xi2, . . . , xip are called the covariates of observation i. Lindsey (1997)[7]

states that the linear predictor ηi describes how the location of the response

distribution changes with the covariates xij’s. If a parameter βj has a known

value, the corresponding term in the linear structure is called an offset. • Link function

The link function links the expected value of each observation to the linear predictor. If µi = E[Yi], the link function g(.) is such that;

ηi= xi1β1+ xi2β2+ . . . + xipβp = g(µi) (1.3)

The link function must be monotonic, which means it must vary in such a way that it either never decreases or never increases, and the function should be differentiable.

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Chapter 1 Introduction 3 Usually, the same link function is used for all observations and the canonical link function is that function which transforms the mean to a canonical location parameter of the Exponential Dispersion Family (EDF) member. The canonical link functions of different EDF members are given below;

Distribution Canonical Link Function Poisson Log ηi= log(µi)

Binomial Logit ηi= log _nµi

i−µi

Normal Identity ηi= µi

Gamma Reciprocal ηi= _µ1_i

Inverse Gaussian Reciprocal2 _η i= _µ12

i

Table 1.1: Canonical Link Functions

1.1.1.2 Exponential Dispersion Family (EDF)

According to McCullagh and Nelder (1989)[9] we assume that each random component Yi from ~Y has a distribution in the EDF taking the form;

fY(y; θ, φ) = exp

hyθ − b(θ)

a(φ) + c(y, φ) i

(1.4) for some specific functions a(.), b(.) and c(.) and here θ is called the natural parameter or the canonical parameter and φ is the dispersion parameter or the scale parameter. If φis known, this is an exponential family model with canonical parameter θ.

The mean and variance of Y can be derived from the well known relations; E ∂l ∂θ ! = 0 (1.5) and E ∂ 2_l ∂θ2 ! + E ∂l ∂θ !2 = 0 (1.6)

where l = l(θ, φ; y) = log fY(y; θ, φ). This will give us results;

E[Y ] = µ = b0(θ) (1.7) and

var[Y ] = a(φ)b00(θ) (1.8) As seen from the two results, the mean of Y only depends on the canonical parameter θ whereas the variance of Y is a product of two functions; one, b00(θ), depends on the

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Chapter 1 Introduction 4 canonical parameter (and hence the mean) only and is called the variancefunction, while the other is independent of θ and only depends on φ. The variance function considered as a function of µ will be written V (µ). The function a(φ) is commonly of the form;

a(φ) = φ

w (1.9)

where φ, also denoted by σ2 _{and called the dispersion parameter, is constant over}

observations, and w is a known prior weight that varies from observation to observation. Thus for a Normal model in which each observation is the mean of m independent readings we have;

a(φ) = σ

2

m (1.10)

so that w = m.

Some of the distributions which fall in the EDF are Poisson, Binomial, Normal, Gamma and Inverse Gaussian and their variance functions are µ, µ(1−µ), 1, µ2_{and µ}3_{respectively.}

1.1.1.3 Measuring the goodness of fit of a model

According to McCullagh and Nelder (1989)[9], the process of fitting a model to data may be regarded as a way of replacing a set of data values ~y = (y1, y2, . . . , yn)T by

a set of fitted values ~ˆµ = (ˆµ1, ˆµ2, . . . , ˆµn)T derived from a model involving usually a

relatively small number of parameters. In general, the fitted values will not be exactly equal to the observed data and the question arises of how discrepant they are, because while a small discrepancy is desirable a large discrepancy is not. In GLMs, the measure of discrepancy or goodness of fit is formed from the logarithm of a ratio of likelihoods between a saturated or full model and the model under study. This measure is called the deviance.

Given n observations we can fit a model to the observations containing up to n parameters. The simplest model, the null model, has one parameter representing a common µ for all the y’s; the null model thus consigns all the variation between the y’s to the stochastic component. The saturated or full model has n parameters, one for each observation and thus the fitted values of ˆµ’s derived from it replicates the observed data. The saturated or full model thus consigns all the variation between the y’s to the systematic component leaving none for the stochastic component.

In practice, the null model is usually too simple and the full model is uninformative because it does not summarise the data but merely repeats them in full. However, the full model gives a baseline for measuring the discrepancy for an intermediate model with p parameters where 1 6 p 6 n.

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Chapter 1 Introduction 5 It is convenient to express the log likelihood in terms of the mean-value parameter ~µ rather than the canonical parameter ~θ. Let l(~ˆµ, φ, ~y) be the log likelihood over ~β for a fixed value of the dispersion parameter φ. The maximum likelihood achievable in a full model with n parameters is l(~y, φ, ~y), which is ordinarily finite. The discrepancy of a fit is proportional to twice the difference between the maximum log likelihood achievable and that achieved by the model under investigation. If we denote ˆθ = θ(ˆµ) and ˜θ = θ(y) the estimates of the canonical parameters under the two models, the discrepancy, assuming ai(φ) = _wφ_i, can be written as;

2Xwi[yi( ˜θi− ˆθi) − b( ˜θi) + b( ˆθi)]/φ =

D(y; ˆµ)

φ (1.11)

where D(y; ˆµ) is known as the deviance and; D(y; ˆµ)

φ = D

∗_{(y; ˆ}_µ) _(1.12)

where D∗_{(y; ˆ}_µ) _{is known as the scaled deviance.}

The forms of the deviances for the distributions in the EDF are; Distribution Deviance

Poisson 2P [y log y_µ_ˆ − (y − ˆµ)]

Binomial 2P [y log y_µ_ˆ + (m − y) log(m−y) (m−ˆµ) ] Normal P (y − ˆµ)2 Gamma 2P [− log y ˆ µ + (y−ˆµ) ˆ µ ]

Inverse Gaussian P(y−ˆµ)2

ˆ µy

Table 1.2: Deviances

1.1.1.4 Comparing different models

Kaas et al (2008)[3] note that from the theory of mathematical statistics it is known that the scaled deviance is approximately χ2 _{distributed, with as degrees of freedom the}

number of observations n less the number of estimated parameters p. Also, if one model is a sub-model of another, it is known that the difference between the scaled deviances has a χ2 _{distribution. Analysis of deviance can be used to compare two nested models,}

one which arises from the other by relaxing constraints on the parameters. Some of the scenarios are if a factor or variate is added as a covariate, if a variate is replaced by a factor and if the interaction between factors is allowed. If the gain in scaled deviance exceeds the 95% critical value of the χ2_(k) _{distribution, with k the number of extra parameters}

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Chapter 1 Introduction 6

1.1.2 Credibility Theory

In insurance practice it often occurs that insurers have to set a premium for a group of insurance contracts for which there is some claim experience regarding the group itself, but a lot more on a larger group of contracts that are more or less related as noted by Kaas et al (2008)[3].

