_] I I Strathmore UNIVERSITY Loss Distributions for Motor Insurance Claim Severity in Kenya Nyairo Frankline Okindo- 101428 Submitted in partial fulfillment of the requirements for the Degree of Bachelor of Business Science in Actuarial Science at Strathmore University Strathmore Institute of Mathematical Sciences Strathmore University Nairobi, Kenya February 2021 Tltis Research Project is available for Library use on the understanding that it is copyright mate1ial and that no quotation from the Research Project may be published without proper acknowledgement. l l I J DECLARATION I declare that this work has not been previously submitted and approved for the award of a degree by this or any other University. To the best of my knowledge and belief, the Research Project contains no material previously published or written by another person except where due reference is made in the Research Project itself ©No part of this Research Project may be reproduced without the petmission of the author and Strathmore University Nyairo Frankline Okindo . .. ..... ... .. .. · · ···· ·~· ·· .. ............. . [Signature] Oq- o2-2o21 .. . ... .. . . . . . . . .. ..... .. .... .... ......................... ... [Date] This Research Project has been submitted for examination with my approval as the Supervisor. ~~~b~~ .. ?. ... ~ ... ~ .. :;:;: ... ~ ....... / ...... ..... [Signature] ~~=ore:J!t:J:~=c~·~~~~::te] Strathmore University ii TABLE OF CONTENTS DECLARATION ....... .... .... ..... .. .... .......... .. .... ....... .... .............. ........ .. .. ... ... ...... ... .. .. ... ........ .ii TABLE OF CONTENTS ... ... .... .... ... ........... ..... .. ......... ....... ............ ...... .. .. ..... .. ..... .. .. .... ... iii CHAPTER ONE: INTRODUCTION ........ .... ... ..... .. .. ..... ......... .. ... ... ... .. ....... .... ....... ... .... 1 1.1 Background Information ........... .. .... ... .. ... .. ... .... .... ...... ... ......... .. ...... .. .. ........ ..... ...... 1 1.2 Problem Statement ....... .. .. .. ... ... .. .... .... ... ...... ..... ....... .... .... ... .. ...... ........... ........ .. ... ... .4 1.3 Research Objectives and Questions .... ... ... ......... .. .. ...... ... .. ... .... ... ... ..... ....... ..... .. .... 5 1.4 Significance of the Resea1·ch ... .... .... .. .. .... .. ......... .. .... .. ... ...... .. ... ... .. ....... ........... .... ... 5 CHAPTER TWO: LITERATURE REVIEW .... ..... .... .. .. .... ............ ....... .. ..................... 7 2.1 Introduction ...... ....... .... ............... .... .. ........... .. ..... ........ ..... ... .. ................ .... ........ .. .. .. 7 2.2 Maximum Likelihood Estimation .... ..... .. .. .. .. .. .......... .. ...... .... .. ...... .. .... ......... .... .... . 9 2.3 Standard Continuous Distributions ... .. .... ...... ........ .. .... .. ... .. .. ........... ..... .. ... .... .. .... . 9 2.4 Goodness-of-fit Test ... ... .. ..... .. ..... ... ....... .......... .... .... .. .. .. .. .. ... ....... ............. ... .... ..... 15 2.5 Model Selection Criteria ... .... .. ........ .. ............ ... .... .... ... ... .. .... ..... .. ......... .. ..... ...... ... 17 CHAPTER THREE: METHODOLOGY. ........ ............. .. ... .... .. ....... .. ........ .... ....... .. ..... 19 3.1 Research Design ... .......... .... .......... .. ..... ... ..... .... ... .. ... .. .. ... .. .. .. .......... ... .. .. .... .. .. .. .. .... 19 3.2 Population and Sampling .. ... ... ..... ..... .... .. ...... .... .......... ... ...... .. .. ... ............ ....... ... ... 19 3.3 Data Collection ........ .. ....... ................. ................. ..... .... .. ..... ............ .. .. ......... .. .. .. ... 19 3.4 Data Analysis ....... ........ ....... .... .. .... ..... .. ..... ....... .. ... .. ..... .. .. .. ........ .... ....... ... ..... .. ... ... 20 CHAPTER FOUR: DATA ANALYSIS .. .. ....... .. .. ..... ... ...... ......... ... .. .. .. .. ...... ..... .. ... ..... . 27 4.1 Intr·oduction ..... .... ..... ....... .. ... ... .. .. ... ... .. .. ... ... .. .... ..... .. ..... ........ .. .. ...... ....... ..... .. .. ... .. 27 4.2 Descl'iptive Statistics ... ...... .. .... .......... .... .. .... .. ...... .... ... ... ..... .. .... ... ... ........ ...... ........ 27 4.3 Parameters Estimation .. ... .. ..... ... .. ....... .. ... .. .......... ....... ..... .. .. .. .... ....... ... .. ... .. .. ... .... 31 4.4 Goodness-of-Fit Test .. ...... .. ..... ........ ........ ...... ... .... .. ... ........... .... .. ......... ... ..... ... ...... 33 J iii 4.5 Information Criteria .......... .... ............... ...... ..................................... ..... ............... 34 4.6 Summary ........ .... ..................................... ........... ........... ....... ................................. 35 CHAPTER FIVE: CONCLUSION ........................................... ........... .... .......... .......... 37 5.1 Introduction ......... ......................... ...... .... .................... ....... ................................. .. 37 5.2 Conclusion ............................... ............................... ............................................... 37 5.3 Recommendations ........................................................................ ............ ..... .. ..... 38 5.4 Limitations of the Study ............................... ............. ................................. .. ........ 38 5.5 Suggestions for Further Research ....................................................................... 39 REFERENCES ....... .. .......................................................................................... .... ........ 40 APPENDICES ......... ...... ..... .. ..... ...... .................. .... ... ............. ........ ......... .... .... .. ... ...... .... . 42 Appendix A: Fitted Distributions for Motor Commercial ........................... ...... ... .42 Appendix B: Fitted Distributions for Motor Private ..................... ...... ............ ....... 44 J iv l l I ~ I J J J CHAPTER ONE: INTRODUCTION This chapter begins with the backgrolmd infmmation of the study in section 1.1 by explaining the key concepts, main developments, and conceptualization of the study. In section 1.2, the problem statement of the study is elaborated at length. Section 1.3 contains the research objectives and questions. Tllis chapter concludes with the significance of the research in section 1.4. 1.1 Baclcground Information 1.1.1 Key Concepts Claim: A legal application made by a policyholder to an insurer for indemnity covered under the policy agreement. Distribution: A function that shows the possible values for a variable and how often they occur. Loss: The basis of a claim for damages under the tenns of an insurance policy. Severity: The cost of a claim. 1.1.2 Main Developments and Conceptualization of Study The insurance industry is one of the oldest industries. Insurance companies exist to provide indemnity and make profits since insurance is a business like any other. The advancement of the insurance market is compelled by the prevailing interest of the public for cover against different forms of risks of tmacceptable arbitrary incidents with a considerable financial effect (Omari et al. , 2018). A policyholder is supposed to pay a premium and make a claim when a certain event occurs witllin a given period as per the policy. The insurer is tl1en obliged to settle the claim, and this is referred to as loss. Insurers are keen with the results oftl1e random outcome of claims instead of the existence of the claims. They are concerned with the loss rather than the circumstances that give rise to the loss (Achieng, 2010). The aggregate amount of claims in a given dmation is a measure that is vital to the operations of an insurance company. 