UNIVERSITY OF GHANA 

 

 

APPLICATION OF BÜHLMANNS-STRAUB CREDIBILITY 

THEORY TO CLAIM HISTORIES OF NON-LIFE MARINE 

INSURERS IN GHANA 

 

BY 

PAUL KWAME ADJORLOLO 

(10599328) 

THIS THESIS IS SUBMITTED TO THE DEPARTMENT OF 

STATISTICS AND ACTUARIAL SCIENCE, UNIVERSITY OF 

GHANA, LEGON IN PARTIAL FULFILLMENT OF THE 

REQUIREMENTS FOR THE AWARD OF MASTER OF 

PHILOSOPHY IN ACTUARIAL SCIENCE 

 

 

JULY, 2019 



 

ii 

 

DECLARATION 

Candidate’s Declaration 

I, Paul Kwame Adjorlolo, hereby declare that this thesis is my own work, 

design and execution, submitted for the award of a Master's degree in 

Philosophy and that, to the best of my knowledge it contains no material that 

has been submitted to this institution or any other university except where duly 

acknowledged in the text. 

 Signature …………………………………  Date: …………………… 

Paul Kwame Adjorlolo 

(10599328) 

 

Supervisors’ Declaration 

We hereby certify that this thesis was prepared from the candidate's own effort 

and supervised in accordance with the University of Ghana's guidelines on 

supervision of theses.  

 

Signature…………………………………  Date……………………… 

Dr. Anani Lotsi 

(Supervisor) 

 

Signature …………………………………   Date……………………… 

Dr. Felix O. Mettle 

(Co-Supervisor) 

 

 



 

iii 

 

ABSTRACT 

The study sought to demonstrate how credibility claim costs without the 

consideration of claim frequency and claim severities underlined by different 

risk profiles underestimate claim costs or premiums charged policyholders by 

non-life insurance companies. We used secondary data of non-life marine 

insurers in Ghana, claim histories that range from the period of 2013 to 2018. 

The claim histories included claim sizes, claim counts and policy counts. 

Bühlmans-Straub Credibility theory model was used in estimating credibility 

weights, credibility claim costs, credibility claim frequencies and credibility 

claim severities and subsequently find the credibility frequency-severity claim 

cost as the product of credibility claim frequency and severity for the 

individual and respective risk classes. We compared the estimates of the claim 

costs or premiums and have observed that the credibility claim costs 

underestimates claim costs or the average claim costs compared to the 

credibility frequency-severity claim costs for most of the risk classes. This is 

an indication of how a lack of consideration for variability or unstableness of 

claim frequencies and severities with different risk profiles undermine claim 

costs estimated through credibility rate makings in insurance. The study 

recommends that credibility ratemaking by insurance companies based on 

inadequate claim history and or with enough class risk variation should include 

credibility risk frequency and severity for the determination of credibility risk 

premiums. 

 

 

 

 



 

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DEDICATION 

To all women who care for children in any state or form. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

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ACKNOWLEDGMENTS 

My sincere gratitude goes to my supervisors; Dr. Anani Lotsi and Dr. Felix O. 

Mettle for their patience and guidance and immeasurable input into this work. 

My sincere gratitude also goes to Dr. Gabriel Kallah-Dagadu for his advice 

and encouragement. My genuine appreciation also extends to all the lecturers 

at the Department of Statistics and Actuarial Science. I'm most grateful to my 

parents (Mr. Edward Newlove Tsatsu and Mrs. Christiana Agbenyegah) for 

their counseling and prayer, siblings for their support especially Dr. Samuel 

Adjorlolo and the wife, Mrs. Rita Adjorlolo and my classmate and friends for 

their supports especially Mr. Freeman Kofi Ahorkonu. Am also grateful to all 

members of the Sogakofe Church of Christ for the love they showed me in 

hard times whilst pursuing academic success.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

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TABLE OF CONTENTS 

DECLARATION .................................................................................................... ii 

ABSTRACT ........................................................................................................... iii 

DEDICATION ....................................................................................................... iv 

ACKNOWLEDGMENTS ....................................................................................... v 

LIST OF TABLES ................................................................................................. ix 

LIST OF FIGURES ................................................................................................. x 

LIST OF ABBREVIATIONS ................................................................................ xi 

INTRODUCTION ................................................................................................... 1 

1.1      Background of the Study ......................................................................... 1 

1.2      An Overview of Marine Insurance .......................................................... 3 

1.2.1      Modern Marine Insurance ................................................................. 4 

1.3     The Ghanaian Insurance Industry ............................................................. 5 

1.4      Statement of the Problem ........................................................................ 6 

1.5      Objectives of the Study............................................................................ 7 

1.5.1      The Specific Objectives .................................................................... 7 

1.6      Research Questions ................................................................................. 8 

1.7      Justification .............................................................................................. 8 

1.8      Scope and Limitations ............................................................................. 9 

1.9      Organization of the Thesis ....................................................................... 9 

LITERATURE REVIEW ...................................................................................... 11 

2.1      Introduction ........................................................................................... 11 

2.2      Insurance Claim Costs ........................................................................... 11 

2.3     Profitability of Underwriters for Assuming Risk ................................... 15 

2.4      Class and Experience Rating of Insurance ............................................ 17 



 

vii 

 

2.5      Credibility Theory ................................................................................. 23 

2.5.1      Application of Credibility theory .................................................... 24 

2.5.2      The Credibility Factor ..................................................................... 25 

2.5.3      Bühlmann and Straub’s Credibility Theory .................................... 27 

CHAPTER THREE ............................................................................................... 30 

METHODOLOGY ................................................................................................ 30 

3.1      Introduction ........................................................................................... 30 

3.2      Type and Sources of data ...................................................................... 30 

3.3      Target Population .................................................................................. 31 

3.4      The Sample data .................................................................................... 31 

3.5      Bühlmann and Straub’s Credibility for Multiple Exposures ................. 32 

3.6      Credibility Estimate of Claim Costs ...................................................... 34 

3.6.1      Assumptions of Model under Claim Cost ....................................... 35 

3.6.2      Estimation of the Structural Parameters .......................................... 37 

3.7      Credibility Estimate of Claim Severities ............................................... 39 

3.7.1      Assumptions of Model under Claim Severities............................... 40 

3.7.2      Estimation of the Structural Parameters .......................................... 43 

3.8      Credibility Estimate of Claim Frequencies ........................................... 44 

3.8.1      Model Assumptions under Claim Frequency .................................. 45 

3.8.2      Estimation of the Structural Parameter ........................................... 48 

3.9      Credibility Frequency-Severity Estimates ............................................. 49 

CHAPTER FOUR ................................................................................................. 50 

DATA ANALYSIS ............................................................................................... 50 

4.1      Introduction ........................................................................................... 50 



 

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4.2      The Sample Period ................................................................................ 50 

4.3      Descriptive Analysis of the Claim History ............................................ 51 

4.4      Buhlmann-Straub Credibility Parameter Estimates ............................... 57 

4.4.1      Variability within Class Risks ......................................................... 57 

4.5      Premium under Claim Frequency and Claim Severity .......................... 64 

CHAPTER FIVE ................................................................................................... 74 

DISCUSSION, CONCLUSIONS AND RECOMMENDATIONS ...................... 74 

5.1      Introduction ........................................................................................... 74 

5.2      Discussion .............................................................................................. 74 

5.3      Conclusion ............................................................................................. 78 

5.4      Recommendations ................................................................................. 79 

REFERENCES ...................................................................................................... 81 

Appendix ............................................................................................................... 86 



 

ix 

 

 LIST OF TABLES 

Table 4.1:  Descriptive Statistics of Claim Sizes for the Non-Life Insurance 

Companies ........................................................................................................ 52 

Table 4.2:  Descriptive Statistics of The policy Count for the Non-Life 

Insurance Companies ........................................................................................ 54 

Table 4.3: Descriptive Statistics of Claim Counts for the Non-Life Insurance 

Companies ........................................................................................................ 56 

Table 4.4: Policy Count and Credibility Factors and Estimates for Claim Cost 

for the Marine Insurance Companies ................................................................ 61 

Table 4.5: Estimated Structure Parameters for Annual Claim Cost ................. 61 

Table 4.6: Claim Count and Credibility factors and estimates of claim 

Severities for the Marine Insurance Companies ............................................... 67 

Table 4.7: Parameter estimates for the Claim Severities .................................. 67 

Table 4.8: Credibility Claim Frequencies Estimates for the Marine Insurance 

Companies ........................................................................................................ 69 

Table 4.9: Parameter estimates for the Claim Frequency ................................. 70 

Table 4.10: Credibility frequency-severity risk premium ................................ 71 

 

 

  

 

 

 

 



 

x 

 

LIST OF FIGURES 

Figure 2.1 Average loss ratio of a portfolio of nine contracts .......................... 18 

Figure 2.2 Yearly loss ratios of the small contracts.......................................... 20 

Figure 2.3 Yearly loss ratios of the medium-sized contracts............................ 20 

Figure 2.4 Yearly loss ratios of the large contracts .......................................... 21 

Figure 2.5 Yearly loss ratios of contracts 1-3 ................................................... 22 

Figure 4.1: Bar Chart of Annual Claim cost for Marine Insurance Companies 59 

Figure 4.2: Bar Chart plot of Average Claim costs and Credibility Estimates of 

risks ................................................................................................................... 63 

Figure 4.3: Bar Chart plot of Average Claim Cost and Credibility Premiums. 72 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

xi 

 

LIST OF ABBREVIATIONS 

MI:                                      Marine Insurance  

IACG:                                  International Awareness of Coordinating Group  

NIC:                                     National Insurance Commission  

GIA:                                    Ghana Insurers Association   

IUMI:                                  International Union of Marine Insurers  

 GIBA:                                 Ghana Insurance Brokers Association  

FPSO:                                  Floating Production Storage and Offloading  

IIM:                                     The Insurance Institute of Michigan 

EPV:                                    Expected Process Variances  

VHM:                                  Variance of the Hypothetical Means 



 

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  CHAPTER ONE 

INTRODUCTION 

1.1      Background of the Study  

Modern insurance is described as a two-party contract normally between an 

insurance company and an insured who an insurance company accepts to  pay 

a settled amount when a loss occurred in exchange for premiums (Mishra, 

2004).  Under the maritime law (Act 1906), marine insurance is described as; 

―A contract whereby the insurer undertakes to indemnify the insured, in the 

manner and to the extent thereby agreed, against marine losses, that is to say, 

the losses incident to marine adventure‖. Insurance reduces the risk of total 

loss to policyholders’ as at when there is an insured hazard. Consequently, 

insurance companies function as managers of premiums gathered from the 

citizens who purchased insurance policy and also operate as allocators after 

offering them insurance, i.e paying for losses to those who encounter 

accidental misfortunes from the premiums collected from insuring citizens.  