Defining ¯X to be the overall mean of the data and ¯Xj to be the average claims in group

j then the premium is given by;

π[X] = zjX¯j + (1 − zj) ¯X

where zj is called a credibility factor and it contains the information of how credible the

individual experience of cell j is and in this case the π[X] is a credibility premium.

1.2 Remark

Some years back, policyholders who had the same risk characteristics to the insurer were charged the same premium on inception and thereafter the premiums remained the same for the next periods regardless of the policyholders’ claims history. This resulted in ex antemoral hazard whereby drivers acted carelessly after the risk transfer hence increasing the probability of accidents. In a bid to reduce ex ante moral hazard, increase efficiency and due to competition in the insurance business, some insurers in developed markets opted to vary the premiums of policyholders from the previous period with regard to the policyholders’ claims history. This pricing system which takes into account the claims experience is known as the Bonus-Malus System (BMS).

Insurance companies bear insurance risk in return for a price known as premium from the policyholders and the motive of shareholders, who are capital providers, is to make profits by taking this risk. By introducing a BMS, more policies will be written especially from good risk and the bad risk may leave for a non-discriminating premium or start to drive carefully. Due to hunger for bonus, policyholders are not likely to file small and unnecessary claims. These dynamics are going to impact the loss ratio of an insurance company which is the ratio of total claims to total premiums written. This paper is going to analyse if there is going to be an improvement in the loss ratio, which is a measure of profitability, following the introduction of a BMS.

Chapter 2 reviews the literature of the BMS, the Markov Chain theory which is the backbone of the development of this system and the Loimaranta Efficiency which tests the efficiency of the system. In Chapter 3 we are going to analyse some insurance data

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Chapter 1 Introduction 7 in R which will help us to simulate the data that we will use for the profitability-testing. Using the information extracted from the analysis in Chapter 3, we will simulate the data that we need to use in Chapter 4 and use this fictitious data to analyse if there will be an improvement in the loss ratio of an insurer following the implementation of a BMS. We will also test for efficiency of the system if the policyholders are being charged premiums equally proportional to their claim frequencies. The thesis is going to be concluded in Chapter 5 where we will indicate the limitations and recommendations if there are any.

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Chapter 2

The Bonus-Malus System

2.1 Definition and some literature

Lemaire (1998)[5] defines a BMS as a merit-rating technique used in non-life insurance whereby policyholders from a given risk cell are subdivided in Bonus-Malus (BM) classes and their claims histories then modify the class upon each renewal. Frangos and Vrontos (2001)[1] acknowledge that a BMS penalizes policyholders who make claims by a premium surcharge (malus) and reward policyholders with claim-free years by awarding a discount of the premium (bonus). In this way, insurers are using prior variables, such as age, sex, marital status, car model and car usage among other covariates, to determine the premium on inception of an automobile insurance policy and thereafter, posterior knowledge of claims experience is used to determine the premiums at policy renewals. In some countries, the BMS is known as the “No-Claims Discount”.

Lemaire (1998)[5] notes that BMSs were introduced in Europe in the early 1960s when the President of France enforced French insurers to introduce BMSs in automobile insurance. Currently, the BMS is used in most parts of Europe, Asia and some Latin American and African countries.

Lemaire (1998)[5] appreciates that the design, evaluation and comparison of the BMSs is based on the Markov Chains Theory.

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Chapter 2 The Bonus-Malus System 9

2.2 Markov Chains

2.2.1 Stochastic processes

Paul and Baschnagel (2013)[11] define a stochastic process as a family (Xt)t∈I of random

variables and the index can either be discrete set, for instance, I = N, which would lead to a discrete time stochastic process, or it can be continuous, for instance, I = R, which would lead to a continuous time stochastic process. The set of values that the random variables Xtare capable of taking is called the state space of the process, which is denoted

by S, and the values in the set I are called the time domain of the process. Examples of stochastic processes in finance and economics include the daily stock prices, the annual inflation rates and the annual economic growth rates.

2.2.1.1 Examples of stochastic processes in insurance • Discrete state space with discrete time domain

The status of each policyholder in an automobile insurance portfolio is reviewed each year on renewal of the policy and there are four possible levels of premium discount which are (0, 20%,40%, 60%) depending on the claims history of the policyholder. In this set-up, the state space of the stochastic process is S = {0, 20, 40, 60}and the time domain is I = {0, 1, 2, . . .} with each interval representing a year. In this case both the state space S and the time domain I are discrete. • Discrete state space with continuous time domain

A life insurer can classify its policyholders as Healthy, Sick or Dead. The state space of the stochastic process is S = {H, S, D} and the time domain is I = [0, ∞) since the policyholder can be ill or die at any time. In this case the state space S is discrete and the time domain I is continuous.

2.2.1.2 Special properties that a stochastic process may possess • Stationarity

A stochastic process is stationary if the joint distributions of Xt1, Xt2, . . . , Xtn and

Xk +t1, Xk +t2, . . . , Xk +tnare identical for all t1, t2, . . . , tnand k+t1, k+t2, . . . , k+tn

in I and all integers n. This clearly shows that the statistical properties of the process remain unchanged as time elapses.

• Independent increments

An increment, Xt+u− Xt, of a process where u ≥ 0 is the amount by which its

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Chapter 2 The Bonus-Malus System 10 if for all t and every u ≥ 0 the increment Xt+u− Xt is independent of all the past

of the process {Xs :{ Xs: 0 ≤ s ≤ t}.

• The Markov property

The “Markov property” is named after the Russian mathematician Andrey (Andrei) Andreyevich Markov. The Markov property states that the future developments of a process can be predicted from its present state alone, without any reference to its previous states. Mathematically, a stochastic process (Xt)t∈N exhibits the

Markov property if;

P[Xt∈ A|X0 = x0, X1 = x1, . . . , Xt−1= xt−1] = P[Xt∈ A|Xt−1= xt−1] (2.1)

for any subset A of the state space S.

2.2.2 Markov processes

Pavliotis (2014)[12] defines a Markov process as a stochastic process that retains no memory of where it has been in the past, only the current state of the Markov process can influence where it will go next. Hence, a Markov process is a stochastic process which possesses the Markov property. An example of a Markov process is a simple random walk which is defined as;

Xt= t

X

j=1

Ij, X0 = 0 (2.2)

and Ij can only take values −1 or +1 with equal probabilities.

2.2.3 Transition matrix and graph

Lee and Lee (2006)[4] define a transition matrix as a square table of transition probabilities which summarize the likelihood of migrating from one state to another state in one period. Transition probability is the likelihood of a Markov chain moving from one state to another and if the one-step transition probabilities are time-independent then the Markov Chain is time-homogeneous.

