# Project Materials

## Loss Function in Actuarial Science and Estimation

ABSTRACT

The non-life insurance consists of establishing a premium or a tariff paid by the insured to the insurance company in exchange for the risk transfer. A key factor in doing that is properly estimating the distribution that the claim and frequency of claim follows.

This thesis aim at having a deep knowledge of loss function and their estimation, several concepts from Measure Theory, Probability Theory, and Statistics were combined in the study of the loss function, and estimating them is illustrated using insurance data set distributed by the Data Sciences website https://www.kaggle.com. The software R is used to obtained our results.

Certification i
Approval iii
Abstract v
Dedication vii
Acknowledgements ix
Chapter 1. Introduction 1
1. Motivation of Study 2
2. Statement of Problem 4
3. Aim 5
4. Objectives 5
5. Basic Definition 6
Chapter 2. Random Variable : A Summary 9
1. Terminology 10
2. Parameters of Random vectors 12
3. Moments of covariances of Real Valued Random
Variable 14
4. Random variables on Rd or Random Vectors 17
5. Independence 25
6. Determining probability laws 29
7. Some Usual Probability Laws and Properties 31
Chapter 3. Loss functions 37
1. Basic Distributional Quantities 39
2. Classifying and Creating Distribution 45
3. Tail Weight 49
4. Discrete Distribution 57
5. The (a, b, 0) Class 60
6. The (a, b, 1) Class 61
7. Compound Frequency s 63
8. Frequency and Severity with Coverage Modification
(Insurance Policies) 67
Chapter 4. Estimation of 77
1. Mathematical formulation 78
2. Intuitive view of Statistical inference 82
3. Intuitive view of Statistical tests 84
4. Validating Hypothesis 85
5. Empirical probability density functions 93
6. Continuous Data ing 99
7. Selecting models 103
8. A General statistical tests for fitting distribution107
Chapter 5. A Case Study 113
Chapter 6. Poisson Processes 125
1. Description by exponential inter-arrival 125
2. Counting function 129
3. Approach of the Kolmogorov Existence Theorem 135
CONTENTS iii
4. More properties for the Standard Poisson Process 138
5. Kolmogorov equations 152
Bibliography 161

CHAPTER ONE

Introduction

science is the discipline that applies mathematical and statistical methods to assess risk in insurance, finance and other industries and professions.

Taking the insurance company as a case study, we realize that the fundamental features of these companies is the concept of risk sharing, also known as risk distribution.

Risk distribution is a concept structure in a way that many pay for the expected losses of the few, this is because the risk measure (and, hence, the capital required to support it) for two risks combined is less than that of the risks treated separately.

If the number of individuals get large enough, the risk might get nearly to zero.

What insurance company do is to organize such a redistribution for the purpose of making profit. In so doing, the insurance companies develop specific insurance products covering specified risk for client(insured) who contribute their own part of investment through premiums charged by the company.

2 1. INTRODUCTION

1. Motivation of Study

In Nigeria, one mandatory insurance product is the third party motor insurance, which indemnifies vehicle driver against third party damages or losses, as specified by the insurance underwriting.

The regulatory framework for insurance companies in Nigeria also places a cap on what amount of premium can be charged. Some vehicle owners take up the comprehensive motor insurance which provides coverage not just for the third party, but for themselves too, as specified in the underwriting. Most often than not, the coverage usually applies to situation such as car theft, accident or fire.

You will no doubt agree that these events have a very low probability of occurrence in motor users. However every motor user frequently struggle with the challenges of routine maintenance cost arising from wear and tear of vehicle through usage. And so a natural question that arises is can we extend motor vehicle insurance product to cover this?

Obvious challenges to answering this question in the affirmative for the insurance companies would be:

(1) What will be the premium? How will it be collected? How will the claims be made?

(2) How will this be designed so that they can avoid losses while maximizing profits and at the same time keeping the premium low?

What is even the likelihood of making profit from this product?

