Modelling Motor Claims Risk

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MOUNT KENYA UNIVERSITY
SCHOOL OF PURE AND APPLIED SCIENCES
RISK MODELLING FOR UNDERWRITING RISK ON MOTOR VEHICLE
INSURANCE COMPANY. A CASE STUDY OF OCCIDENTAL INSURANCE NAIROBI,
KENYA
BY
DOMINIC ONDIGO
REG NO: BAS/2013/51749
TEL: +254728207265
dominicondigo@live.com
A RESEARCH PROJECT SUBMITTED TO THE DEPARTMENT OF MATHEMATICS
AND PHYSICAL SCIENCES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
OF THE BACHELORS DEGREE IN ACTUARIAL SCIENCE IN MOUNT KENYA
UNIVERSITY
PROJECT IN ACTUARIAL SCIENCE
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2016

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DECLARATION
I hereby declare that this research project is my original work and has not been submitted for a
degree in other university.
Signature……………………………………….. Date………………………………………..
SUPERVISOR
Mr. Peter Chacha
Contact: +254727351534
Email: pchacha@mku.ac.ke
Signature………………………………………… Date…………………………………………

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DEDICATION
I dedicate this project to my supportive family and my friends for their substantial contributions.
May the Almighty reward you abundantly.

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TABLE OF CONTENTS
DECLARATION......................................................................................................................................... ii
DEDICATION............................................................................................................................................ iii
DEFINITIONS........................................................................................................................................... vi
ABSTRACT............................................................................................................................................... vii
INTRODUCTION.......................................................................................................................................1
1.1 Background of the study ......................................................................................................................1
1.1.1Overview of Occidental Insurance Company and insurance Industry..........................................1
1.1.2 Management of risk in regards to the insurance industry .............................................................2
1.1.3 Underwriting risk in Kenya ..............................................................................................................3
1.2 Statement of the problem .....................................................................................................................3
1.3 Objectives of the study..........................................................................................................................4
1.3.1 Main Objective...................................................................................................................................4
1.3.2 Specific Objectives .............................................................................................................................4
1.4 Significance of the study.......................................................................................................................4
LITERATURE REVIEW ..........................................................................................................................5
2.1 Introduction...........................................................................................................................................5
2.2 Empirical Review..................................................................................................................................5
2.2.1 An overview of Foreign Insurance companies.................................................................................5
2.2.2 The Kenyan case ................................................................................................................................6
2.3 Theoretical Framework........................................................................................................................6
2.3.1 Credibility theory ................................................................................................................................6
2.3.2 Financial Economic Theory ..............................................................................................................6
2.4 Summary of the Literature Review.....................................................................................................7
CHAPTER THREE....................................................................................................................................8
RESEARCH METHODOLOGY ..............................................................................................................8
3.1 Introduction...........................................................................................................................................8
Claims count random variable ................................................................................................................10
Claims severity distribution ....................................................................................................................10
3.4 Data Processing...........................................................................................................................13
DATA ANALYSIS AND RESULTS .......................................................................................................14
4.1 Introduction.........................................................................................................................................14

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4.2 General Information...........................................................................................................................14
4.2.1 Underwriting losses..........................................................................................................................14
4.2.2 Number of claims/policies ...............................................................................................................15
4.3 Fitting of the data................................................................................................................................17
4.3.1 Fitting the number of motor vehicles claims data.........................................................................17
4.3.2 Goodness-of-Fit tests for the number of motor vehicle claims data............................................18
4.4 Determining the estimated credibility premium amount for the number of claims.....................19
CHAPTER FIVE ......................................................................................................................................24
SUMMARY, CONCLUSION AND RECOMMENDATIONS.............................................................24
5.1 Summary..............................................................................................................................................24
5.2 Conclusion ...........................................................................................................................................24
5.3 Limitations of the Study .....................................................................................................................24
5.4 Suggestions for further research .......................................................................................................24
REFERENCES..........................................................................................................................................26
APPENDICES...........................................................................................................................................28
6.0 R-Programming Codes.......................................................................................................................28
6.1 Codes for deviance residuals and histogram ....................................................................................28
6.2 Codes used to represent the claims data and Poisson distribution.................................................28
6.3 Codes used represent the negative binomial distribution ...............................................................29
6.4 Testing the goodness of fit (𝝌𝟐).........................................................................................................30

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DEFINITIONS
‘Risk modeling’ means use of risk models that estimate risk measures and possibly carry out stress
tests;
‘Underwriting risk’ means inaccurate premium computation, incorrect modeling of aggregate
claims and inadequate reserves set aside by insurance companies;
‘Risk’ means the possibility that the actual outcome of an investment will differ from expected
outcome;
‘Insurance’ means a co-operative device to spread the loss caused by a particular risk over a
number of persons who are exposed to it and who agree to insure themselves against that risk;
‘Insurance company’ means the firm that offers insurance cover;
‘Aggregate loss’ means the total amount of losses in one period of time;
‘Insurance Regulatory Authority (IRA)’ means body which regulates insurance companies in a
state. It formulates and enforces standards, issue licenses to all persons, prompt settlement of
claims, investigate and prosecute all insurance fraud.
‘Selection of risk’ means process whereby inferior lives are “weeded out”. The selection process
is used to determine whether the degree of risk presented by applicant for insurance is
commensurate with the premium established for persons in his category or some additional
premium should be charged or the applicant should be rejected the insurance;
‘G.D.P’ means the Gross Domestic Product; the monetary value of all finished goods and services
produced by a state;
‘Solvency’ means assessing company’s financial position;