Given that pij is the transition probability then it is the the likelihood of migrating from

state i to state j in one time period. Mathematically, it can be represented as;

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Chapter 2 The Bonus-Malus System 11 Assume that we only have two possible states, state 1 and state 2, and that the transition probabilities from state 1 to state 2 and from state 2 to state 1 are p12 and p21

respectively. From basic probability theory we know that probabilities should sum up to 1 hence, if the process is in state 1 and it does not move to state 2 then it will stay in state 1 with probability p11 which can be written as;

p11= 1 − p12 (2.4)

Analogously, if the process is in state 2 and it does not move to state 1 then it will stay in state 2 with probability;

p22= 1 − p21 (2.5)

The transition graph can be shown as below;

1 2

1 − p12

p12

1 − p21

p21

The transition matrix P of this Markov chain can be written as; P = " 1 − p12 p12 p21 1 − p21 # (2.6) and each row should sum up to one. If we have 30 insurance policies in state 1 and 70 insurance policies in state 2, the expected number of policies in each state in the next period can be calculated as follows;

h 30 70 i " 1 − p12 p12 p21 1 − p21 # = h 30 − 30p12+ 70p21 30p12+ 70 − 70p21 i (2.7)

2.2.4 Long-run behavior of Markov chains

2.2.4.1 Limiting distributions

According to Privault (2013)[13], a Markov chain (Xt)t∈N is said to have a limiting

distribution if the limits;

lim

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Chapter 2 The Bonus-Malus System 12 exist for all i, j ∈ S and form a probability distribution on S, that is;

X

j∈S

lim

t→∞P(Xt= j|X0= i) = 1 (2.9)

For a regular transition matrix P, whose entries are all non-zero, on the finite state space S = 0, 1, . . . , N, the Markov chain admits a limiting distribution π = (πi)0≤i≤N given by

π_j = limt→∞P(Xt= j|X0 = i), 0 ≤ i, j ≤ N

2.2.4.2 Stationary distributions

Paul and Baschnagel (2013)[11] state that a finite, irreducible and aperiodic Markov chain possesses a unique invariant (stationary) probability distribution πi > 0, ∀i. Irreducible

means that every state can be reached from every other state. A Markov chain is periodic if the chain can return to its initial state only in multiples of some integer larger than 1. A chain is therefore aperiodic if it is not periodic.

A probability distribution on S, that is a family π = (πi)i∈S in [0,1] such that;

X

i∈S

πi = 1 (2.10)

is said to be stationary if, starting X0 at time 0 with distribution (πi)i∈S, it turns out

that the distribution of X1 is still (πi)i∈Sat time 1. In other words, (πi)i∈Sis stationary

for the Markov chain with transition matrix P if, letting;

P(X0 = i) = πi, i ∈ S (2.11)

at time 0, implies;

P(X1= i) = P(X0 = i) = πi, i ∈ S (2.12)

at time 1. This also means that;

πj = P(X1= j) =X i∈S P(X1 = j|X0= i)P(X0= i) =X i∈S π_ipij (2.13)

that is the distribution π is stationary if and only if it is invariant by the matrix P that means;

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Chapter 2 The Bonus-Malus System 13 Using transition matrix P from (2.6), we can compute the stationary distribution as follows; h π1 π2 i =hπ1 π2 i " 1 − p12 p12 p21 1 − p21 # = h π1(1 − p12) + π2p21 π1p12+ π2(1 − p21) i (2.15) Two simultaneous equations can be extracted from the matrix equation as follows;

π₁(1 − p12) + π2p21= π1

π1p12+ π2(1 − p21) = π2

(2.16) We can either discard the first or the second line in (2.16) and use constraint (2.10) to get the following stationary distribution;

π=h p21 p12+p21 p12 p12+p21 i (2.17) Stationary and limiting distributions are related concepts because if the limiting distribution exists, it has to be stationary distribution. If the Markov chain is started in the stationary distribution then it will remain in that distribution at any subsequent time step and on the other hand, in order to reach the limiting distribution the Markov chain can be started from any fixed given state, and it will converge to the limiting distribution if it exists.

The concept of stationary distribution is important in developing the BMS because in the long-run, the system will reach an equilibrium where the same number of policyholders would be in each discount level at any point in time. This does not necessarily mean that each policyholder stays put in their discount levels, but that, even though each policyholder moves around, the process will achieve an equilibrium. In the long-run, insurers who predicts the stationary distribution accurately will make profits because their BMS will be efficient.

2.2.4.3 Mean stationary premium

The expected value of the asymptotic premium to be paid when a BMS achieves a stationary distribution is called the mean stationary premium and it depends on the claim frequency λ. Leimaire and Zi[6] point out that any BMS cannot escape the continuous decline of average premium levels as a large number of policyholders will be concentrated in higher premium discount levels. A BMS will encourage drivers to drive more carefully as no policyholder wants to pay more in premiums when they can easily avoid that.

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Assuming a BMS with stationary distribution in (2.17) and respective premiumsha bi, the mean stationary premium will be;

M ean Stationary P remium = a p21 p12+ p21 + b p12 p12+ p21 (2.18)

2.3 Loimaranta Efficiency

Kaas et al (2008)[3] state that the ultimate goal of a BMS is to make everyone pay a premium that reflects as closely as possible the expected value of his annual claims. If we consider two drivers, where Driver 1 has a claim frequency of 0.12 per year and Driver 2 has a claim frequency of 0.144 per year, then in the long-run we expect Driver 2 to pay 20% more premiums than Driver 1 if the BMS is perfectly efficient. Practically this is not the case as Driver 2 may only pay 10% more premiums than Driver 1 which shows inefficiency of a lot of BMSs. Considering a case where a relative increase in the claim frequency of 12% results in a relative increase in the mean premium level of 3% then the efficiency of the BMS will be 25%.

To investigate the efficiency of a BMS, we have to look at how the mean stationary premium depends on the claim frequency λ. We assume that the random variation around the theoretical claim frequency is a Poisson process such that the number of claims in a year is a Poisson(λ) variate. This means that the probability of a year with one or more claims is;

p = 1 − e−λ (2.19)

Assuming that the stationary distribution π =hp p(1 − p) (1 − p)2i and the premiums arehc c ai for the Bonus-Malus classes 1, 2 and 3 respectively then the mean stationary premium will be;

b(λ) = cp + cp(1 − p) + a(1 − p)2

= c(1 − e−λ) + ce−λ(1 − e−λ) + ae−2λ = c(1 − e−2λ) + ae−2λ

(2.20)

This is the premium the policyholder pays on average each year in the long-run and in principle, this premium should be proportional to the claim frequency λ, since the average of the total claims for a policyholder with claim frequency λ is equal to λ times the average size of a single claim which we assume to be independent of claim frequency.