 

In view of the growing hazards on our seas, the need for specialized sea-related 

insurance has become more critical. The catastrophic effects of happenings at 

sea may be similar or more heartrending than those in the air. The need for 

insurance emerged from the desire of business persons to take calculated risks. 

Hitherto, such risks were encountered when sailing ships were sent off on 

hazardous sea voyages to trade with other countries far from their sources of 

destination without any form of financial compensation. This gave birth to 

marine insurance as the mother of all insurances. Marine Insurance (MI) 

intended to cover loss or damage to ships, cargo, terminals and any other type 



 

2 

 

 

of ocean transport for which property is transferred, obtained or kept between 

the point of origin and the final destination also protects importers and 

exporters against loss, theft or damage to goods or property conveyed by sea 

including other  waterways and rivers. Ship-owners are also protected against 

loss or damage to hull, machinery of the vessel and legal liability of the ship-

owners, while on the high seas (Zogbenu, 2018). 

 

Africa's marine insurance penetration nonetheless stays disappointingly low. 

Whilst worldwide changes have crept up by way of 2% as indicated with the 

aid of the global International Union of Marine Insurers' (IUMI's); Africa's 

numbers are still at a baffling 2.4%. Local players must take advantage of the 

lucky breaks which accompany the increasing worldwide exchange. Africa 

must also desire to position resources into growing the capacity in marine 

protection. There are not many markets like South Africa and Mauritius at the 

mainland with enough chronicles of marine products and local talents (IUMI, 

2018). Ghana, in recent years has made striking effort to improving the marine 

insurance industry; the International Awareness of Coordinating Group 

(IACG) which comprises representatives of the National Insurance 

Commission (NIC), the Ghana Insurers Association (GIA), the Ghana 

Insurance Brokers Association (GIBA), held a workshop on marine insurance 

market. Among other relevant stakeholders who participated in an all-

important workshop were freight forwarders, cargo haulers, trans-ocean 

importers and exporters, trade regulators and policymakers, among others, to 

deliberate on the way forward for marine insurance.  

 



 

3 

 

 

The workshop which was held on June 2018 seeks to provide a single platform 

on which concerns and perceived misgivings from all stakeholders would be 

addressed; right from ‘the anchor to the masthead’ until premiums are duly 

collected and claims paid when they fall due’. This would to a very large 

extent streamline and strengthen the marine insurance sector in Ghana. The 

procurement of specialized vessels like the Floating Production Storage and 

Offloading (FPSO) Kwame Nkrumah together with other cargo vessels, there 

is no way one could gloss over the need for various types of marine insurance 

policies to manage risks. And in Ghana, Marine Laws are in place to 

essentially protect local importers who will be disadvantaged in the event of 

offshore claims - especially in the event of litigation. 

 

 1.2      An Overview of Marine Insurance 

For people to have a right knowledge of the outlook on maritime protection, it 

becomes important to hint the events of company transactions and some of the 

international locations of the sector. Seas have been the fundamental transport 

systems of alternate used within Man's early records evidently observed 

throughout in sacred and secular history. A couple of nations have claimed the 

honour of getting advanced the association of compensation as insurance or 

risk management. Be that as it is able to, the fine evidence proves that the Jews 

during the season when they were expelled from France in the remaining 

pieces of work of the 12th century presented this type of insurance plan for the 

protection of their belongings among their deportation from France. Preferred 

average and Bottomry Bonds have been typically the two  oblique varieties of 



 

4 

 

 

cutting-edge coverage practiced and had been utilized early inside the records 

of mass-marketplace (Winter, 1919). 

 

A lawful widespread of maritime laws by which all parties on an ocean journey 

have an extraordinary proportion of any losses incurred from a deliberate 

forfeiture of part of  shipments or the ship or part of those to save lives is 

known as the law of the general average. While the general average draws its 

beginnings in the ancient oceanic law, some aspects of the admiralty regulation 

form part of most countries’ laws (Johnson, 2013). Bottomry  is an out-dated 

bond agreement described as a loan in which a vessel is mortgaged in one of 

these manners that if the vessel had been lost, money lend by a lender is lost 

too, but if the ship arrived securely at the port of vacation spot, the borrower 

pay back the mortgage loan to the lender and with an interest agreed upon in 

advance (Winter, 1919). Besides the loans, a ship can be promised as a 

Respondentia Bond and cargo hypothecated and the documents carrying out 

the terms of the contract were known as Bottomry Bonds.   

 

1.2.1      Modern Marine Insurance  

In fact, a Bottomry bond was an inverse type of the current marine insurance 

scheme; contemporary time marine insurers receive premium against loss of 

the ship and shipments, plus interest and other costs, which amount the 

underwriter agreed to pay the insured as compensation in the case of a ship or 

cargo being lost as a consequence of the risks of insurance or marine 

adventure.  In addition, under current commercial use, credits used on ships are 

produced through the implementation of hypothecated bonds on ships , Which, 



 

5 

 

 

on the other hand, is covered by an insurance policy that must be paid to the 

lender (Winter, 1919).  Since its advent in the 13th century, modern marine 

insurance has experienced tremendous growth and has had legal corroboration 

throughout the globe. For instance, the Hanseatic backing across countries 

including Northern Europe's emerging merchants had an insurance centre in 

Bruges, known as the first insurance chamber. The town of Barcelona also laid 

down the first statute recorded for insuring vessels in 1432. The Lombards 

started marine insurance by progressing amounts on Bottomry loans. The 

creation of Lloyd's of London in the 16th century became the insurers, traders, 

ship owners ' first meeting place and others interested in insuring their cargoes 

and boats including those that are ready to trade in the marine enterprises and 

at the time, Lloyds of London became the world's biggest insurance industry, 

particularly marine protection (Kyriaki, 2007). 

 

1.3     The Ghanaian Insurance Industry  

In Ghana, the National Insurance Commission (NIC) is the only establishment 

commanded to control and supervise insurance against future losses in the 

nation according to the insurance laws (Act 724, 2006). Ghana's insurance 

companies are now divided into three categories: general and composite and 

life insurance. Loss adjusters, insurance brokers, actuarial firms and dealers are 

also in Ghana. Globally, insurance provided the risk transfer mechanism in the 

early stages of the insurance industry, but now the industry is helping to 

channel resources appropriately to promote business operations in the 

economy (Curtis & Eric, 2013). 

 



 

6 

 

 

one element of the non-life insurance sector in Ghana that's also beginning to 

grab attention is the maritime insurance industry due to the recent enforcement 

of section 37 of the Insurance Act, 2006 (Act 724) ; this act compelled local 

dealers to ensure their products locally for the industry to take advantage of  

premium income gains (NIC, 2015). The implementation of this amendment 

has gone all the way to positively influence the country’s economic business 

industries. 

 

1.4      Statement of the Problem 

The non-life insurance companies like any other type of insurance business do 

not only act as risk managers but also serve as intermediaries that mobilize 

funds for banks which are also lend to business firms who also invest in the 

economy. The economy of Ghana like any other country will be affected if 

insurance companies do not charge risks the right premium in a competitive 

insurance business because insurance companies would not have enough funds 

to pay for losses or enough to invest in the economy(Mazviona, Dube, & 

Sakahuhwa, 2017). This can see many insurance companies out of business 

and the collapse of the insurance business can be disastrous to any economy, 

particularly in developing countries. As a consequence of these problems faced 

by insurance companies especially when there is little claim history available it 

is necessary to determine the best procedures of estimating fair premiums to be 

charged policyholders by these insurance companies. 

The researcher’s interest in this study is to use Bühlmans-Straub credibility 

theory as an empirical approach to estimating and comparing credibility 

frequency-severity premiums or claim cost to credibility premiums (claim 



 

7 

 

 

cost) for policyholders of some selected leading maritime insurance companies 

in Ghana taking into consideration the heterogeneous and limited past claim 

histories.  

 

 1.5      Objectives of the Study 

The primary target of this research or study is to find the credibility frequency-

severity premiums or claim costs for policyholders for each of the non-life 

marine insurance companies (risk classes) using claim frequencies and 

severities experienced by applying Bühlmans-Straub Credibility Theory model 

and also compare the result with credibility claim cost or premiums using 

annual claim costs estimated based on the same method and condition.  

 

1.5.1      The Specific Objectives 

The specific objective of the study is to; 

1. estimate the average claim cost  of the individual non-life marine 

insurance  companies 

2.  estimate the factor of credibility for each non-life insurance companies 

3. estimate credibility premiums based on claim costs and credibility 

frequency-severity premium for policyholders of the risk classes or 

individual non-life  marine insurance companies for the period 2018 

using claim histories between the period of 2013 to 2018. 

4. Compare the two credibility estimates; the credibility frequency-

severity risk claim costs and the credibility risk claim cost.  

 



 

8 

 

 

1.6      Research Questions 

Three questions seem crucial in finding the credibility claim costs. Thus,  

1. How much variation is there in the claim costs or frequencies or 

severities of the companies? 

2. How homogeneous are the risk classes in terms of claim frequency and 

claim severity? 

3. How significant are the credibility factors of the Bühlmans-Straub 

model in doing experience rating? 

 

1.7      Justification 

This research will open the debate on the need to integrate the frequency and 

severity of claims in a suitable and fair determination of Ghanaian non-life 

insurance policies ' credibility premiums. The Study can also serve as a point 

of reference for decision-making in evaluating and defining equilibrium 

between premiums granted based on experience rating and collective risk 

rating organizations. The results of this research will be helpful to players and 

stakeholders in the industry of maritime business, in particular the National 

Insurance Corporation (NIC) in its quest to determine premiums among 

competitive non-life insurance firms taking into consideration the intrinsic 

random risk files of each non-life marine insurance company that influence the 

size of claim costs, frequency and severity of these claims. The 

implementation of Bühlmanns-Straub's credibility methodology to selling 

maritime insurance products based on the frequency and severity of claims that 

includes individual risk profiles making the marine insurance company more 



 

9 

 

 

competitive, leading to general industrial growth and also helping to reduce the 

insolvency of maritime industrial businesses.  

 

1.8      Scope and Limitations 

This study aimed at determining the best balance between experience ratings 

and class ratings incorporating the frequency and severity of claim histories. 

The data considered for this research is claim amounts, claim counts and policy 

counts for policyholders of maritime insurers reported by the marine insurance 

companies to the NIC from 2013 to 2018.  Bühlmans-Straub credibility theory 

model was used to estimate the credibility claim cost using only annual claim 

cost with their respective policy counts and credibility frequency-severity 

claim counts or premiums using claim frequencies and severities with their 

respective policy counts and claim counts respectively. From these two 

estimates the best balance was observed. One of the limitations of this study is 

that the claim histories used for the analysis did not cover all non-life marine 

insurance companies in Ghana but limited to only 15 non-life marine insurance 

companies. The study did not factor inflation and other macroeconomic factors 

that might have been having an effect on risk rating into the analysis.  