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ABSTRACT
Underwriting risk on motor vehicle insurance companies in recent years has increased gradually.
Occidental insurance for example as at June 2015 made the highest loss of 686 million. Managing
this risk has become the subject due to the financial crises of 2002 to 2014 and use of inappropriate
methods in pricing of premiums and claim calculation. Greek for example has faced challenges
whereby there is weaknesses of risk management leading to loan obligation becoming greater than
G.D.P. Despite the existence of risk modeling in occidental insurance by use of deterministic
models they have not yet employed the use of stochastic or simulation models leading to these
losses. United Kingdom (UK), USA and India have practiced risk management whereby the UK
has employed Solvency II model and stochastic models. This has assisted in the eradication of risk
in the UK. Risk modeling is rarely used in computation of aggregate claims. The Kenyan insurance
industry has not yet employed this modeling. The main objective of this study was to apply risk
modeling as an underwriting risk management strategy in Occidental insurance company and the
insurance industry at large.
The study relied on secondary data obtained from IRA for Occidental Insurance Company Limited.
The data reviewed was motor vehicle claims data for 2015. Exploratory statistical tests were
carried out before modelling the number of claims.

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CHAPTER ONE
INTRODUCTION
1.1 Background of the study
1.1.1Overview of Occidental Insurance Company and insurance Industry
Occidental insurance company limited was incorporated in 1984 and started transacting insurance
business in 1987. During the first year the company did a gross premium of Ksh9 million rising
steadily to premium of over Ksh1 billion in 2010. It has general insurance business whose range
of products are personal lines, commercial lines, fire domestic and industrial insurances, marine,
liabilities insurance and lastly motor vehicle insurance which is the main. A report from Insurance
Regulatory Authority(IRA) showed that the company made the highest underwriting losses of
Ksh686 million. In regard to this CIC had a loss of Ksh349 million and Ksh211 million for British
American Insurance.
Insurance is a co-operative device to spread the loss caused by a particular risk over a number of
persons who are exposed to it and who agree to insure themselves against that risk. It is a means
of indemnity against a future occurrence of an uncertain event. Also it is a business whereby a
promise must be kept. The insurance company offers insurance cover and is also known as the
underwriter. The insured is the one who has taken an insurance policy. Three parties are involved
in such an arrangement: policyholders, shareholders and the management of the insurance
company. Policyholders are the buyers of an insurance policy. They expect to be paid from the
insurance fund whenever they make a claim. Shareholders are owners of insurance companies.
They expect to make profits from the insurance fund. The management is involved in the day to
day running of the insurance company. They hold the policyholders money in trust in order to
indemnify them in case of a loss.
According to Pithadia (2008), insurance is a growing business in India. India’s life insurance
penetration is at 19% whereas most countries grappling with 3%-5% penetration. The banking and
insurance sectors add about 7% to the country’s (GDP).Life insurance in India is said to have been
started in 1818.Currently there are several insurance companies in India. Inadequate reserving is

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viewed as a great risk in relation to the general insurance industry. This is seen to expose the
insurers to large losses if their claims experience deteriorates.
According to the Insurance Annual Report (2013) by Insurance Regulatory Authority (IRA),
insurance industry penetration in Kenya has increased from 2.5%to3.16% in the last seven years.
Insurance performance relative to the GDP at market prices has been increasing since 2008.The
Kenyan insurance sector constitutes forty nine operating insurance companies as at the end of
2014, twenty four insurance companies writing non-life business, eleven writing life business only
and thirteen composite insurance companies. Non-life insurance covers property and casualty risks
which in Kenya are divided into fifteen broad classes.
Life insurance is a long-term contract between the policy holder and the insurer that facilitates
long term savings. In the event of death of the policyholder, the beneficiaries are paid the agreed
sum assured. The broad classes of life insurance are ordinary life assurance, group life assurance,
deposit administration scheme and investment.
The Insurance Regulatory Authority (IRA) was started 2007 through an Act of Parliament as an
autonomous Government Institution. IRA took up the functions of the former Department of
Insurance .IRA is charged with regulating, supervising and developing Kenyan insurance industry.
Other roles of IRA include: formulating and enforcing standards, issue licenses to all persons,
prompt settlement of claims, investigate and prosecute all insurance fraud.
1.1.2 Management of risk in regards to the insurance industry
Risk is the possibility of losing all or some of an original investment. A risk could also be defined
as a probability or threat of damage, injury, liability, loss, or any other negative occurrence that is
caused by internal vulnerabilities, and that may be avoided through pre-emptive action .According
to finance, risk is the probability that an actual return on an investment will be lower than the
expected return. There are various types of financial risk for example exchange rate risk, liquidity
risk, political risk, reinvestment risk and underwriting risk among others.
In 2009 a survey conducted by American Academy of Actuaries, categorized the various risks
experienced in the insurance industry as underwriting risks, market risks, default risks, credit risks,
and liquidity risks among other risks. Underwriting risks were defined as the risks that occur when
premiums are computed inaccurately, aggregate claims are incorrectly computed and claim

