UNIVERSITY OF GHANA 
 
 
 
 
 
 
EFFECTS OF DEPENDENT CLAIMS ON THE PROBABILITY OF RUIN, THE 
TIME TO RUIN GIVEN RUIN OCCURS 
 
 
BY 
 
SEFA MICHAEL ADDO 
(10308147) 
 
 
 
THIS THESIS IS SUBMITTED TO UNIVERSITY OF GHANA, LEGON IN 
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE AWARD OF 
MPHIL ACTUARIAL SCIENCE DEGREE 
 
 
                                                                   JULY, 2019
 i 
 
DECLARATION 
Candidate’s Declaration 
I, Michael Addo Sefa hereby declare that apart from references to other people’s 
publications, which have been duly acknowledged, this thesis is a result of my independent 
ideas, thought, deliberations and has not been submitted for the award of any degree at this 
institution and other universities elsewhere. 
 
 
SIGNATURE: ……………………………………   DATE: ……………… 
MICHAEL ADDO SEFA  
(10308147) 
 
Supervisors’ Declaration 
We hereby certify that this thesis was prepared from the candidate’s own work and 
supervised in accordance with guidelines on supervision of thesis laid down by the 
University of Ghana. 
 
 
SIGNATURE: ……………………………………   DATE: ……………… 
PROF. KWABENA   DOKU-AMPONSAH 
 (PRINCIPAL SUPERVISOR) 
 
SIGNATURE: ……………………………………   DATE: ……………… 
DR. E. N. N. NORTEY 
(CO-SUPERVISOR) 
i 
 
ABSTRACT 
Ruin basically occurs to an insurance company when the claims paid out supersedes its 
initial capital and total premiums accumulated. In the classical theory of risk, the surplus is 
a significant model that deals with how long an insurance company's capital or surplus 
evolves. The first time ruin occurs is very crucial and the business must try to prevent it 
from happening again because it makes the business inefficient and inoperable. The time to 
ruin is so much a function of the initial capital and the way in which the insurance company's 
business books are priced. An insurance company has no control over how claims are issued, 
can attempt and handle its excess to assess the number of claims that will arise over time. 
The hypothesis that individual claims occur separately is one of the assumptions of the 
excess method. But this premise of independence is no longer true and sensible, as 
individual risks are generally homogeneous and share comparable features that claims from 
one eventuality could cause claims from another. In this thesis, the impacts of dependent 
claims on the likelihood of ruin and the time-to-ruin was examined also proposed premium 
adjustment when there is claim dependence. the objectives of this study is to determine how 
dependent claims, affect the likelihood of ruin and time to ruin of insurance companies in 
Ghana. And also to determine the time left for an insurance company to go to ruin when the 
assumption of dependency claims hold. The study employed Pollaczek-Khinchine formula, 
which was used to estimate probability of ruin and time to ruin, copulas and Pearson 
correlation was used to determine claims dependence.  The study found out that the moment 
of ruin occurs more quickly in the presence of dependent claims. Therefore, the study 
recommended an adjustement of premims to hedge against negative cash flows in the 
assumption of claims dependent.  
 
ii 
 
DEDICATION 
This study is devoted to my parents and all the lecturers at the Department of Statistics at 
the University of Ghana. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
iii 
 
ACKNOWLEDGEMENT  
My first gratitude goes to the Almighty God for his grace and mercy on my existence 
throughout my studies at this University. My profound appreciation also goes to my 
supervisors, Prof. K. Doku Amponsah and Dr. E. N. N. Nortey whose suggestions and 
criticisms helped to enrich my work. I also thank all the lecturers of Statistics Department, 
especially Dr. F.O. Mettle, Dr. E. N. N. Nortey, Dr. L. Asiedu, Mr. E. N. B. Quaye and Mr. 
A.A. Asare-Kumi for their services and pieces of advice throughout my two years of study 
in this University. 
My special and sincere gratitude goes to my parents Mr. and Mrs. Sefa and my Uncle Mr. 
Emmanuel Mensah for their financial support and advice. May the excellent Lord keep 
blessing all of you. 
 
 
 
 
 
 
 
 
 
 
  
iv 
 
TABLE OF CONTENTS                                
Content                                     Page 
DECLARATION ................................................................................................................... i 
ABSTRACT ......................................................................................................................... ii 
DEDICATION .................................................................................................................... iii 
ACKNOWLEDGEMENT ................................................................................................... iv 
TABLE OF CONTENTS ...................................................................................................... v 
LIST OF TABLES ............................................................................................................ viii 
LIST OF FIGURES .............................................................................................................. ix 
 
CHAPTER ONE ................................................................................................................... 1 
INTRODUCTION ................................................................................................................. 1 
1.1 Background of the study ............................................................................................. 1 
1.2 Statement of the Problem ............................................................................................ 3 
1.3 Research Questions ..................................................................................................... 3 
1.4 Objectives of the study ................................................................................................ 4 
1.5 Significance of the study ............................................................................................. 4 
1.6 Limitations of the Research ......................................................................................... 5 
1.7 Organisation of the study ............................................................................................ 5 
 
CHAPTER TWO ................................................................................................................... 6 
LITERATURE REVIEW ...................................................................................................... 6 
2.0 Introduction ................................................................................................................. 6 
2.1 Definition and Concept of Ruin Probability ............................................................... 6 
2.2 The Aggregate Claims Process.................................................................................... 7 
2.3 Background of Insurance claims in Ghana ............................................................... 10 
2.4 The Image of the claims in the Insurance Industry ................................................... 12 
2.5 The Evolution of Claims ........................................................................................... 13 
2.6 Claims Administration, Payment and E-Commerce ................................................. 13 
2.7 The Outsourcing of Claims service ........................................................................... 14 
2.8 Insurance claims fraud .............................................................................................. 15 
2.9 Premium Adjustment Mechanism ............................................................................. 16 
2.10 Benefits of Premium Adjustment Mechanisms ....................................................... 17 
v 
 
2.11 Appropriate and Inappropriate Use Of Premium Adjustment Mechanism Payments.
 ......................................................................................................................................... 18 
2.12 Mandatory Motor Insurance .................................................................................... 19 
2.13 Pricing In Insurance ................................................................................................ 20 
2.14 Previous works on Probability of Ruins and Time to Ruin ..................................... 21 
2.15 Previous works on adjustment of premiums ........................................................... 24 
 
CHAPTER THREE ............................................................................................................. 26 
METHODOLOGY .............................................................................................................. 26 
3.1 Introduction ............................................................................................................... 26 
3.2 Data Source and Structure ......................................................................................... 26 
3.3 Method of Data Analysis ........................................................................................... 26 
3.3.1 Copula Test of Independent ............................................................................... 26 
3.3.2 Estimating Probability of Ruin and Time to Ruin .............................................. 27 
3.3.3 Adjustment of Premiums .................................................................................... 28 
3.3.4 Probability of Ruin ............................................................................................. 29 
3.3.5 Maximum Likelihood Estimator (MLE) ............................................................ 29 
3.3.6 Chi Square Goodness of Fit Test ........................................................................ 31 
3.3.7 Wilcoxon matched-pairs signed-ranks test......................................................... 32 
3.3.7 Pearson Correaltion ............................................................................................ 33 
 
CHAPTER FOUR ............................................................................................................... 34 
DATA ANALYSIS AND DISCUSSION ........................................................................... 34 
4.1 Introduction ............................................................................................................... 34 
4.2 Descriptive Analysis ................................................................................................. 34 
4.3 Modelling Amount of Claims ................................................................................... 35 
4.4 Modelling Number of Claims ................................................................................... 36 
4.5 Estimating Dependency between Claims Amount and Number of Claims .............. 37 
4.6 Probability of Ruins .................................................................................................. 38 
4.7 Time to Ruin .............................................................................................................. 39 
4.8 Relationship between level of independency and Time to Ruin ............................... 40 
4.9 Relationship between level of dependency and Probability of Ruin ........................ 41 
4.10 Premiums Adjustment to meet Claims dependency ................................................ 41 
vi 
 
CHAPTER FIVE ................................................................................................................. 43 
SUMMARY, CONCLUSION AND RECOMMENDATION ........................................... 43 
5.1 Introduction ............................................................................................................... 43 
5.2 Summary ................................................................................................................... 43 
5.3 Conclusions ............................................................................................................... 44 
5.4 Recommendations ..................................................................................................... 44 
 
REFERENCES .................................................................................................................... 45 
 
APPENDIX: R codes .......................................................................................................... 49 
 
 
 
 
 
 
 
 
 
 
 
  
vii 
 
LIST OF TABLES 
Table 4.1: Descriptive Analysis of Premiums, Claims Paid, Number of Claims ............... 34 
Table 4.2: Chi-square goodness of fit of Exponential on Amount of Claims ..................... 35 
Table 4.3: Modelling Amount of Claims ............................................................................ 36 
Table 4.4: Model Adequacy for Number of claims ............................................................ 37 
Table 4.5: Modelling the number of claims ........................................................................ 37 
Table 4.6: Estimating Dependency between Claims Amount and Number of Claims ....... 38 
Table 4.7: Probability of Ruins ........................................................................................... 39 
Table 4.8:  Time to Ruin ..................................................................................................... 39 
Table 4.9: Adjusted Premiums ............................................................................................ 42 
 
 
 
 
 
 
 
 
  
viii 
 
LIST OF FIGURES 
Figure 4.1: Relationship between level of Dependency and Time to Ruin ......................... 40 
Figure 4.2: Relationship between level of independency and Probability of ruin .............. 41 
ix 
 