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Chapter 2 The Bonus-Malus System 15 The efficiency e(λ) of the Bonus-Malus system is defined as;

e(λ) = db(λ)/b(λ) dλ/λ = λ b(λ) db(λ) dλ = d log b(λ) d log λ (2.21)

This is known as the Loimaranta efficiency which was introduced by Loimaranta (1972)[8]. The last term including logarithms follows from the chain rule. The Loimaranta efficiency, e(λ), is the elasticity of the mean stationary premium with respect to the claim frequency. It can also be defined as the change in the mean stationary premium relative to a change in the claim frequency.

For small h, if the claim frequency λ increases by a factor 1+h to λ(1+h) then the mean stationary premium b(λ) increases to b(λ(1 + h)) by a factor of approximately 1 + e(λ)h since;

b(λ(1 + h)) = b(λ) + db(λ) = b(λ) + b(λ)e(λ)h = b(λ)(1 + e(λ)h)

(2.22)

Ideally, the Loimaranta efficiency should satisfy e(λ) = 1 so that the relative change in the claim frequency λ will be the same as the relative change in the mean stationary premium, but no BMS achieves that for all λ.

From the mean stationary premium b(λ) in (2.18), db(λ)

dλ = 2(c − a)e

−2λ _(2.23)

and hence the Loimaranta efficiency will be given by; e(λ) = 2λe

−2λ_{(c − a)}

c(1 − e−2λ_{) + ae}−2λ (2.24)

2.3.1 Determining the mean stationary premiums and the Loimaranta efficiency

Let n denote the number of BM classes and tij(k)describe the transition rules, as follows;

tij(k) = 1 if a risk of BM class i moves to class j when k claims have occurred in the

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tij(k) = 0 if such a risk goes to a BM class different from j

where i, j = 1, 2, . . . , n

Given that the claims frequency is λ, the probability to go from BM class i to BM class j is; pij(λ) = ∞ X k=0 pk(λ)tij(k) (2.25) where pk(λ) = λ k_e−λ k! .

Defining l(t) to be the distribution of automobile insurance contracts at time t then l(0) is the initial distribution at time 0 and l(∞) is the long-run distribution which is the stationary distribution π, defined earlier on. Also, lj(t) is the probability of finding a

contract in BM class j at time t, such that the vector of probabilities to find a policyholder in BM class j at time t + 1 can be expressed as follows;

lj(t + 1) = n X i=1 li(t)pij(λ) (2.26) for t=0,1,2,. . .

Given the stationary distribution l(∞) = (l1(∞), . . . , ln(∞)) and that bj is the premium

for BM class j, the mean stationary premium as a function of the claim frequency λ can be derived from the following expression;

b(λ) =

n

X

j=1

lj(∞)bj (2.27)

The Loimaranta efficiency e(λ), can now be computed since we now have an algorithm to compute b(λ). Since lj(∞) is a function of the the claim frequency λ, we can write

lj(∞) = lj(λ) and we can denote;

gj(λ) = dlj(λ) dλ (2.28) and therefore; db(λ) dλ = n X j=1 bj dlj(λ) dλ = n X j=1 bjgj(λ) (2.29)

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Chapter 2 The Bonus-Malus System 17 The gj(λ) can be found as follows;

gj(λ) = n X i=1 gi(λ)pij(λ) + n X i=1 li(λ)p0ij(λ) (2.30)

where the derivative of pij can be found as;

p0_ij(λ) = d dλ ∞ X k=0 e−λλ k k!tij(k) = ∞ X k=0 e−λλ k k![tij(k + 1) − tij(k)] (2.31)

Using the fact that P_jgj(λ) = 0, the Loimaranta efficiency e(λ) can be computed for

every λ by solving the resulting system of equations. In this way, one can compare various BMSs regarding efficiency.

To compute the Loimaranta efficiency for the BMS with R, there are three possible approaches;

Approximation: Compute the stationary distribution using any row of Pn, where n is any large integer power, for λ(1 − ε) and λ(1 + ε) and use e(λ) ≈ ∆logb(λ)

∆logλ .

Exact: The stationary distribution arises as a left-hand eigenvector of P, and b0(λ) is computed by solving a system of linear equations derived from (2.25) to (2.31). Simulation: Use the simulated BM positions after T years for M policyholders to

estimate the stationary distribution, and from the average premiums paid in year T, approximate e(λ) like in the Approximation method.

2.4 The average optimal retention level

The average optimal retention level is the average loss amount that policyholders are willing to retain out of their own interest in order to avoid future premium increases. Each policyholder has his own optimal retention level whereby he sees it beneficial to self-finance claims below this level rather than paying more premiums. Hence, a severe system will induce “hunger for bonus” which will result in policyholders not filing small claims with the insurer in order to continue paying low premiums. Clearly, the “hunger for bonus” will reduce the chances of small claims being filed to the insurer by the policyholders.

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2.5 Design of a good Bonus-Malus System

A comparative analysis of different BMSs that was done by Leimaire and Zi[6] concluded that the following should be taken into account in the construction of a good BMS;

• The BMS should have many classes;

• First claims should be penalized as severely as commercially possible especially in the lower classes;

• The system should not give amnesty to first claims;

• There should be no special transition rules to erase maluses faster;

• A prior surcharge for young drivers should not be introduced but rather use a high access class especially in countries where malus evasion is easy. However the system should not be a pure bonus system.

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Chapter 3

The Model

In this chapter we analyse a real automobile insurance portfolio, “dataCar”, provided in R under the “insuranceData” Package. This data set comprises one-year car insurance policies taken out in 2004 or 2005. In the data there are 67,856 policies and 4624 of the policies made some claims during the period. The data consists of the following variables: vehicle value, occurrence of claim indicator, number of claims, claim amounts, type of vehicle body (vb), vehicle age (va), gender of policyholder (sx), residential area of the policyholder (re) and the age category of the policyholder (age). The analysis will assist us to extract some realistic information that we will use for the simulation of the data that we need for the research.

3.1 Nature of number of claims

Numbers of claims are discrete in nature and positively skewed which means the claim numbers are heavily populated to the left and lightly populated to the right of their mean value. On average, a large number of drivers have claim-free years, some make one or two claims in a year and a very few make three or more claims in a year hence the density function for the number of claims is concentrated to the left and the right tail is thin and longer. The number of claims can be nicely described by a positively skewed and discrete probability distribution.

We are going to assume a Poisson distribution, with parameter λ, for modeling the number of claims and the λ will vary with each risk cell because the frequency of claims is different with each risk cell. Using GLMs, we analyse how the number of claims in “dataCar” depends on the given covariates and this will enable us to estimate how our λ for the Poisson Distribution should vary in each risk cell that we are going to use.