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reserves are understated. Understatement of claims and incorrect computation of aggregate claims
could result in future company losses. Inaccurate calculation of premiums has adverse financial
implications to the company especially in relation to payment of claims
Dowd et al (2007) defines risk management strategies for the financial industry to constitute
clearly set out risk policies, an independent risk management function headed by a Chief Risk
Officer, risk modeling and timely communication of risk matters. The risk management function
would be responsible for formulation and implementation of risk control systems. Risk modeling
involves use of risk models that estimate risk measures and possibly carry out stress tests.
1.1.3 Underwriting risk in Kenya
As seen previously, underwriting risk stems from inaccurate premium computation, incorrect
modeling of aggregate claims and inadequate reserves set aside by insurance companies. Between
2014 and 2015 most underwriters have been the victims of inadequate reserves due to hefty court
awards given for certain claims.
A second risk factor would be in the lack of risk modeling of aggregate claims. Modeling of
aggregate claims based on the past claims enables an insurance company to reserve for claims and
to price for premiums. Lack of risk modeling of past claims exposes a firm to insurance risk. A
large number of insurers have not done a risk model for aggregate claims. However, a good number
of insurers affirm that claims is the riskiest area in running an insurance company especially those
relating to motor vehicle claims.
1.2 Statement of the problem
This study has been necessitated by the by use of risk modeling methods that exist internationally
and the lack thereon of an appropriate underwriting risk modelling in Kenya. Due to the financial
scandals and market failures that were experienced between 2002 and 2007, risk management
keeps regaining its relevance.
The Kenyan insurance industry reveals the lack of risk modelling as a risk management measure
and hence the main objective of this study. Risk modelling that relate to underwriting risk have
not been employed in Kenya. However, use of rate method has been seen to undervalue the claims
paid and result in very high risk taken as compared to the solvency of the company. This in turn

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results in a company that is not able to honor the insurance promises and hence the collapse of
such a company.
1.3 Objectives of the study
1.3.1 Main Objective
To employ claims risk modeling as a risk management strategy for a motor vehicle class of
business.
1.3.2 Specific Objectives
i. To determine the credibility premium chargeable for every event of loss.
ii. To study Kenyans risk management strategy via the strategies that are used internationally.
iii. To provide viable solutions to the risk modeling questions in the Kenyan insurance
industry.
1.4 Significance of the study
This study is important to Occidental Insurance Company and other insurance companies as it
shows the benefits of risk management in underwriting risk. Adequately priced insurance products
will also benefit the insurance industry in terms of unwarranted competition. Occidental Insurance
Company will also be able to cover risks of which it can be able to afford and pay in case of a
claim which may be profitable to the company.

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CHAPTER TWO
LITERATURE REVIEW
2.1 Introduction
This chapter will review the existing literature on risk management strategies. The chapter reviews
the past and current Kenyan insurance with specific interest in risk management strategies.
2.2 Empirical Review
2.2.1 An overview of Foreign Insurance companies
Different studies on risk management and prevention of such risk have been done. Harrington and
Neihaus (2003) date the origin of risk management to 1955-1964.They further associate risk
management with the use of market insurance to protect individuals and companies from various
risks associated with accidents.
Georges (2013) highlighted the emergence of pure risk management as an alternative to market
insurance in mid 1950s.1970s and 1980s saw the development and use of derivative instruments
which were further discredited due to their risky and ambiguous nature. He further demonstrates
that despite the development of financial risk models and capital calculation formulas by the
financial institutions, financial crises of 2002 and 2007 are inevitable.
A survey conducted by Everis in 2009, on the risk management in the insurance industry in Europe
and South America drew various conclusions. Portugal was considered more advanced in terms of
risk management. 90% of the Portuguese companies under consideration dealt with their own risk
as a separate unit within the organization, tasked to handle the affairs of risk within the
organization.
In Spain, 73%of the companies under consideration had a reserve allocated to the risk
management, 18% did not have such a reserve and the remaining 9% were not even intending
to put up such a reserve.
According to Jason Thacker (2011), the European insurance industry adopted the Solvency II
risk management model, which was developed from the Basel II and Basel III framework of the
Banking Industry. The risk based requirements of the Solvency II model include technical

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provisions in the balance sheet and minimum capital requirements, among others. The solvency
II model has been faulted to reduce foreign insurance and long-tailed business exposure and
hence a shift of foreign business to United States (US).
2.2.2 The Kenyan case
Non-life insurance in Kenya has adopted a rate approach to risk management. Through the years,
it has been noted that there exists cut-throat competition in the insurance industry. In specific,
motor vehicle comprehensive insurance rates have been changing over the years. In 2007, the rate
employed on motor vehicle comprehensive insurance was 7.5% of the motor vehicle value,
whereas as at June 2015, this rate has gone as low 3.5%of the motor vehicle value. Motor vehicle
third party insurance rates have been constant at a rate of Ksh3500 per month for a small vehicle
and Ksh8500 per year for the same motor vehicle. Premium pricing especially for the motor vehicle
class of insurance is not risk based and hence not in tandem with insurable risk for example
expected claims.
2.3 Theoretical Framework
2.3.1 Credibility theory
According to Actuarial Education Company (2013), credibility theory is a branch of actuarial
science used to quantify how unique a particular outcome will be compared to an outcome deemed
as typical. It was developed originally as a method to calculate the risk premium by combining the
individual risk experience with the class risk experience. It consists of a series of statistical skills
aimed to calculate the insurance premium based on the individual claim experience of the product.
The application of this theory is based on rating a posterior and which attempts to the heterogerity
within each risk factor. Credibility can be calculated using two approaches, Bayesian and
Empirical Bayes.
2.3.2 Financial Economic Theory
Financial Economics approach to financial risk management builds upon the classic Modigliani
and Miller (1958) which states the conditions for irrelevance of financial structure. This was later
extended to risk management which stipulates that hedging leads to lower volatility of cash flows
and therefore lower volatility of the firm value. Hedging is a form of risk management which was