 

 1.9      Organization of the Thesis 

The study is structured into five major sections. The first section gives 

background of the research and brief overview of marine insurance. The 

second chapter focuses on literature and model assessment of other associated 

papers. The methodology and fundamental hypotheses are discussed in the 

third chapter.  Analysis and presentation of results including interpretation of 



 

10 

 

 

results are completed in the fourth chapter and discussion of research findings, 

conclusions and recommendations in chapter five. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

11 

 

 

CHAPTER TWO 

LITERATURE REVIEW 

 2.1      Introduction  

This chapter includes the review of insurance claim costs, underwriters’ 

profitability for assuming risk, class and experience rating of insurance as well 

as credibility theory. The section also includes an appropriate empirical 

evaluation of the theory of credibility and its creation, determination of the 

factors of credibility and application of the credibility theory to the pricing of 

insurance products and the assessment of credibility premiums. The evaluation 

of the credibility views expressed by other authors, such as the credibility 

model of Bühlmann, the merits, and demerits or constraints of these models, is 

considered in this chapter with the aim of adequately using this understanding 

to determine claim costs of maritime insurance in Ghana. 

 

2.2      Insurance Claim Costs 

To be allowed to enter into an agreement with an insurer, an insured must pay 

a premium. The price of claim or premium may vary among individuals or 

insureds depending on how dangerous they are or how dangerous the social 

group to which they belong is perceived. Information is essential for insurance 

organizations to be able to determine the appropriate premium for individuals 

or groups of individuals to insure. Underwriters pooled the premiums from  

policyholders and use this pool of money received by the insurers is used  to 

pay loss cases as they occur (Thiery & Schoubroeck, 2006).   



 

12 

 

 

―An insurance claim is a formal request to an insurance company for coverage 

or compensation for a covered loss or policy event. The insurance company 

validates the claim and, once approved, issues payment to the insured or an 

approved interested party on behalf of the insured ‖ (Kagen, 2018). In principle 

for an insurance provider to fulfil and cover all outstanding claims before the 

end of the accounting period, it is necessary for the insurer to render 

arrangements for claims from monies earned within such an accounting period. 

Shapland (2007) defined outstanding claim provision as ―a sum conveyed in 

the liability area of a hazard or risk bearing element's balance sheet as a result 

of claims incurred preceding a given bookkeeping date, where liability, as 

utilized in the definition, is the actual sum of losses insureds made on an 

insurance company and that will, at last, be paid by a risk bearing company 

before the end of a given bookkeeping date‖. 

 

Insurance is the transfer of potential losses from policyholders to insurance 

companies or insurers for a predetermined claim price, which in effect offers 

the economic insurance coverage service in the event of a loss under policy 

cover. The desire of a reasonable premium is of paramount relevance for an 

insurer regarding the retaining of present clients, gaining new enterprise and 

also to have competitive advantage when the policy targets a big population 

and also extends over a significant region of property (Emanuelsson, 2011). 

Claim payment could be very crucial to the execution of any insurance 

coverage, which is why the NIC requires insurance companies to have an 

extremely good claim provision on their accounting books. Regulators are 

particularly concerned about the economic situation of insurers, given that the 



 

13 

 

 

security of insureds  and the security of investors is of paramount importance 

to regulators and stakeholders by ensuring that the security net supplier gives 

an accurate picture of their solvency situation (Asare & Opokutakyi, 2017).  

 

According to Asare and Opokutakyi ( 2017) in determining the solvency 

factors of non-life insurance firms in Ghana, the absence of appropriate 

provision for enough claim reserves is significantly the primary mechanistic 

explanation of the bankrupt insurance industry, exposing investors and various 

associates to enormous losses and being unable to fulfil their share of insured 

contracts. Genuinely, insolvent insurance firms are not allowed to continue 

offering protection because they do not have the financial support to maintain 

their legally binding obligations to their clients (Cheung, 1997).  

Notwithstanding the relevance of this concern, little is learned about all the 

concepts of the topic in the non-life insurance market in Ghana, particularly in 

the maritime insurance sector. 

 

  Andoh and  Aboagye (2014) also researched the factors that influence  

property insurance, market-based claims provision errors. The research used 

the information on published and paid-out claims for the 2000-2010 period. 

The authors discovered that the claim reserves or provision errors of the 

underwriters are random cross-sectional over companies, meaning that insurers 

behave separately and are not affected by trends of sector and competition. 

Insurance firms bankruptcy could be avoided if the pricing or assessment 

of  claim costs for pricing insurance products for subsequent instant periods 

dependent on past knowledge of personal  hazards for insurers taking 



 

14 

 

 

considering the risks’ features of each business and the relationship between 

individual risk premiums and the collective claim cost (Asare et al., 2017). 

 

 Many researchers used distinct techniques for exactly estimating insurance 

firms' prices required to charge policyholders in order to fully cover claim 

reserves when risks occur. The actuarial accuracy measures are one of these 

distinct techniques and are normally employed depending on the data available 

on claims and based on what assumptions are allowable. Zhang (2004) 

suggested a Bayesian non-linear hierarchical model that tends to tackle some 

of the major problems faced by insurance companies in anticipating the 

outstanding claim sizes for which they will finally be required to redeem. This 

approach requires taking into account previous understanding and expert and 

technical opinion to be included in the assessment by judgmentally selected 

priori distributions. Viaene et al., (2007) similarly, in their investigation, it was 

found out that the assessment of the likelihood of claim is critical since a loss 

protection company can use these assessments to allocate claim reserve 

boundaries and discounts based on the threat features of an individual client or 

to develop strategies to detect false claims. Consequently, Meulbroek (2000) 

argued that loss protection Companies must treat corporate governance as 

development of associated variables and opportunities. Boland (2007) in his 

research, he said, in order to cope with allegations incidental events emerging 

in advance underwriters must utilize prognostic techniques for managing the 

level of loss, indicating that back-up plans are required to identify techniques 

for anticipating claims and appropriately charge a premium for covering claims 

created by those risks. The recovery schemes need to search for suitable 



 

15 

 

 

solutions to capturing the risk features of policyholders influencing claim 

frequencies and severities and, consequently, distinguishing persons with poor 

risk, with a greater affinity to periodic claims on insurance companies.  

 

As stated by Weisberg (1983), insurance organizations are endeavouring to 

assess sensitive expenses of insurance arrangements depending on the 

announced losses for specific types of hazards. The estimates must be based on 

previous claims histories; claim severity and the frequency of losses so as to 

determine the trends that occur in order to be able to take appropriate measures 

to mitigate losses and the consequences associated with the losses. 

 

2.3     Profitability of Underwriters for Assuming Risk 

Fixed premium firms are provided by insurers as a commercial insurance 

service with the main objective of profiting. The fixed premium is a 

predetermined insurance premium on the basis of agreements between 

underwriters and policyholders. In these agreements, an insurer continues the 

hazard section via charging a fixed price to the insured relying on the regular 

assessment of losses. The hazard is either accepted or reinsured or backed up 

or securitized or mixed of the two against the regular losses of these 

operations. Additionally, the peril risk claim cost is needed to pay 

policyholders, shareholders and investors. Because the insurance firms are 

required to make a profit, in theory of economics the charges to the 

policyholders would be larger than that of mutual insurance premiums that are  

cover only the full cost of losses (Ravichandran, 2001). 

 



 

16 

 

 

Drewry (1998), Haiss (2006) and Marrewijk (2002) all, in their study have 

made significant attempt to enumerate some problems and the causes 

undermining the performance and the profitability of the marine insurance 

industry. King (2008) suggested that the marine insurance industries must 

charge competitive and reasonable premiums as a solution to the problems of 

marine insurance. However, the suggestion by King (2008) did not really 

provide enough information on how policy or underwriters must go about the 

rate making in order to achieve competitive and reasonable premiums 

especially in situations where there is only a little information on risks to be 

considered or rated. Petersson (2010) stated that, because of the uneven nature 

of the hazards concerned, it is most hard to decide on maritime insurance claim 

costs. Hsrrington and Niehaus (2003) also mentioned that for insurance 

companies to price practical rates due to competition, they should rate 

premiums that may contend with the expected cost of claims and 

administrative prices. The authors also pointed out that if Policyholders ' risk is 

homogeneous, the company can prospectively do equal treatment. Potential 

equal treatment with a variable degree of risk for insured persons, however, 

may generate adverse selection for insurance companies. In this manner, less 

dangerous policyholders instead of paying same premium as higher-risk 

policyholders they moved to a more cautious treatment scheduled organisation. 

It is obvious that insurance firms cannot just enough rely on experience and 

expertise or by mere direct calculation of pure premiums as reasonable 

premiums for the competitive market without proper statistical analysis of risks 

facing individual exposures of the insurance policy. The study of risks leading 

to premium determination must include the study of claim severities as well as 



 

17 

 

 

claim frequencies which all affect the financial position of insurance firms. 

Lawrence et al., (2017) emphasized in summary that bankruptcy of insurers 

could be avoided if suitable statistical and mathematical techniques are used to 

estimate premiums priced by insurers for selling insurance products in the 

future based on individual or class risks’ claim information and other 

comparable risks deemed relevant. 

 

2.4      Class and Experience Rating of Insurance  

Risks with almost uniform characteristics or heterogeneous but with restricted 

reliable claim information, have been difficult to rate. Therefore, an additional 

and related claim data from similar risks that are not necessarily the same may 

be useful in assessing those risks whose estimated premiums may lie between 

the average claim cost and collective claim cost. This is very necessary for 

estimating fair premiums especially for contracts with limited historical 

information and cannot be depended upon for estimating premium for the next 

immediate year.  Bhulman and Gisler (2005) demonstrated with diagrammatic 

illustrations of the difficulties to rely on limited available data and the problem 

of allocating collective or grand risk premium to less risky policyholders of 

non-life insurance contracts instead of doing experience rating or determining 

premiums by attaching some level of credibility to policyholders of a certain 

class of risks.  

 

The authors assumed a claim information including gross premiums on a 

portfolio comprising of nine health contracts (appropriate and applicable to 

marine insurance contracts) over a period of 5 years. Thus, contract 1, 4 and 7 



 

18 

 

 

were assumed to have had a premium volume of 2000 per year, and contract 2, 

5 and  8 had a premium volume of 20000 per year whilst the contracts 3, 6, and 

9 had a premium volume of 200000 per year all in CHF(Swiss franc). Their 

loss ratios were calculated as the ratio of the aggregate claim  to the gross 

premium of each contract over the five years expressed as a percentage. 