1 J J A general insurance actuary of a company needs to understand different risk models comprising of the aggregate claim amount overdue in a ce11ain period. These models enlighten a company and allow it to decide on things such as anticipated profits, premiums to be charged, required reserves that will guarantee profitability with a high likelihood, and the effect of reinsurance and policy excess (Boland, 2006). Actuaries are tasked with the responsibility of developing cashflow models for insurance companies. TI1ese models are crucial since they are used to assist in the day-to-day work of insurers and to provide checks and controls on the business. An actuary may decide to develop a stochastic model and estimate one ofthe parameters e.g. claim size by assigning it a probability distribution. If the assigned probability distribution is not appropriate for the claim size parameter in the stochastic cash flow model, it will lead to adverse future experience by the insurance company due to parameter error which results in a model error. An actuary will, therefore, be concemed with objectively assigning an approp1iate probability distribution to the value of the claim size parameter. In the general insurance business, there is heightened concern in motor insurance because it involves the control of many risk events. These include fire, theft, third party bodily injmy and accidental damage to the vehicle. In most COlmtries, the motor insurance industiy is growing rapidly due to legislations that make motor insurance compulsory for all vehicles. The insurance industry is driven by data, and insurers engage a lot of analysts to comprehend claims data (Boland, 2006). The claims data contains among other things, the frequency and size of claims that a company has received within a given period. Based on the claims data, mathematical methods can be applied to model individual claims. The mathematical models are known as loss distributions. Loss distributions are vital in the insurance industry since they are used for many purposes which include: premium rating (deciding the premium rates to be paid by policyholders), reserving (determining the required amount of funds to be retained to offset the cost of claims), reviewing reinsurance anangements and testing for solvency (evaluating the insurer' s financial health). This explicitly highlights the importance of having a good estimate of an insurance company' s loss distributions. 2 l l J J A loss distribution is the associated probability distribution of a claim-size variable. The claim-size is a non-negative continuous random vruiable since the claim arising fi:om a covered incident cru1 be measured in the lowest unit of cmTency e.g. cents. Loss distributions are usually positively skewed and long-tailed. To model the size of insurance claims, it is pmdent to fit a continuous parametric claim-size distribution to a discrete sample of claim data. This involves employing a variety of parametric families of continuous distiibutions which include: Gamma, Lognormal, Exponential and Pareto distributions, among others. The gamma and lognonnal distributions are among the common distiibutions that have been applied for modelling claim seveiity. The exponential, Pareto, Weibull, and Burr distributions are also used to model claim sevetity. This is primarily because all these distributions are positively skewed. Omaii et al. (2018) suggested that a lognonnal distiibution is suitable to model claim severity based on a sample of the automobile pmifolio datasets obtained from the insurance Data package in R. Achieng (2010) concluded that the lognormal distiibution was a suitable model for the motor comprehensive policy claim seveiity of First Assurance Company Limited, Kenya. Nduwayezu (2016) fom1d out that the exponential distiibution is suitable to model insurance data; though he left the research open for fhrther studies to be carried out to detennine the distiibutions that are most suitable for each class of insurance. Most research papers have provided a good framework to use when modelling loss distributions for motor insurance claims severity. However, it can be noted that the statistical distributions suggested for modelling claims severity are general and not exhaustive since they are based on the sample data that was being used by the various researchers. In the Kenyan enviromnent, there are no specific statistical distributions that have been recommended to be used for modelling motor insurance claim severity based on the motor insurance claims data in Kenya. Tlus creates a need for research on suitable statistical distiibutions that can be used within the Kenyan environment as this will enhance the motor insurance industry: 3 I l 1 j J J 1.2 Problem Statement One of the major challenges that general insurers face is to precisely estimate the likely prospective claims experience and therefore charging suitable premiums and setting aside sufficient reserves (Omari et al., 2018). For these companies to overcome the challenge of accurately forecasting future claims experience, they need to have a good estimate of loss distributions. A good estimate of a loss distribution entails selecting a suitable statistical distribution that fits the claims data. Determining motor insurance claims distributions often comprises associating the value of claims with two elements: the occtmence of an accident and the claim ammmt in case of an accident (Frees & Valdez, 2008). Records from insurance databases show the following claim types: third-pa11y liability claims, and damage claims to the policyholder, comprising property damage, injury, theft, and fire . This, therefore, implies that for every accident, it is probable for multiple types of claims to be incurred; thus, increasing the claim severity of an insurance company for eve1y single accident. This creates a need for having good models ofloss distlibutions that will enable an insmer to plan accordingly to lower the probability of incurring such a loss and reducing the claim severity incurred. Motor insurance is the biggest segment out of the 14 distinctive classes of the non-life insurance market in Kenya accmmting for over KES 46 billion gross written premiums representing close to 35.8% of the entire non-life insurance market in Kenya in 2018. It is not surprising that motor insurance is a huge business in Kenya given that more than 7000 cars are imported monthly. This can be credited to the fact that motor insurance is compulsory in Kenya, and thus, for every new vehicle purchase in the country, a motor insurance policy is added to the existing motor insurance policies for the vehicles on the Kenyan roads. To tllis end, the huge role played by motor insurance to the Kenyan insurance industry and the economy at large cannot be overlooked. This implies that motor insurance companies need to have good models that will enable them to accurately forecast future claims expeiience and thus be able to set aside enough reserves. According to h1surance Outlook Report 2019/2020, East Africa by Deloitte, motor insurance in Kenya is one of the largest general insurance classes togetl1er with medical 4 l l l J J J J insurance. However, they are also among the top loss-making businesses. This could be partly attributed to the fact that motor insurance companies in Kenya do not have good models for loss distributions, and thus cannot be able to conectly forecast future claims experience. Tllis leads them to undergo huge losses because of failing to plan accordingly to lower the probability of incuning such losses. This research paper will , therefore, seek to address this gap in the Kenya motor instrrance indust:Iy by providing a good model of loss distribution for claim severity. This will, in tum, help insurers to precisely estimate prospective claims experience and thus plan accordingly to reduce their huge losses and the chances of them making such losses. 1.3 Research Objectives and Questions 1.3.1 Research Objectives The objective of tllis research is to determine the appropriate statistical distribution that fits claim severity data of motor insmance in Kenya and can be used to accurately forecast future claims experience. 