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preferred in the 1970s and 1980s through derivatives Gorges(2013).The ultimate result of hedging
was indeed beneficial to the firm and resulted in a higher value –a hedging Premium. Evidence to
support the predictions of financial economics theory approach to risk management is poor.
However, according to Jin and Jorion (2006), risk management does lead to lower variability of
corporate value.
2.4 Summary of the Literature Review
Risk modelling for underwriting risk for Occidental insurance company is a dynamic issue. As
years go by, risk modelling for risk management strategies need to be enhanced.US adopted the
solvency II model of risk management in 2012.The United Kingdom had already adopted the
solvency II model. Actuaries have also been faulted for relying on the deterministic models as
opposed to stochastic models.
The risk management strategies have not been dwelt on in the insurance industry. This is a new
concept and viewed with great resistance by the insurers. It would be interesting to see the uptake
and effectiveness of risk management in Kenya.

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CHAPTER THREE
RESEARCH METHODOLOGY
3.1 Introduction
This chapter sets out mathematical model that was used in analysis of data and its application.
Various derivations as well some theorems proved. It explains in details the steps that were
encountered in the risk modeling process which includes the data processing and data analysis that
were used.
3.2 Scope of Data
Secondary data was used from Occidental Insurance Company regarding their motor
comprehensive policy (June 2014-June 2015).Two assumptions were made on the data before use,
these included:
1. All the claims came from the same distribution (they were independent and identically
distributed)
2. There was a zero claim meaning no claim for motor vehicle registered under the policy.
3. All the future claims were to be generated from the same distribution.
3.3 Risk modeling Process
This section will describe the steps that were followed in fitting a statistical distribution to the
claim severity, that is, the steps that were taken in the risk modeling process. (Kaishev, 2001)
These steps are:
 Selecting a model to model individual and aggregate losses.
 Estimate the Event frequency and severity distribution.
 Fitting the data.
 Testing the Goodness-of-Fit tests.
 Determining the credibility premium.
3.3.1 Selecting a model to model individual and aggregate losses.
This is the first step in the risk modeling process. Therefore two claim risk models were used, these
included: collective and individual risk model. Still in this step, it was necessary to do some

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descriptive analysis of the data to obtain its salient features for occidental insurance and other
companies. This involved finding the mean, median etc of the underwriting losses. Histogram was
plotted using R Statistical Programme to show the representation of the data of underwriting losses.
3.3.1.0 Modeling Individual and Aggregate losses by use of collective and individual risk
model
An aggregate loss refers to the total amount of losses in one period of time, which is often
encountered in the analysis of a portfolio of risks such as a life insurance. An aggregate loss is
similar to an aggregate payment. It is the total amount paid or the total losses on all claims
occurring in a fixed period on a defined set of insurance contracts. Aggregate loss models inform
us and allow us to make decisions on expected profits, premium loadings, reserves necessary to
ensure high probability of profitability and the impact of reinsurance and deductibles.
There are two standard situations for modeling aggregate losses: when there is collective risk and
when there is individual risk.
Collective Risk Model
This model represents the aggregate loss of a sum, S, of a random number which represents annual
number of events modelled as a random compound process, N, of an individual payment
amounts(X1, X2, … , XN). Hence
S=∑ 𝑋𝑖
𝑁
𝑡=1
Where 𝑋𝑖 is the total amount paid. The random variables are assumed to be independent and that
Xi’s have identical probability distribution.
Independence assumptions are of the form:
i. Conditional on N=n, the random variables𝑋1, 𝑋2,…,XN are identical and
independently distributed (iid) random variables, where 𝑋1, 𝑋2, … is the individual
claim size distribution.
ii. The distribution of N does not depend in any way on the value of X1,…,XN
The term S could be the total amount claimed in one year.
Individual Risk Model

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An individual risk model represents the aggregate loss as the sum of the amounts paid on each
component of the portfolio of risks. Therefore
S=∑ 𝑌𝑗
𝑛
𝑗=0
𝑌𝑗 is the amount paid on the 𝑗 𝑡ℎ
contract and it is assumed that 𝑌1, 𝑌2, … , 𝑌𝑛are independent .It is not
assumed that the 𝑌𝑗′𝑠 have identical distributions. Every contract produces losses according to its
own provisions and the underwriting characteristics of the policy holder.
The assumptions of this individual risk model are that at most, one claim may arise from a policy
whereas in the collective risk model multiple claims may arise from a single policy or policy
holder. The practical advantage of this model is that the factors affecting claim numbers may well
be different. For this project, the collective Risk model was used as there are multiple claims that
arise from a single policy holder.
3.3.2 Estimating the event frequency and severity distribution
It involved estimation of the frequency data for policies for Occidental using the claims data. Once
the claims number and observed policies were estimated, then a fitted distribution was available
for further analysis.
Claims count random variable
This refers to the number of claims, N, as a random variable which when modeled is referred to as
claims count distribution. Another term that could be used is the frequency distribution.
The number of claims is a discrete random variable.
Claims severity distribution
Severities refer to the individual or single loss random variables, Xi ’s and the Yj ’s, which when
modeled become the claims severity distribution. Another frequently used term is the loss. An
assumption is made here as to the claims severity being a continuous random variable. The
claims severity and claims count were modelled separately.
Modeling the severity and the frequency has very distinct advantages:

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i. The expected number of claims changes as the number of insured policies change.
Growth in business volume needs to be accounted for in forecasting the number of claims
in future years based on past years’ data.
ii. The effects of general economic inflation and additional claims inflation are reflected in
the losses incurred by insurance parties and the claims paid by insurance companies.
iii. It is easier to study the effects of changing individual deductibles and policy limits for
example reinsurance.
iv. It is easier to understand the effects of changing deductibles on claims frequencies.
v. Models which are developed for non-covered losses to policyholders, claim costs to
insurers and the claim costs to reinsurers can be mutually consistent. This is useful for a
direct insurer who wants to study the consequence of shifting losses to a reinsurer.
vi. The shape of the distribution of S depends on the shape of the distribution of N and X.
Understanding these distributions become important for modifying policy details.
Derivation of mean and variance
From above information
𝑠 = 𝑋1 + 𝑋2 + ⋯ + 𝑋 𝑁
Using the property 𝐸(𝑠) = 𝐸N(𝐸(𝑆|𝑁)) then:
𝐸[𝑆] = 𝐸[𝐸[𝑆|𝑁]] = 𝐸[𝐸[𝑋1 + 𝑋2 + ⋯ + 𝑋 𝑁|𝑁]]
= 𝐸[𝐸[𝑋1|𝑁]] + 𝐸[𝐸[𝑋2|𝑁]] + ⋯ + 𝐸[𝐸[𝑋 𝑁|𝑁]]
= 𝐸[𝑁𝐸[𝑋1]] = 𝐸[𝑋1]𝐸[𝑁]
Similarly, we have the variance as Var[S] as 𝑉𝑎𝑟[𝐸[𝑆|𝑁]] + 𝐸[𝑉𝑎𝑟[𝑆|𝑁]]:
𝑉𝑎𝑟[𝑆] = 𝑉𝑎𝑟[𝐸[𝑆|𝑁]] + 𝐸[𝑉𝑎𝑟[𝑆|𝑁]] = 𝑉𝑎𝑟[𝑁𝐸[𝑋1]] + 𝐸[𝑁𝑉𝑎𝑟[𝑋1]

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= (𝐸[𝑋1])2
𝑉𝑎𝑟[𝑁] + 𝑉𝑎𝑟[𝑋1]𝐸[𝑁]
3.3.3 Fitting the data by checking model fit
The motor vehicle claims data was fitted to various distributions in order to determine the
distribution that would be suitable for modeling such data. (Shanker R, Mishra A (2015))Poisson
and negative binomial distribution were mainly used. Therefore an assessment was made on how
good this distribution fitted the claims data using graphical representation.
3.3.4 Testing the Goodness-of-fit
A chi-square distribution was used with n degrees of freedom and at ∝= 0.05 level of
significance.
3.3.5 Determining the credibility premiums
This was the last step in the risk modelling process. Credibility theory is a set of quantitative tools
that allow an insurer to perform prospective experience rating on a risk or group of risks. Generally,
the rate pricing is designed to reflect the experience of the entire rating class and implicitly assumes
that the risks ere homogenous. Credibility is motivated by various considerations:
i. Competition may force insurers to give the policyholders’ full credibility in order
to retain the business.
ii. The more past information the insurer has on a given policyholder, the more
credible the policyholder’s own experience.
iii. Other classes of insurance may lack actual past experience and thus application of
credibility may be difficult.
Partial credibility premium is determined through a weighted average:
P=Z𝑋̅+ (1-Z)µ
Z denotes the credibility factor and this lies between 0 and 1.Z is determined by actuarial
technique shown below
μ denotes the mean which is given by 𝐸(𝑋𝑗) and the variance, 𝑉𝑎𝑟(𝑋𝑗) is given by formulae
below:

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𝑉𝑎𝑟(𝑋𝑗) = 𝐸[𝑉𝑎𝑟(𝑋𝑗|𝜃)] + 𝑉𝑎𝑟[𝐸(𝑋𝑗|𝜃)]
= 𝐸[𝑣(𝜃)] + 𝑉𝑎𝑟[𝜇(𝜃)]
𝐶𝑜𝑣(𝑋𝑖, 𝑋𝑗) = 𝑎
𝑘 =
𝑣
𝑎
Therefore; Z=
𝑛
𝑛+𝑘
3.4 Data Processing
The study mostly involved the use of computer statistical packages to perform the tests and to plot
the graphs. MS-Excel was used for representation of the findings, R programme was the main
toolbox used for fitting the motor vehicle claims data, computation of means, testing goodness of
fit and in plotting the histogram, Poisson and negative binomial plots. The R program codes are
provided in the appendix.