Figure 2.1 demonstrates the average loss ratio observed over the five years. 

From the Figure, the average loss ratio for contracts is 50 per cent for 1 to 3, 

75 per cent average loss portions for contract 4 to 6 and 100% average loss 

proportion for contracts 7 to 9.  

 

 

 

Figure 2.1 Average loss ratio of a portfolio of nine contracts 

     Source:  Bühlmann and Gisler (2005). 

  

The 75 per cent average loss proportion is the make back premium for 

covering claims whereas 25 per cent of the average loss ratio of insurance costs 

is needed for administrative costs as well as for various charges. Estimates of 

new claim costs for each policyholder had to be determined for the insurance 

policy for the next immediate period. In studying these agreements on the basis 



 

19 

 

 

of their individual loss ratios over the last five years, two deductions can be 

made.  

a) The variations observed in the loss ratios among the policyholders or 

individual contracts are completely as a result of the randomness of the 

claims and not due to the inherent different risk profiles of each these 

of contracts. This deduction makes no contract better or risky than the 

other. Hence, the best estimated loss ratio that can be made for the next 

immediate period for a given contract, is the 75 per cent average loss 

proportion. Therefore, a change to the current premium level of 

individual contracts is unnecessary. 

 

b) The variations in the measured ratios of losses are not due to the 

random nature of claims but are systematic. The differences are due to 

individual risk profiles fluctuating over the agreements. This deduction 

makes some contracts better than others. Hence premiums for the first 

three contracts ought to be reduced by one-third of the 75 per cent 

average loss proportion, while the last three ought to be increased by 

one-third of the 75 per cent average loss proportion. The authors also 

considered the ratios of annual losses over the five-year period 

independently for the small, medium and large contracts for an 

increasing examination. 

 

 Figure 2.2 shows the annual loss ratios of agreements 1, 4 and 7, each of 

which has a premium amount of 2 000 Swiss francs per year. 

 



 

20 

 

 

 

Figure 2.2 Yearly loss ratios of the small contracts 

   Source: Bühlmann and Gisler (2005). 

 

From Figure 2.2, the authors identified a high variability in the annual average 

loss proportions and wonder whether the differences observed in the contracts 

of the 5-year loss ratios, irregular fluctuations should be attributed as they 

were, this truth decreases trust in the average reliability as a long-term average.  

 

Figure 2.2 shows the annual loss ratios of contracts 2, 5 and 8, each of which 

has a premium amount of 2 000 Swiss francs per year. 

 

Figure 2.3 Yearly loss ratios of the medium-sized contracts 

     Source:  Bühlmann and Gisler (2005). 



 

21 

 

 

Figure 2.3 also shows a similar picture as figure 2.2. The annual fluctuation is 

still significant, but it is difficult to accept the causes of the differences 

between the proportions of the contract to be by random nature of claims. 

 

Figure 2.4 shows the yearly or annual loss ratios for the biggest contracts 3, 6 

and 9, all having 200000 Swiss francs as their annual premium volume. 

 

 

Figure 2.4 Yearly loss ratios of the large contracts 

  Source:  Bühlmann and Gisler (2005). 

 

 From Figure 2.4, it seems certain that the risk profiles of the three contracts 

show real and clear contrasts. Taking into account contracts 1, 2 and 3 shown 

in Figure 2.5, the average loss ratio for each contract over the course of 5 years 

is 50 per cent. 

Figure 2.5 demonstrates the annual loss proportions of the contracts 1, 2 and 3. 



 

22 

 

 

 

             Figure 2.5 Yearly loss ratios of contracts 1-3 

Source:  Bühlmann and Gisler (2005). 

 

Notwithstanding, consideration of the accessible data without further critical 

examination, the best prediction for each of the three agreements should not be 

the same. Thus, should a forecast of an average loss proportion of 50 per cent 

for the individual contracts or 75 per cent over the whole portfolio be utilized 

as an estimate? For contract 1, there is little certainty that its own measured 

average loss ratio is accurate as an estimate owing to the elevated variability in 

its annual loss ratios; we would be inclined to use the observed average loss 

ratio over a total of nine contracts ; 75 per cent as the estimate.   For contract 3, 

the information from its claim history proves to be reliable and one would feel 

favourably arranged to use as an estimate a projected value in the 

neighbourhood of its 50 per cent average loss ratio. In addition, our prediction 

for contract 2 would most probably be between agreements 1 and 3.  

 

 Rate makings that are not based on statistical or mathematical reasoning may 

inherently involve a great deal of bias in the operation that may cast doubt on 



 

23 

 

 

the insurers' forecasted and charged premiums. This is because such 

procedures do not adequately incorporate or account for variation or the 

different risk levels that are associated with each contract neither do the 

procedures determine the reliability of the data used for forecasting premiums. 

The solution to the issue is to determine and use suitable statistical and 

mathematical techniques to capture and describe the contract variation as well 

as the reliability of previous claim documents used in forecast premiums. The 

credibility theory, based on the literature of other authors, offers a sound 

mathematical and statistical foundation for finding a solution to the practical 

challenge of estimating reasonable insurance premiums and establishes 

guidelines for finding a solution to the arguments described in figure 2.1, 2.2, 

2.3, 2.4 and 2.5.The methodology employed allowed the researcher to estimate 

and compare the claim severity and frequency including premiums of 

policyholders of some selected leading marine insurance companies in Ghana. 

  

2.5      Credibility Theory  

Credibility model is a set of processes for estimating claim costs for claims of 

contracts in advance before they occurred. The process establishes claim cost  

risks utilizing two fixings: information from individual risks and that of the 

collateral or collective information, for instance, information from distinct 

sources considered relevant (Waters, 1994). Klugman and Panjer (1949) also 

explained the concept of credibility as a collection of quantitative tools 

enabling an insurance firm to assess the risk or group of hazards (modify 

future premiums based on prior experience). The authors added that if a 

policyholder's experience is reliably higher than assumed in the fundamental 



 

24 

 

 

manual rate, the policyholder may request a rate decrease. The authors 

explained that the insured may claim that the grand rate reflects the anticipated 

experience of the entire rating class and that the risks are definitely 

homogeneous. However, no rating technique is flawless or perfect, and risk 

levels are somewhat heterogeneous after all the endorsement criteria are 

shown. Thus, few policyholders could be considered as better hazards and 

preferred in the basic manual or collective rate over those projected rate. A 

comparable rationale suggests that a rate increase should be linked to a bad 

risk, but in this situation the policyholder will certainly not ask for a rate 

increase. Due to value thoughts and the economics of the situation, an 

expansion may be crucial in a period of time.  

 

 2.5.1      Application of Credibility theory 

The theory of credibility was originally developed in the mid-twentieth century 

by actuaries from North America for quite a while. Mowbray (1914) puts it 

into a practical used for computing premium and is presently known as the 

American theory of credibility. Whitney (1918) and different authors 

scrutinized a great deal the concept of the theory. The author suggested that 

insurance losses are of an erratic or random nature and subsequently the 

supposition of a "fixed effect" model was not valid. Moreover, the model 

further is met with the question of partial legitimacy as it was hard to decide on 

the assessment of weights for claim histories and after the World War II 

revolution; "Whitney's random impact model" came into being. 

 



 

25 

 

 

Bühlmann (1967) suggested credibility claim costs with a empirical structure 

that formalized the credibility theory assumption and norms and also proposed 

approaches to record Bühlmann's credibility variables with equal exposure 

units in the credibility display.  Bühlmann and Straub (1970) established the 

formula of credibility estimates with no underlying distribution with the 

ultimate objective that no previous distribution as an assumption of claim 

history and further clarified in his job the few aspects of using the equation of 

credibility estimates. Given the result that Bühlmann (1967) produced, there 

have been wide-ranging efforts by scientists over the past few decades to 

expand the model to increasingly expanding instances and gradually handle 

complex circumstances. Gerber and Jones (1975) for example, were dedicated 

to models of credibility with time dependence. Hachemeister (1975), Jewell 

(1975) used the approach and showed that exponential family distributions are 

the best straight choice for Bayesian credibility estimates obtained using the 

mean square error estimates. In addition, the real leap forward noted a large 

portion of the research moving to Jewell (1975) development of Bayesian 

estimation techniques. The generalized linear model approach was 

subsequently used by Nelder and Verall (1997) to create credibility features 

incorporating the model of random effects. In actuarial science, this has 

provided a broad range of apps, including claim reservation and risk rating.   

 

2.5.2      The Credibility Factor 

The factor of credibility is a percentage of how much reliance the risk firm is 

set up to bring on the individual's own data. It indicates how valid an item or 

estimate may be or may be suitable. Whitney (1918) gave a weighted average 



 

26 

 

 

of risk class claims and collective risk class claims for a future estimated claim 

known as a credibility claim cost.  The load associated with the risk groups 

under account was tended to be the credibility. Dorweiler (1934) and Bailey 

(1945) had an objective of achieving credibility for experience rating using 

risk-related weights, but Bühlmann (1967 suggested methods for processing 

Bühlmann's credibility variables with equivalent exposure units and equal time 

frames in the credibility model. Bühlmann and Straub (1970)  improved the 

method of Bühlmann by allowing an unbalanced unit presentation. The authors 

also displayed their model for evaluating credibility by considering and 

comparing weights for each risk depending on claim frequencies or the number 

of exposures for each of the time periods. Jewell (1973) tended to increase the 

credibility of a multivariate version by minimizing the mean square error 

(MSE). Hachemeister (1975) presented Bühlmann and Straub's credibility for 

regression extension where he tended to a credibility matrix. For experimental 

design models, Bühlmann and Jewell (1987) showed different levels or 

hierarchical credibility. Couret & Venter (2008) introduced multi-dimensional 

credibility for an anticipated danger as one factor is accepted as a linear 

mixture of more than one variable over a few time frames in perspective of the 

covariance conditions between variables, the Expected Process Variances 

(EPVs) and the Variance of Hypothetical Means (VHMs) for exposure units. 

 

An actuary is fascinated by a revised premium for the next instant time frame. 

All things considered, for the next time frame, the actuary must locate the 

premium for each risk using past claim histories. Credibility finds an arched 

combination with a weighted average between the class risk premium and the 



 

27 

 

 

collective risk premium (Finan, 2016). The question here is how much weight 

of credibility should be given to the risk premiums of the individual class and 

the collective risk premium?  This weight or factor that normally referred to as 

credibility factor lies in the ranges of zero to one. Furthermore, in credibility 

theory, zero credibility implies that the data provided is too little to be used for 

the rating of experience. The projected credibility premium is known as the 

credibility premium. The vital aspect of estimates of credibility is that it has a 

linear function of class mean and collateral or collective risk. The update of 

premium is often taken into consideration as more risk information are 

subsequently obtained.  