1.3.2 Research Questions Throughout this research, the paper will aim to answer the following questions: 1. What is the most appropriate statistical distribution for claim severity? n . How well does tllis loss distribution fit the claims data? 1.4 Significance of the Research Tllis research will provide motor insurance companies in Kenya with the most suitable loss disu·ibution for claim severity. This will enable them to accmately forecast future claims experience and thus be able to correctly: rate premiums, reserve, review reinstrrance anangements and test for solvency. Given that risk associated with claim seve1ity would have been mininlised, motor insurance policyholders will consequently 5 l J pay reduced premiums. The research will enhance understanding of the complexity of the extensive volume of claims that is usually concealed in a large amount of data. 6 l l J J ) CHAPTER TWO: LITERATURE REVIEW This chapter discusses the theoretical and empirical framework. It begins with a brief introduction of various studies that have been previously carried out on claims data modelling in section 2.1. In section 2.2, the Maximum Likelihood Estimation is then presented as a method of estimating parameters. Subsequently in section 2.3, various continuous distributions are discussed. This chapter closes by presenting Kolmogorov­ Smimov and Anderson-Darling as goodness-of-fit tests in section 2.4, and Akaike Infonnation Criterion and Bayesian Information Criterion as model selection criteria in section 2.5. 2.1 Introduction A loss distribution is a mathematical method of modelling individual claims. It involves fitting statistical distributions to observed claims data and then testing for the goodness of fit. The fitted loss distributions can then be used to estimate probabilities. The main assumption in all the distributions in tllis study is that tl1e amount of a claim and its occurrence can be considered independently. Thus, a claim arises according to some elementmy model for incidents ensuing in time, then the claim amount is selected from a distribution representing the claim amatmt. Ignatov et al. (200 1) provided a statistical process of fitting a suitable model to claims data. 111e process begins with the selection of a family of distributions for the claims model and then estimating the parameters for the model. A selection criterion to determine tl1e appropriate distribution from the family of distributions should be specified. Finally, a goodness-of-fit test should be cruried out on the selected appropriate distribution. Achieng (201 0) modelled the claim sevetity motor comprehensive data of First Assurance Compru1y Limited, Kenya (June 2006- June 2007). The estimates for the parameters were obtained using the Maximum Likelihood Estimation method. The Akaike Infonnation Criteria and Quantile-Quantile plots were further utilised to cany out a goodness-of-fit test. The finding oftl1e study was that the lognonnal distribution was a suitable model for the claims data. 7 l l J J Mazviona and Chiduza (2013) used four distributions (Gamma, Pareto, Exponential and Lognonnal) to model the claims for a motor portfolio in Zimbabwe. Their study used Maximum Likelihood Estimation and Method of Moments Estimation to estimate parameters for the models. The Chi-Square and Kolmogorov-Smimov tests were used as goodness-of-fit tests for the models. The finding of their study was that the Lognonnal distribution is suitable for smaller claims while the Pareto distribution is appropriate for larger claims. Packova and Brebera (2015) used Gatmna, Weibull, Lognonnal and Pareto distributions to model data obtained from a Czech insurance company for compulsory motor third-party liability insurance. The Maximmn Likelihood Estimator method was used to estimate the parameters of the selected parametric distributions. They further used the Anderson­ Darling, Chi-Square and Kolmogorov-Smimov tests to determine whether the chosen distribution provides a good fit to the data. The finding of the study was that the Pareto distribution can be assumed to be a good model for the losses. Omari et al. (2018) modelled a sample of the automobile portfolio datasets obtained from the insurance Data package in R with variables; Auto Collision, data Car, and data Ohlsson used. They used the Maximmn Likelihood Estimation method to obtain parameter estimates for the fitted models. TI1e Anderson-Darling and Kolmogorov-Smirnov tests were then used as goodness-of-fit tests for the claim severity models. The Akaike Information Criterion and Bayesian Infonnation Ctiterion were further applied to choose between competing distributions. The finding of their study was that the lognonnal distribution provides a good model for claims sevetity on a shmt-tenn basis. The study recmmnended that for a long-tenn basis, insurers should adjust the distributions accordingly based on insurer-specific claims experience. To this end, this study seeks to apply the theoretical and empirical frameworks of past studies at1d extend it in the Kenyan economy. The aim is to fit a suitable loss distribution to the claim severity data for motor insurance companies using the company-specific data set. In line with the research objectives of this study, it is pmdent to explain the theoretical framework that will be applied throughout this paper. TI1is will include parameter 8 J J estimation, standard continuous distributions, goodness-of-fit test, and model selection criteria. These will then fonn the basis of the subsequent parts of the study that will eventually yield an appropriate loss distribution model. 2.2 Maximum Likelihood Estimation The method of maximum likelihood is generally considered as the best general method of finding estimators. Maximum likelihood estimators have excellent and nonnally simply detennined asymptotic prope1ties and so are especially good in the large-sample situation. The likelihood fi.mction of a random variable, X, is the probability (or probability density fi.mction) of observing what was observed given a hypothetical value of the parameter, e. The maximmn likelihood estimate (MLE) is the one that provides the highest probability (or probability density function), i.e. that maximises the likelihood fi.mction . 2.3 Standard Continuous Distributions The claim-size is a non-negative continuous random variable since the claim arising from a covered incident can be measured in the lowest unit of currency e.g. cents. To model the size of insurance claims, it is prudent to fit a continuous parametric claim-size dishibution to a discrete sample of claim data. Due to the infrequency of relatively large claims which are of concern, Boland (2006) suggested the use of relatively fat-tailed distributions. Kaas et al. (2008) stated that claim-size is best modelled using continuous distributions that are positively skewed and long-tailed. This is because very large claims occur at the upper­ right tails of the distribution. The following parametric families of continuous distributions will be considered: Exponential, Gamma, Lognormal, Pareto, Weibull, and Burr. In this section, the dish·ibution fimctions and probability density fimctions will be gi.Ven. 9 J J _j I u 2.3.1 Exponential Distribution The exponential dist1ibution is one of the elementary models for claim severity. A random vmiable X has the exponential distribution with parmneter A. > 0 if it has distribution function : F(X)=l- e-AX,X>O In that case, we write X- Exp (A.). The probability density function is: f(x) = A.e-.i!.x ,x > 0 Achieng (2010) used the exponential disttibution due to its heavy-tailed and highly skewed nature to model claim severity motor comprehensive data of First Assurance Company Limited, Kenya (June 2006 - June 2007). The distribution was not a good fit due to its low log-likelihood value and low-density value of its probability density fimction graphical plot. Mazviona and Chiduza (2013) used the exponential distribution to model a motor dataset. Their study fmmd out that this distiibution failed to fit the data very closely based on the critical value for the chi-square test and thus rejecting the null hypothesis. Omari et al. (2018) used exponential distribution to model an automobile dataset. The study rejected the null hypothesis for this distribution because it had the largest values among the distributions used for the Kolmogorov-Smimov and Anderson-Darling tests. 2.3.2 Gamma Distribution The random variable X has a gmnma distribution with pm·mneters a > 0 and .A > 0 if it has probability density fimction: x>O The parameter a changes the shape of the graph of the probability density function, and the parameter A. changes the x-scale. In that case, we write X- Ga (a, A.). 