CHAPTER ONE 
INTRODUCTION 
1.1 Background of the study 
Insurance firms are risk-takers. Thus, they are in the business of pooling people’s risk 
together to create a lesser shock for their customers in times of loss. They basically are into 
the business of creating a cushion to relieve its customers of unfortunate outcomes when 
they occur (Charpentier, 2010). These customers (insured) are rewarded to decrease their 
economic burden by insurance companies. To simplify this, an insurance contract shall grant 
the policyholder the right to claim all or part of the loss if certain unforeseen events happen. 
The policyholder agrees in writing to pay a stipulated quantity called premium. In  order to 
make this agreement binding, the insurer is required to honour its commitments when such 
events occur (Dickson, 2016). 
To ensure that they can pay as specified in the contract, the insurance company sets aside a 
reserve or surplus amount from which it can withdraw when the claims are due. This surplus, 
set aside by the insurance company, does not arise in a day, but takes a long time from 
possible excesses of premiums over claims paid. Additional sources of excess accumulation, 
such as investment revenue, are accessible, but the traditional strategy to risk theory has 
been to disregard interest effects, although literature on this topic is increasing. A very 
significant stochastic structure for knowing how the capital or surplus of the insurance 
company evolves over time is the surplus process (Bladt, Nielsen & Samorodnitsky, 2015). 
To begin with the first surplus 𝝁𝟎 which is added to the total premiums obtained until the 
time limit t 𝜋(𝑡) and the overall claims 𝑆(𝑡) subtracted to ascertain the reserve or surplus 
𝜇(𝑡). 
 One of the main objectives of an insurance company is the so-called likelihood of ruin or 
probability of ruin. This  gives a measure of how certain the insurance company can support 
1 
 
its business books. The time to ruin is described as the first moment the surplus becomes 
negative.  
In classical model, the aggregate claims process and the total amount of claims paid from 0 
to 1 represent the claim number process, and this is because the claim amount is generally 
assumed to be independent and identically random variables. This study examines the 
deviation from the assumptions and their impact on the likelihood of ruin by the insurer. 
The latest discussion of concern has proven or validated that the independence no longer 
holds. The rise in interest is understandable because it no longer seems realistic to assume 
that risks are independent. In practical terms, risk dependence exists in several 
circumstances. For instance, a unique disastrous event such as an earthquake or fire outbreak 
can influence the risk of an entire portfolio. Another instance is that an insurance portfolio 
can cover the life of people of common interest (insurable interest) such as a family or group 
(religious community members, company staff) whose mortality is to some extent 
dependent.  
This study aims to evaluate the probability of ruin of the company when using the hypothesis 
of dependence in the event of claims. Assuming that, this is represented by adding each 
claim as a product of a Bernoulli claim indicator and the size of the ruin should be 
acknowledged  
within a period that a company has fixed 𝑁 policyholders. The researcher assumed that the  
incidence of claims among policyholders is no longer independent but dependent. We also 
assumed that the claim sizes are independent when claims are made.   
 
 
2 
 
1.2 Statement of the Problem 
Insurance companies anticipate the claim size of policyholders due to the assumption of 
independence of claims (Kasumo, Kasozi, & Kuznetsov, 2018). Due to the assumption, 
insurance companies can indicate the time to ruin emphatically. Since the claims are 
unrelated, the severity of the claims paid is easily absorbed by accumulated premiums most 
often than not. This problem makes it unlikely for most insurance firms to go into ruin 
(Milevsky, 2016). But in reality, independent claims are not realistic. An insurance portfolio 
can cover the life of people of common interest whose mortality depends on a particular 
case of event happening. For instance, when there is a fire outbreak, the individuals affected 
who have fire policies will all go for claims. In actual sense, these claims are dependent but 
it’s the claim sizes that are independent. With the assumption of independency, they record 
a bigger reserve or surplus amount which in the actual sense is not feasible.” 
In this light, this study aims at analysing the likelihood of ruin when removing the hypothesis 
of independence in the case of claims. Therefore, the study proposed adjusted premiums 
process to accommodate claims dependency against ruin occuring. 
 
1.3 Research Questions 
The study is guided by the following questions 
 What is the exact time of the insurance company's ruin indicator?  
 What is the impact of dependency claims on insurance companies ' reserves? 
 What is the size of premiums accumulated to dependent claims? 
 
3 
 
1.4 Objectives of the study 
The primary goal is to investigate the effect of dependent claims on the likelihood of ruin 
and the time-to-ruin that occurs given ruin.  
The specific objectives are; 
 To determine how dependent claims, affect the likelihood of ruin and time to ruin of 
insurance companies in Ghana. 
 To determine the time left for a given insurance company to go to ruin when the 
assumption of dependency of claims hold. 
 To determine how premiums can be adjusted to accommodate claims dependency. 
  
1.5 Significance of the study 
Ruin probability is a key characteristic for an insurance company. However, due to low level 
of expectations of dependent claims, many insurance companies do not anticipate ruin 
before it occurs. 
 To inform policy makers to factor in strategies to capture and manage dependent 
claims. 
 To also inform management of insurance companies the actual value of the reserve 
or surplus considering dependent claims.  
 The study will serve as a literature in research areas of actuarial, pension and 
insurance. 
 
4 
 
1.6 Limitations of the Research 
Despite the efforts to minimize all limitations, there were certain constraints within which 
the research was done. The main limitation was the data collections, the researcher was only 
privileged to obtain annual data of the selected  
 
1.7 Organisation of the study    
The first section is the introduction of the research focusing on the background, problem 
statement, study questions, objectives and significance of the study. The second chapter 
gives an examination of a relevant literature of dependent claim, ruin theory and the surplus 
theory. Chapter three discusses the research methods which includes the data sources and 
structures and also specified the models used. Chapter four provides the analysis and 
discussion and the study's conclusion and recommendations are presented in chapter five. 
 
 
 
 
 
 
 
 
5 
 
CHAPTER TWO 
LITERATURE REVIEW 
2.0 Introduction 
This section provides an examination of the study's literature. It includes concept and 
definition of ruin probability, aggregate claims process, joint distribution of claim 
occurrence, and previous studies on dependent claim and ruin time. 
 
2.1 Definition and Concept of Ruin Probability 
Mathematical models portray the vulnerability of an insurer to insolvency in actuarial 
science and applied probability ruin theory (sometimes called risk theory). Key number of 
interests in such models are the likelihood of ruin, excess allocation immediately before 
ruin, and deficit at ruin moment. 
In 1903, the Swedish actuary Filip Lundbeg introduced the theoretical basis of the ruin 
hypothesis known as the Cramer-Lundberg model or classical compound Poisson model 
(Panjer, & Willmott, 2015). The model depicts an insurance company with two adverse cash 
flows; incoming premiums and outgoing claims. Premiums come from clients at a constant 
pace 𝑐 > 0 and claims arrive according to an intensity Poisson method and are independent 
and identically distributed non-negative random variables with 𝐹  distribution and mean 
𝜇 (they form a Poisson compound process). So, for an insurer that begins with the original 
capital surplus 𝑋 
𝑋𝑡 = 𝑋 + 𝑐𝑡 − ∑
𝑁𝑡
𝑖=1 𝜉𝑖     for 𝑡 ≥ 0                (2.1) 
The main purpose of the model is to study whether the excess level of the insurer will 
eventually fall to zero (which will make the company bankrupt). This amount is known as 
the probability of ultimate ruins is defined as 
6 
 
                                          𝜓(𝑥) = 𝑝𝑥{𝑟 <  ∞}                                       (2.2) 
where the time to ruin is 
                      𝛾 = 𝑖𝑛𝑓{𝑡 > 0 ∶ 𝑥(𝑡) < 0}                                           (2.3) 
with the convention that 𝑖𝑛𝑓𝜙 = ∞.  
The Pollaczek-Khinchine Formula can be used to compute this precisely in line with the tail 
function of the stationary waiting time distribution in 𝑀 / 𝐺1 queue.  
𝜆𝑢 𝜆𝑢 𝑛
                         𝜓(𝑥) = (1 − ) ∑∞𝑛=0 ( ) (1 − 𝐹𝑙(𝑥))                            (2.4) 𝑐 𝑐
1 𝑥
Where 𝐹𝑙(𝑥) = ∫ (1 − 𝐹(𝑢)) 𝑑𝑢.    𝜇 0
If the claim sizes are distributed exponentially, this will be simplified to 
                                    
1 𝜆
𝜆𝑢 ( − )𝑥
                                       𝜓(𝑥) =   𝑒− 𝜇 𝑐                                                       (2.5) 
𝑐
 
 
2.2 The Aggregate Claims Process 
This chapter provides an alternative representation of the method of aggregated claims. 
Similar to the risk model for each person, each claim must be expressed as the product of a 
claim and the quantity connected with a claim. It enables us to impose on the occurrence of 
the claim the dependence framework. Consider 𝑛(𝑡) risk portfolio 𝑌1, 𝑌2. . . , 𝑌𝑛(𝑡)  for [ 0, 𝑡] 
period. The sum of these claims is the amount of the overall claims as: 
𝑛(𝑡) 𝑛(𝑡)
𝑆(𝑡) = 𝑌1 + 𝑌2 + ⋯ + 𝑌𝑛(𝑡) = ∑ 𝑌𝑘  = ∑ 𝐼𝑘𝐵𝑘                               (2.6) 
𝑘=1 𝑘=1
7 
 