 

2.5.3      Bühlmann and Straub’s Credibility Theory  

The Bayesian concepts and processes were introduced into actuarial practiced 

in the early 1960s when Bühlmann Straub (1967, 1969) laid the basis for the 

empirical Bayes approach to legitimacy, which is still widely used in the 

insurance business. Within the actuarial field, Bayesian method is used in 

distinct areas. The minimum value of the quadratic loss function  is the 

credibility premiums such as the Bühlmann or Bühlmann-Straub credibility 

premiums and the others. Bühlmann (1967) displayed the strategy to 

credibility as a direct ability to assess and predict the anticipated claims for 

future periods, using previous   information of claim histories for class risk or 

collective class of risks. 

 

The Bühlmann model is simplest of the credibility models since it viably 

necessitates that the past model is the simplest of models of credibility since 



 

28 

 

 

past claims of a policyholder require independent and identically distributed 

parts compared to each prior year. Vital useful trouble with this supposition is 

that it does not take into account varieties of exposure or size  (Klugman, 

Panjer & Willmot, 2004). The accompanying inquiries by these authors further 

uncover the down to earth trouble of the Bühlmann model; for instance, What 

if the policyholder's first year claims experienced represented only a part of a 

year owing to an uncommon policyholder's anniversary? What if a shift in 

benefits happened partly through a policy year? For group insurance, what if 

the group size has altered over time? The Bühlmann and Straub model enables 

the inclusion of weights or volumes in the variance expressions which is a 

solution to the constraints of the Bühlmann model. This is a first generalization 

of the contract-related independent and identically distributed hypothesis, 

extending the field of application.  

     

In this study, the researchers’ interest is to apply Buhlmann-Straub model to 

determine and predict credibility premium of each marine insurance company 

in Ghana for the next immediate year ( 1)n  based on claim frequencies and 

severities given past records (1, 2, 3,…….., n) by combining the weighted 

average of the mean of the class risks or individual premiums and the manual 

or grand mean premium considering the risk profile of these companies. This 

will help insurance companies and other players including investors have a 

broader view of risks in terms of frequency or severity. This will also offer 

players and stakeholders the opportunity of comparing the cost of insurance 

among marine insurance companies in Ghana. 

 



 

29 

 

 

In summary the Review torched on these broad areas of the study such as; 

Insurance Claims and Premiums which is very key to risk bearers in 

determining the prospective premium for insureds. Another area considered 

was Underwriters’ Profitability for Assuming Risk of insureds which is one of 

the major reasons for insurance business which is always considered for setting 

rates. The review also considered rates for homogeneous and heterogeneous 

risks; the Class and Experience Ratings of Insurance including Credibility 

Theory and its applications. Finally the researcher considered Bühlmann and 

Straub Credibility model for estimating credibility frequency-severity premium 

which considered the variation or unstableness of claim frequencies and 

severities in estimating premium for risks. 

 

 

 

 

 

 

 

 

 

 

 



 

30 

 

 

CHAPTER THREE 

METHODOLOGY 

3.1      Introduction 

The chapter explains the methodology employed in achieving the objective of 

the study. In this chapter the type and sources of data as well as the size of data 

used in the evaluation were explained. The Bühlmann and Straub’s Credibility 

theory model employed in the analysis and estimation of credibility premiums 

as well as the assumptions, theorems and lemma underlying the model were 

also discussed. R-software was employed in analysing the claim data. The data 

and codes written by the researcher can be found at the appendix.  

 

3.2      Type and Sources of data 

The study depends considerably on secondary data acquired from the NIC, the 

body in Ghana mandated by way of law to make sure high quality 

administration, supervision, rules and the management of insurance 

commercial enterprises. The information obtained from the NIC was claimed 

histories of policyholders disclosed through the various non-life marine 

insurance companies operating in the country. Generally, secondary data most 

often benefit from cost savings and overcomes some problems in achieving a 

wider population connected with collecting the most significant information. 

The advantage of using secondary data is less cost of obtaining the data and 

less consuming time for accessing the data.  However, the use of secondary 

data exposes this research to the risk of replicating any defective data reported 

due to human or system errors. The resources and time available for this 

research mean cannot be eliminated completely. The researcher however, finds 



 

31 

 

 

comfort in using the data from the National Insurance Commission due to the 

reputation of the commission.  

 

3.3      Target Population 

The marine insurance claim histories produced by policyholders of non-life 

marine insurance companies in Ghana is the target population considered for 

this study. The claim histories include the claim amounts or sizes, the exposure 

units or policy counts (the number of individual or group of policyholders for 

policy years) and finally claim count by the exposure units in the year reported 

to the National Insurance Commission by the non-life insurance companies in 

Ghana.  

 

3.4      The Sample data 

Purposive and convenient sampling methods of non-life insurance companies 

were considered. There are 26 non-life insurance institutions licensed by the 

National Insurance Commission as at 2018, and out of this number 15 met the 

prescribed standards of working and reporting annual claims continuously for a 

minimum of six years to the National Insurance Commission. The Years of 

operation in the non-life marine insurance business in Ghana played a principal 

role in determining which insurance organizations to be covered in the sample 

for the research analysis. The proposed model employed for this research 

requires understanding about the variability existing in the claim quantities; 

within and across the individual insurance companies. The claim sizes of the 

marine insurance organizations in relation to the total non-life portfolio or 

policy count, the number of claims made by policyholders were additionally 



 

32 

 

 

considered. The researcher considered reported marine insurance claim 

histories to the National Insurance Commission from the period 2013 to 2018.   

In other to ensure that variability within the reported claim amount was largely 

due to the associated risks of the marine insurance portfolio, the researcher 

employed a technique to reduce the exogenous effects on the claim data 

reported by the insurance company by dividing the claim sizes or counts by 

their respective risk volumes. All the 15 sampled non-life marine insurance 

companies reported claim experiences consecutively for six accounting years 

to the NIC. 

  

 3.5      Bühlmann and Straub’s Credibility for Multiple Exposures 

The researcher estimated the basic average or risk premium per policy count or 

exposure units for policyholders of each marine insurance company. The 

researcher also used Bühlmann and Straub’s Credibility model in estimating 

credibility premium for policyholders using their annual claim costs as well as 

credibility premium based on credibility claim severities and credibility claim 

frequencies by assuming that risk profiles or parameters of claim frequencies 

follow a poison distribution.  

 

The Bühlmann and Straub’s Credibility model generally estimates premium 

for class risks of companies for the next immediate period ( 1n  ) given 

observed aggregate claims and their respective exposure units for the periods 

1,  2,  3, .,j n   for each risk class or company. A risk’s exposure to loss 

may vary from period to period and the number of years of observations may 

also differ over various risks. For these situations, let 
ijS  be the aggregate 



 

33 

 

 

claim amount in the year j  for a risk class or company i  such that for each of 

the observed aggregate claims of the company i  at period j , there is a 

corresponding weight 
ijW  representing exposures or exposure units in the year 

j  for risk i  and 
ijN  representing a number of claims in a year j  for risk or 

company i . The Bühlmann and Straub’s Credibility theory also known as 

Empirical Bayesian Credibility Theory (II) makes use of a risk parameter   

which is a random variable and does not observe any specific statistical 

distribution. In this estimation process, the risk classes stand for the individual 

companies ( 1,  2,  3, .,i I  ) which experiences or pay claims over n-years, (

1,  2,  3,  ..,  j n  ). 

 

 Given the fixed I  risk classes (companies), we use the notation 
ijY for claim 

cost and 
ijX for claim severities and 

ijF  for claim frequencies for the class risk 

(insurance company) i  in the year j , whilst 
ijW  is the associated weight for

ijY  

and 
ijF  but 

ijN  is the associated weights for
ijX . Our interest is to estimate and 

compare the credibility risk premiums for policyholders of each insurance 

company whose claims’ history and exposures units and claim counts are 

under study. Thus, the researcher seeks to factor severity and frequencies of 

claims made by policyholders on each of the insurance companies, which are 

the two major underlying risks that undermine the fair determination of 

premium for policyholders by insurance companies.  

 

In the calculation of premiums, the expected values of claim frequencies and 

severities are estimated or determined and modeled separately. Conditional 



 

34 

 

 

independence is an assumption which is frequently appropriate in many 

situations in insurance practice. We can now estimate credibility frequency’ 

1[F / ]n iE    and credibility severity; 1[X / ]n iE    separately and then multiply 

the two estimators to get the frequency-severity credibility estimate; 
1( )n iP   . 

 

3.6      Credibility Estimate of Claim Costs 

The assumptions we do make for Bühlmann and Straub’s Credibility model are 

most conveniently expressed in a manner which makes the model less 

restrictive than was the case for Bühlmann Credibility model itself and these 

assumptions are made not about the claims’ variables themselves, but about the 

variables representing aggregate gross claim sizes per unit weight, 
ijY . The

ijY  

can also be described as claim cost for company i at year j  scaled to account 

for the average claim size for policyholders of company i   made in year j .  

Thus,   

 
ij

ij

ij

S
Y

W


          

  1,  2,  3, . ,  ,    1,  2,  3, ,i I j n     

3.1 

        Where  

ijY   Aggregate claim per policy count of company i  in the year j  

ijS Aggregate claim for company i  in the year j  

ijW  Policy count for company i  in the year j  

 



 

35 

 

 

3.6.1      Assumptions of Model under Claim Cost  

(1) Given the risk profile i  associated with company i , the ,ijY  j 1, 2, 3,…., 

n are independent but not identically distributed either conditionally and 

unconditionally 

(2) [Y | ]ij iE 
 
does not depend on j 

(3) [Y | ]ij ij iW Var 
 
does not depend on j. 

(4) For different class risks or companies i d the pairs of claim costs (
,i ilY ) 

and ( ,Yd dk ) are independent 

(5) The risk parameters i  ( i 1, 2, 3,…….., I) are independently and 

identically distributed.  

Under the assumptions 1, 2, and 3 for company i  we defined, 

   E[Y /( ) [Y] | ]i i j iiim   E
 

 

2( ) Y | ] [ij iji is W Var  

3.2 

3.3 

The points, (2) and (3) under assumption (3.6.1) for Bühlmann and Straub 

showed that the claim sizes  per unit of exposure  from year to year for a 

particular company have constant mean but non constant variance. 

 Point (5) implies that [m( )] E[m( )]iE     and var[m( )] var[m( )]i    or 

2 2[s ( )] E[s ( )]iE      because none of the structure parameters, [m( )],iE   

var[m( )],i  and 2[s ( )]iE   depends on i . Thus, none of them is risk or 

company specific. 