A solid foundation that will help in furnishing answers to those questions would be the concept of loss functions.
This Thesis contributes in great details to a thorough exposition and understanding of loss functions as applied in insurance.

What are loss functions and how do they contribute to insurance product development?
Suppose an insurance company has a revenue R and client made claim X the revenue of the company will become R􀀀X. So X is a loss to the insurance company and is called the Loss function.

Suppose the insurance company has an initial premium u, let us suppose that all premiums are received at a constant rate c so that
Pt = u + ct;

4 1. INTRODUCTION

Suppose at a time t there are N(t) number of claims up to time t, X1;X2;X3 XN(t), the revenue(surplus) of the company at time t is
St = u + ct 􀀀
XN(t)
i=1
Xi:

this gives a relationship between premium and claims.

Usually the stochastic process fN(t); t 0g is the counting process of a Poisson process of intensity > 0. For reader who want to read more on such stochastic processes, we presented it in the Appendix in Chapter 6.

What did we see?

The number of Claims and the Claims are all random.

2. Statement of Problem

A key factor in calculating the premium of an insurance company to avoid ruin and maximize profit is knowing the distribution that the claims and frequency of claims follows. An Actuary is presented with data from the field, a natural question that arise is how can we estimate these distributions from data?

4. OBJECTIVES

3. Aim

This Thesis aim at estimating loss function from data. This can be achieve through the following objectives

4. Objectives

(1) To have a deep knowledge on real valued random variables and their characterization as well as the related vocabulary in actuarial science.

(2) To have a deep knowledge on Loss Modification

(3) To have deep knowledge on data modeling.

Accordingly to the objectives described above, we organize the body of the dissertation as follows.

A loss function is merely the probability law of a real-valued random variable. Hence, in Chapter 2, we make a round on such laws from a probabilistic approach.

We mainly follow the presentations of Lo () and Lo (), but the fundamental books of Lo`eve (1997), Chung (1974), Gutt (2005) might be useful. In that chapter, we present the characterization of such laws in a global and comprehensive way, That chapter, of course, goes beyond the scope of the thesis.

In Chapter 3, we present a specific presentation of the probability laws introduced in Chapter 2 in the specialized way they are used in Sciences and in their terminology. We

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6 1. INTRODUCTION

Study in it very detailed and particular properties that are very important in the profession. Our main source in that chapter is Klugman et al. (2008).

In Chapter 4, we deal with estimations of the loss functions.

It constitutes an initiation of statistical estimation and tests, using data and the software R. We followed the book’s project of Lo et al. (2019) which is under development.

In Chapter 5, we applied the knowledge to real-data as a case study of modelling loss data.
Finally, we conclude by final chapter devoted to conclusions and perspectives.

5. Basic Definition

(1) Insurance: An economic device transferring risk from an individual to a company and reducing the uncertainty of risk through pooling.
(2) Insured: Party(ies) covered by an insurance policy.
(3) Insurer: An insurer or reinsurer authorized to write property and/casualty insurance under the law of any state.

5. BASIC DEFINITION 7

(4) Premium: Money charged for the insurance coverage, reflecting expectation of loss

(5) Actuary: A business professional who analyzes probabilities of risk and risk management including calculation of premiums, dividends and other applicable insurance policies held.

(6) Claim: A request made by the insured for insurer remittance of payment due to loss incurred and covered under the policy agreement

(7) Coinsurance: A clause contained in most property insurance policies to encourage policy holders to carry a reasonable amount of insurance. If the insured fails to maintain the amount specified in the clause(Usually at least 0.8), the insured shares a higher proportion of the loss.

(8) Policy: A written contract ratifying the legality of an insurance agreement.

(9) Ruin Time: This is the time when the insurance company become bankrupt.

(10) Probability space: A probability space is a measure space (;A;m) where the measure assigns the unity value to the whole.

8 1. INTRODUCTION

Space, that is m() = 1. Such a measure is called a probability measure. Probability measures are generally denoted in blackboard font P, Q etc.

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