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Chapter 3 The Model 20 The summary R output of fitting GLMs to the number of claims in “dataCar” is as shown below;

> anova(g1, test="Chisq") Analysis of Deviance Table

Model: poisson, link: log

Response: n/expo

Terms added sequentially (first to last)

Df Deviance Resid. Df Resid. Dev Pr(>Chi)

NULL 67855 25507 sx 1 1.641 67854 25505 0.2001828 age 5 90.783 67849 25414 < 2.2e-16 *** re 5 11.945 67844 25403 0.0355474 * vb 12 38.796 67832 25364 0.0001137 *** va 3 30.134 67829 25334 1.293e-06 *** ---Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 There were 50 or more warnings (use warnings() to see the first 50)

From the R output above, we see that age, re, vb and va are the covariates that have greater significance with more than 95% confidence. However, in simulating our data we are not going to consider the vb which is the vehicle body type because it has 13 levels which will result in too many risk cells when combined with other significant covariates. The drawback of too many risk cells is that each risk cell is likely to contain only a few policies and this will contravene the Law of Large Numbers (LLN) which states that the average loss will become more predictable as the pool of risks expands. Hence, by considering only covariates age, re and va we will have 144 risk cells as shown in the R output below;

> #number of levels for each significant covariate

> length(levels(age));length(levels(re));length(levels(va)) [1] 6

[1] 6 [1] 4

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Chapter 3 The Model 21

> #number of risk cells given the significant covariates > length(levels(age))*length(levels(re))*length(levels(va)) [1] 144

Considering the 3 covariates, we can fit a GLM to get the following parameters; > #Determining coefficients of the GLM using the significant covariates > glm(n/expo ~ age+re+va, poisson, wei=expo)

Call: glm(formula = n/expo ~ age + re + va, family = poisson, weights = expo)

Coefficients:

(Intercept) age2 age3 age4 age5 age6

-1.563102 -0.162994 -0.213458 -0.244530 -0.460825 -0.448979

re2 re3 re4 re5 re6 va2

0.048562 0.001318 -0.109862 -0.034803 0.081710 0.042760

va3 va4

-0.077004 -0.146727

Degrees of Freedom: 67855 Total (i.e. Null); 67842 Residual Null Deviance: 25510

Residual Deviance: 25380 AIC: Inf

There were 50 or more warnings (use warnings() to see the first 50)

The value of the intercept represents a driver who is in the youngest age category, lives in region 1 and whose car has the youngest age as well. Since we used the Poisson as our fitting distribution, to estimate the Poisson parameter for each cell we will use the formula;

λage,re,va= exp(β0+

X

βixi) (3.1)

where β0 denotes the intercept term, βi is the coefficient estimated for covariate i, and

xi is the dummy variable indicating presence of covariate i.

Hence the base level of our λ will be 0.21 which is Poisson parameter for the intercept. As the number of claims depends on the risk cell, then for the Poisson distribution of our simulation we are going to use the following λ;

lambda <- 0.21*c(1,0.85,0.81,0.78,0.63,0.64)[age]*

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Chapter 3 The Model 22 If a BMS is introduced, drivers will have a “hunger for bonus” and this means that generally the number of claims will decrease with more drivers having claim-free years and only reporting a few claims which are above their average optimal retention level. Policyholders are only willing to self-finance losses if it is financially profitable than the increased future premiums. We are going to use the same λ for the simulation of the number of claims after introduction of the BMS but we would reduce the the number of claims for every policy with claim amounts less than the optimal retention level which we will assume.

3.2 Nature of claim amounts

Claim amounts are continuous in nature because policyholders can claim to the nearest cent. Moreover, similar to the number of claims, claim amounts are positively skewed which means they are heavily populated to the left and lightly populated to the right of their mean. Small and medium claim amounts have high chances of occurring whereas very large claim amounts rarely occur. Therefore the claim amounts of a car insurance can be described using a positively skewed and continuous probability distribution. We are going to determine the best model between the Lognormal and the Weibull to see which best fit the “dataCar” in R using the Maximum Likelihood method. When comparing different models, the best model will have the lowest Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). To get the correct parameters for the distribution of the claim amounts, we use average claim amounts per policyholder which will be obtained by dividing the total claim amount per policyholder by the number of claims made by the respective policyholder in that given period. We will remove the information for policyholders with zero claims as the Maximum Likelihood method can only fit non-zero claims. The summary of fitting the Lognormal distribution to the data is given in the R output below;

> fln <- fitdist(Aver.AmtCl, "lnorm"); summary(fln)

Fitting of the distribution ’ lnorm ’ by maximum likelihood Parameters :

estimate Std. Error meanlog 6.764581 0.01723612 sdlog 1.172056 0.01218774

Loglikelihood: -38574.7 AIC: 77153.4 BIC: 77166.28 Correlation matrix:

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meanlog 1 0

sdlog 0 1

The summary of fitting the Weibull distributionis also given in the R output below; > fw <- fitdist(Aver.AmtCl, "weibull"); summary(fw)

Fitting of the distribution ’ weibull ’ by maximum likelihood Parameters :

estimate Std. Error shape 0.7863614 0.008132899 scale 1606.2226337 31.952428800

Loglikelihood: -39257.41 AIC: 78518.82 BIC: 78531.7 Correlation matrix:

shape scale shape 1.0000000 0.3403111 scale 0.3403111 1.0000000

By comparing the AIC and BIC of both models, we see that the Lognormal has lower values and this gives assurance that the Lognormal has the better description of the distribution of the claim amounts of “dataCar”. To simulate claim amounts for the data that we need, we will use a Lognormal Distribution with parameters µ = 6.76 and σ = 1.17to simulate a single claim amount for each policyholder. If the policyholder has made more that one claim, the total claim amount will be obtained by multiplying the number of claims and the simulated single claim amount.

However, the emergence of an average optimal level is inevitable when a BMS is introduced because of “the hunger for bonus” by the policyholders. The behavior of policyholders is that they tend to self-finance small claims and not report them to the insurer when a BMS is introduced. Under a BMS, we are still going to assume the same Lognormal distribution but for any claim amounts below the average optimal retention level we will make them zero because the policyholders will not report these to the the insurer.

3.3 Proportions of covariates

The analysis of the significant co-variates in the raw “dataCar” gives us the following proportions;

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Chapter 3 The Model 24 > round(table(age)/sum(table(age))*100) age 1 2 3 4 5 6 8 19 23 24 16 10 > round(table(re)/sum(table(re))*100) re 1 2 3 4 5 6 24 20 30 12 9 5 > round(table(va)/sum(table(va))*100) va 1 2 3 4 18 24 30 28

In generating the automobile insurance portfolio we are going to use the same proportions for the covariates such that;

• age six levels of age categories in the proportion 8:19:23:24:16:10 • re six levels of regions of residence in the proportion 24:20:30:12:9:5 • va four levels of vehicle age categories in the proportion 18:24:30:28

3.4 Assumptions of the model

We are going to make the following assumptions for the model;

• It is going to be a closed book portfolio of 10,000 automobile insurance policies. This means that no new business will be written and that no policies will be surrendered along the way.