10 _l _l Achieng (2010) used the gamma distribution due to its heavy-tailed and highly skewed natme to model claim severity motor comprehensive data of First Assurance Company Limited, Kenya (Jtme 2006 - Jtme 2007). The study concluded that gamma distribution was not a suitable model for the claims data based on its Q-Q plot. Mazviona and Chiduza (2013) used the gamma distribution to model a motor dataset Their study found out that this distribution failed to fit the data vety closely based on the critical value for the chi-square test and thus rejecting the null hypothesis. Packova and Brebera (2015) used gamma distribution to model data obtained fimn a Czech insurance company for compulsory motor third-party liability insurance. The reason for using tllis distribution was because it is specifically applicable for modelling of claim severity. The study found out that the gamma distribution failed to be a suitable model for the losses based on the Anderson-Darling test value. Omari et al. (2018) used gamma distribution to model an automobile dataset. The study found out that the gamma distribution is a better model among the others based on the log­ likelihood value. 2.3.3 Lognormal Distribution X has a lognom1al disttibution if log X has a nonnal distribution. If X represents, for example, claim size and Y = log X has a nonnal distribution, then X is said to have a lognonnal distribution. When log X- N (/1 , <52 ), X- LN (fl, <5 2 ). The probability density function of the lognonnal distribution is defined by: 1 l(logX-JL) 2 f(X) = e 2 -rr- for 0 0 and A. > 0 if it has distribution fimction: A a F(X) = 1- C+x) ,x > o In that case, we write X- Pa (a, A.) . 12 l I 1 _] .1 The probability density function is given by: a.Aa f(X) = (il. + X)a+l ,X> 0 Mazviona and Chiduza (20 13) used the Pareto distribution to model a motor dataset. TI1eir study found out that this distribution fit the data very closely and based on the graphical plot of its probability density fimction, concluded that it provided the best fit for larger claims. The study further recommended the Pareto distribution because it does not undervalue the probabilities for larger claims. Packova and Brebera (20 15) used Pareto distribution to model data obtained from a Czech insurance company for compulsory motor third-party liability insurance. This was because this distribution is frequently used as a model for insurance losses required to obtain well­ fitted tails. The study concluded that the Pareto distribution is a good model for large claims based on the tests carried out. Omari et al. (2018) used Pareto distribution to model an automobile dataset. This distribution was used since it has been shown to sufficiently mimic the tail-behaviour of claims amount thereby providing a good fit. However, the study discarded the Pareto distribution since its values were extremely out ofrange based on the tests canied out. 2.3.5 WeibuU Distribution This is a very flexible distribution which can be used to model claim severity. It is a modification of the Pareto and exponential distributions usually with y < 1. A random variable X has a Wei bull distribution with parameters c > 0 and y > 0 if it has distribution fimction: F(x) = 1- exp( -cxY), x>O In that case, we write X - W (c, y). The probability density fimction of theW (c, y) disttibution is: f(x) = cyxY-l exp( -cxY), x>O 13 l l I l I J 1 1 1 I . J J I/ I Achieng (2010) used the Weibull distribution due to its heavy-tailed and highly skewed nature to model claim severity motor comprehensive data of First Assurance Company Limited, Kenya (June 2006 - June 2007). The study established that the Weibull distribution is not a suitable model for the claims data based on the tests carried out. Packova and Brebera (20 15) used Wei bull distribution to model data obtained from a Czech insurance company for compulsory motor third-party liability insurance. The reason for using tllis distribution was because it is specifically applicable for modelling of claim severity. The study found out that the Weibull distribution failed to be a good model for the losses based on tl1e Anderson-Darling test value. Omari et al . (2018) used Weibull distiibution to model an automobile dataset. The study discarded this distribution as the appropriate distribution since it failed to meet the selection criteria based on the tests canied out. 2.3.6 Burr Distribution The disn·ibution function of the Pareto Distribution Pa (a, y) is: A a F(x) = 1- (A.+ x)a ,x > 0 A further parameter y > 0 can be inn·oduced by setting: A a F(x) = 1- (il.+xY)a ,x > 0 This is the distribution ftmction of the transformed Pareto or BtuT distribution . The extra parameter provides additional flexibility when a fit to data is needed. The probability density ftmction is given by: Bmnecki et al. (2010) used the Burr distribution to model fire losses dataset. This distribution was chosen since it is a typical candidate for claim size distributions considered in application. The study failed to suggest the Blm distribution as a good model since it failed to pass the applied tests therein. 14 J J ~ I J It is worth to mention that while this distribution can be used to model claim size distributions, most sh1dies fail to utilize it among their chosen distributions. It would be interesting to conduct a sh1dy armmd the same to detennine the rationale behind the exclusion of this distribution in major sh1dies. 2.4 Goodness-of-fit Test Tllis refers to verifying whether a particular Joss distribution provides a good model for the observed claim amounts i.e. whether a model provides a "good fit" to the data. This involves detennining the quantitative "compatibility between the estimated theoretical distributions against the empitical distributions of the sample data" ( Omari et al., 20 18). TI1is enables one "to detennine whether the observed sample was drawn from a population that follows a particular probability distribution" (Dodge, 2008). Mylmg (2003) argued that even though a good fit is required, it is not an adequate requirement for one to conclude that one model provides a closer approximation to data than does another model simply because the former model fits the data better than the latter. A better fit (larger value of the maximized log-likelihood) simply places the model in a series of competing models for additional considerations such as goodness-of-fit tests. The Kolmogorov-Smimov and Anderson-Darling tests are used because they are suitable for performing an exact test on continuous distributions. 2.4.1 Kolmogorov-Smirnov Test The Kolmogorov-S1nirnov test is a nonparamettic goodness-of-fit test and is used to detennine whether an underlying probability distribution differs from a hypothesized disttibution. Consider an independent random sample (x1 , x2 , ... , Xn), a sample of size n with unknown distiibution function F(x) coming from a population with a specific and known distribution function F0 (x ). The hypothesis to test is as follows: H0 : F(x) = F0 (x) 15 J J j If F(x) is the empilical distribution function ofthe random sample, then the statistical test T11 is defined as the greatest vertical distance between F0 ( x) and F (x) : sup Tn = IF0 (x)- F(x)l X The decision rule is to reject H0 at the significance level a if T71 is greater than the value of the Kolmogorov table having for the parameters nand 1-a, which is denoted by t 11 i-a• i.e., if: 2.4.2 Anderson-Darling Test The Anderson-Darling test is a goodness-of-fit test which allows controlling the hypothesis that the distribution of a random variable observed in a sample follows a cettain theoretical distribution. Consider a random variable X, which follows a pruticular distribution, and has a distribution ftmction F0 (x; 8), where 8 is a parameter (or a set of parameters) that detennine, F0 • We ftnther assume 8 to be known. An observation of a sample of size n issued :fi·om the variable X gives a distribution function F(x). The Anderson-Darling statistic, denoted by A2, is then given by the weighted sum of the squared deviations F0 (x; 8)- F(x): Starting from the fact that A2 is a random variable that follows a certain distribution over the interval [0; +oo ], it is possible to test, for a significance level that is fixed a ptiori, whether F(x) is the realization of the random variable F0 (X; 8); that is, whether X follows the probability distribution with the distribution ftmction F0 (x; 8). 