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CHAPTER FOUR
DATA ANALYSIS AND RESULTS
4.1 Introduction
Secondary data was mainly used for the purpose of addressing the main objective of these research
objectives of this project. Data in regard to Occidental insurance company based in Nairobi Kenya
was used. This data is sourced from the Insurance Regulatory Authority (IRA).The nature of data
was claims paid by the company for the motor vehicle line in business in June 2014 and June 2015
which consists of motor commercial (psv) and private. Analysis of data was done through Excel
and R-Statistical programme. Computational results were used to represent the results.
4.2 General Information
4.2.1 Underwriting losses
Representation of underwriting losses of some insurance companies is shown below:
Source: IRA (2015)
Deviance Residuals for underwriting losses:
Min 1st
Qu Median Mean 3rd
Qu Max
0
100
200
300
400
500
600
700
800
Occidental
Insurance
Cic Insurance British American
Insurance
Gateway UAP Insurance
lossinshsmillions
insurance company
Underwriting losses as at June 2015
losses in millions

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100.0 110.0 211.0 291.2 349.0 686.0
Table 4.1: Deviance residuals for underwriting losses.
Loss ratio=
𝑖𝑛𝑐𝑢𝑟𝑟𝑒𝑑 𝑐𝑙𝑎𝑖𝑚𝑠
𝑒𝑎𝑟𝑛𝑒𝑑 𝑝𝑟𝑒𝑚𝑖𝑢𝑚𝑠
i.e. Occidental Insurance company
Incurred claims=53585
Net earned premium=381020
Loss ratio=
53585
381020
= 0.140635662
A Graphical representation of underwriting losses fitted to an histogram which is normalized as
shown below:
4.2.2 Number of claims/policies
Number of claims Observed frequencies

16
0 1050
1 715
2 62
3 3
4 1
5+ 12
Total 1843
Table 4.2: Table number of claims
The table represents the number of claims against the observed frequencies.1843 policies were
issued between June 2014 and June 2015.Out of this 1050 made no claims in the year 2015 whereas
715 claims were made once. A graphical representation is shown below:
The claims frequencies reduces drastically as the number of claims increases and vice versa. The
mean of claims frequency is 0.5822029 and the variance is 4.86.

17
4.3 Fitting of the data
4.3.1 Fitting the number of motor vehicles claims data.
Two methods were used to fit the data using various distributions whereby motor vehicle claims
data if fitted to determine the distribution which will be suitable in modeling such type of data.
A graphical representation of expected frequencies using negative binomial distribution is shown
below:
Graphically, the fitted data does not seem to follow the same trend as the observed data. However,
a goodness-of-fit test is necessary to statistically evaluate the fitted data. A graphical representation
of the expected frequencies fitted using Poisson distribution is shown below:

18
The fitted data graphically seems to follow the same trend as observed data. However, a goodness-
of-fit test is necessary to statistically evaluate the fitted data.
4.3.2 Goodness-of-Fit tests for the number of motor vehicle claims data
A 𝜒2
goodness of fit test was conducted with the following hypothesis:
𝐻0: The Negative Binomial distribution provides a good fit for the number of claims.
𝐻1: The Negative Binomial distribution does not provide a good fit for the number of claims.
The obtained results indicated that the test-statistic 𝜒2
=1.968313e+164=1.968 ∗ 10164
;the degrees
of freedom(d.f) were 74 table value is equal to 0.We therefore reject the null hypothesis at level
of significance 5% and we conclude that the Negative Binomial Distribution does not provide a
good fit for the number of claims.
A second 𝜒2
goodness of fit test was conducted with the following hypothesis:
𝐻0: The Poisson distribution provides a good fit for the number of claims.

19
𝐻1: The Poisson distribution does not provide a good fit for the number of claims.
The obtained results indicated that the test-statistic 𝜒2
=7.786961e+121=7.787 ∗ 10121
and the
degrees of freedom=74, table value=0.This implies that we reject the null hypothesis and conclude
that the Poisson distribution does not provide a good fit for the number of claims.
However, despite the goodness of fit test failing we opt for a fit better than the other. The Poisson
fit provides a better fit as it is closer to the region where we fail to reject 𝐻0 .
A study done by Shanker R, Mishra A (2015) reveals that the Poisson-Lindley Distribution model
provides a fit in the number of claims. At 5% level of significance it is seen that the Negative
Binomial Distribution and Poisson distribution provide good fit. The number of claims are thus
concluded to follow a Poisson distribution and thus it can be considered as an important tool to
model the above data.
4.4 Determining the estimated credibility premium amount for the number of claims
We denote that we have a total of 1843 policyholders which is our r variable using the data that
was obtained in table 4.2 above. We denote that we have 1 year past experience of policyholders
i.e. 𝑛𝑖=1,𝑚𝑖𝑗=1
.
Where (i=1,…………………….., 1843), for policyholder i we make an assumption 𝑋𝑖1/𝜃𝑖=𝜃𝑖 is
Poisson distributed with mean 𝜃𝑖 such that 𝜇(𝜃𝑖) = 𝑣(𝜃𝑖) = 𝜃𝑖.
𝑋̅ =
1
1843
∗ [∑ 𝑋𝑖1
1843
𝑖=1
]
𝑋̅ =
1
1843
∗ [1073] = 0.58220293
𝑉𝑎𝑟(𝑋𝑖1) =
1
𝑛 − 1
∗ ∑ (𝑋𝑖1 − 𝑋)̅̅̅2
1843
𝑖=1
𝑉𝑎𝑟(𝑋𝑖1) =
1
1843
∗ 8954.29 = 4.86