It is apparent that 𝑛(𝑡) refers to the complete amount of exposures to claims and that 
𝑛(1)  ≤  𝑛(2)  ≤ · · · ≤  𝑛(𝑡)  for any 𝑡, with severe disparities primarily due to fresh 
policies that have been in place over the era. Therefore, each insurance risk can be 
described as the product of the indicator in a typical set-up of the individual risk model, 
1, if  claim occurs
 if claim occurs Ik       (2.7) 
0, otherwise
. 
and the benefit level, denoted by Bk., if there is a claim. We suppose that I and B are 
autonomous, for simplicity and for reasons we believe that is realistic. Ik has a Bernoulli 
distribution with the random variable indicator, 
 𝑃𝑟𝑜𝑏(𝐼𝑘  =  0)  =  𝑝𝑘  and 𝑃𝑟𝑜𝑏(𝐼𝑘  =  1)  =  𝑞𝑘  =  1 − 𝑝𝑘.       (2.8) 
The gain level Bk is assumed to have a distribution function  
                                           𝐹𝐵𝑘 (𝑏) =  𝑃𝑟𝑜𝑏(𝐵𝑘 ≤
 𝑏).                         (2.9)                                   
We will also suppose that its generating function exists and is 
𝑡
                                            𝑀𝐵 (𝑡)  =  𝐸(𝑒
𝐵𝑥 )                               (2.10) 
𝑘
and its mean and variance are 
                                       µ𝑘  =  𝐸(𝐵𝑘)𝑎𝑛𝑑 𝑉𝑎𝑟(𝐵
2
𝑘) =  𝜎𝑘                      (2.11) 
    
respectively. It is straightforward to find the mean and variance of the aggregate claims: 
8 
 
n(t )
E S t  E(Yk )
k1
n(t )
(2.12) 
E E Yk | Ik 
 k1           
n(t )
E Yk | Ik 1qk
k1
n(t )
qkk
k1
and 
n(t )
               Var S t  Var Yk  2Cov Yi ,Yj   (2.13) 
k1 i j
where,  
Var Y   E E Y 2k  k | Ik   E Y k 
2
 E qk B
2 
k  qkk 
                                              
 q E B2  q2 2k k kk
 q  2k k  q
2
k 1 qk k
(2.14) 
Cov Yi ,Yj   E  IiBiI j B j E Yi E Y j 
                                     E  I  iI j i j  qiiq j j
 Cov  I , I i j  i j for i  j. (2.15) 
Likewise, the moments generating function of Yk can be obtained as follows.: 
(2.16) 
                                    
where 𝑀𝐼𝐾 is the moment generating function of the indicator 𝐼𝑘. The moment 
generating function of the sum can also immediately be evaluated using 
9 
(2.17) 
    
Equations (2.6) to (2.16) holds despite if the individual risks are not independent. The 
findings are already well known in the event of autonomy. For instance, see Bowers al. 
(1997), & Klugman al. (1998). It is well-known that the sum of independent Bernoulli trials 
has a binomial distribution. 
 
2.3 Background of Insurance claims in Ghana 
Claims settlement goes way back as the history of insurance. In the actual sense, it is the 
ultimate reason customers buy insurance products. The very first insurance was in the field 
of marine. History shows that modern insurance was practiced as early as in the year 1347. 
In this era, ships with cargo would be pledged against a loan and given the condition that 
should the vessel not successfully reach its destination, the loan would not be refunded. 
(Irukwu, 1977). 
An ancient maritime practice that made it through generations literally unchanged is that of 
“general average”. 
 Its operation involved when a cargo is“thrown overboard during a journey in an attempt to 
save the voyage. If the journey is successful, the owners of the cargo that was not thrown 
overboard would receice a claim (Fisher, et al, 2005). 
In recent times, Ghana has a bit of growth in the insurance industry seeking to the needs of 
the local and foreign stakeholders, thus the need to uphold the customer interest and the 
need to attend to their requirements with speed and efficiency. Insurance has even been 
hailed as a possible solution to the catastrophic food crises affecting third world nations like 
Ghana. Insurance claims arise from the occurrence of an event insured against in a policy 
10 
taken by an insurance policyholder. Customers of insurance companies tend to be delayed 
in receiving their claims due. Thus, when an event insured for in a policy has happened, 
insurance companies delay in the payment. In Ghana, to the layman insurance companies 
are seen as profit making venture who are in the system to take people’s money and not 
return them. This notion should be reversed because the idea of insurance is not to defraud 
but to create a cushion of relief for their customers who have suffered a loss. Claims payment 
in Ghana is also seen as a tussle between the insurance company and the customers. This is 
due to the requirements needed to issue a claim, which include a police report, a driver’s 
licence or birth certificate in the case of a general insurance. In the case of life insurance, a 
doctor’s report and a death certificate is needed to receive a claim. This may take about two 
weeks to a month to receive settlement. Most claims are settled by means of negotiation 
between the parties without the need of any formal procedures as litigation. This is, of 
course, the fastest and most economical method of adjustment. In most claims, there may 
be nothing over which to negotiate and the claim may be paid almost immediately. When 
negotiation does not work out, the contract may itself prescribe some other procedure to be 
followed to be followed, such as arbitration and litigation. When activities with adjustment 
of the loss are completed and the amount is determined and agreed upon, the insured is 
entitled to receive payment. There are at least four methods of payment, which insurers can 
employ in providing claim settlements. They are as follows; Cash, Repairs, Replacement 
and Reinstatement.  
The method of payment is normally given to the insured by writing in the policy. In spite of 
the above and all other policyholders who have contributed to the fund. Although the insured 
is entitled to be paid in accordance with the promise of the  insurance contract, the fund 
should be protected against payment of unearned claims. There are certain prohibiting 
11 
factors like Average and Excess/ franchise/ deductibles inherent in the practice of the 
insurance that makes it possible for clients not to receive their full payment. 
 Average is a condition in the policy which provides that the amount of premium paid by 
the insured is only for a smaller proportion of the total value at risk, since that is what has 
been disclosed by the insured , any claims settlement under this policy will recognize this 
fact and the amount payable to the insured will be proportionately reduced. 
  Excess/Franchise/Deductibles are amounts of money (decided at inception of the policy) 
that are subtracted from each claim to be settled. It is immediately observed that, should the 
total amount of the claim be less than the amount of the excess, the claim will not be paid 
(Wildman, et al, 2005). 
 
2.4 The Image of the claims in the Insurance Industry 
It is commonly recognized even among insurance professionals throughout Western African 
nations that today, unlike in other areas of the globe, the insurance sector does not enjoy a 
favourable government picture. Insurance individuals are considered fraudsters in some 
fields who exploit society without providing much in exchange, except for the occasional 
allegations they are forced to pay either out of fear of losing their clients to another business. 
(Irukwu, 1977) 
To all intends and purposes, the claim department can be seen as the shop window of the 
insurance company. It does not matter how cheap an insurance company’s premiums are, 
or how efficient they conduct their underwriting administration, if a claim is not properly 
and fairly dealt with, this is where an insurer will be judged (Roff, 2004). 
Lijadu (2002), said the insurance industry is conscious of the insurance sub-sector's 
misconduct picture of the public. He stressed this by stating that, the insurance sector is 
12 
viewed as fast collecting premium, slow paying claims, using tiny prints to confuse you, 
offering bad services, and participating in sharp procedures. 
It is important to note that, in the minds of the Ghanaian people this perception is still 
perceived while the tale stays different for the other areas of the globe. 
 
2.5 The Evolution of Claims 
Over the years, claims have evolved. The Chartered Insurance Institute Claims Faculty 
reports, the way claims are handled, regarded, and managed has changed over latest years. 
These modifications are incremental and accumulative. It is agreed that claims are much 
nearer to the industry's core than ever before and are thought to be the greatest trigger for 
profit and loss from an organization in many instances. So, it's not surprising that a claim 
has a higher presence in  this ever-competitive sector. (Faculty of Claims, CII 2007). 
 
2.6 Claims Administration, Payment and E-Commerce 
 The involvement of new and enhanced technology, better processes and administrative 
services all contributed to the development of claims. The internet plays a major role in 
reducing costs. It also provides  better customer services to their customers. 
An instance is internet claims and fast communication through standardized formats to 
request for data. It allows insurers to distinguish their brand from their rivals and provide 
their clients with better service (Tovstik, 2013). 
The processing of claims is programmed to guarantee that claims are registered in the 
documents of the insurer and to establish a reserve as rapidly and as easily as appropriate 
for the prospective liabilities involved in the claims. A feature that has been solely manual 
for many years, one or one that had the help of certain billing devices. The dependence on 
technology has been greatly enhanced by the emergence of technology where telephone 
13 
dependence exists and the need for fast and effective service is of paramount significance 
(Rudź, 2015).  
The e-business revolution is gaining converts among the growing number of insurers and 
brokers of insurance companies around the world. According to Milevsky(2016), a study 
undertaken on more than 60 major insurance and financial organisations in the UK, a 
staggering 90 percent of the business processing recently stays based on paper. The later 
price of this with others, amounted to as much as 35% of the net premium, mainly owing to 
administrative duplication between sides. 
 