• For premium calculation we are going to use the same covariates that we found out to be significant.

• The BMS is going to include the following nine BM Classes: 48.22%, 57.87%, 69.44%, 83.33%, 100%, 120%, 144%, 172.8% and 207.36%. This system has been designed as a geometric sequence with a common ratio of 1.2. The system is not currently being used in any country at the moment but it is more close to the new system in Germany.

• The average optimal retention level is going to be estimated as the excess premiums paid over and above the average base premium when a policyholder makes a claim up until he starts to pay the base premium again.

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3.5 Transition rules of the Bonus-Malus System

The following transition rules of the BMS are designed to achieve a good system, where the first claim should be penalized as severely as commercially possible and there are no special transition rules to erase maluses faster;

• At the start of the BMS, every policyholder will start in BM class 100% which is the base premium for each risk cell

• The policyholder will move down 1 class on policy renewal in the case of a claim-free year

• The policyholder will move up 3 classes on policy renewal if he makes 1 claim in a year

• The policyholder will move up 4 classes on policy renewal if he files 2 claims in a year

• The policyholder will move up 5 classes on policy renewal if he files 3 or more claims in a year

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Chapter 4

Analysis of the Bonus-Malus System

4.1 Analysis of the claims data

Using the information extracted from Chapter 3, we simulate the claims data for 20 years assuming that the claims frequency for each policyholder does not change and the distribution of claim amounts is the same for every policyholder. The R output below gives the total number of claims and total claim amounts per year before adjusting for the average optimal retention for the last 10 years of the 20 years when a BMS will have been introduced;

> colSums(aggr[,5:44])

nCl Yr1 nCl Yr2 nCl Yr3 nCl Yr4 nCl Yr5 nCl Yr6 nCl Yr7

1526 1563 1627 1557 1544 1590 1521

1485 1469 1532 1495 1628 1621 1541

nCl Yr15 nCl Yr16 nCl Yr17 nCl Yr18 nCl Yr19 nCl Yr20 AmtCl Yr1 1534 1614 1546 1549 1632 1615 2531145 AmtCl Yr2 AmtCl Yr3 AmtCl Yr4 AmtCl Yr5 AmtCl Yr6 AmtCl Yr7 AmtCl Yr8 2728413 3012337 2639668 2680921 2795001 2606381 2535465 AmtCl Yr9 AmtCl Yr10 AmtCl Yr11 AmtCl Yr12 AmtCl Yr13 AmtCl Yr14 AmtCl Yr15 2221678 2719076 2519960 2697477 2675171 2618250 2522788 AmtCl Yr16 AmtCl Yr17 AmtCl Yr18 AmtCl Yr19 AmtCl Yr20

2881322 2763017 2648941 2925294 2700107

If the BMS is introduced in the 11th year and we estimate the average optimal retention level per policyholder to be D609.03 then it means that all claim amounts less than the average optimal level will be discarded as the policyholders will self-finance them instead

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Chapter 4 Analysis of the Bonus-Malus System 27 of reporting them to the insurer. The average optimal level is estimated by adding the premium increases to be paid over and above the initial premium when a policyholder makes a claim. If we adjust this development from the the 11th year to the 20th year we will obtain the following column totals for the number of claims and claim amounts; > colSums(aggr[,5:44])

1526 1563 1627 1557 1544 1590 1521

1485 1469 1532 939 985 986 908

nCl Yr15 nCl Yr16 nCl Yr17 nCl Yr18 nCl Yr19 nCl Yr20 AmtCl Yr1

956 1001 964 939 1031 1015 2531145

AmtCl Yr2 AmtCl Yr3 AmtCl Yr4 AmtCl Yr5 AmtCl Yr6 AmtCl Yr7 AmtCl Yr8 2728413 3012337 2639668 2680921 2795001 2606381 2535465 AmtCl Yr9 AmtCl Yr10 AmtCl Yr11 AmtCl Yr12 AmtCl Yr13 AmtCl Yr14 AmtCl Yr15 2221678 2719076 2342148 2491397 2470262 2409981 2340739 AmtCl Yr16 AmtCl Yr17 AmtCl Yr18 AmtCl Yr19 AmtCl Yr20

2683090 2580676 2457814 2728546 2513631

From these results we can see that the total number of claims and the total claim amounts from the the 11th year to the 20th year have decreased due to the “hunger for bonus” which is induced by the BMS.

4.2 Premium income

The premium is going to be determined using the standard deviation principle given by the formula below;

π_{[X] = E[X] + ασ(X)} (4.1) where α ≥ 0 measures the degree of the risk posed by the policyholder to the insurer. Therefore the higher the α, the higher the premium that will be paid and the α will be determined for each risk cell. E[X] and σ(X) are the mean and standard deviation of the claim amounts respectively. The mean measures the average claim amount per policyholder if the total claim amount is redistributed equally among the policyholders and the standard deviation measures by how much the claim amounts deviate from the mean. Using the claims data in the 1st year, to estimate these parameters we get the following mean and standard deviation;

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Chapter 4 Analysis of the Bonus-Malus System 28

[1] 253.11 [1] 1234.54

Since we assumed that the claim amounts follow the same distribution for every policyholder, it is reasonable to assume that the degree of risk α will be estimated by the claim frequency λ for each risk cell. Therefore all the policyholders in the same risk cell will pay the same premium.

Without a BMS in the first 10 years, the aggregate premium income to be collected will be constant each year at D4,451,976 and thereafter the the premium income each year will depend on the claims history of each policyholder as shown in the R output below; > Premiums

[1] 4451976 4451976 4451976 4451976 4451976 4451976 4451976 4451976 4451976 [10] 4451976 4300163 3876317 3509643 3197964 3093836 3002183 2913736 2820229 [19] 2780995 2764117

As assumed in Chapter 3, all policyholders start at 100% level which is the base premium for each risk cell and from the 11th year onwards they start to move between classes according to their claims history until they achieve stationarity. The continuous decrease in the premium income from the 11th year to the 20th year is coherent with what Leimaire and Zi[6] noted in their paper. According to the analysis done by the duo, an apparently inescapable consequence of the implementation of a BMS is a progressive decrease of the observed aggregate premium income due to a concentration of policyholders in the high-discount classes. However, we would expect the aggregate premium level to stabilise after about 30 years which is the maximum period most BMSs seem to take to reach stationarity.

4.3 Profitability, loss ratios and Loimaranta Efficiency

The main motive behind capital providers establishing insurance companies is to get a return on their capital by taking insurance risk. So, when the management of an insurance company are setting up strategies they should ensure that they enhance the profitability of the firm otherwise the shareholders may lose confidence and disinvest their money from the insurance business. Insurance is one of the riskiest businesses and if not properly managed there is a high probability of insurance companies going bankrupt because the premium rates are deterministic but the claim amounts are stochastic.

