ANALYSIS OF MOTOR VEHICLE ACCIDENT CLAIMS USING PROBABILITY DISTRIBUTIONS

CHAPTER ONE

INTRODUCTION

1.1 Background to the Study

Motor insurance is the most widely held class of general insurance business in Nigeria, partly because third-party motor cover is compulsory under law. In the 2025 financial year, motor insurance accounted for 16.1 per cent of all non-life gross written premium, ranking third after oil and gas and fire business (National Insurance Commission [NAICOM], 2026). It is also the class in which the insurer's exposure is most directly tied to a physical, observable and statistically well-recorded phenomenon: the road traffic crash.

Nigeria's road safety record makes that exposure substantial. The Federal Road Safety Corps (FRSC) records tens of thousands of crashes annually, with the corresponding casualty figures placing Nigeria among the countries with the highest road traffic fatality burdens in Africa (Federal Road Safety Corps, 2024). Yet the proportion of that exposure which is insured remains low: of roughly twelve million vehicles on Nigerian roads, only about 3.11 million were insured at the end of 2023, a coverage rate of approximately 25 per cent, and the insured population had actually fallen from 3.70 million in 2022 (Nigerian Insurers Association, 2024). The portfolio that insurers do carry is therefore not a random sample of the vehicle population, which has direct implications for the distributional properties of the claims arising from it.

Actuarial analysis of motor claims proceeds from the recognition that the number of claims arising from a portfolio in a period, and the amount of each claim, are random variables governed by probability distributions. Claim counts are conventionally modelled by the Poisson distribution when the mean and variance are approximately equal, and by the negative binomial when overdispersion is present, as is typically the case where drivers differ systematically in risk. Where a large proportion of policies generate no claim at all, zero-inflated or hurdle formulations are required. Claim amounts, being non-negative, continuous and strongly right-skewed with occasional very large losses, are modelled by the gamma, lognormal, Weibull, Pareto, or generalised Pareto distributions, with the last of these being particularly appropriate for the tail of the loss distribution (Klugman, Panjer, & Willmot, 2019). The aggregate claim amount is then obtained as a compound distribution, and it is from this that the insurer derives the pure premium, the risk loading and the required reserve.

Recent methodological work has broadened this toolkit. Clemente, Guerreiro and Bravo (2023) applied gradient boosting to motor claim frequency and severity, Meng, Gao and Huang (2022) incorporated telematics-derived driving-behaviour features into frequency models, and Gao, Huang and Meng (2023) used telematics data to interpret and evaluate driving risk. In the African context, work on non-life claims in Ghana has demonstrated the applicability of the discrete generalised Pareto distribution to reported and settled claims (Akinyemi & Doku-Amponsah, 2020). Nigerian studies, however, have concentrated largely on the causes and socio-economic consequences of road traffic accidents rather than on the probabilistic structure of the resulting insurance claims.

This study addresses that imbalance by fitting and comparing candidate probability distributions to Nigerian motor accident claim frequency and claim severity data, and by using the fitted models to estimate aggregate claim liabilities and risk premiums.

1.2 Statement of the Problem

Nigerian motor insurers face a claims environment characterised by high accident exposure, low insured penetration, rapid cost inflation in vehicle repair and replacement, and a documented incidence of fraudulent or inflated claims. Pricing in this environment is nevertheless dominated by flat administrative tariffs most visibly the fixed third-party premium that bear no explicit relationship to the underlying claim distribution of the insured vehicle.

Three difficulties follow. First, where premiums are set without reference to a fitted probability model, the insurer cannot know the probability that aggregate claims will exceed premiums plus reserves, which is the central solvency question. Second, motor claim data are typically overdispersed and zero-inflated, so that the Poisson assumption implicit in simple average-based pricing understates the variability of claim counts. Third, motor claim severity in Nigeria exhibits a heavy tail driven by total-loss and third-party bodily injury claims; fitting a light-tailed distribution to such data leads to systematic underestimation of extreme loss and hence of required capital.

Compounding this, there is very little published Nigerian research establishing which distributions actually fit local motor claims data. Practitioners therefore rely on assumptions imported from foreign markets whose vehicle age profiles, road conditions, repair costs and litigation environments differ materially from Nigeria's. The problem this study addresses is the absence of empirically validated probability models for Nigerian motor vehicle accident claims, and the pricing, reserving and solvency errors that this absence produces.

1.3 Aim and Objectives of the Study

The aim of this study is to analyse motor vehicle accident claims in Nigeria using probability distributions in order to improve the estimation of claim liabilities and premiums.

The specific objectives are to:

1. examine the pattern and descriptive characteristics of motor vehicle accident claims in Nigeria;

2. fit candidate discrete probability distributions (Poisson, negative binomial, geometric and zero-inflated variants) to motor claim frequency data;

3. fit candidate continuous probability distributions (gamma, lognormal, Weibull, Pareto and generalised Pareto) to motor claim severity data;

4. select the best-fitting distributions using formal goodness-of-fit and information criteria;

5. estimate the aggregate claim distribution and the pure premium for the motor portfolio.

1.4 Research Questions

1. What are the descriptive characteristics and patterns of motor vehicle accident claims in the study portfolio?

2. Which discrete probability distribution best describes motor claim frequency?

3. Which continuous probability distribution best describes motor claim severity?

4. What aggregate claim distribution and pure premium are implied by the fitted models?

5. Is there a significant relationship between reported road traffic crashes and motor insurance claim frequency?

1.5 Research Hypotheses

The following null hypotheses will be tested at the 5% level of significance:

H₀₁: Motor vehicle accident claim frequency does not follow a Poisson distribution.