16 l l j J J The computation of A2 Statistic is as follows: Anange the observations x1 , x2 , ... , xn in the sample obtained from X in ascending order i .e., x1 < x2 < ··· < x,1 . The A2 is then computed as: where zi = F0 (xi; 8), (i = 1, 2, ... , n) The null hypothesis is rejected beyond the limiting values of A2 depending on the significance level based on the Anderson-Darling Test Table. 2.5 Model Selection Criteria An information ctiterion measures the quality of a model by analysing how well the model fits the data and the simplicity of the model. Information criteria are used to analyse different models that are fitted to the same data set. Ceteris paribus, a model with a smailer value is preferable to one with a larger value. 2.5.1 Akaike Information Criterion The Akaike Information Criterion (AIC) was formulated by and named after Akaike (1974). The AIC is an estimator of out-of-sample prediction error that yields a teclmique for model selection. AIC estimates the relative ammmt of information lost by a given model: the fewer information that is lost by a model, the better the model's quality. In a seiies of models, the most appropriate model that is selected is the one with the least AIC value. The AIC value for a model is calculated as follows: AIC = 2k- 2ln(L) where k is the munber of estimated parameters in the model and L is the maximmn value of the likelihood ftmction of the model. 17 l l I l J J 2.5.2 Bayesian Information Criterion l11e Bayesian Information Crite1ion (BIC) is also known as the Schwarz Infonnation Criterion (SIC). It is a criterion used to select models among a definite series of models. BIC was fonnulated by Schwarz (1978) and given a Bayesian argument for its adoption. It has been widely used for model selection and can be applied to any set of maximtm1- likelihood based models. The appropriate model that is preferred is one that has the lowest BIC value since it implies a lower penalty te1m. It is similar to the Akaike lnfonnation Criterion and it is based to some extent on the likelihood function . BIC introduces a larger penalty term than AIC. The BIC value for a model is calculated as follows: BIC = k ln(n)- Zln (L) where k is the number of estimated parameters in the model, n is the munber of observations and Lis the maximized value of the likelihood fimction of the model. 18 J CHAPTER THREE: METHODOLOGY This chapter outlines the methodological framework that will be used in the study. Section 3.1 discusses the design of the research; section 3.2 discusses the population and sample ofthe study while section 3.3 discusses the collection oftl1e study's data. Finally, section 3.4 explains the data analysis processes that will be cruTied out by the study. 3.1 Research Design This study adopts a quantitative method which is focusing on modelling an appropriate loss distribution for motor insurance claim severity in Kenya. The selected appropriate loss distribution will then be tested to check on its goodness-of-fit before being recommended as an appropriate model. The variable of interest is the claim size in the motor insurance industry. This study will use data for Kenya motor insurance companies from 2014 to 2018. The choice of this period is to make the study to be appropriate, relevant, reliable, ru1d applicable as possible. 3.2 Population and Sampling The focus of the study is motor insurance companies in Kenya. As of 2018, there were 36 licensed insmance companies in Kenya which were providing motor insm·ance services in Kenya. These companies will fonn the population of the study consequently. This study will focus on the whole population since the data is readily available to obtain and the population size is relatively small. It is worth to note that this study will not focus on reinsurance companies since they need to be studied independently. 3.3 Data Collection TI1is study seeks to apply the theoretical and empirical frruneworks of past studies and extend it in the Kenyan economy. The aim is to fit a suitable loss distribution to the claim severity data for motor insurance companies using the company-specific data set. This study is focused on claim sizes for motor insmance companies in Kenya from 2014 to 19 l l l 1 _] I 2018. This implies the type of data to be used within the study to be quantitative continuous ratio panel data. The population that is being focused on by the study is made up of 36 licensed motor insurance companies that are regulated by the Insurance Regulatory Authority (IRA). IRA produces annual reports highlighting activities within the insurance industry by various insurance companies. This study will use the data that is contained within these annual reports that have been published by the regulator. The study will thus use seconda.Iy data provided by IRA that includes among other things the claim sizes of various licensed insurance companies providing motor insurance services. The annual repmts provided by IRA are available in Microsoft Excel Binary File Format. Tllis will make it easier for the study to extract the relevant data from the report using programs such as Microsoft Excel and R. TI1ese prograills will then be further used to analyse the extracted data according to the research objectives. 3.4 Data Analysis In compliance with the research objectives, this study seeks to find an approp1iate model that fits claims size of motor insurance companies. lgnatov et al. (2001) provided the following steps to be followed when fitting a suitable model to claims data: a. Select a fa.Inily of distributions for the claims model. b. Estimate the parameters for the model. c. SpecifY a selection criterion to determine the appropriate distribution from the family of distributions. d. CaiTy out a goodness-of-fit test on the selected appropriate distribution. This study will, therefore, follow the above steps in line with the research objectives while analysing the data. Microsoft Excel and R computer prograiTis will be used together for data analysis within the study. The first step that will be carried out on the data is to find its descriptive statistics such as mean, variance a11d skewness. These values will come in 20 l l I J _! handy when comparing them with the results obtained from the various models to select an appropriate loss distribution. In line with the research objectives of this study, it is pmdent to explain the framework that will be applied in the process of analysing data. This will include parameter estimation, standard continuous distributions, goodness-of-fit test, and model selection criteria. These will then form the basis of data analysis that will eventually yield an appropriate loss distribution model. It is important to note that sections 3.4.1 to 3.4.4 below are outlining the theoretical framework that will be applied in analysing the data using Microsoft Excel and R computer programs. 3.4.1 Maximum Likelihood Estimation The parameters of the chosen loss distribution in this study will be estimated using the Maximum Likelihood method. The most important stage in applying the method is that of writing down the likelihood: n L(8) = n f(xi; 8) 1 for a random sample x1, x2 , ... , x11 from a population with density or probability function f(x; 8). In most cases taking logs greatly simplifies the determination of the maximum likelihood estimator (MLE) {j. The following steps are used when detennining a maximum likelihood estimate (MLE): 1. Specify the likelihood function for the available data. n L(8) = n f(xi; 8) 1 2. Simplify the algebra using natural logs. n I (8) = logL (8) = L log f (xi 18) i=l 21 l l 1 l I 1 - J 1 J 3. Maximise the log-likelihood ftmction by differentiating the log-likelihood ftmction with respect to each ofthe unknown parameters and equating the resulting expression(s) to zero. :,:/(8) = 0 4. The MLEs of the parameters are obtained by solving the resulting equation(s). To ensure that the obtained values maximise the likelihood ftmction, differentiate a second time. 3.4.2 Standard Continuous Distributions Tins sntdy will employ the following parametric families of continuous disttibutions: Exponential, Gamma, Lognormal, Pareto, and Wei bull. In this section, the mean, variance, and the maximum likelihood estimates for the parameters will be given. A. Exponential Distribution The distribution function of an exponential distribution is given by: The probability density ftmction is: f(x) = A.e-A.x ,x > 0 The mean and variance of X are: T11e likelihood ftmction is: 1 E(X) =I n 1 var(X) = ;tz L = n A.e-AXi = ;tne-ALXi = jl_1te-ilnx i=1 h - 1 '\"'1l w ere x = - L..i=l xi n The log-likelihood ftrnction becomes: 22 l ! _l j j J J log L = n log il. - il.nx Determine stationary points by differentiating: Setting this to zero gives : B. Gamma Distribution a n -logL = -- nx aA A ~ 1 il.