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Chapter 4 Analysis of the Bonus-Malus System 29 Loss ratio is a determinant of profitability in insurance business as it measures the total claim amounts as a proportion of total premium income. From the data simulated, the loss ratios obtained are shown in the R output below;

> cbind(Year,Claims,Premiums,Loss_Ratio1) Year Claims Premiums Loss_Ratio1 [1,] 1 2531145 4451976 0.57 [2,] 2 2728413 4451976 0.61 [3,] 3 3012337 4451976 0.68 [4,] 4 2639668 4451976 0.59 [5,] 5 2680921 4451976 0.60 [6,] 6 2795001 4451976 0.63 [7,] 7 2606381 4451976 0.59 [8,] 8 2535465 4451976 0.57 [9,] 9 2221678 4451976 0.50 [10,] 10 2719076 4451976 0.61 [11,] 11 2342148 4300163 0.54 [12,] 12 2491397 3876317 0.64 [13,] 13 2470262 3509643 0.70 [14,] 14 2409981 3197964 0.75 [15,] 15 2340739 3093836 0.76 [16,] 16 2683090 3002183 0.89 [17,] 17 2580676 2913736 0.89 [18,] 18 2457814 2820229 0.87 [19,] 19 2728546 2780995 0.98 [20,] 20 2513631 2764117 0.91

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Chapter 4 Analysis of the Bonus-Malus System 30 As shown in the R output and the plot above, in the first few years after the BMS implementation the loss ratios will be maintained at the prevailing loss ratios before the BMS was implemented because the initial decline in the premium income as many policyholders move to the discount classes is compensated by an almost proportionate decline in total claim amount as policyholders do not file claim amounts less than the average optimal retention. Once policyholders adjust their claims with regards to the average optimal retention level, only the stochastic component will have impact on the total claim amounts.

However, due to the “hunger for bonus” a lot of policyholders keep on moving to the higher discount classes and this means that the premium income will keep on decreasing each year up until the process reaches stationarity.

Stationarity of of a BMS may take up to 30 years but the rate of movement of policyholders to higher discount classes is very high in the first few years and lower thereafter. Hence, the loss ratio deteriorates heavily in the first few years and stabilises in the long-run as the system achives an equilibrium. In R, stationary distributions have been calculated for the 144 risk cells because they have different claim frequencies. Since the stationary distribution, the base premium, the number of contracts and the BM percentages are known for each of the 144 risk classes, the total premium income for the automobile insurance portfolio when the BMS reaches stationarity will be given by the R output below;

> sum(stst.Prem) [1] 3225470

Thereafter, the premium income on this portfolio will remain roughly constant at D3,225,470 and therefore deterministic. This means that the loss ratios will now only depend on the total claim amount which is stochastic each year. Therefore, the loss ratio on the automobile insurance portfolio will deteriorate significantly in the first few years then settles down between 70-75% as in the 13th and 14th years which is however a worse position than before the implementation of the BMS.

The efficiency of the BMS has been tested by determining the Loimaranta Efficiency for each of the 144 risk cells using the exact approach described in Chapter 2. Since every risk cell has its own claim frequency, λ ,then we can determine the Loimaranta Efficiency for each risk cell and the R output below shows the first few values of the Loimaranta Efficiency and their corresponding risk cells and claim frequencies;

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Age Re Va Claim_Freq Efficiency 1 1 1 1 0.2100 0.7096 2 2 1 1 0.1785 0.6256 3 3 1 1 0.1701 0.5967 4 4 1 1 0.1638 0.5735 5 5 1 1 0.1323 0.4418 6 6 1 1 0.1344 0.4511

As seen in the R output above, the Loimaranta Efficiency roughly increases as the claim frequency increases. The graph below shows the complete behavior of the Loimaranta Efficiency with respect to the claim frequency for all the 144 risk cells; For this BMS,

Figure 4.2: Loimaranta for the original BMS (black line)

the maximal Loimaranta Efficiency is 0.750526 where λ = 0.238056

4.4 Reducing the number of Bonus Classes

If the number of Bonus Classes are reduced such that we will have the following BM percentages: 83.33%, 100%, 120%, 144%, 172.8% and 207.36% then the minimum premium to be paid by the best risk is 83.33% of the base premium. Having reduced the number of Bonus Classes, we will obtain the following loss ratios;

> cbind(Year,Claims,Premiums,Loss_Ratio2) Year Claims Premiums Loss_Ratio2 [1,] 1 2531145 4451976 0.57 [2,] 2 2728413 4451976 0.61 [3,] 3 3012337 4451976 0.68

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Chapter 4 Analysis of the Bonus-Malus System 32 [4,] 4 2639668 4451976 0.59 [5,] 5 2680921 4451976 0.60 [6,] 6 2795001 4451976 0.63 [7,] 7 2606381 4451976 0.59 [8,] 8 2535465 4451976 0.57 [9,] 9 2221678 4451976 0.50 [10,] 10 2719076 4451976 0.61 [11,] 11 2342148 4300163 0.54 [12,] 12 2491397 4358865 0.57 [13,] 13 2470262 4391193 0.56 [14,] 14 2409981 4392937 0.55 [15,] 15 2340739 4322830 0.54 [16,] 16 2683090 4334927 0.62 [17,] 17 2580676 4355023 0.59 [18,] 18 2457814 4357852 0.56 [19,] 19 2728546 4341692 0.63 [20,] 20 2513631 4362431 0.58

Figure 4.3: Loss Ratios with reduced number of Bonus classes (red line)

The results above clearly show that if we reduce the number of Bonus classes then the rate at which premium income declines is very low hence maintaining the loss ratios at almost the same levels as before the implementation of the BMS. The red plot shows the development of the loss ratios of a BMS with fewer Bonus Classes.

Moreover, if the number of Bonus Classes is reduced, the premium income to be collected when the BMS reaches stationarity will be as shown in the R output below;

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[1] 4853655

Therefore, by reducing the number of Bonus Classes, the profitability of the insurer is improved at the expense of the the Loimaranta Efficiency which declines as shown by the R output and graph below;

> head(Eff_Summary)

Figure 4.4: Loimaranta Efficiency with reduced number of Bonus classes (red line)

4.5 Reducing the BM Factor

If the BM Factor is reduced from 1.2 to 1.05, the BM percentages will be adjusted from 48.22%, 57.87%, 69.44%, 83.33%, 100%, 120%, 144%, 172.8% and 207.36% to 82.27%, 86.38%, 90.7%, 95.24%, 100%, 105%, 110.25%, 115.76% and 121.55% respectively. This will reduce the average optimal retention level to roughly D138.07 and it will have an impact on the claim amounts.