2.7 The Outsourcing of Claims service 
More insurers in the developed world now place their claims service in the hands of third 
parties more than ever before. According to Dickson (2016), this is a revolution in the 
handling of claims. Claims management is regarded to be one of any insurer's most precious 
property and the issue to be asked is whether he will be willing to trust it to an outsider. It 
seems that many insurers are willing to do just that, as they enable their clients to be cared 
for outside businesses. 
Claims outsourcing means an insurance company uses an external company to handle its 
claims. The outsource company is paid a fee and can handle the claim from start to finish 
but customer loyalty is the key to insurer. It costs far more to win business than to retain an 
existing client. So, is outsourcing really such a good idea, and why is it growing so rapidly? 
Cost is a significant factor, according to Yang, J (2015). In her words, using outside 
companies is far cheaper than employing large numbers of individuals in-house involving 
wage bills and overheads. But it's not the bottom line alone. The outsource suppliers would 
agree that they can deliver better service, leading to higher customer loyalty. 
14 
While outsourcing allegations are deemed viable, there is still some caution when it comes 
to signing cheques. While aid firms say they are willing to accept validation of claims, 
insurers are unwilling to give up control. 
There is always the fear of paying dubious claims. And at the same moment, customers 
loyalty could be at risk if an outsource business were to become too hard when negotiating 
a claim settlement. 
Therefore, outsourcing seems to be an unstoppable force in the insurance industry, as it 
becomes essential to survive the need to cut expenses and deliver enhanced services. For 
the insurance sector in West Africa, with specific regard to Ghana, the outsourcing of claims 
is an area that still needs to be considered for broad implementation 
 
2.8 Insurance claims fraud 
Insurance fraud remains a problem, but while it becomes more effective in detecting the 
cheats, it is not a picture that it wants to project. 
The British Insurers Association report, June 2003, estimated that such costs of fraud in the 
insurance industry exceed £ 1bn per year. The study says that insurance fraud occurs not 
only when people make false claims for a non-occurring accident, but also when people 
inflate claims for real occurrences. This highlights the importance of putting in place 
effective strategies to detect fraud but also raises some interesting issues around prevention. 
(Stears, 2003). 
Financial crime costs at least £ 13.9 billion, up to £ 20 billion when account is taken of 
income tax and EU-related fraud. For every man, this amounts to £ 330; female and baby in 
the UK. Individuals lost at least £ 2.75 billion while businesses lost £ 3.7 billion in 2005. In 
2005-2006, at least £ 6.8 billion was defrauded from the public sector. (Police Officers 
Association, March 2007). 
15 
Fraud is everybody's a concealed tax. It raises the price of products and services, 
impoverishes both tiny and corporate shareholders, hits the future of private pensioners, and 
jeopardizes employment and saps faith in the distinctive position of the country at home and 
overseas. It damages the development and investment of businesses. (Bulletin of fraud, June 
2006). 
Fraud-dominated countries ' governance defines transnational organized crime and fraud as 
a threat to national security, reflected in the cabinet declaration below; Potential impacts 
include: undermining lawful cross-border trade; threatening economic markets ' integrity 
through large-scale money laundering and threatening businesses operations.  
Although insurance companies join other bodies in launching a significant crackdown on 
fraudulent insurance proposals and claims such as fake robbery reports, staged accidents, 
private profit burning claims, numerous claims on the same property and material falsehood 
at the proposal point. Needless to say, even the most honest claims may sometimes have 
been denied their compensation due to the trouble of saying if the claim was real. 
 
2.9 Premium Adjustment Mechanism 
Premium Adjustment Mechanisms (alternately referred to as Insurance Experience 
Adjustments, Premium Equalization, Claims Experience Discounts, and Profit Sharing 
Discounts) are given as processes in group insurance contracts maintained by members ' 
superannuation funds, to return a part of premiums to employees where the experience of 
claims arises more favourably than expected in pricing.The mechanisms may, in some 
conditions, be two-way payments and may allow for enhanced future premiums reflecting 
bad knowledge of underlying claims. 
Premium adjustment mechanisms have existed since the earliest agreements for group life 
(which were used in employer agreements). As more data about the experience of claims 
16 
becomes known, they are typically evaluated annually over a number of years. Premium 
adjustment mechanisms are separate from volume bonuses. They are not commissions, but 
a mechanism for adjusting premiums over time to align more tightly with the contract's real 
claims experience. For superannuation insurance, any payments arising from Premium 
Adjustment Mechanisms are usually payable to the fund for the benefit of members (usually 
in the insurance reserves of the fund) and are not paid to the trustee company, its directors, 
fund managers, sponsoring employers or unions. 
Members are beneficiaries of these Premium Adjustment Mechanisms where cumulative 
modifications are used to decrease future membership premiums or to provide members 
with extended insurance services such as extra assistance for claimants, enhanced insurance 
management, online insurance instruments and insurance benefits enhancements. These 
adjustment mechanisms can be paid in two ways. They can be either an insurer's payment 
to the fund or the insurer's fund. 
 
2.10 Benefits of Premium Adjustment Mechanisms 
Premium Adjustment Mechanisms are a way to address the uncertainty associated with 
group insurance policy setting premium rates. They are intended to generate higher 
alignment between superannuation funds. It helps guarantee that insurers are not making 
excessive profits at the cost of members of the superannuation fund. Premium Adjustment 
Mechanisms are a way to address the uncertainty associated with group insurance policy 
setting premium rates.  Adjustment Mechanisms, the insurer will set premiums based on its 
anticipated future level of claims expenses and will seek to include a contingency margin in 
the premium prices set to represent uncertain future experiences. By using Premium 
Adjustment Mechanisms, for the benefit of members, excess premiums can be returned to 
the superannuation fund where the future level of claims cost is less than expected. This 
17 
enables insurers and funds to smooth members' premium results over time and prevent using 
conservative margins of contingency in setting premium prices. As a consequence, insurers 
must hold less capital against the company, thereby lowering premium expenses. These 
mechanisms have been especially crucial in recent years where the incidence of claims has 
increased, including an rise in late reported claims. It was hard to predict in this setting when 
the growing number of claims would stabilize. 
Premium adjustment mechanisms provide the insurer with a level of security in that the 
premiums paid over time will align with the price of the experience of claims. This also 
helps to make superannuation funds more competitive in the insurance industry where 
insurers otherwise may not be involved in contract tenders. At the same moment, rates paid 
by employees in funds with Premium Adjustment Mechanism agreements are often more 
stable compared to those without them (especially when coupled with lower price 
guarantees) as the processes enable more gradual adjustments to reflect experience rather 
than important step changes at the end of a longer guarantee period. Dependent claims 
continue to impact the insurance company's reserve. The business must, however, take 
certain steps to prevent payment of various claims. 
 
2.11 Appropriate and Inappropriate Use Of Premium Adjustment Mechanism 
Payments.  
Industry rules should govern the correct use of payments from the Premium Adjustment 
Mechanism and identify forbidden or inappropriate uses. Premium adjustment mechanisms 
must be used in accordance with the ' sole purpose test ' in the SIS Act only for reasons. 
Without Premium Adjustment Mechanisms, the effect of accepting lower (or greater) 
volumes of claims will be postponed until the next reset of the premium prices. However, 
when a Premium Adjustment Mechanism is in place, lower (or greater) claim volumes lead 
18 
to a more instant shift in insurance rates for employees – this does not conflict with the 
obligation of the trustee to do everything to pursue a fair claim. 
 
 2.12 Mandatory Motor Insurance 
Compulsory motor insurance requires all those operating a motor vehicle purchase insurance 
to ensure or guarantee some compensation to those injured in an automobile accident. In the 
United States, compulsory insurance was introduced in Massachusetts in 1927 known then 
as Compulsory Automobile Liability Insurance Law (Cohon et al., 2004). The Road Tra_c 
Act 1930 was enacted to commence compulsory insurance in the United Kingdom. 
Compulsory motor insurance observed in many countries is backed by some legislature with 
a minimum of third party insurance. This is known as Third Party Personal Injury Insurance 
(EU, 2015). 
He proposed that mandatory third-party insurance would require adequate public oversight 
to produce the economic growth required in the motor insurance industry. This rise may 
lead from an rise in the total premium received by insurance companies, but it can also boost 
the general claim made on insurance companies. However, according to Lawrence (2001), 
in order to match the increased competition and understand the possibilities that 
liberalization will bring, there is a need to inject some liberalization in phases. 
Hoffer et al. (1991) proposed alternative funding scheme for uninsured motorist coverage 
in order to tackle the danger associated with car use in public space. In their studies, they 
suggested that premiums for uninsured motor vehicle coverage should be gathered by a tiny 
universal motor fuel surcharge and therefore distributed to insurers based on the amount of 
cars they insure in that jurisdiction. Despite the benefit of reduced costs for the average 
motorist using this strategy over the different other funding system for uninsured motorists, 
Hoffer et al. (1991) created a lot of assumptions about the fuel consumption pattern of cars, 
19 
the driver's driving abilities and experience as well as the hazards connected with the 
highway mostly used by the driver. However, these are very critical parameters which 
determine the likelihood of the claim being ever present and ignore them in the price of the 
motor insurance premium eliminates the actuarial precision of the estimate. 
 
2.13 Pricing In Insurance 
Several scientists have suggested different methods to correctly calculate the claims 
insurance companies need to pay the insured when the danger happens. The actuarial 
precision of these different methods depends on the accessible information and on the 
admissible assumptions. 
Edward et al. (2008) used detailed micro-level records that included experience at the 
individual level of the vehicle, the type of insurance claim and the corresponding amount of 
claims and used a hierarchical model for the three components. These elements include 
claim frequency, claim form, and claim seriousness. Edward et al. (2008) used a negative 
binomial regression model to evaluate the frequency of claims. The model used the gender, 
age, no claim discount of the owner as well as the age and type of the car to forecast a claim 
occurrence. As proposed by other scientists, these factors showed a powerful connection to 
the likelihood of a claim. The next model used to predict the type of insurance claim that is 
whether third party injury, third party property damage, insured own damage or some 
combination is the multinomial logit model. This model also considered year, vehicle age 
and vehicle type as important predictors for this model.The third model which this research 
is about is the severity component. Edward et al., (2008) proposed the use of a generalized 
beta of the second kind of long-tailed distribution for claim amounts and also incorporate 
predictor variables. This model also found year and vehicle age and insurers age to be 
important predictors of this model. They found in their research a significant dependence 
20 
among the different claim types and concluded that the three component models proficient 
prediction of automobile claims compared with the traditional methods. This however 
cannot be said for countries where data integrity cannot be confirmed. Despite the actuarial 
accuracy in computing motor insurance premium using this approach, a lot of the 
information needed is provided by the insured and some audit or double checking is 
required, which means more time and cost. A more detailed personal information approach 
was suggested by Vlad et al.,(2006) in estimating motor insurance premium. 
 