20
Claims no Frequency Claims*frequency P*(claims no-mean
claims)^2
0 1050 0 355.91
1 715 715 124.81
2 62 124 124.63
3 3 9 17.54
4 1 4 11.68
5 2 10 39.03
6 1 6 29.35
7 1 7 41.19
8 1 8 55.02
9 1 9 70.86
10 1 10 88.69
11 0 0 -
12 1 12 130.37
13 0 0 -
14 0 0 -
15 0 0 -
16 0 0 -
17 0 0 -
18 1 18 303.38
19 0 0 -
20 0 0 -
21 0 0 -
22 0 0 -
23 0 0 -
24 0 0 -
25 0 0 -

21
26 0 0 -
27 0 0 -
28 0 0 -
29 0 0 -
30 0 0 -
31 0 0 -
32 1 32 987.08
33 0 0 -
34 0 0 -
35 1 35 1184.58
36 0 0 -
37 0 0 -
38 0 0 -
39 0 0 -
40 0 0 -
41 0 0 -
42 0 0 -
43 0 0 -
44 0 0 -
45 0 0 -
46 0 0 -
47 0 0 -
48 0 0 -
49 0 0 -
50 0 0 -
51 0 0 -
52 0 0 -
53 0 0 -
54 0 0 -
55 0 0 -

22
56 0 0 -
57 0 0 -
58 0 0 -
59 0 0 -
60 0 0 -
61 0 0 -
62 0 0 -
63 0 0 -
64 0 0 -
65 0 0 -
66 0 0 -
67 0 0 -
68 0 0 -
69 0 0 -
70 0 0 -
71 0 0 -
72 0 0 -
73 0 0 -
74 1 74 5390.17
1073 8954.29
Table 4.3: Table showing the credibility premium computation
Values between 37 and 71 have values carrying zero observations.
𝑉𝑎𝑟(𝑋𝑖1) = 𝑉𝑎𝑟[𝐸(𝑋𝑖1|𝜃𝑖1)] + 𝐸[𝑉𝑎𝑟(𝑋𝑖1|𝜃𝑖)]
= 𝑉𝑎𝑟[𝜇(𝜃𝑖)] + 𝐸[𝑉𝑎𝑟(𝜃𝑖)]
But 𝑎 = 𝑉𝑎𝑟[𝜇(𝜃𝑖)] 𝑎𝑛𝑑 𝑣 = 𝐸[𝑉𝑎𝑟(𝜃𝑖)]𝑡ℎ𝑢𝑠 𝑤𝑒 𝑔𝑒𝑡 𝑡ℎ𝑎𝑡
𝑉𝑎𝑟(𝑋𝑖1) = 𝑎 + 𝑣 = 𝑎 + 𝜇
An unbiased estimator for a and v is the sample variance. The following observations are made:

23
â = 4.86 − 0.5822029 = 4.2777971
𝑘 =
0.5822029
4.2777971
= 0.136098764
𝑧 =
1
1+0.136098764
= 0.880205165 for n=1
These values are fitted into partial credibility premium formulae given by
Pc=Z𝑋̅+ (1-Z)µ
Estimated credibility premium for the number of claims for each policyholder is given by
0.88 ∗ 𝑋𝑖1 + (0.12) ∗ (0.5822029)
𝑤ℎ𝑒𝑟𝑒 𝑋𝑖1 = 0,1,2,3, … … … … . ,74 depending on the policyholder.

24
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATIONS
5.1 Summary
With increase in claims number there is an increase in credibility premium. This means that
insurance companies are expected to increase the amount of premium as the inflation rate increases
to coup up with underwriting losses which are expected to occur.
5.2 Conclusion
In the study above, we conclude that risk modeling as a risk management strategy yields higher
premiums for the insurer. The higher premiums yielded will assist the underwriter to cover the
claims when they arise. This will reduce the underwriting losses accompanied by the insurance
company’s non-life business.
We explore the use of poisson-lindley model estimators. Using negative binomial distribution the
value we get under 𝜒2
is 1.968313e+164 and for Poisson we get 7.786961e+121 yields a positive
relationship and variance(> 1) .At 5% level of significance it is seen that the Poisson distribution
provide good fit using the poisson-lindley model.
5.3 Limitations of the Study
Secondary data was used to conduct the research. This type of data could have a lot of errors due
to estimation of other values. Other errors may occur through the application of smoothing
technique which is being applied. The data may not represent the true feeling of the ground.
Lack of enough international as well as local studies on the emerging underwriting risk where by
the study relied on the risk management strategies in Kenya and abroad
5.4 Suggestions for further research
There is need for further study especially in the reserving of aggregate claims. The creation of
reserves assists the company to reduce the underwriting loss which is brought up by underwriting
risk.

25
It is important to include the effect of inflation and deflation in the computation of premiums. In
the recent years the inflation rate has been in the upward trend so we should focus on future years
on the economy.
A further study should be conducted through the use of other loss aggregate models.