Thus, for each risk class or company, the aggregate claims per policy count in 

any given year have mean ( )im  and variance 2 ( )is   where, 
i  is the risk 

parameter or profile for a risk class or an insurance company. To find the 



 

36 

 

 

credibility estimator, 
1( )n im   which is a linear function of the observations 

,Y ,  . . . ,Yi i inY1 2
with minimum mean square error from the pure premium; 

there is the need to choose 
0a ,a ,. . . ,ai i in1

 that will optimize the estimator of

1( )n im   given by 

 

2 011( ) ·· · i i i i

n

n i i ijn ji n ii i

j

m Y Y Y Y        0 1 2  

3.4  

Lemma 3.1 With the set-up and notation of Bühlmann and Straub’s credibility 

model given the company or risk i  and time periods 1,  2,  ..,   j n   we 

have 

(i) [Y /[Y [ )]]] (ij i iE E E m  
 

(ii) 
2[Y ( )] [ ( )]ij iiE m E m   

(iii) 
2[Y ] [ ( )]ij ikE Y E m   For j  k; 

(iv) 
2 2 2[Y ] [ ( )]

1
.[ ( )]ij

ij

E E s E m
W

    

In the case of Bühlmann Straub’s model, the constants 1 2, ,  . . . ,i i in    are not 

equal because 
ijY  ( 1,  2,  ,  j n  ) are not identically distributed.   

 Theorem 3.1 Let 1 2,Y ,  . . . ,Yi i inY  be a sequence of random variables, each of 

whose distribution depends on a risk parameter
i , and which given 

i  are 

independent with 

 2[Y |  ( )W ]ij ij ii Var s   3.5  

Then the mean square error of the estimator, 0

1

n

i ij ij

j

Y 


 from ( )im  given 

by   



 

37 

 

 

 

 
2

2

1 0

1

( ) m ( ) ( )
n

i n i i i ij ij

j

E m E m Y 



  
               

  

3.6  

 and is minimized and given by   

 
1 (( ) 1  ) [ ( )]n i i i im Z Y Z E m       3.7  

where 

iY Average claim or hypothetical mean for company i  

iZ Credibility factor for claim cost histories of company i  

[ ( )]E m  =Collective risk mean for claim costs 

To obtain the values of 0i and
ij , ( j 1, 2, ….., n) that  minimize the 

estimate in equation (3.4), there is the  need to take partial derivative of 

equation (3. 6)  and  equating them to zero and solving for 0i and 
ij

simultaneously by applying lemma (3.1) and theorem (3.1)  to obtain equation 

(3.7). 

 

 The estimate 1( )n im    in equation (3.3) is the credibility premium or claim 

cost for the company; i  for the next immediate period 1n . We now consider 

how to estimate the three structural parameters E[m( )] , 2E[s ( )]  and 

var[m( )]  using data from a collective of I (fixed) comparable risks or 

companies. 

 

3.6.2      Estimation of the Structural Parameters 

The formula for inhomogeneous credibility estimator involves the three 

structural parameters, E[m( )] , var[m( )]  and 2E[s ( )] . These three 



 

38 

 

 

parameters are also unknown and must be estimated from the data of the 

collective risks. Various articles are to be found in the actuarial literature about 

the estimation of such parameters example (Frees & Wang, 2005) and (Gisler, 

2005). However, Bühlmann’s approach of estimating parameters is convenient 

and widely used by researchers in estimating parameters since it provides an 

estimate that has small expected square deviations from its parameters. The 

Bühlmann’s credibility estimators are based on the variance measures named 

Expected Process Variance; 2E[s ( )]  and Variance of the Hypothetical 

Means; var[m( )] . These two variance measures are squared functions. 

According to literature and course work done on ―Actuarial Applications‖ by 

Bühlmann and Gisler (2005), the Variance of the Hypothesized Means must 

not be negative but in case it happened, the credibility factor is assumed to be 

zero for that risk classes.  

  

NOTE: the researcher adopted the statistical convention of using a clear point 

( ) in place of a subscript to indicate summation over that subscript. 

The credibility factors iZ  for the experience of each company can be estimated 

as, 

 

1

n

ij ij

n
i

i

W Y

Y
W








 

3.8  

 

 
2[ ( )]

var[m( )]

i
i

i

W
Z

E s
W











 

[0,1]iZ   

3.9  



 

39 

 

 

 

 

1[ ( )]

I

i i

i

W Y

Y E m
W







  


 

3.10  

 

where 

 

 

1

n

i ij

j

W W




 

1 1

I n

ij

i j

W W

 

  

  

The parameters are therefore estimated as  

 
2 21

[s ( )] (Y )
( 1)

I n

ij ij i

i j

E W Y
I n

  

  

3.11  

 
2 2

*
1 1 1 1

1 1 1 1
var[ ( )] { W (Y ) W (Y ) }

1 1

I n I n

ij ij ij ij i

i j i j

m Y Y
W In I n   

    
 
    

3.12  

where    

 
*

1

1
(1 )

1

I
i

i

i

W
W W

In W




 

 

  

  

 

3.7      Credibility Estimate of Claim Severities 

The assumptions made under claim costs for Bühlmann and Straub’s 

Credibility model remained the same under claim severities. 
ijX represents the 

aggregate claim experience per claim count for company i  over n years. The

ijX , also known as claim severity for company i at time j and is scaled to 

account for the average claim size by policyholders of company i  who made 

claim in year j.  



 

40 

 

 

Thus,  

 
,

ij

ij

ij

S
X

N


         

  1,  2,  3, . ,  ,    1,  2,  3, ,                                                 i I j n     

 

3.13 

Where   

ijX   Annual claim severity for company i  in year j   

ijS Aggregate claim size for company i  in year j  

ijN Claim count for company i  as a risk class in year j  

 

3.7.1      Assumptions of Model under Claim Severities  

(1) Given the risk profile i  associated with company i , the ,ijX  j 1, 2, 

3,…., n are independent but not identically distributed either conditionally and 

unconditionally 

(2) [ | ]ij iE X 
 
does not depend on j 

(3) [ | ]ij ij iN Var X  does not depend on j. 

(4) For different risks or companies i d the pairs of claim severities (
,i ilX ) 

and ( ,d dkX ) are independent 

(5) The risk parameters i  ( 1,  2,  3, ..,  j I  ) are independently and 

identically distributed.  

Under the assumptions 1, 2, and 3 for company i  we defined, 

   E[X /( ) | ]] [i i iji iX   E                                                                                   

2( ) [ | ] ii ij ijs N X  Var  

 3.14 

3.15 



 

41 

 

 

Point (2) and (3) under Assumption (3.7.1)  showed under this model that the 

claims per  claim count from year to year for a risk or particular company have 

constant mean but non constant variance. Point (5) of assumption (3.7.1) 

implies that [( )] E[ ( )]iE      and var[ ( )] var[ ( )]i     or 

2 2[s ( )] E[s ( )]iE      because none of the structure parameters, [ ( )],E    

var[ ( )],   and 2[s ( )]E   depends on i . Thus, none of them is risk or 

company specific. Thus, for each risk class or company, the aggregate claims 

per claim counts in any given year have mean ( )i  and variance 2 ( )is  where, 

i  is the risk parameter or profile for a risk class or non-life marine insurance 

company. 

 

To find an estimator, 1( )n i   which is the linear function of the observations 

, ,  . . . ,i i inX X X1 2
with minimum mean square error from the average claim 

severity; ( )i   there is the need to choose , ,  . . . ,i i ina a a0 1
 that will optimise 

the estimator of 1( )n i   given by; 

 
2 011( ) ·· · i i i i

n

n i i i ij in ji in

j

a a X a X a X a a X      0 1 2                                                
 3.16 

 Applying Lemma 3.1, Bühlmann and Straub’s credibility model given the 

company or risk i  and time periods, 1,  2,  ..,   j n   we have 

(i) [ /[ ] [ ( )] ]ij i iE X E X E   
 

(ii) 
2[ ( )] [ ( )]iij iE X E    

(iii) 
2[ ] [ ( )]ij ikE X X E    For j  k; 



 

42 

 

 

(iv) 
2 2 2[ ] [ ( )]

1
[ ( )].ij

ij

E X E s E
N

    

Under Bühlmann-Straub’s model, the constants 1 2, ,  . . . ,i i in    are not equal 

because 
ijX  ( 1,  2,  ,  j n  ) are not identically distributed.   

Applying Theorem 3.1 let 1 2, ,  . . . , nX X X be a sequence of random variables, 

each of whose distribution depends on a parameter
i , and which, given

i , 

are independent, with 

 2[ | ] ( ) Nij ij iVar X s    3.17 

Then the mean square error of the estimator, 0

1

n

ij ij

j

X 


 from ( )i  for 

which 

 

 
2

2

1 0

1

( ) ( ) ( )
n

i n i i i ij ij

j

E E X    



  
               

   

  

3.18 

is minimized and given by 

 
1 (( ) 1  ) [ ( )]n i i i iZ X Z E         3.19 

Where 

iX The average claim per claim count for company i  

iZ Credibility factor of claim history for company i

      [ ( )]E   Collective risk mean for claim costs

 To obtain the values of 0i and
ij , ( j 1, 2, ….., n) that will minimize the 

estimate in equation (3.16) and (3.18) there is the  need to take partial 

derivative of equation (3.18)  and  equating them to zero and solving for 0i



 

43 

 

 

and 
ij simultaneously by applying lemma (3.1) and theorem (3.1)  to obtain 

equation (3.19). 

 The estimate 1( )n i    in equation (3.11) is the credibility claim severity for 

company i  for the next immediate period 1n  

 Now, the estimation of the three structural parameters E[ ( )]  , 2E[s ( )]  and 

var[ ( )]   using data from a collective of I (fixed) comparable risks or 

companies. 

 

 3.7.2      Estimation of the Structural Parameters 

The formulation for the heterogeneous credibility estimator entails the three 

structural parameters, E[ ( )]  , var[ ( )]   and 2E[s ( )] . These three 

parameters are also unknown and must be estimated from the data of the 

collective. Bühlmann Staub’s approach of estimating parameters is convenient 

and widely used by researchers in estimating parameters since it provides an 

estimate that has small expected square deviations from their parameters.    

The Bühlmann’s credibility estimators are based on the variance measures 

named Expected Process Variance; 2E[s ( )]  for the claim severities and 

Variance of the Hypothetical Means var[ ( )]   for the claim severities. These 

two variance measures are squared functions and must not be negative else the 

credibility factor is assumed to be zero for the risk classes. 

 

To be able to determine credibility factors iZ  for the claim histories of each 

company. 