2.14 Previous works on Probability of Ruins and Time to Ruin 
Rudź (2015), studies the exact ruin probabilities in discrete time models. To determine the 
precise formulae for ruin probabilities, he applies an integral operator produced by the 
discrete time risk process. The methodology is based on discovering an operator's fixed 
point and checking if it is identical to the probability of ruin. He derived the exact 
probabilities of ruin for both an absolute continuous distribution of claims and a discrete 
amount. 
Livshits (1999), examines the likelihood of a Poisson model insurance company's ruin. This 
is because there are issues in estimating an insurance company's probability of ruin over an 
infinite interval and the associated conditional average time before the ruin. He discovered 
that an insurance company's likelihood of ruin over an infinite interval and the conditional 
average time before the insurance company's ruin for insurance premium and payment flows 
from Poisson.” 
Kasumo., Kasozi and Kuznetsov (2018), in their study entitled “On minimizing the ultimate 
ruin probability of an insurer by reinsurance,” consider an insurance company whose 
dynamic reserves follow a diffusion-disturbed model of risk. They discovered that the 
business chose reinsurance using proportional or excessive reinsurance to decrease its risk. 
21 
They used the Hamilton-Jacobi-Bellman (HJB) strategy to achieve a second-order Volterra 
integrodifferential equation (VIDE) and use the block-by-block technique for ideal 
reinsurance strategy to minimize the ultimate probability of ruin for the parameters selected. 
They also found that proportional reinsurance increases the company's survival for the 
Cramér-Lundberg and diffusion-perturbed models in both light- and heavy-tailed 
distributions. 
Tovstik (2014), “examined the probabilities of ruin for an insurance business based on 
certain stochastic risk models. With autonomous random claims and premiums, he 
concentrated on stochastic risk model. Recurrence formulas for an insurance company's ruin 
probabilities when paying claims are obtained and assumed that both the random premiums 
and the insurance damages are independent and distributed identically. The number of 
claims and premiums are independent processes of Poisson, both independent of the size of 
premiums and claims. He also presumed the exponential distribution of the random 
premiums and insurance damages. They were able to calculate the probabilities of ruin on 
infinite and finite time periods based on the probabilities acquired. 
In his studies, Charpentier (2010), considers ideal reinsurance from the point of perspective 
of an insurer. This is because insurers may want to discover the ideal risk transfer system 
given a (small) likelihood of ruin. He derives an effective algorithm from Monte Carlo to 
connect the probability of ruin with the parameter of reinsurance.  In the event of 
nonproportional reinsurance, he also obtained a methodology estimating the ruin likelihood 
target. 
Panjer (2015), examines an insurance company's excess generally modelled on the classical 
Poisson compound risk model and the risk model of Sparre-Andersen. The claim quantities 
and inter-claim times under these models were presumed to be distributed separately, which 
is not always suitable in practice. Risk models relaxing the hypothesis of autonomy have 
22 
attracted extensive study attention in latest years. He integrated small fluctuations in the 
underlying excess mechanism, the dependent insurance risk is further generalized to a 
disturbed version. Together with apps and examples, explicit solutions for the Gerber-Shiu 
function were deduced. Finally, he brought the disturbed structure of reliance into the dual 
risk model and study the issue of ruin time. Exact solutions are acquired for the 
transformation of the Laplace and the first moment of ruin with an arbitrary gain-size 
distribution. He also gave numerical examples to demonstrate the effect of the structure of 
reliance and the disturbance. 
Yang, (2015), discusses how it is possible to relax this often-unwarranted hypothesis in a 
straightforward manner while integrating rating variables into the model. His strategy 
consisted of fitting generalized linear models to the marginal frequency and the conditional 
effect elements of the complete claim price; reliance is caused by treating the amount of 
claims as a covariate in the average claim size model. In addition to being simple to enforce, 
this modelling approach has the benefit that when Poisson counts are assumed for the 
conditional severity model together with a log-link, the resulting pure premium is the result 
of a marginal mean frequency, a modified marginal mean magnitude, and an readily 
interpreted correction term that represents the reliance. 
Wang, (2013), for insurance claims and dimensions, presented a joint copula-based model. 
To accommodate the reliance between these amounts, she utilizes bivariate copula. Without 
the restrictive premise of independence, she derived the overall distribution of the political 
loss. She demonstrated that this distribution tends to be skewed and multi-modal, and that 
an assumption of independence can lead to significant bias in policy loss assessment. 
Furthermore, by merging marginal generalized linear models with a copula, she extended 
the structure to regression models.  She demonstrated that this strategy leads to a flexible 
model class and that maximum likelihood can be used to estimate the parameters effectively. 
23 
To select the ideal copula family, she suggested a test operation. A simulation research and 
an assessment of car insurance policies illustrated the usefulness of her strategy. 
Yang, J (2015), explored how the heterogeneity between probabilities of incidence and 
severity of claim impacts aggregate claim figures and aggregate claim quantity for a 
portfolio of insurance. He demonstrated that greater heterogeneity (and reliance) between 
probabilities of occurrence outcomes in both lower aggregate claim amounts and aggregate 
claim quantity in the mean remaining lifetime order. He also demonstrated that as the 
heterogeneity among the claims rises, the aggregate claim quantity rises in the sense of the 
usual stochastic order when the probability of occurrence vector remains weakly stochastic 
arrangement. These theoretical results were applied to characteristics of binomial random 
variables convolutions, providing upper limits for the mean residual lifetime features of 
aggregate claim figures and amounts, and comparing stop-loss premiums and risk capital of 
various insurance portfolios. 
 
2.15 Previous works on adjustment of premiums 
Fred Jawadi (2009) studied  the adjustment dynamics of the non‐ life insurance premium 
(NLIP) and its dependence on the financial markets. He also examined the relationship 
between the NLIP, the interest rate, and the stock price using the recent developments of 
nonlinear econometrics. He therefore suggested a strong evidence of significant linkages 
between insurance and financial markets. He found that the NLIP adjustment toward 
equilibrium is time varying, with a convergence speed that varies according to the insurance 
disequilibrium size. Therefore adjusting premiums becomes very necessary in the insurance 
business to hedge against negative cashflow and prolonging the ruin time of an insurance 
companies. 
24 
Wynand P.M.M (1994) explained there exists a consumer information surplus that may 
result in adverse selection. He indicated that insurers can greatly reduce this surplus by risk-
adjusting the premium. He therefore concluded that there need not be any substantial 
unavoidable consumer information surplus if consumers can choose whether to take a 
deductible for a one- or two-year health insurance contract with otherwise identical benefits. 
Therefore, adverse selection should not be a problem in a competitive insurance market with 
risk-adjusted premiums. 
According to Boikov (2003), the adjusted premium process is a linear function of time in 
the classic Cramer--Lundberg model. The premium process is stochastic and it is also 
independent of a risk process. The nonruin probability is chosen as a measure of the payment 
ability. Integral equations and exponential bounds are obtained similarly with the classic 
Cramer--Lundberg model. He also proposed that adjusted premiums should be a function 
of the numder of claims  and the claim size. 
 
 
 
 
 
  
25 
CHAPTER THREE 
METHODOLOGY 
3.1 Introduction 
The methodology of the study is presented in this chapter. It includes data source and 
structure, methods of data analysis and the models adopted for the study.  
 
3.2 Data Source and Structure 
Data that was used for the study was secondary and was extracted from Annaly reports of 
National Insurance Commission (NIC), from 2010 to 2017. The researcher randomly 
selected four general insurance companies. The data obtained includes annual claims paid, 
premiums received and the number of claims reported.  
 
3.3 Method of Data Analysis 
To ensure accuracy in the data processing, data editing and clearing of the data were done 
before analysing the data. The data were coded in order to make possible inputting into the 
data processor. The codes were transformed into units to facilitate their description and 
analyses. Diagrammatic presentation by means of tables and graphs were done. Although 
there will be several electronic means of analysing data R were used for the analysis. The 
study employed Pollaczek-Khinchine formula, which was used to estimate probability of 
ruin and time to ruin, copulas and Pearson correlation was used to determine claims 
dependence.  
 
3.3.1 Copula Test of Independent 
Copulas are mathematical objects that fully capture the dependence structure among random 
variables and hence, offer a great flexibility in building multivariate stochastic models. In 
26 
statistics, a copula is used as a general way of formulating a multivariate distribution in such 
a way that various general types of dependence can be represented. The Frank Copula Test 
of dependence is presented as  
 e x1 1e y 1
c (x, y)   ln 1                            (3.1) 
  e
 1 
where  𝑥 is the number of claims and 𝑦 is the amounts of claims, 
𝜃 ∈ [1, ∞) control the degree of dependence between 𝑥 and 𝑦. 
 
3.3.2 Estimating Probability of Ruin and Time to Ruin 
In classical risk theory, the company’s surplus process is given as  
N (t )
U t   0  t  X k                                   (3.2) 
k1
N (t )
where the aggregate claims process  X  consists of the sum of individual claim amounts k
k1
assumed to be independent and identically distributed random variables, together with a 
random number of claims {N (t)} also assumed to be independent of the claim sizes.  
If the number of claims and claim amount assumed to be dependent, the company surplus 
process is given as  
N (t )
U t   0  t Yk                                     (3.3)                
k1
N (t )
where the aggregate claims process Yk  consists of the sum of individual dependent 
k1
claims. 
 