This will result to the following changes to the loss ratios of the insurer; > cbind(Year,Claims,Premiums,Loss_Ratio3)

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Year Claims Premiums Loss_Ratio3 [1,] 1 2531145 4451976 0.57 [2,] 2 2728413 4451976 0.61 [3,] 3 3012337 4451976 0.68 [4,] 4 2639668 4451976 0.59 [5,] 5 2680921 4451976 0.60 [6,] 6 2795001 4451976 0.63 [7,] 7 2606381 4451976 0.59 [8,] 8 2535465 4451976 0.57 [9,] 9 2221678 4451976 0.50 [10,] 10 2719076 4451976 0.61 [11,] 11 2512161 4374402 0.57 [12,] 12 2688564 4275852 0.63 [13,] 13 2667647 4186519 0.64 [14,] 14 2610909 4104761 0.64 [15,] 15 2514979 4100049 0.61 [16,] 16 2873925 4082093 0.70 [17,] 17 2753790 4062886 0.68 [18,] 18 2640315 4032306 0.65 [19,] 19 2918064 4026780 0.72 [20,] 20 2691300 4025763 0.67

Figure 4.5: Loss Ratios with reduced BM factor (blue line)

Reducing the BM factor will reduce the claim amounts slightly and simultaneously the premium income will reduce slightly but on a continuous basis due to the progressive concentration of policyholders into higher discount classes. The loss ratios will therefore deteriorate slightly each period up until the system is stationary and hence the premium income will become roughly constant.

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Chapter 4 Analysis of the Bonus-Malus System 35 If the BM factor is reduced, the stationary premium income to be collected will increase from the stationary premium income in the original BMS as shown in the R output below;

> sum(stst.Prem) [1] 4006232

However, the stationary premium income increases at the expense of the the Loimaranta Efficiency which declines as shown by the R output and graph below;

> head(Eff_Summary)

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Chapter 5

Conclusion

5.1 Summary

The main aim of the thesis was to understand the impact on profitability of introducing a BMS to a portfolio of automobile insurance. When the BMS was first introduced in 1958 in France, it was an act of law and therefore every insurer had to comply. Insurers had to carefully design good BMSs which would either maintain or improve their profitability otherwise they would become insolvent in the long-run.

The thesis covers the concept of Markov chains and how they behave in the long-run as they are the grassroots to the design of the BMS. Furthermore, some literature on the Loimaranta efficiency is reviewed which is an important concept as well under the BMS as an efficient system will discourage adverse selection because it fairly charges policyholders according to their claim frequency.

Some automobile insurance data was simulated using some realistic information extracted from some insurance data. The simulated insurance data was used to analyse the impact on profitability after implementing a BMS. The analysis shows that for a BMS with geometric progressive percentages and with an equal number of malus classes and bonus classes, profitability decreases as there is a deterioration of loss ratios after the implementation of the BMS.

The deterioration of the loss ratios is caused by the fact that the rate at which the premium income declines is higher than the rate at which claim amounts are reduced. The decline in the premium income results from the concentration of policyholders in higher discount levels as trend shows that many policyholders have claim-free years. Simultaneously, the decline in claim amounts is caused by the “the hunger for bonus” whereby policyholders are willing to retain and self-finance claim amounts that are below

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Chapter 5 Conclusion 37 their average optimal retention level in order to prevent future increases in premiums. However, the rate at which the loss ratio deteriorates also declines and will be roughly constant when the BMS reaches stationarity whereby the premium income will no longer be reducing because there will be a steady proportion equilibrium of policyholders in each BM class.

If the BM factor is reduced, the profitability will still deteriorate but this time at a reduced rate but it will finally be maintained when the BMS is stationary. However, if the number of Bonus classes is reduced as compared to the number of Malus classes this improves profitability as the premium income is not reduced as much as it would be reduced when there are many lower discount Bonus classes.

5.2 Recommendations

From the results of this thesis, in order for insurers to at least maintain their profits when they design a BMS to implement for their automobile insurance portfolio, they should ensure that;

• the number of Bonus classes are fewer than the number of Malus classes

• the Malus classes are so severe that the average optimal retention level is increased hence reducing total claim amounts

• the Bonus classes are not too generous such that it heavily reduces the total premium income

5.3 Limitations

In simulating the data used for the analysis, some assumptions used are not realistic and this may distort the results slightly.

• The assumption that the automobile insurance portfolio is a closed book is not realistic because in practice policyholders have the choice of movement from one insurer to another.

• The number of claims after implementing the BMS was assumed to be pure Poisson distributed but it would then be adjusted by reducing the number of claims for any claim amount below the average optimal retention level. In a BMS, a high number of zero-values is often observed and this can be properly modelled by a Zero-Inflated Poisson distribution.

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Chapter 5 Conclusion 38 • A same average optimal retention level was assumed for every policyholder but this is not realistic because each policyholder has a different average optimal retention level depending on the policyholders’ personal characteristics, impatience rate and income levels.

• In the thesis, the claim amounts after implementing a BMS are assumed to follow the same Lognormal distribution as before the implementation of the BMS but however adjusting for all claim amounts below the average optimal retention level. This assumption is not realistic because when a BMS is implemented, policyholders will drive carefully. Not only the number of claims are reduced by careful driving but also the claim severity.

• Also the assumption that the claim amounts for every policyholder follows the same Lognormal distribution is not realistic because claim severity may also depend on certain covariates such as use of car and the age of the policyholder.

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Appendix A

R Code

rm(list=ls(all=TRUE)) #Remove objects from Workspace

install.packages("insuranceData") #To enable us download insurance data library(insuranceData)

data("dataCar") #Download insurance data

#Change covariates columns from numeric to factors dataCar$veh_age <- as.factor(dataCar$veh_age)

dataCar$gender <- as.factor(as.numeric(dataCar$gender)) dataCar$area <- as.factor(as.numeric(dataCar$area)) dataCar$agecat <- as.factor(as.numeric(dataCar$agecat)) dataCar$veh_body <- as.factor(as.numeric(dataCar$veh_body)) #Name the columns of the dataframe

names(dataCar) <- c("VehVal","expo","IndicatorCl","n","AmtCl", "vb","va","sx","re","age","X_OBSTAT_") attach(dataCar)

#Fitting GLM to find significant covariates to the claim numbers g1 <- glm(n/expo ~ 1+sx+age+re+vb+va,fam=poisson(link=log),wei=expo) anova(g1, test="Chisq")

rm(g1) #remove large objects to free memory space #number of levels for each significant covariate

length(levels(age));length(levels(re));length(levels(va)) #number of risk cells given the significant covariates length(levels(age))*length(levels(re))*length(levels(va))

#Determining coefficients of the GLM using the significant covariates glm(n/expo ~ age+re+va, poisson, wei=expo)

coeff <- coef(glm(n/expo ~ age+re+va, poisson, wei=expo)) #Estimation of lambda for each risk cell

Profitability-testing of a Bonus-Malus System on a portfolio of automobile insurance