 

44 

 

 

 

1

n

i j ij

j

i

i

N X

X
N








 

  

3.20 

The factor 

 
2[ ( )]

var[m( )]

i
i

i

N
Z

E s
N









  

[0,1]iZ 
 

1[ ( )]

I

i i

i

N X

E
N








 


                                  

 

3.21 

 

 

 

 3.22 

we estimate the parameters as; 

2 21
[s ( )] N (X ) 3.23

( 1)

I n

ij ij i

i j

E X
I n

  

  

2 2

*
1 1 1 1

1 1 1 1
var[ ( )] { N (X x) N (X X ) } 3.24

1 1

I n I n

ij ij ij ij i

i j i jN In I n


   

    
 
  

 

Where 
*

1

1
(1 )

1

I
i

i

i

N
N N

In N




 

 

 . 

 

3.8      Credibility Estimate of Claim Frequencies 

Claim frequencies play a useful role in calculating a multivariate tariff or 

listing charges for policyholders of unique businesses in relation to Bonus— 

Malus structures. A common hypothesis is that the declare numbers can be 

represented as random variables with a conditional Poisson distribution 

(Buhlmann and Gisler, 2005). 

Consider a portfolio of risk groups and relevant observations, as shown below: 



 

45 

 

 

 
ij

ij

ij

N
F

W


 

3.25 

 

ijN Annual claims frequency for company i  in year j  

ijN   Number of claims of risk i  in year j  

ijW   Associated weight 

 

3.8.1      Model Assumptions under Claim Frequency 

Similar to the Bühlmann-Straub model we have; 

The risk i  is characterized by its individual risk profile i , which is itself the 

realization of a random variable
i , and we assume that: 

(i) Given 
i  the 

ijN  (j = 1, 2,….…,n) are independent and Poisson 

distributed with Poisson parameter, 

 ( )ij i ij iW   

 

3.26 

(ii) The pairs ( 1 1, N ), ( 2 2, N ) . . . are independent, and 1 , 2 ….., are 

independent and identicall  y distributed with E [
i ] = 1. 


 
reflects the frequency structure between the risks, whereas   is the overall 

frequency level for all class risks or companies. 

Given the model assumptions (3.8.1) we have; 

 [F / ]ij i iE  

 var[F / ]ij ij i iw    

3.27 

3.28 

                                                                                                          

The conditions of the Bühlmann-Straub model (Model Assumptions 3.6.1) are 

therefore satisfied for the claim frequencies with, 



 

46 

 

 

 ( )i i   

 
2s ( ) ( )i i i      

3.29 

3.30 

Just as it was the case under claim costs or severities for each risk class or 

company, the aggregate claim Number or count per exposure in any given year 

have mean ( )i  and variance ( )i   where 
i  is the risk parameter or profile 

for a risk class or an insurance company i . 

To find an estimator, 
1( )n i   which is the linear function of the observations 

,F ,  . . . ,Fi i inF1 2
with minimum mean square error from the ( )i  ; there is the 

need to choose , ,  . . . ,i i in 0 1
 that will optimize the estimator; 1( )n i   given 

by 

 
2 011( ) ·· · i i i i

n

n i i in i i ij in j

j

F F F F         0 1 2

 

3.31 

Applying   Lemma 3.1 in conjunction with the assumptions under claim 

frequency, With the set-up and notation of Bühlmann -Straub’s Credibility 

model with regard to company i  and time periods j 1, 2, …….., n  we have 

(i) [F ] [ ( )][F / ]ij i iE E E     
 

(ii) 
2[F ( )] [ ( )]i ijE E    

(iii) 
2[F ] [ ( )]ij ikE F E    For j  k; 

(iv) 
2 22 2[F ] [s ( )] [ ( )]

1 1
[ ( )]ij

ij ij

E E E E
W W

         

Under Bühlmann-Straub’s model, the constants 1 2, ,  . . . ,i i in    are not equal 

because 
ijF  ( j 1, 2, ……, n) are not conditionally or unconditionally 

identically distributed.   



 

47 

 

 

Applying Theorem 3.1 ,Let 1 2,F ,  . . . ,FnF be a sequence of random variables, 

each of whose distribution depends on a parameter
i , and which, given

i ,are 

independent, with 

 [F | ] ( W ,  j 1,  . . . ) ,iij ijVar n  

 

3.32 

Then the mean square error of the estimator 1 0

1

( )
n

n i ij ij

j

F  



   of ( )i 

given by 

 

 
2

2

1 0

1

( ) ( ) ( )
n

i n i i i ij ij

j

E E F    



  
               


 

3.33 

is minimized and given by 

 
1( ) (1 )n i i i iZ F Z     

 

3.34 

where 

iF The average claim frequency for company i                 

iZ Credibility factor of claim frequencies for company i  

 The collective mean for claim frequencies  

 

1
( )

n

i j ij

j

i i

i

W F

F
W






  



 

1

I

i

i

wi F

w











 

i
i

i

w
Z

w k






  

[0,1]iZ 

 

3.35 

 

 

3.36 

 

3.37 

   

where 

 
2

[ ] 1

var[ ] var[ ] var[ ]

E
k

 

  


  

  
 

 



 

48 

 

 

To obtain the values of 
0i and

ij , ( j 1, 2, ….., n) that minimized the 

estimate in equation (3.31), there is the  need to take partial derivative of 

equation (3.33) with respect to 0i and 
ij  equating the partial derivative to 

zero and solving for 
0i and 

ij simultaneously by applying lemma (3.1) and 

theorem (3.1)  to obtain equation (3.34). The estimate 
1( )n i    in equation 

(3.34) is the credibility claim frequency for company i  for the next immediate 

period 1n

 
Now considering how to estimate the three structural parameters and 

var[ ( )]i   using data from a collective of I (fixed) comparable risks or 

companies. 

 

3.8.2      Estimation of the Structural Parameter 

The formula for inhomogeneous credibility estimator under poison model also 

involves the three structural parameters; 

  Properties  

(1)
 
E[ ( )] [ ( )]E        

 (2) 2[s ( )] [ ( )]E E     
 

 (3) 2var[ ( )] var[ ( )] var[ ]i i i         

These three parameters under the properties above are also unknown and must 

be estimated from the data of the class risks using the Bühlmann’s parameter 

estimation approach. 

 
2 2 2

1

1
var[ ( )] var[ ] ( (F F) )

I

i

iI
  



    
 

3.38 

where  
1

1 I

i

i

F F
I 

   



 

49 

 

 

3.9      Credibility Frequency-Severity Estimates 

Premium estimation in terms of claim frequency and severity help factor the 

risk components that directly or indirectly affect the investment position of 

insurance companies, claim reserves and also help reduce adverse selection in 

insurance. According to Gisler (2005) factorization or product of claim 

frequency and claim severity is allowable. Based on this principle the 

credibility frequency-severity premium or claim count is estimated. 

Thus, 

 
1 1 1( ) ( ). ( )n i n i n iP       

 

3.39 

where 

1( )n iP    the credibility frequency-severity estimate for company i  for the 

next period 1n  obtained as the product of the credibility frequency and 

severity all for the period 1n . 

1( )n i   Credibility frequency for company i  in period 1n . 

1( )n i   Credibility severity of company i  in the period  1n .  

 

 

 

 

 

 

 

 

 



 

50 

 

 

CHAPTER FOUR 

DATA ANALYSIS 

 4.1      Introduction 

This chapter discusses and explains the descriptive analysis of claim sizes, 

policy counts and claim counts including estimation of structural parameters 

under claim frequencies and claim severities and subsequently estimate 

credibility weights and premiums (claim or loss cost) including credibility 

frequencies and severities and by extension credibility frequency-severity 

premiums for policyholders of each non-life marine insurance company. The 

researcher under this chapter presents bar charts of estimated premiums 

showing the extent of variation in the pure premiums of individual class risks 

and also sort to compare the three different estimated premiums(the pure 

premiums, credibility risk premiums and credibility frequency-severity 

premiums) and thereby projecting the importance of estimating premiums 

based on claim frequencies and severities since this incorporated inherent risks 

both in terms of the claim frequencies and severities by policyholders who 

experience these risks and to whom obligation have to be met by insurers. 

  

4.2      The Sample Period 

The greater facts we have got as a result of an increase in the sample period for 

the relevant risks, the greater emphasis is placed on the precise hazard. In this 

study, the risk studied is the claim sizes, policy counts and claim counts made 

by marine insurance policyholders on marine insurers in Ghana. The number 

of relevant risk classes used in this analysis is 15 non- life marine insurance 

businesses that had reported claim history to the National insurance 



 

51 

 

 

commission in Ghana. Ghana's economy recorded notably low and strong 

inflationary figures, interbank interest rate and exchange rates for the period 

2013 to 2018 hence the outcome of this research will not be much affected by 

these external economic factors.  

 

4.3      Descriptive Analysis of the Claim History 

The sample information collected from the National Insurance Commission for 

the period 2013 to 2018 include claim sizes, claim counts and policy counts for 

policyholders of 15 marine insurance enterprises in Ghana. The descriptive 

assessment of sample information, claim size, policy count and claim count are 

provided in Tables 4.1, 4.2 and 4.3 respectively. 

 

Table 4.1 displayed the mean, maximum and minimum observation, the 

standard deviation of claim counts data of each insurance company. Skewness 

and kurtosis, which are indications of symmetry and peakness respectively of 

the distribution of the reported claim size data for the companies, are also 

displayed. 

 

 

 

 

 

 

 

 



 

52 

 

 

Table 4.1:  Descriptive Statistics of Claim Sizes for the Non-Life Insurance 

Companies 

Company Mean Max. Min.   Std. sk'nes Kurt. 

Activa Int. Insurance 402517 582619 121728 163371 -0.555 -1.232 

Allianz Insurance  349172 1829787 24704 725733 1.358 -0.089 

Donewell 68563.5 139653 1158 49424.6 0.040 -1.640 

Enterprise Insurance  279881 374831 212357 61554.6 0.267 -1.661 

Ghana Union Assurr.  302029 393340 206921 90239.3 -0.020 -2.249 

Glico General Ins.  188107 717007 25818 263361 1.264 -0.240 

Hollard Insurance  220037 595226 30414 207852 0.757 -1.042 

NSIA Ghana Ins.  58656.7 133006 2002 55805.5 0.299 -1.964 

Phoenix Insurance  22135.7 52212 8693 19070 0.614 -1.719 

Provident Ins. 19900 75561 3000 27671.6 1.272 -0.224 

Quality Insurance  39416.3 65972 6597 27390 -0.052 -2.204 

RegencyNem Ins.  558878 921565 24221 411294 -0.496 -1.952 

SIC Insurance  1815258 4437325 311473 1536010 0.578 -1.354 

Star Assurance  244300 427874 2023 171708 -0.123 -1.885 

Unique Insurance  38604.3 89320 9131 36539.6 0.463 -1.939 

From Table 4.1, SIC Insurance Company Limited shows the highest average of 

GHC1, 815, 257.7 whilst Provident Insurance Company Limited had GHC19, 

900 as the lowest average claim size. The maximum claim size of GHC4, 437, 

325 is recorded by SIC Insurance Company Limited. The risk classes with a 

positive coefficient of skewness show that the distributions of their 

observations are skewed to the right whilst those with negative skewness have 

most of their observations skewed to the left. However, the negative kurtosis 



 

53 

 

 

depicted a platykurtic distribution for the claim sizes history of the risk classes 

or non-life marine insurance companies. 