27 
Time to ruin 
The time to ruin is defined to be the first time that the   t becomes negative 
   𝛾 = 𝑖𝑛𝑓{𝑡 > 0 ∶ 𝑥(𝑡) < 0}                      (3.4) 
 
Premium Process 
The premium process typically has the form 
                                 (t)  c                                         (3.5) t
where c, the rate of premium per unit of time, is expressed as 
                     c  E N E X 1                                  (3.6) 
with θ denoting the relative security loading.  
3.3.3 Adjustment of Premiums  
From equation 3.3, the surplus fuction with the assumption of cliam dependent is given as  
N (t )
U t   0  t Yk  
k1
making  t   the subject 
N (t )  
 t  U t  0 Yk
k1
U t  0  S t 
E   t   E U t   0  E S t 
Now to prevent ruin for occurring it is expexted that the positive surplus process 
E U t  |U t   0  should be added to the premiums (Trufin & Loisel, 2009)  
Adj t   E  t   E U t  |U t   0             (3.7) 
28 
3.3.4 Probability of Ruin 
We employed Pollaczek-Khinchine Formula to compute for probability of ruin. This is 
because when data are tailed, Pollaczek-Khinchine precisely compute the stationary waiting 
time distribution in 𝑀 / 𝐺1 queue.  
𝜆𝑢 𝜆𝑢 𝑛
                         𝜓(𝑥) = (1 − ) ∑∞𝑛=0 ( ) (1 − 𝐹𝑙(𝑥))                            (3.8) 𝑐 𝑐
1 𝑥
where 𝐹𝑙(𝑥) = ∫ (1 − 𝐹(𝑢)) 𝑑𝑢.    𝜇 0
 
3.3.5 Maximum Likelihood Estimator (MLE) 
A value θ̂𝑛 {where θ̂𝑛 is a function of the observations 𝑥1, 𝑥2, … 𝑥𝑛; say θ̂𝑛 =
𝑡(𝑥1, 𝑥2, … 𝑥𝑛) which maximizes the likelihood function 𝐿(θ; 𝑥1, 𝑥2, … 𝑥𝑛) over all θ ∈ Θ 
is called a maximum likelihood estimate (MLE) for 𝜃. 
Hence, for all θ ∈ Θ; 
𝐿[𝑡(𝑥1, 𝑥2, … 𝑥𝑛); 𝑥1, 𝑥2, … 𝑥𝑛] ≥ 𝐿(θ; 𝑥1, 𝑥2, … 𝑥𝑛)                (3.9) 
The Random Variable 𝑇𝑛 = 𝑡(𝑥1, 𝑥2, … 𝑥𝑛) is called a Maximum Likelihood Estimator for 
𝜃.  
Procedure: 
1. Find (obtain) the likelihood function 𝐿(𝜃; 𝑥) 
2. Find the natural log of 𝐿, i.e. 
𝑙((𝜃; 𝑥) = ln {𝐿(𝜃; 𝑥)} 
3. Obtain the score function 𝑆(𝜃; 𝑥) 
𝛿𝑙
𝑆(𝜃; 𝑥) =  
𝛿𝜃
4. Obtain the information function 𝐼(𝜃; 𝑥) 
29 
𝛿2𝑙
𝐼(𝜃; 𝑥) =  
𝛿𝜃2
Let 𝜃 be the MLE for 𝜃, then 𝜃 is a solution to the equation 
𝑆(𝜃; 𝑥) = 0 
i.e. Solve 𝜃 in the 𝑆(𝜃; 𝑥) = 0 
       5. Finally check that 𝐼(𝜃, 𝑥) ≥ 0 
 
MLE for Exponential 
Let 𝑥1, 𝑥2 … . , 𝑥𝑛  be identical and independent random variables that use exponentially 
distributed as 𝑋~ exp( 𝜆) 
                            𝑓(𝑥) = 𝜃𝑒−𝜃𝑥        𝜃 > 0 , 𝑥 > 0 
𝑛
𝐿(𝑃(𝑥)) = ∏ 𝜃𝑒−𝜃𝑥 
𝑖=1
𝐿(𝑃(𝑥)) = 𝜃𝑛𝑒−𝜃 ∑ 𝑥𝑖  
                                     = 𝑙𝑜𝑔𝜃𝑛 + 𝑙𝑜𝑔𝑒−𝜃 ∑ 𝑥𝑖 
                                = 𝑛𝑙𝑜𝑔𝜃 −  𝜃 ∑ 𝑥𝑖 
𝑙(𝑃𝑥) 𝑛
               = −  ∑ 𝑥 = 0 
𝑑𝜃 𝜃 𝑖
𝑛
      =  ∑ 𝑥𝑖 𝜃
𝑛
𝜃 =   
∑ 𝑥𝑖
1
𝜃 =   
?̅?
Where ?̅? is the mean 𝑓{𝑥𝑖} 
30 
𝑁
1
?̅? = ∑ 𝑋  
𝑁 𝑖
𝑖=1
By the invariant property of the MLE, the MLE estimator ?̂? is simply ?̂? =  ?̅? 
 
MLE Poisson 
Let 𝑥1,   𝑥2,…𝑥𝑛 identically independent random variable, the likelihood function is defined 
as 
𝐿(𝜆) = ∏ 𝑓(𝑥𝑖 |𝜆) 
𝜆∑ 𝑥𝑖𝑒−𝑛𝑥
𝐿(𝜆) =  
∏ 𝑥!
                                             𝐿(𝜆) = 𝑙𝑜𝑔𝜆∑ 𝑥𝑖 + 𝑙𝑜𝑔𝑒−𝑛𝜆 − 𝑙𝑜𝑔 ∏ 𝑥𝑖 
                                                    =  ∑ 𝑥𝑖 𝑙𝑜𝑔𝜆 − 𝑛𝜆 − 𝑙𝑜𝑔 ∏ 𝑥! 
1
                  𝑙′(𝜆) =  ∑ 𝑥𝑖 − 𝑛 = 0 𝜆
∑ 𝑥𝑖 = 𝑛?̂? 
∑ 𝑥𝑖
    ?̂?  =   
𝑛
?̂?   =  ?̅? 
 
3.3.6 Chi Square Goodness of Fit Test 
Assumptions 
 The data available for analysis consist of a random sample 𝑛 
independentobservations. 
 The measurement scale be norminal. 
31 
 The observation can be classified into 𝑟 nonoverlapping categoriesthat exhaust all 
classification possibilities. The number of observations falling into a given category 
is called the observed frequency of the category. 
 
Hypothesis 
     𝐻0: 𝑇ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 ℎ𝑎𝑠 𝑏𝑒𝑒𝑛 𝑑𝑟𝑎𝑤𝑛 𝑓𝑟𝑜𝑚 𝑎 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑡ℎ𝑎𝑡 𝑓𝑜𝑙𝑙𝑜𝑤𝑠 𝑎 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑  
        𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛. 
     𝐻1 : 𝑇ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 ℎ𝑎𝑠 𝑛𝑜𝑡 𝑏𝑒𝑒𝑛 𝑑𝑟𝑎𝑤𝑛 𝑓𝑟𝑜𝑚 𝑎 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑡ℎ𝑎𝑡 𝑓𝑜𝑙𝑙𝑜𝑤𝑠 𝑡ℎ𝑒  
        𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 
Test statistics 
The test statistic Chi Square Goodness of Fit test 
𝑟
(𝜃 2
2 𝑖
−  𝐸𝑖)
𝑋 = ∑          (3.10) 
𝐸𝑖
𝑖=1
where 𝜃𝑖 is the observed frequency and 𝐸𝑖 is the expected frequency. 
 
3.3.7 Wilcoxon matched-pairs signed-ranks test 
The Wilcoxon signed-ranks test is a non-parametric equivalent of the paired t-test. It is most 
commonly used to test for a difference in the mean (or median) of paired observations - 
whether measurements on pairs of units or before and after measurements on the same unit 
(MacFarland & Yates, 2016).  
 
  
32 
Procedure 
 Determine the sign of the difference Di  between each pair of observations. 
Examine the distribution of the differences.  
 Rank the differences in order of absolute size with a rank of 1 assigned to the smallest 
difference. Differences of zero are (usually) dropped from the analysis. The rank 
assigned to tied ranks is the mean of the ranks that would have been given if the 
observations had not been tied. 
 Reassign the signs of the differences to their respective ranks Ri  . 
Test Statistics   
Then calculate the appropriate test statistic. 
Calculate 𝑆+ and 𝑆− which are the sums of positive and negative ranks respectively. 
The test statistic is given as  
𝑆 = min(𝑆−, 𝑆+)                                  (3.11) 
. 
 
3.3.7 Pearson Correaltion 
Peasrson correlation is a method used to study the strength of a relationship between two 
numerical variables. The correlation analysis was used to established the relationship 
between claims amount and claims count in the study.  
n n n
n xi yi  xi yi
r x, y  i1 i1 i1                                      (3.12)  
2 2
n  n  n  n 
n x2i  xi  n y
2
i  yi 
i1  i1  i1  i1 
where  𝑥 is the number of claims and 𝑦 is the amounts of claims. 
 
33 
CHAPTER FOUR 
DATA ANALYSIS AND DISCUSSION 
4.1 Introduction  
This chapter presents the analysis and discussion pertaining to effects of dependent claims 
on the probability of ruin, the time to ruin given ruin occurs. The analysis includes modelling 
the amount of claims, modelling number of claims and estimating probability of ruins, and 
time to ruin. 
 