26
REFERENCES
Y Jin, P Jorion-The Journal of Finance, 2006-Wiley Online Library.
M.B. Finan, 𝐴 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝐶𝑜𝑢𝑟𝑠𝑒 𝑓𝑜𝑟 𝑡ℎ𝑒 𝐴𝑐𝑡𝑢𝑎𝑟𝑖𝑒𝑠: 𝐴 𝑃𝑟𝑒𝑝𝑎𝑟𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝐸𝑥𝑎𝑚 P/1,2007,
Arkansas Tech University.
Arunga, A.N. (2012).Risk Management Practices: A survey of micro-insurance service providers
in Kenya.𝐼𝑛𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝐽𝑜𝑢𝑟𝑛𝑎𝑙 𝑜𝑓 𝐹𝑖𝑛𝑎𝑛𝑐𝑖𝑎𝑙 𝑅𝑒𝑠𝑒𝑎𝑟𝑐ℎ.
al, D. e.(2007).Risk Management in the United Kingdom Insurance Industry: The changing state
practice.𝐶𝑒𝑛𝑡𝑟𝑒 𝑜𝑓 𝑅𝑖𝑠𝑘 𝑎𝑛𝑑 𝐼𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒 𝑆𝑡𝑢𝑑𝑖𝑒𝑠.
Dickson, D.C.,Tedesco, L.M. Zehnwirth,B, (1993).Predictive aggregate claim
distribution.𝐽𝑜𝑢𝑟𝑛𝑎𝑙 𝑜𝑓 𝑅𝑖𝑠𝑘 𝑎𝑛𝑑 𝐼𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒, 65,689 − 709.
Gonzalez-Fragueiro, C.,Vilar,J.M Cao,R.,and Au,(sin,M.C2006).Analysis of an Aggregate loss
model.𝐷𝑖𝑠𝑐𝑢𝑠𝑠𝑖𝑜𝑛 𝑗𝑜𝑢𝑟𝑛𝑎𝑙.
Panjer,H.H.& Willmot,G.E.(1983).Compound Poisson models in Actuarial Risk
Hogg,R.V.&Klugman,S.A. (1984).𝐿𝑜𝑠𝑠 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛𝑠.Wiley, Toronto.
http://www.mrc-bsu.cam.ac.uk/bugs
Embrecths, & Frei, M.(2010).Panjer;s recursion for compound distributions.
𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠
Everis(2009).𝑅𝑖𝑠𝑘 𝑚𝑎𝑛𝑎𝑔𝑒𝑚𝑒𝑛𝑡 𝑖𝑛 𝑡ℎ𝑒 𝐼𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒 𝐵𝑢𝑠𝑖𝑛𝑒𝑠𝑠 𝑆𝑒𝑐𝑡𝑜𝑟.United Kingdom: Everis.
Vijay,P.(2006).An overview of the Insurance Sector in India.
𝐴𝑛 𝑜𝑣𝑒𝑟𝑣𝑖𝑒𝑤 𝑜𝑓 𝑡ℎ𝑒 𝐼𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒 𝑠𝑒𝑐𝑡𝑜𝑟 𝑖𝑛 𝐼𝑛𝑑𝑖𝑎.

27
Report,A.o. (2014).𝐴𝐾𝐼 𝑅𝑒𝑝𝑜𝑟𝑡.www.akinsure.com.
Kunrether,& al,e.(2001).Ambiguity and underwriter decision processes.
Shi, E.W. (2008).Actuarial Applications of a hierarchal insurance
model. 𝐴𝑐𝑡𝑢𝑎𝑟𝑖𝑎𝑙 𝐴𝑝𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑎 ℎ𝑖𝑒𝑟𝑎𝑟𝑐ℎ𝑎𝑙 𝑖𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒 𝑚𝑜𝑑𝑒𝑙.
Insurance Regulatory Authority(IRA)
Warutere,P.,2014.KenyaOverview.[Online]http://www.worldbank.org/en/country/kenya/overvie
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Kreps, Rodney, “Riskiness Leverage Models,” accepted to the Proceedings of the Casualty
Actuarial Society, 2005.
Available online at http://www.casact.org/library/05pcas/kreps.pdf.

28
APPENDICES
6.0 R-Programming Codes
6.1 Codes for deviance residuals and histogram
insurancecompany=seq(1,5,1)
insurancecompany
insurancecompany=c(Occidental,CIC,British,UAP,Gateway)
underwritingloss=c(686,349,211,110,100)
summary(underwritingloss)
hist(underwritingloss)
sam<-rnorm(1000)
sam
hist(sam)
sam<-rnorm(500,2,2)
hist(sam, 20,prob=T,col="red",main="HISTOGRAM OF UNDERWRITING
LOSSES",xlab="UNDERWRITING LOSSES")
6.2 Codes used to represent the claims data and Poisson distribution
claimsdata=seq(0,74,1)
claimsdata
observed.frequency=c(1050,715,62,3,1,2,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)
observed.frequency
sum(observed.frequency)

29
product=claimsdata*observed.frequency
product
mean=sum(product)/sum(observed.frequency)
mean=0.5822029
expected.frequency=c(dpois(claimsdata,lambda=mean)*1843)
expected.frequency
plot(claimsdata,expected.frequency,'line')
6.3 Codes used represent the negative binomial distribution
claimsdata=seq(0,74,1)
claimsdata
observed.frequency=c(1050,715,62,3,1,2,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)
observed.frequency
sum(observed.frequency)
product=claimsdata*observed.frequency
product
mean=sum(product)/sum(observed.frequency)
mean=0.5822029
expected.frequency=c(dnbinom(claimsdata,p=0.007762705333,size=75))#since the observations
is 75 and the mean is 0.5822029,then p=0.007762705333
expected.frequency
plot(claimsdata,expected.frequency,’line’)

30
6.4 Testing the goodness of fit (𝝌 𝟐
)
X2<-sum(((observed.frequency-expected.frequency)^2)/expected.frequency)
#test statistic
X2
gdl<-74
1-pchisq(x2,gdl)

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