 

Table 4.2 displayed the summary statistics for the policyholders of each Non-

life marine insurance company. It presents the mean, the maximum and 

minimum observations, the standard deviation, including skewness and 

kurtosis that depict the symmetry and peakness of the policy count data 

respectively for each risk class or non-life marine insurance company. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

54 

 

 

Table 4.2:  Descriptive Statistics of The policy Count for the Non-Life 

Insurance Companies 

Company Mean Max. Min. Std. sk'nes Kurt. 

Activa Int. Insurance 108.333 158 72 30.982 0.371 -1.522 

Allianz Insurance  63.5 154 30 45.654 1.19 -0.357 

Donewell 25.833 48 11 13.906 0.378 -1.598 

Enterprise Insurance  1647.67 2702 1163 561.28 0.898 -0.748 

Ghana Union Assurr. 166.333 225 54 63.108 -0.691 -1.105 

Glico General Ins.  135.833 163 93 34.354 -0.399 -1.999 

Hollard Insurance  126.333 207 43 84.455 -0.003 -2.298 

NSIA Ghana Ins. 28.667 54 13 14.828 0.532 -1.28 

Phoenix Insurance  106.833 202 54 59.647 0.597 -1.665 

Provident Insurance  39.333 82 24 21.538 1.182 -0.357 

Quality Insurance  23.167 47 3 15.613 0.204 -1.587 

RegencyNem Ins.  54.667 145 11 51.644 0.667 -1.255 

SIC Insurance  1297.5 1896 790 458.35 0.087 -2.075 

Star Assurance  177.333 222 85 54.021 -0.695 -1.394 

Unique Insurance  79.833 114 24 31.928 -0.638 -1.176 

Table 4.2 showed that, among the sampled Insurance companies, Star 

Assurance Company Limited shows the highest average of 177.333 policy 

count whilst Quality Insurance Company Limited has recorded 23.167 as the 

lowest average policy count. The maximum policy count, 2702 is recorded by 

Enterprise Insurance Company Limited whilst Quality Insurance Company 

Limited recorded 3 as the lowest minimum value among all the risk classes or 

non-life marine insurance companies. Risk classes with a positive coefficient 



 

55 

 

 

of skewness suggest that their observations are skewed to the right whilst those 

with negative skewness have their observations skewed to the left. The 

negative kurtosis for all risk classes depicted a platykurtic distribution for the 

policy counts for all the marine insurance companies. 

 

Table 4.3 displayed the descriptive statistics; the mean, maximum and 

minimum observations, the standard deviation, for claim counts data for 

insureds of each of the marine insurance companies. The symmetry and 

peakness of the distributions of the claim count given by the values of 

skewness and kurtosis respectively are also displayed in Table 4.3. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

56 

 

 

Table 4.3: Descriptive Statistics of Claim Counts for the Non-Life Insurance 

Companies 

Company Mean Max. Min. Std. sk'nes Kurt. 

Activa Int. Insurance 9 25 1 8.809 0.79 -1.002 

Allianz Insurance  1.333 2 1 0.516 0.538 -1.958 

Donewell 1.5 2 1 0.548 0 -2.306 

Enterprise Insurance  138 154 121 13.416 -0.097 -1.997 

Ghana Union Assurance  8.167 11 1 3.764 -0.989 -0.693 

Hollard Insurance  5.5 8 4 1.643 0.451 -1.757 

NSIA Ghana Insurance  2.667 4 2 0.817 0.476 -1.583 

Phoenix Insurance  2.333 5 1 1.751 0.51 -1.799 

Provident Insurance  1.667 3 1 0.817 0.476 -1.583 

Quality Insurance  2.167 4 1 1.169 0.371 -1.618 

RegencyNem Ins.  Ghana  8 18 1 7.043 0.192 -1.932 

SIC Insurance  28.333 117 3 44.889 1.189 -0.413 

Star Assurance  3.167 4 1 1.329 -0.67 -1.621 

Unique Insurance  1.833 4 1 1.169 0.881 -0.904 

From Table 4.3, Enterprise Insurance Company Limited recorded the highest 

average of 138 claim count whilst Allianz Insurance Company Limited 

recorded 1.333 as the lowest average claim count. The maximum claim count, 

154 claim count is recorded by Enterprise Insurance Company Limited. The 

risk classes with a negative coefficient of skewness show that their 

observations are skewed to the left whilst those risk classes with a positive 

coefficient of skewness are positively skewed. The negative kurtosis of the 



 

57 

 

 

claim counts of the risk classes depicted a platykurtic distribution for the claim 

counts of the risk classes or marine insurance companies. 

 

4.4      Buhlmann-Straub Credibility Parameter Estimates 

The Buhlmann and Straub credibility model made no assumption about any 

type of statistical distribution for the risk parameters ( i ) for claim cost and 

claim severity as well as claim frequency. It however uses the linear 

combination of the experience rating and class rating shown by equation (3.7), 

(3.19) and (3.34) with credibility factors; iZ  for the claim cost, Severity and 

Frequency gave by equation (3.9), (3.21) and (3.37) respectively. 

 

4.4.1      Variability within Class Risks 

Two elements appear crucial in finding proper balance between collective class 

rating and individual class ratings: How homogeneous are the class risks? If all 

of the risks in a category are equal and have identical expected price for losses 

then the class rating is preferred. Thus, using class rate known as collective or 

collateral risk means recorded under this research for claim costs as [ ( )]E m   

and claim severities as [ ( )]E    and for claim frequencies as . On the other 

side, if there is significant variation within the anticipated results of 

risks within the class, then the individual experience rating has to receive 

distinctly more weight. Each risk in the class has its own individual risk mean 

called its hypothetical means recorded for claim costs as ( )im   claim severities 

as ( )i  and claim frequencies as ( )i   respectively. The Variance of the 

Hypothetical Means (VHM), recorded for claim costs as var[ ( )]m  , claim 



 

58 

 

 

severities as var[ ( )]   and claim frequencies as var[ ( )]i  respectively 

across risk classes in the class are statistical measures for the homogeneity or 

heterogeneity, within the class. 

 

A small variance of the Hypothetical means (VHM) indicates more class 

homogeneity and consequently, more weight for the class rate. A large VHM 

indicates more class heterogeneity and consequently, less weight going to the 

class rate. 

 

Secondly, how much variation (Process Variance) represented as 2 ( )is   is 

there in an individual risk loss experienced? If there is a large amount of 

variation expected in the actual losses experienced by individual class risks, 

then the actual losses observed as 
ijY , 

ijX  or 
ijF are  far from their expected 

values; ( )im  , ( )i  and ( )i   respectively and not very useful for individual 

experience rating. Therefore, less weight should be allocated to the individual 

experience rating in this situation. That is, the process variance (PV), which is 

the variance of a risk's random experience of its expected value, is a measure 

of variability in the losses experienced by individual risk. The Expected Value 

of the Process Variance (EPV) which is the average value of the process 

variance over the entire class of risks denoted under this study as 2[ ( )]E s  . 

The smaller the process variances of the risks of a class ( 2 ( )is  ) the smaller 

the Expected value of the Process Variance; 2[s ( )]E   and the higher the 

credibility factor or weight of the individual risk experience or risk means 

compared to the collective risk mean of the class ratings. 



 

59 

 

 

The result of experience rating implies a100% credibility to the means of 

individual risk classes or companies. This could be the case if the expected 

process variance within all risk is very small and the variance of the 

hypothetical means between the risk classes or the companies is large. The 

opposite, which is the class rating, implies 100% credibility to the collective 

risk means or 0% credibility to the individual class risk means. 

 

Figure 4.1 displayed the annual risk means or annual claim costs the period 

(2013-2018) for the individual risks or companies including the collective risk 

mean.   

 

Figure 4.1: Bar Chart of Annual Claim cost for Marine Insurance Companies. 

 

 Figure 4.1 shows the variation that existed in the annual claim cost for 

policyholders of risk classes from the collective risk mean. RegencyNem 

Insurance Company Limited, Hollard Insurance Company Limited, Allianz 



 

60 

 

 

Insurance Company Limited and Activa Insurance Company Limited have 

shown that there exists a wide variation in their annual claim costs. This 

variation is also observed in class risks below the collective risk means. The 

insurance companies cannot set the same premium nor allow policyholders to 

pay the same premium since others may have a high tendency of making 

claim. 

 

Table 4.4 displayed total policy count, the weight of average claim cost or 

experience rating and the credibility premium for the year, 2019 for 

policyholders of the non-life marine insurance companies. Whilst the estimated 

structure parameters of the claim cost are presented in Table 4.5. 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

61 

 

 

Table 4.4: Policy Count and Credibility Factors and Estimates for Claim Cost 

for the Marine Insurance Companies 

Marine Insurance company  iW    ( )im    iZ   1( )n im    

Activa Int. Insurance  650 3715.54  0.169 1567.248 

Allianz Insurance  381 5498.77 0.107 1595.708 

Donewell 

 

155 2654.07 0.046 1199.711 

Enterprise Insurance  9886 169.865 0.756 403.6203 

Ghana Union Assurance  998 1815.8 0.239 1292.788 

Glico General Insurance  815 1384.84 0.204 1181.078 

Hollard Insurance  758 1741.72 0.192 1246.722 

NSIA Ghana Insurance  172 2046.16 0.051 1175.936 

Phoenix Insurance  641 207.198 0.168 974.5657 

Provident Insurance  236 505.932 0.069 1085.988 

Quality Insurance  139 1701.42 0.042 1152.887 

RegencyNem Insurance  328 10223.4 0.093 1977.841 

SIC Insurance  7785 1399.04 0.710 1320.609 

Star Assurance  1064 1377.63 0.250 1191.212 

Unique Insurance  479 483.562 0.131 1044.605 

 

 

Table 4.5: Estimated Structure Parameters for Annual Claim Cost  

 

 

In Table 4.4, the total policy counts ( iW  ), credibility weights ( iZ ), average 

claim cost; m( )i and the Credibility claim costs or premiu