4.2 Descriptive Analysis  
The descriptive statistics such as mean, standard deviation, skewness, kurtosis of the 
premiums, claims paid and number of claims for four selected insurance companies. The 
results were presented in Table 4.1. 
Table 4. 1Descriptive Analysis of Premiums, Claims Paid, Number of Claims 
Std. 
  Minimum Maximum Mean Skewness Kurtosis 
Deviation 
Premiums       
A 10017757.00 69660874.00 45062120.38 22523078.63 -0.19 -1.32 
B   64042765.00 161928878.00 117461352.88 35145129.99 -0.14 -0.94 
C  28168710.00 129593555.00 74252986.38 36866587.64 0.40 -1.09 
D  11559429.00 35671068.00 19101680.75 9152316.98 1.22 -0.01 
Claims Paid       
A 2823494.08 19633842.73 12700710.37 12700710.37 1.19 -1.32 
B   12565341.00 47818099.00 31099659.13 31297659.82 1.11 -1.39 
C  2560791.82 25505000.00 10587097.95 10588079.42 0.98 -0.77 
D  2311885.80 10176799.33 4408686.43 4402286.33 1.56 1.62 
Number of 
      
claims 
A 3589.00 7092.00 5749.50 75.41 -0.71 -1.30 
B   4507.00 8000.00 6856.25 82.55 -1.27 -0.03 
C  2911.00 6404.00 5050.50 71.01 -0.71 -1.28 
D  1004.00 4497.00 3158.00 56.38 -0.74 -1.16 
 
From Table 4.1, on the average company A received a premium of 45,062,120.38 annually 
with a standard deviation of 22,523,078.63. The premiums ranges from 10,017,757.00 to 
34 
69,660,874.00. With regards to claim paid by Company A, on the average company A paid 
claims amounted to 12,700,710.37 annually. Also, Company B on the average received a 
premium of 117,461,352.88, with claims paid annually as 31,099,659.13 and the number of 
claims recorded annually was 6856.25 and ranges from 4,507 to 8,000. With regards to 
Company C the mean premiums received annually was 74,252,986.38, the amount of claims 
paid annually was 10587097.95 and mean the number of claims recorded annually 5050.50. 
In Company D, it was found that the average premium was 19,101,680.75 and the number 
of claims recorded was 3158.00. 
 
4.3 Modelling Amount of Claims 
From the descriptive statistics it was found that the mean and the standard deviation of the 
amount of claims paid were almost similar. This implies that the suggested model for the 
amount of claims paid is exponential distribution. Hence a chi-square of goodness of fit test 
was done to determine how well the data was fit to exponential distribution.  
Table 4.2: Chi-square goodness of fit of Exponential on Amount of Claims 
Company Chi-Square DF P-Value 
A 8.458 6 0.2064 
B 5.652 6 0.4632 
C 7.067 6 0.5781 
D 6.7154 6 0.6522 
Overall 27.243 23 0.7542 
 
From Table 4.2, it can be seen that all the p-value of the chi-square test is greater than zero. 
This implies that the null hypothesis that the found that the amount of claims follows 
exponential was failed to be rejected. Hence the data follows an exponential distribution. 
35 
 Since the amount of claims follows both exponential, Maximum Likelihood Estimator 
(MLE) was used to estimates the parameters of the distributions. The results can be seen in 
Table 4.3.  
Table 4.3: Modelling Amount of Claims 
Company 
 Rate (?̂?) Stand. Error 
 
A 0.000000079 0.000000028 
B 0.000000032 0.000000011 
C 0.000000087 0.000000033 
D 0.000000243 0.000000020 
Overall 0.000000363 0.000000080 
 
From Table 4.3, it can be found in Company A, the amounts of claims follow exponential 
distribution with parameter 0.000000079 with a standard error of 0.000000028, whiles 
Company B follows an exponential distribution with parameter 0.000000032 with a 
standard error of 0.000000011. With respect to Company C and D the amounts of claims 
follow exponential distribution with parameters 0.000000087 and 0.000000243 
respectively. Putting all the companies together, it was found that the amounts of claims 
follows distribution with parameter 0.000000363 with a standard error of 0.000000080.   
 
4.4 Modelling Number of Claims 
From the descriptive, it was found that average number of claims, variance (square of 
standard deviation) of the number of claims are similar. Hence, it is likely to follow a 
Poisson distribution. Tovstik (2014), in his study also found out that the number of claims 
observed follows Poisson distributions. Therefore, chi-square goodness of fit test was done 
to determine whether the number of claims recorded in the insurance companies follows 
Poisson distribution. 
36 
Table 4.4: Model Adequacy for Number of claims 
Company Chi-Square DF P-Value  
A 10.245 6 0.1147 
 
B 9.452 6 0.1497 
C 8.845 6 0.0576 
D 7.452 6 0.2811 
Overall 29.425 23 0.1667 
 
From the chi-square test of goodness fit, since the p-value of the test statistic is greater than 
zero. It implies that the number of claims follows a Poisson distribution. As a result of that, 
MLE was used to estimate the parameter of the Poisson distribution. The results of MLE 
were presented in Table 4.5. 
Table 4.5: Modelling the number of claims 
 
Company 
Rate (?̂?) Stand. Error 
 
 
A 5739.500 26.785 
B 6647.500 28.826 
C 5051.500 25.128 
D 3144.500 19.826 
Overall 5739.500 26.785 
 
From Table 4.5, it can be seen that the number of claims modelled for Company A, B, C 
and D follows Poisson distribution with parameter 5739.500, 6647.500, 5051.500 and 
3144.500 respectively.  
4.5 Estimating Dependency between Claims Amount and Number of Claims 
The dependency between claims paid and number of claims recorded, was estimated using 
Pearson correlation and Frank Copula test of dependence.   
  
37 
Table 4.6: Estimating Dependency between Claims Amount and Number of Claims 
Company Pearson Correlation 𝑟 Test of dependence 𝜂 
A 0.5377329 0.3622671 
B 0.5703038 0.3296962 
C 0.5913531 0.3086469 
D 0.604668 0.195332 
Overall 0.7119262 0.0880738 
 
From Table 4.6, it was found that there is a positive correlation, between claims amount and 
number of claims. This implies that as the number of claims increases, amounts of claims 
also increases. From the Copulas test of Independent it was also found that the level of 
independency between claims amount and number of claims was low. Company D, recorded 
the highest dependency among the four selected companies (𝑟 = 0.60;  𝜂 = 0.1956). The 
overall results imply that there is a higher dependency between claims amount and number 
of claims in Ghana companies (𝑟 = 0.7;  𝜂 = 0.1956).  
 
4.6 Probability of Ruins 
The probability of rule was estimated for two cases, firstly when the dependency between 
number of claims and amount of claims is considered in the model and when the dependency 
is not considered. Pollaczek-Khinchine formula was used to estimate the ruin probabilities 
given there is no initial surplus and the results can be seen in Table 4.7.  
 
 
  
38 
Table 4.7: Probability of Ruins 
Probability of Ruin 
Company Assumptions of  Assumptions of 
Dependence Independence 
A 0.288 0.0808 
B 0.382 0.0935 
C 0.423 0.1285 
D 0.509 0.1737 
Overall 0.597 0.1836 
 
From Table 4.7, it was revealed in all the companies that when the dependency between 
number of claims and amounts of claims is considered, the ruin probability is higher than 
when the dependency is not considered. This implies the insurance companies can 
underestimate their ruin probabilities when the dependencies are not considered.  
 
4.7 Time to Ruin  
The time it takes for the insurance company to ruin was estimated using equation 3.4 and 
the reults were presented in Table 4.8. 
 
Table 4.8:  Time to Ruin  
Time to Ruin (days) 
Company Assumptions of Assumptions of 
Dependence Independence 
A 30.29 92.94 
B 23.67 84.12 
C 21.60 55.11 
D 19.28 42.13 
Overall 16.03 39.77 
 
From Table 4.8, it can be found that the time to ruin for all companies when dependency 
between number of claims and amount of claims is taken into consideration are lower than 
when the dependency are ignored. This implies that it will take shorter time for ruin to occur 
when dependency between number of claims and amount of claims is considered than when 
39 
it is ignored. Therefore, a company may have shorter time to ruin than the actual time 
estimated when the dependency is ignored.   
 
4.8 Relationship between level of independency and Time to Ruin  
The level of correlation was plot against time to ruin, in order to display the relationship 
between the relationship between level of independency and Time to Ruin. 
 
Figure 4.1: Relationship between level of Dependency and Time to Ruin 
Figure 4.1, it can be seen that there is a negative relationship between dependency and time 
to ruin. This implies that the higher the dependency the lower the time to ruin. Therefore, 
when the dependency between number of claims and amount of claims are higher, the 
insurance company ruin faster.  
40 
4.9 Relationship between level of dependency and Probability of Ruin  
A line plot was done to examine the relationship between the relationship between level of 
dependency and probability of ruin. The level of dependency from correlation was plot 
against probability of ruin and results were displayed in Figure 4.2. 
 
Figure 4.2: Relationship between level of independency and Probability of ruin 
Figure 4.2, it can be seen, there is a positive relationship between probability of ruin and 
level of dependency. This implies that the higher level of dependency, the higher probability 
of ruin and vice versa. Hence given the same initial surplus, insurance companies that 
exhibit higher relationship between amount of claims paid and number of claims are more 
likely for ruin to occur than companies with relatively lower dependency. 
4.10 Premiums Adjustment to meet Claims dependency 
From Table 4.7, it was found that the probability of ruin based on assumption of claim 
dependency is significantly higher than the assumption of claim independency. This is 
because with the assumption of claim dependency, claims amount are higher. Hence to 
41 
ensure solvency, premiums must be adjusted. This section, present results of how the 
premiums are adjusted to to prevent ruin occuring in case claim dependency. The adjustment 
is based on adding the expected positive surplus process to the initial premiums as shown 
in equation 3.7.The results of the adjusted premiums can be seen in Table 4.9.  
 
Table 4.9: Adjusted Premiums 
 
 Adjusted 
Company  Premiums  E U t  |U t   0  Premiums 
 
            
A      2,577,499.28  
45,062,120.38  47,639,619.66  
        
B      5,155,099.96  
117,461,352.88  122,616,452.84  
            
C      1,112,070.20  
74,252,986.38  75,365,056.58  
            
D         110,513.96  
19,101,680.75  19,212,194.71  
 
The premiums are adjusted by the proposed formula in equation  3.7. For examples the 
adjusted premiums for company A was computed as 
45062120.382577499.28  47639619.66 . 
 
 
  
42 
CHAPTER FIVE 
SUMMARY, CONCLUSION AND RECOMMENDATION 
 
5.1 Introduction 
This chapter presents the summary, conclusions and recommendations on effects of 
dependent claims on the probability of ruin, the time to ruin given ruin occurs.  
 