H₀₂: The negative binomial distribution does not provide a significantly better fit to claim frequency data than the Poisson distribution.

H₀₃: Motor vehicle accident claim severity does not follow a lognormal distribution.

H₀₄: There is no significant difference in goodness of fit among the candidate severity distributions.

H₀₅: There is no significant relationship between reported road traffic crashes and motor insurance claim frequency.

1.6 Significance of the Study

The study will assist motor insurers in setting technically justified premium rates and in quantifying the capital needed to support the motor account, which is directly relevant under the risk-based capital regime of the Nigerian Insurance Industry Reform Act, 2025. It will assist NAICOM in reviewing the adequacy of prescribed third-party premium levels and the reserves held against motor liabilities. It will be of value to the Federal Road Safety Corps, by demonstrating quantitatively how crash frequency translates into financial loss and thereby strengthening the economic case for road safety intervention. Vehicle owners and the motoring public benefit indirectly, since actuarially sound pricing distributes cost more fairly between high- and low-risk drivers and reduces the incidence of disputed or unpaid claims. Finally, the study contributes to the actuarial literature by supplying Nigerian empirical evidence on distributional fit, an area in which African data are under-represented relative to European and North American portfolios.

1.7 Scope of the Study

The study covers motor insurance claims data obtained from one or more insurance companies licensed by NAICOM, over a period of not less than five consecutive years. It also draws on published road traffic crash statistics from the FRSC Statistical Digest and the National Bureau of Statistics for the corresponding period. The claim variables examined are the number of claims per policy per year and the amount of each settled claim. The analysis is restricted to comprehensive and third-party motor business; it excludes marine, aviation and other classes of general insurance, and it does not examine the engineering or behavioural causes of accidents except insofar as aggregate crash counts are used as an explanatory variable.

1.8 Limitations of the Study

(i) Under-insurance bias because only about a quarter of Nigerian vehicles are insured, insured claims data represent a self-selected sub-population and may not describe the risk profile of the whole vehicle fleet. (ii) Data confidentiality insurers may restrict access to policy-level records, limiting the covariates available for risk classification. (iii) Inflation in claim amounts rapid increases in the naira cost of spare parts and repairs during the study period distort nominal severity comparisons across years and will require deflation using an appropriate index. (iv) Under-reporting of crashes FRSC statistics capture reported crashes, and minor incidents are widely known to go unrecorded, weakening the crash claim linkage analysis. (v) Claim settlement lag claims incurred late in the study period may remain unsettled at the data cut-off, truncating the observed severity distribution.

1.9 Operational Definition of Terms

Probability distribution: A mathematical function that assigns probabilities to the possible values of a random variable.

Overdispersion: The condition in count data where the observed variance exceeds the mean, violating the equidispersion assumption of the Poisson distribution.

Goodness of fit: The extent to which a fitted theoretical distribution reproduces the observed data, assessed here by the chi-square test for discrete data, the Anderson Darling and Kolmogorov Smirnov tests for continuous data, and the Akaike and Bayesian Information Criteria for model comparison.

Maximum likelihood estimation (MLE): A method of estimating distribution parameters by maximising the likelihood of observing the given sample.

Third-party motor insurance: The statutorily compulsory class of motor cover indemnifying the insured against liability for death, bodily injury or property damage caused to third parties.

Total loss: A claim in which the cost of repair exceeds a defined proportion of the vehicle's value, so that the insurer settles the full insured value.

Pure premium: The expected claim cost per policy per period, before expense, profit and contingency loadings.

References

Akinyemi, S., & Doku-Amponsah, K. (2020). Assessing the performance of the discrete generalised Pareto distribution in modelling non-life insurance claims (arXiv:2004.06150). arXiv. https://doi.org/10.48550/arXiv.2004.06150

Clemente, C., Guerreiro, G. R., & Bravo, J. M. (2023). Modelling motor insurance claim frequency and severity using gradient boosting. Risks, 11(9), 163. https://doi.org/10.3390/risks11090163

Federal Road Safety Corps. (2024). FRSC statistical digest, first quarter 2024. Federal Road Safety Corps.

Gao, Y., Huang, Y., & Meng, S. (2023). Evaluation and interpretation of driving risks: Automobile claim frequency modeling with telematics data. Statistical Analysis and Data Mining, 16(2), 97 119. https://doi.org/10.1002/sam.11599

Klugman, S. A., Panjer, H. H., & Willmot, G. E. (2019). Loss models: From data to decisions (5th ed.). John Wiley & Sons.

Meng, S., Gao, Y., & Huang, Y. (2022). Actuarial intelligence in auto insurance: Claim frequency modeling with driving behavior features and improved boosted trees. Insurance: Mathematics and Economics, 106, 115 127. https://doi.org/10.1016/j.insmatheco.2022.06.001

National Insurance Commission. (2026). Bulletin of the insurance market performance: Fourth quarter 2025. NAICOM.

Nigerian Insurers Association. (2024). Nigeria insurance digest 2023. Nigerian Insurers Association.

Shi, P., Feng, X., & Ivantsova, A. (2015). Dependent frequency severity modeling of insurance claims. Insurance: Mathematics and Economics, 64, 417 428. https://doi.org/10.1016/j.insmatheco.2015.07.006

World Health Organization. (2023). Global status report on road safety 2023. WHO.