=­x The probability density function of the gamma distribution is given by: The mean and variance of X are: a E(X) =­ A x>O a var(X) = p The moment estimators are used as initial estimators for the MLEs since they cannot be obtained in closed fonn (i.e. in tenns of elementary functions) . C. Lognormal Distribution The probability density fimction of the lognormal dist:Iibution is defined by: 1 l(logX-!l) 2 f (X) = e 2 -u- for 0 < X < oo XC5{2-; The mean and variance of X are: Estimating the MLEs is simple since f1 and C5 2 may be estimated using the log-transfonned data. Let x11 x 2 , ..• , Xn be the observed values and let Yi =log xi· The MLEs will be given by: 23 l _j l J j n (1 = y = ~ L Jog xi i=l where the subscript y signifies a sample variance computed on they values D. Pareto Distribution The distribution ftmction of the Pareto distribution is defined as: A a F(X) = 1- C.+ x) ,x > o The probability density function is given by: ail. a f(X) = (i!. + X)a+l ,X> 0 The mean and variance of X are: i!. E(X) =--(a> 1) a-1 The likelihood ftmction is: ai!.z var(X) = (a_ 1)Z(a _ Z) (a> 2) Ti n ai!.a L = ----a+! ,0 < i!.:::;; min(xi),a > 0 X· i=l t The log-likelihood ftmction becomes: 24 l l ., ! _! _! n log L = n log( a)+ an log(A.)- (a+ 1) L log (xi) i=l To maximize the log-likelihood function, set X =min (xi), such that A. is less than the least xi · Differentiating and setting to zero: 11 8logL n ~ aa = ~ + nlog(A.)- L log(xJ = o This will result in: E. Weibull Distribution i=l n a=-----::-:-- Ir=liog(j) The disttibution fimction of the Weibull disttibution is: F(x) = 1- exp( -cxY) 1 The probability density function is: f(x) = cyxY-l exp( -cxY) 1 x>O x>O The method of maximum likelihood is not simple to apply ifboth candy are unknown. Nevertheless, the equations are elementary when a computer is used. In the case where y has the known value y*, maximmn likelihood is easy enough. We use the data transfonnation Yi = x?. The y values will now have an exponential distribution. The MLE analysis can now be done easily. 3.4.3 Goodness-of-fit Test It is impmtant to test whether a particular loss disttibution is a suitable model for the observed claim an10unts i.e. whether a model provides a "good fit" to the data. This enables one "to determine whether the observed sample was drawn fi·om a population that follows a particular probability distribution" (Dodge, 2008). In this paper, both the Kolmogorov-Smimov and Anderson-Darling tests will be applied because they are 25 l l I l J J j suitable for perfonning an exact test on continuous distributions. For all the goodness-of­ fit tests, the hypotheses will be fonnulated as follows: H0 : The claim severity data follows a particular distribution [F(x) = F0 (x)] H1 : The claim severity data does not follow the particular distribution [F(x) =1: F0 (x)] where F(x) is the unknown distribution ftmction of the claim seveiity data (sample) wlule F0 (x) is a specific and known distribution ftmction (population). 3.4.4 Information Criteria For all the selected claim seveiity disnibutions that pass the goodness-of-fit test, both the Akaike lnfonnation Criterion (AIC) and the Bayesian lnfonnation Criterion (BIC) will be used to select the best model for the claim severity data. It is possible to increase the likelihood by adding parameters when fitting models. However, this may lead to overfitting. Both AIC and BIC introduce a penalty term for the number of parameters in the model in a bid to resolve the problem of overfitting. BIC introduces a larger penalty term than AIC. Even though BIC value is always higher than AIC value, the lower the value of these two criteria the better a model is. 3.4.5 Model Selection Criteria The research objective is to detennine an appropiiate loss distribution that provides a good fit to the claim severity data of motor insurance companies in Kenya. Based on the data analysis procedures that have been elaborately outlined in sections 3.4.1 to 3.4.4 above, the loss distribution that will be selected as being an appropiiate model is one that has: 1. The maximum MLE value subject to passing the goodness-of-fit tests, and 2. The minimum AIC and BIC value The study will focus on finding the loss distiibution that meets the above requirements. 26 I l I 1 1 J l .J _I _I J CHAPTER FOUR: DATA ANALYSIS 4.1 Introduction The data required for this study was motor insurance claim severity for Kenya. This data was readily available :fi:om publications ofthe Insurance Regulatory Authority (IRA) in fonn of annual reports. The annual reports were obtained in Microsoft Excel format and the required data was extracted therein. The data contained claim severity for 36 insurance companies that were licensed and regulated by IRA :fi:om 2014-2018. The data for Kenya Motor Insurance incmTed claims was provided in tenns of motor commercial and motor private. These two sets of data for the petiod 2014-2018 were analysed separately to obtain suitable models for each categmy of motor insurance. The data was analysed using R software. 4.2 Descriptive Statistics The first step of data analysis was to determine the descriptive statistics of the data to get a general overview and allow a simpler interpretation of the data. Table 1 shows the descriptive statistics of the motor insurance incurred claims. The descriptive statistics in Table 1 confirm the positive skewness of the claims data and as such, positively skewed and long-tailed distributions being appropriate to model tlris data. 27 l J J J Table 1. Kenya Motor Insurance Incurred Claims Descriptive Statistics for 2014- 2018 Motor Commercial Motor Private No. of observations (n) 170 174 Mean 281,703,882.35 386,762,402.30 Standard Error 23,826,723.55 30,253,278.83 Median 178,457,000.00 216,641 ,000.00 Standard Deviation 310,662,467.00 399,068,156.03 Sample Vruiance 9.65E+16 1.59E+17 Kurtosis 3.7716 2.6583 Skewness 1.9078 1.6815 Range 1,470,681,000.00 1,991,252,000.00 Minimmn 89,000.00 994,000.00 Maxirmun 1,470,770,000.00 1,992,246,000.00 Smn 47,889,660,000.00 67,296,658,000.00 Figure 1 shows the histograms of the miginal data of motor commercial and motor private claim sizes. The nonnal curves superimposed on the histograms show that the claim sizes are skewed to the right. The purpose of carrying out the normality test was to detennine whether to use either parametric tests or non-parametric tests on the data after fitting distributions. Given that the data was not following the nonnal curve, this was a confirmation that non-parametric tests would be applied to the distributions that would be fitted on the data. 28 l J l I J I J J J J Histogram of Motor Commercic 0 co 0 <.0 0 N 0 O.Oe+OO 1.0e+09 Claim Size >- (.) c IV :::J o- ~ lL Histogram of Motor Private 0 co 0 <.0 0 ~ 0 N 0 O.Oe+OO 1.0e+09 2.0e+09 Claim Size Figure 1 Figure 2 shows the Q-Q plots of the original data of motor commercial and motor private claim sizes. Given that the upper ends of the Q-Q plots are deviating more :fi·om the straight line than the lower ends, this confinns that the data is positively skewed. Tliis implied the need to use continuous distributions that are positively skewed to fit the data. VI IV ~ ro :::J a Q) Ci E ro (f) Q·Q Plot of Motor Commercial Q·Q Plot of Motor Private (J) 0 + I]) q 0 0 + I]) 0 0 -2 -1 0 1 2 Theoreti cal Quantiles (J) 0 + IV 0 VI I]) ~ ro :::J a I]) Ci E ro (f) q N (J) 0 + Q) q 0 0 + I]) q 0 Figure 2 29 -2 -1 0 1 2 Theoretical Quantiles l l J J The data was transformed using the cube root function to make it simpler to work with since the miginal data contained very large values. Figure 3 shows the histograms of the cube-root transfonned data of motor commercial and motor private claim sizes. >­ <..) c Q) ::J 0" Q) u:: Histogram of Motor Commercii 0 400 800 1200 Claim Size Histogram of Motor Private 0 c0 >- (j c 0 Q) ::J N 0" (J) u:: 0 0 0 400 800 1200 Claim Size Figure 3 Figure 4 shows the Q-Q plots of the cube-root transformed data of motor commercial and motor private claim sizes. (/) Q) ~ ro ::J a (!) 