5.2 Summary  
Probability of ruin and time to ruin are mostly estimated by the insurance companies without 
considering the dependent claims. In practice, the amount of claims paid are dependent on 
number of claims recorded. Therefore, this study determines how dependent claims, affect 
the ruin probability companies in Ghana. Also, how fast a given insurance company goes to 
ruin when the assumption of dependency of claims hold.  
Secondary data was used and was extracted from Annaly reports of National Insurance 
Commission (NIC), from 2010 to 2017. The researcher randomly selected four general 
insurance companies. The data includes annual claims paid, premiums received and the 
number of claims reported. 
The amount of claims was modelled using exponential distribution and maximum likelihood 
estimator (MLE) was used to estimates the parameters. The study found out that amount of 
claims are well fit using exponential distribution. Also, with regards to number of claims, 
the data follows Poisson distribution. The study also found out that there is a positive 
correlation, between claims amount and number of claims and from the Copulas test of 
Independent it was also found that the level of independency between claims amount and 
number of claims was low.  
43 
With regards to the relationship between the level of independency and time to ruin and 
probability of ruin. The study found out there is a positive relationship between level of 
independency and time to ruin and a negative relationship between level of independency 
and probability of ruin. This means that as the level of dependency between number of 
claims and amount of claims is high, ruin occurs faster with higher probability of ruin. 
 
5.3 Conclusions 
The following conclusions were drawn from the study:  
Findings from the study indicate that when there is claim dependency it plays an important 
role in the insolvency insurance companies as it significantly increases the likelihood of ruin 
of insurance companies.   
Based on the results, it can be concluded that when the assumption of dependency of claims 
hold, time left for a given insurance company to go to ruin decreases.  
From the study it can be concluded that using the proposed adjusted premium, the impact of 
claim dependencies  on ruin occurring can be neutralised.  
 
5.4 Recommendations    
Based on the findings from this study, it is recommended that insurance companies 
with high claim dependencies should adjust their premiums in order to enhance their 
chances of survival.   
Computation of time to ruin without the assumptions of claims dependency may be 
misleading results and this will lead to wrong decision making. Hence, any policies on 
the solvency of the insurance companies should be supported by the level of claim 
dependencies.  
44 
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48 
APPENDIX: R codes 
datac=read.delim("clipboard") ##loading the data 
names(datac) ### Variables names 
library(MASS)  
 
claimfit<- dexp(datac$Glico_C, rate = 1, log = FALSE) 
AIC(claimfit) 
#### number of Claims ###### 
#### Poisson ##### 
172.600000/12 
AADC<-fitdistr(datac$Claim_Amount, "weibull")  
AIC(AADC) 
BIC(AADC) 
AADC 
AB<-fitdistr(datac$SIC_nc, "Poisson")  
AIC(AB) 
BIC(AB) 
AC<-fitdistr(datac$Star_nc, "Poisson")  
AIC(AC) 
BIC(AC) 
AC 
AD<-fitdistr(datac$Donewell_nc, "Poisson")  
AIC(AD) 
BIC(AD) 
49 
AD 
 
OV<-c(datac$Glico_nc,datac$SIC_nc,datac$Star_nc,datac$Donewell_nc) 
 
OV1<-fitdistr(datac$Glico_nc, "Poisson")  
AIC(OV1) 
BIC(OV1) 
OV1 
OV2<-fitdistr(datac$SIC_nc, "negative binomial")  
AIC(OV2) 
BIC(OV2) 
OV2 
com.fit(OV2) 
AC<-fitdistr(datac$Star_nc, "Poisson")  
AIC(AC) 
BIC(AC) 
AC 
AD<-fitdistr(datac$Donewell_nc, "Poisson")  
AIC(AD) 
BIC(AD) 
AD 
 
 
#### negative binomial##### 
50 
 
BA<-fitdistr(datac$Glico_nc, "negative binomial")  
AIC(BA) 
BIC(BA) 
BA 
BB<-fitdistr(datac$SIC_nc, "negative binomial")  
AIC(BB) 
BIC(BB) 
BB 
BC<-fitdistr(datac$Star_nc, "negative binomial")  
AIC(BC) 
BIC(BC) 
BC 
BD<-fitdistr(datac$Donewell_nc, "negative binomial")  
AIC(BD) 
BIC(BD) 
BD 
 
CA<-fitdistr(datac$Glico_nc, "compo")  
AIC(CA) 
BIC(CA) 
CA 
library(compoisson) 
MA<-matrix(datac$SIC_nc, nrow = 1, ncol =8 ) 
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DDC<-com.fit(MA) 
DDC 
 
MAM<-matrix(OV, nrow = 1, ncol =32 ) 
DDC<-com.fit(MAM) 
DDC 
 
CC<-as.numeric(DDC$lambda) 
AIC(CC) 
com.fit 
?fitdistr 
compoisson 
compoisson::compoisson:: 
  Glico 
WA1<-fitdistr(datac$Glico_C, "exponential")   
AIC(WA1) 
BIC(WA1) 
WA1 
 
WA2<-fitdistr(datac$SIC_C, "exponential")  
AIC(WA2) 
BIC(WA2) 
WA2 
 
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WA3<-fitdistr(datac$Star_C, "exponential")  
AIC(WA3) 
BIC(WA3) 
WA3 
 
 
WA4<-fitdistr(datac$Donewell_C, "exponential")  
AIC(WA4) 
BIC(WA4) 
WA4 
 
OVC<-c(datac$Glico_C,datac$SIC_C,datac$Star_C,datac$Donewell_C) 
GA<-fitdistr(OVC, "exponential")   
AIC(GA) 
BIC(GA) 
GA 
 
 
dpois(x, lambda, log = FALSE) 
mean(datac$Glico_nc) 
lambda=5749.5, log = FALSE) 
dpois(datac$Glico_nc, lambda=5749.5, log = FALSE) 
rpois(1000,lambda=5749.5) 
 
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as.matrix(datac$Glico_nc) 
com.fit(dglico) 
 
datac$Glico_nc 
dglico<-matrix(datac$Glico_nc, nrow = 1, ncol = 8) 
dsic<-matrix(datac$SIC_nc, nrow = 1, ncol = 8) 
dstar<-matrix(datac$Star_nc, nrow = 1, ncol = 8) 
ddonwell<-matrix(datac$Donewell_nc, nrow = 1, ncol = 8) 
aa<-com.fit(dglico) 
ab<-com.fit(dsic) 
ac<-com.fit(dstar) 
ad<-com.fit(ddonwell) 
## fitdistr()included in package MASS 
fitdistr(datac$Glico_C, "gamma") 
 
 
x <- rgamma(100, shape = 5, rate = 0.1) 
fitdistr(x, "gamma") 
set.seed(123) 
x3 <- rweibull(100, shape = 4, scale = 100) 
fitdistr(x3, "weibull") 
 
set.seed(123) 
x4 <- rnegbin(500, mu = 5, theta = 4) 
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fitdistr(x4, "Negative Binomial") 
options(op) 
mean(x4) 
poisson(datac$Glico_nc) 
 
####  
library(copula) 
library(IndepTest) 
 
cor.test(OVC, OV) 
indepTest {copula} 
 
indepTestSim(n, p, m = p, N = 1000, verbose = interactive()) 
indepTest(datac$Donewell_C, dp, alpha=0.05) 
dependogram(test, pvalues = FALSE, print = FALSE) 
ggraph-tools {copula} 
pairwiseIndepTest(datac$Donewell_C,datac$Donewell_nc) 
pairwiseIndepTest 
 
###  copulas 
 
 
## Case with an explicit formula: exponential claims and exponential 
## interarrival times. 
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psi <- ruin(claims = "e", par.claims = list(rate = 5), 
            wait   = "e", par.wait   = list(rate = 3)) 
psi 
psi(0:10) 
plot(psi, from = 0, to = 10) 
 
ruin(claims = c("exponential", "Erlang", "phase-type"), par.claims, 
     wait = c("exponential", "Erlang", "phase-type"), par.wait, 
     premium.rate = 1, tol = sqrt(.Machine$double.eps), 
     maxit = 200L, echo = FALSE) 
 
# S3 method for ruin 
 
plot(x, from = NULL, to = NULL, add = FALSE, 
     xlab = "u", ylab = expression(psi(u)), 
     main = "Probability of Ruin", xlim = NULL, …) 
psi <- ruin(claims = "exponential", 
            par.claims = list(rate = c(3, 7),weights = 0.5), 
            wait = "exponential", 
            par.wait = list(rate = 3), 
            premium.rate = 1) 
 
ruin::ruin_probability(claims = "exponential", 
                       par.claims = list(rate = c(3, 7),weights = 0.5), 
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                       wait = "exponential", 
                       par.wait = list(rate = 3), 
                       premium.rate = 1) 
 
ruin_probability {ruin} 
### INDEPENDENCY 
model <- CramerLundberg(initial_capital = 0, 
                        premium_rate = 63969535.09375, 
                        claim_poisson_arrival_rate =, 
                        claim_size_generator = rexp, 
                        claim_size_parameters = list(rate = 0.000000223)) 
ruin_probability(model = model, 
                 time_horizon = 10, 
                 simulation_number = 4, 
                 return_paths = TRUE, 
                 parallel = FALSE) 
 
pchisq(27.245, df=40, lower.tail=FALSE) 
pchisq(25.4253, df=40, lower.tail=FALSE) 
pchisq(11.245, df=10, lower.tail=FALSE) 
pchisq(14.452, df=10, lower.tail=FALSE) 
pchisq(12.243, df=10, lower.tail=FALSE) 
pchisq(13.845, df=10, lower.tail=FALSE) 
pchisq(11.452, df=10, lower.tail=FALSE) 
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pchisq(12.458, df=10, lower.tail=FALSE) 
pchisq(12.652, df=10, lower.tail=FALSE) 
pchisq(14.06714,df=7) 
58