0.. E ro (f) Q-Q Plot of Motor Commercial 0 0 0 0 0 (!) 0 0 N -2 -1 0 2 Theoretical Quantiles (/) 0 Q) 0 ~ 0 ro ::J 0 0 Q) 0 a.. (!) E ro (f) 0 0 N Figure 4 30 Q-Q Plot of Motor Private 0 -2 -1 0 2 Theoretical Quantiles l l I I j J J The cube-root transfonned data seemed to be more suitable for purposes of fitting the distributions compared to the original data as shown by their respective histograms and Q-Q plots above. With the data being transformed, the next step was fitting distributions to obtain an appropriate model. Subsequently, goodness-of-fit tests were carried out and information ctiteria applied on the fitted distributions as outlined in the sections below. All these steps were canied out using the R software package "fitdistrplus". 4.3 Parameters Estimation The parameters for the various fitted distributions were obtained using the MLE method. Table 2 shows the parameter estimates, their corresponding standard errors and the maximmn Ioglikelihood ftmction (LLF). The most appropriate distribution is the one witl1 the highest loglikelihood function. 31 l l l J J Table 2. Estimated Parameters for fitted distributions Distribution Parameter Motor Commercial Motor Private Rate 0.0021 0.0020 Exponential std. error 0.0001 0.0001 LLF -1218.773 -1254.143 Shape 5.4827 7.3252 std. eiTor 0.5377 0.7165 Gamma Rate 0.0096 0.0113 std. enor 0.0010 0.0011 LLF -1165 .534 -1192.602 Meanlog 6.2576 6.4079 std. error 0.0360 0.0296 Lognonnal SD1og 0.4696 0.3898 std. error 0.0255 0.0209 LLF -1176.499 -1197.959 Shape 5094930 3221289 (std. enor) - - Pareto Scale 2925993718 2095806234 (std. enor) - - LLF -1249.768 -1301.128 Shape 2.7097 3.0010 std. en·or 0.1617 0.1747 Weibull Scale 644.7977 728.9534 std. enor 19.2325 19.4038 LLF -1161.697 -1193.441 The parameter values in Table 2 are the ones that maximised the loglikelihood ftmction for each distribution. For motor commercial, Weibull distribution has the maximum loglikelihood function (-1161.697) followed by Gamma (-1165.534), Lognormal (- 32 l l J J 1176.499), Exponential (-1218.773) and Pareto (-1249.768) distributions. For motor ptivate, Gamma distribution has the maximum loglikelihood function (-1192.602) followed by Weibull (-1193.441), Lognonnal (-1197.959), Exponential (-1254.143) and Pareto ( -1301.128) disnibutions. Based on the LLF values, Wei bull and Gamma distributions were the most appropriate for motor commercial and motor private, respectively. The plots for the various fitted distributions are provided in the appendices section. 4.4 Goodness-of-Fit Test The main aim of carrying out the goodness-of-fit test was to measure the distance between the fitted paran1en·ic distribution F(x) and the empirical distribution Fo(x). The Kolmogorov-Smirnov (K-S) and Anderson-Darling (A-D) tests were used to detennine the appropriateness of the fitted distt·ibutions to the claim size data. TI1ese two tests helped to determine the most suitable continuous distribution for the claim severity. Table 3 shows the K-S and A-D test statistic values for the disn·ibutions fitted on the claim severity data. Table 3. K-S and A-D test statistic values for fitted distributions Test Statistic Distiibution Motor Commercial Motor Private Exponential 0.3089418 0.3601346 Grumna 0.07988259 0.04042086 K-S Lognormal 0.09186256 0.04602873 Pareto 0.3084464 0.3600815 Weibull 0.05444920 0.08136149 Exponential 27.6395653 33.0098233 Gatmna 0.67351891 0.37752203 A-D Lognormal 1.68105273 0.59667217 Pareto 27.5714722 33 .0018593 Wei bull 0.41538833 1.08746132 33 I J J I The K-S and A-D test statistic values were used to compare the fit of the distributions to the data as opposed to an absolute measure of how a particular distJibution fits the data. Smaller K-S and A-D test statistic values indicate that the distribution fits the data better. For motor commercial, Weibull distribution had the smallest K-S and A-D test statistic values (0.05444920, 0.41538833) followed by Gamma (0.07988259, 0.67351891), Lognormal (0.09186256, 1.68105273), Pareto (0.3084464, 27.5714722) and Exponential (0.3089418, 27.6395653) distributions. For motor private, Gamma distribution had the smallest K-S and A-D test statistic values (0.04042086, 0.37752203) followed by Lognormal (0.04602873, 0.59667217), Weibull (0.08136149, 1.08746132), Pareto (0.3600815, 33.0018593), and Exponential (0.3601346, 33.0098233) distributions. This implied that Wei bull and Gamma distributions fitted the data of motor commercial and motor private better, respectively. 4.5 Information Criteria The Akaike and Bayesian Information Critetia were applied to determine the appropriate distribution among the fitted distributions. Table 4 shows the AIC and BIC values for the fitted distributions . The most appropriate distribution is one which has the minimum AIC and BIC values. 34 l l J J J J Table 4. AIC and BIC values for fitted distributions Information Distribution Motor Commercial Motor Private Criterion Exponential 2439.547 2510.286 Gamma 2335.067 2389.204 AIC Lognonnal 2356.999 2399.918 Pareto 2503.536 2606.255 Weibull 2327.393 2390.881 Exponential 2442.682 2513.445 Gamma 2341.339 2395.522 BIC Lognormal 2363.270 2406.236 Pareto 2509.808 2612.573 Weibull 2333.665 2397.199 For motor commercial, Weibull distribution had the minimum AIC and BIC values (2327 .393, 2333 .665) followed by Gamma (2335 .067, 2341 .339), Lognormal (2356.999, 2363.270), Exponential (2439.547, 2442.682) and Pareto (2503.536, 2509.808) distributions. For motor private, Gamma distribution had the minimtun AIC and BIC values (2389.204, 2395.522) followed by Weibull (2390.881, 2397.199), Log:nmmal (2399.918, 2406.236), Exponential (2510.286, 2513.445) and Pareto (2606.255, 2612.573) distributions. This implied that Weibull and Gamma distributions were the most appropriate for motor commercial and motor private, respectively . 4.6 Summary The most appropriate distribution is one which has the maximmn LLF, minimum K -S and A-D test statistic values, and minimum AIC and BIC values. Based on the data analysis canied out, as shown above, the most appropriate distiibutions to model claim severity 35 _j l .l J J _j 1 data for motor commercial and motor private are Weibull distribution and Gamma distribution, respectively. Based on past studies that were carried out as discussed in the literature review section, the lognonnal distribution is fronted as being the most suitable to model motor insurance claim severity. However, given the findings of this research, Weibull and Gamma distributions are the most appropriate to model motor commercial and motor p1ivate claim severity, respectively. The lognonnal distiibution is the third most suitable model for tlris purpose based on the findings of the study. 36 I J j _j CHAPTER FIVE: CONCLUSION 5.1 Intmduction l11e objective of this research was to determine the appropriate statistical distribution that fits claim severity data of motor insurance in Kenya and can be used to accurately forecast futm·e claims experience. To achieve tltis, the study used the steps to be followed when fitting a suitable model to claims data as provided by lgnatov et al. (2001). The first step entailed selecting a family of distributions for the claims model whereby the Exponential, Gamma, Lognonnal, Pareto and Weibull distiibutions were selected given that they are continuous positively skewed distributions. The parameters for these distributions were estimated using the MLE method. The K-S and A-D tests were applied as goodness-of-fit tests on the fitted distributions. The AIC and BIC infonnation criteria were used to determine the appropriate distribution among the fitted distributions. 5.2 ConcJusion The study carried out an analysis of Kenya motor insurance claim severity data from 2014- 2018 based on the above steps. The finding of the study was that the Weibull and Gamma distributions are suitable for modelling motor commercial and motor private data, respectively. Tltis was because they had the maximum LLF, minimum K-S and A-D test statistic values, and minimmn AIC and BIC values among the fitted distributions. Findings of past studies as ltighlighted in the Literature Review section [Achieng (2010) and Omrui et al . (2018)] :fi·onted the Lognormal distribution as being t11e most suitable for modelling claim severity. Gamma, Weibull, Pareto, and Exponential distributions were also :fi·onted as being appropriate as well in tl1at decreasing order of preference [Mazviona and Chiduza (2013), and Packova and Brebera (2015)]. The findings of this study are different from past studies given that tile Weibull and Gamma distiibutions have been selected as being the most suitable to model claim severity for motor commercial and motor private data respectively . Lognonnal, Pareto and Exponential distributions ru·e also appropriate in that decreasing order of preference. 37 J J The objectives of this study were achieved given that the Weibull and Gamma distributions have been selected as being the most appropriate to model claim severity of motor commercial and motor private, respectively. These distiibutions were selected after the data analysis steps used to achieve the research objectives were successfully followed and meeting the selection criteria of the study and thus being selected as the most suitable after emerging the best among the other competing distributions. To this end, Weibull ru1d Gamma distributions fit the claim severity data of motor commercial and motor private respectively and can be used to accurately forecast future claims experience; thus, achieving the objectives of the sn1dy. 5.3 Recommendations The appropriate distlibutions obtained under this sn1dy are suitable for forecasting shmt­ term claims experience given the period covered by the study. It is recommended that insurers use their own claims expelience to adjust the distributions accordingly for long­ tenn forecasts . This would allow insurers to consider their specific financial objectives and expected changes in their investment portfolios. These proposed claim severity distributions would also be useful to insurance regulators while testing for solvency and assessing the required levels of reserves for various insurance companies. 5.4 Limitations of the Study Tltis study focused on statistical distributions only as opposed to consideting other approaches such as non-parametric methods, machine leruning and deep learning. These approaches are also suitable for modelling claim severity. Non-parrunetric methods do not assume underlying statistical distributions in the data and thus do not rely on any distribution. This approach may serve well in certain circumstances whereby data fails to follow any statistical distribution and thus be well modelled using this approach. Technological advancement in teims of machine and deep learning has improved 38 l l l J J efficiency. TI1is can be applied by insurers in the modelling process by predicting patterns of claim volume and augmenting loss analysis using artificial intelligence. This study did not determine the impact of modelling claim severity on the business of insurance companies. Insurers need to know tllis impact to make suitable adjustments in their modelling process. Modelling claim severity involves the use of resources and thus, insurers need to know whether it is worth it or not for them to comnlit resources towards modelling given tl1e value being added to the business in terms of increased or decreased profitability. If modelling has a positive impact on the business in tenns of increased profitability, insurers will strive to commit high-quality resources towards the modelling process in a bid to ensure they increase their profitability. 5.5 Suggestions for Further Research Interested parties may investigate otl1er approaches that may be used to model claim severity such as non-parametric methods, machine learning and deep learning. Further studies may also be earned out to detennine the impact of modelling claim severity on the business of insurance companies. 39 l l " l J J j REFERENCES 1. Omari, C. , Nyambura, S. and Mwangi, J. (2018). Modeling the Frequency and Severity of Auto Insurance Claims Using Statistical Dist:Jibutions. Journal C?l Mathematical Finance, 8, 137-160. doi: 10.4236/jmf.2018.81012 2. Achieng, 0. M. (2010). Actuarial modeling for insurance claim severity in motor comprehensive policy using industrial statistical distributions. 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The Annals of Statistics, 6(2), 461-464. doi: 1 0.1214/aos/1176344136 19. Research & Statistics: Annual Reports. Retrieved from https ://www.ira.go.ke/index.php/publications/statistical-repmts/annual -repmts 20. Actuarial Education Company. (2014). Retrieved from https ://acted.co. uk/docs/2014/CMP%20Upgrade/CT6-PU- 1 4.pdf?cv= 1 41 l l J J J APPENDICES Appendix A: Fitted Distributions for Motor Commercial Appendix Al : Exponential Distribution u.. 0 () 0 Empirical and theoretical dens. 0 200 400 600 800 1000 Data Empirical and theoretical CDFs 0 200 400 600 800 1000 Data Appendix A2: Gamma Distribution Empirical and theoretical dens. >. -- 3F~ (J) c 0 oZ> 0 0 0 0 c:i 0 200 400 600 800 1200 Data Empirical and theoretical CDFs 0 LL 0 () 0 0 0 200 400 600 800 1200 Data Ul Q) c ro ~ -m u ·.::: ·o_ E w "' "' .L:l rn -"' 0 0.. rn u 0.. E w (J) "' c:: "' o- ro u 0.. ~ ill Ul Q) 0.. E w 42 Q-Q plot ~;:::wvV: 0 -, 0 0 <'J 0 500 1500 2500 Theoretical quantiles 0 ;~¢~~ 0.2 0.4 0.6 0.8 TI1eoretical probabilities Q-Q plot 0 0 N 200 600 1000 1400 Theoretical quantiles P-P plot 0.0 0.2 0.4 0.6 0.8 1.0 TI1eoretical probabilities l l l I _j _j _I Appendix A3: Lognormal Distribution >. -- rJ) c Q) 0 LL 0 u 0 0 0 0 6 Empirical and theoretical dens. ~~ 0 200 400 600 800 1200 Data Empirical and theoretical CDFs 0 200 400 600 800 1200 Data Appendix A4: Pareto Distribution Empirical and theoretical dens. >. 3~9 2 0. -ro 0 -;:: 0. E w 43 0 0 N 0 0 0 0 0 N 0 0 0 Q-Qplot 500 1000 1500 Theoretical quantiJes P-P plot 0.0 0.2 0.4 0.6 0_8 Theoretical probabilities Q-Q plot c I 0 500 1500 2500 Theoretical quantiles P-P plot ~ 0.2 0.4 0.6 0.8 Theoretical prob-abil ities 1 I l J J J j J Appendix AS: Weibull Distribution LL 0 (_) 0 Empirical and theoretical dens. 0 200 400 600 800 1200 Data Empirical and theoretical CDFs 0 200 400 600 800 1200 Data 0. E w (J) Q) J:l ro J:l e 0. "' 0 : ~ a_ E w Q-Q plot 200 400 GOO 800 1200 Theoretical quantiles P-P plot ~ ~r 0 c:i 0.0 0.2 0.4 0.6 0.8 1.0 Theoretical probabil ities Appendix B: Fitted Distributions for Motor Private Appendix Bl: Exponential Distribution Empirical and theoretical dens. C/) Q-Q plot Q) E "" ro ·- ~~ :::::> ~¢ ~:o~x:oooo v Ql C/) t:T c 0 ro 0 Q) 0 0 0 9 (.) 0 0 · ;:: N I I I I c:i a_ 0 200 400 600 800 1000 E 0 500 1500 2500 w Data Theoretical quantiles C/) Empirical and theoretical CDFs OJ P-P plot 0 "' ~ ~~~COVQ~ ~ .n i-~:::J LL 0 0 a. (_) 0 -,;; 0 c:i 0 c:i ·;:: 0 200 400 600 800 1000 ·a.. 0.2 0.4 0.6 0.8 E w Data Theoretic al probabil it ies 44 l l J Appendix B2: Gamma Distribution Empirical and theoretical dens. >- ~ ~r--:r, "' c 0 Q) 0 0 0 0 r 0 0 200 600 1000 Data Empirical and theoretical CDFs 0 ~ I of~ I u.. 0 u 0 0 0 200 600 1000 Data Appendix B3: Lognormal Distribution lL 0 u Empirical and theoretical dens. 0 200 600 1000 Data Empirical and theoretical CDFs 0 200 600 1000 Data "' ~ c ~ cr- -ro u CL E w "' "' ·- ..0 "' ..0 e CL "' :~ CL E w c "' :J ;:r "' u ·.:::: ·a.. E w "' aJ ..0 "' ~ c.. 45 0 0 N 0 0 c) Q-Q plot iF I :;:UQ I I Ql 200 600 1000 1400 TI1eoretical quantiles P-P plot ~f re:e& I r;r»:~ I 0.0 0.2 OA 0.6 0.8 1_0 TI1eoretical probabilities Q-Q plot 500 1000 "1500 Theoretical quantiles P-P plot 0.0 0.2 0.4 0.6 0.8 1.0 Theoretical probabilities l -1 -1 I I I I _I J I J J J Appendix B4: Pareto Distribution Empirical and theoretical dens. >.. ~~ (/) c: 0 Q) 0 0 0 0 0 0 200 600 1000 Data Empirical and theoretical CDFs 0 ~r~:1 u.. 0 u 0 0 0 200 600 1000 Data Appendix BS: Weibull Distribution Empirical and theoretical dens. >.. ~F~ ·r;; c: 0 Q) 0 0 0 0 0 0 200 600 1000 Data Empirical and theoretical CDFs ~ ~ 0 f~~:::· I 1..:... 0 u 0 c:) I 0 200 GOO 1000 Data (/) Q-Qplot Q) c: "' :::; ~~x;ooov v ~I :::r ro 0 0 0 : ~ N I 0.. c w 0 1000 2000 3000 Tl1eoretical quantiles (/) - ~ P-P plot :i5 "' 0 ..0 ~ 0.. ro 0 0 0 : ~ 0.. 0.2 0.4 0.6 0.8 E w Tl1eoretical probabilities (/') Q-Q plot .!£ c ~ ~~ · ·:: Ql c- ro 0 ; 0 0 N I ·c.. c LU 200 600 1000 Theoretical quantiles (/) ~ P-P plot :E ro ~ .0 0 0.. -rn 0 -~ 0 ·c.. 0.0 0.2 0 .. 4 0.6 0.8 1.0 E w Theoret ical probabilities 46