MODELLING HEALTH INSURANCE CLAIMS USING STATISTICAL METHODS

CHAPTER ONE

INTRODUCTION

1.1 Background to the Study

Health insurance operates on the principle of risk pooling, in which the uncertain and potentially catastrophic medical expenditure of an individual is transferred to an insurer in exchange for a comparatively small and certain premium. The financial viability of any such arrangement depends almost entirely on the insurer's ability to predict, with acceptable accuracy, how often claims will arise and how large they will be. This predictive exercise is fundamentally statistical: claim occurrence is a counting process, claim size is a continuous and typically right-skewed random variable, and the aggregate liability of the insurer is a compound distribution formed from the two (Klugman, Panjer, & Willmot, 2019).

The classical actuarial approach decomposes aggregate claims into a frequency component and a severity component. Frequency is conventionally modelled using the Poisson, negative binomial, or zero-inflated count distributions, while severity is modelled using the gamma, lognormal, Weibull, or Pareto families. Generalised Linear Models (GLMs) extended this framework by allowing rating factors such as age, sex, plan type, and provider category to be incorporated as covariates. A long-standing simplifying assumption in this framework is that frequency and severity are independent, an assumption that is theoretically convenient because it permits the two components to be estimated separately. Empirical evidence has, however, repeatedly contradicted it. Frees, Gao and Rosenberg (2011) demonstrated dependence between the number and size of health-care expenditure claims, while Erhardt and Czado (2012) modelled dependent yearly claim totals, including zero claims, in private health insurance. Shi, Feng and Ivantsova (2015) formalised a two-part dependent frequency severity framework, and more recent contributions have relaxed distributional and functional-form assumptions altogether through machine-learning estimators such as gradient boosting and copula-based hierarchical models (Power, Côté, & Duchesne, 2024; So & Valdez, 2024; Xu, Manathunga, & Hong, 2023).

In Nigeria, the statistical modelling of health insurance claims has assumed policy urgency. The National Health Insurance Authority (NHIA) Act, 2022, which repealed the National Health Insurance Scheme Act of 2004, made health insurance mandatory for all residents and created a Vulnerable Group Fund, thereby signalling a decisive shift towards universal health coverage (Adewole & Adebowale, 2022; Ipinnimo, Durowade, Afolayan, Ajayi, & Akande, 2022). The reform was driven by a persistent coverage failure: after more than two decades of operation, the old scheme had enrolled less than ten per cent of the population, and survey evidence indicates that only about two to four per cent of Nigerian adults reported any health insurance cover in recent national surveys (Aloh, Onwujekwe, Aloh, & Nwankwo, 2020; Okoroafor et al., 2025). As enrolment expands under the new mandate, Health Maintenance Organisations (HMOs) and the Authority itself will confront claim volumes and claim-cost distributions for which there is little accumulated Nigerian actuarial experience.

This matters because the pricing of capitation and fee-for-service arrangements, the setting of provider tariffs, the estimation of technical provisions, and the detection of fraudulent or inflated claims all rest on distributional assumptions about the claims process. Where those assumptions are imported wholesale from foreign experience or set administratively rather than estimated from data, the result is either under-pricing, which threatens solvency, or over-pricing, which suppresses enrolment. Fraudulent and exaggerated claims have already been identified as a material drain on Nigeria's health financing architecture (Ogunbekun & Adeleye, 2023). A rigorous, data-driven model of the Nigerian health insurance claims process is therefore not an academic luxury but a precondition for the sustainability of the reform.

This study responds to that need by fitting, comparing, and validating a set of candidate statistical models for claim frequency and claim severity on Nigerian health insurance claims data, and by using the selected models to estimate aggregate claim distributions and pure premiums.

1.2 Statement of the Problem

Despite the centrality of claims modelling to health insurance solvency, Nigerian insurers and HMOs largely set premiums and capitation rates on the basis of administrative negotiation, historical averages, and competitor benchmarking rather than fitted probability models. Three problems follow.

First, the use of simple arithmetic averages ignores the heavy right tail characteristic of medical claim severity, so that the probability of extreme aggregate loss is systematically understated. Second, the conventional assumption of independence between claim frequency and claim severity, still embedded in most practical rating exercises, has been shown to be empirically false in health portfolios (Erhardt & Czado, 2012; Frees et al., 2011), leading to biased estimates of aggregate loss. Third, the Nigerian literature on health insurance is dominated by studies of enrolment, awareness, willingness to pay, and policy evaluation, with comparatively little quantitative modelling of the claims process itself using local data.

The consequence is a knowledge gap: it is not currently established which probability distributions best describe the frequency and severity of health insurance claims in Nigeria, whether dependence exists between the two components, or how much pricing error results from ignoring it. Without this evidence, the expansion of coverage envisaged by the NHIA Act, 2022 risks being built on actuarially unsound foundations. This study addresses that gap.

1.3 Aim and Objectives of the Study

The aim of this study is to model health insurance claims using appropriate statistical methods in order to improve the estimation of claim liabilities and premium rates.

The specific objectives are to:

1. examine the distributional characteristics of health insurance claim frequency and claim severity data;

2. fit and compare competing count models (Poisson, negative binomial and zero-inflated variants) to claim frequency data;

3. fit and compare competing severity models (gamma, lognormal, Weibull and Pareto) to claim amount data;

4. determine whether a statistically significant dependence exists between claim frequency and claim severity;

5. estimate the aggregate claim distribution and the pure premium using the best-fitting frequency and severity models; and

6. assess the predictive accuracy of the selected models using appropriate validation criteria.

1.4 Research Questions

1. What are the distributional characteristics of health insurance claim frequency and severity in the study portfolio?

2. Which count distribution provides the best fit to observed claim frequency?

3. Which continuous distribution provides the best fit to observed claim severity?

4. Is there a statistically significant dependence between claim frequency and claim severity?

5. What aggregate claim distribution and pure premium result from the fitted models?

6. How accurately do the selected models predict out-of-sample claim experience?

1.5 Research Hypotheses

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

H₀₁: The Poisson distribution does not provide a significantly better fit to health insurance claim frequency than the negative binomial distribution.

H₀₂: There is no significant difference in goodness of fit among the gamma, lognormal, Weibull and Pareto distributions for claim severity.

H₀₃: There is no significant dependence between claim frequency and claim severity.

H₀₄: Claimant characteristics (age, sex, plan type and provider category) have no significant effect on expected claim cost.

1.6 Significance of the Study

The study will benefit several groups. For insurers and HMOs, it provides an empirically grounded basis for premium loading, capitation rate-setting and reserve estimation, replacing rule-of-thumb pricing with fitted models. For the National Health Insurance Authority and state health insurance agencies, it supplies evidence on the cost structure of claims that can inform benefit-package design and the sizing of the Vulnerable Group Fund. For regulators (NAICOM and the NHIA), the fitted distributions offer a benchmark against which submitted technical provisions can be reviewed, which is directly relevant under the risk-based capital regime introduced by the Nigerian Insurance Industry Reform Act, 2025. For scholarship, the study extends the international frequency severity literature to a Nigerian data setting that has been largely unexamined, and tests in that setting the dependence findings reported by Frees et al. (2011) and Erhardt and Czado (2012). Finally, for policyholders, more accurate pricing reduces the risk of insurer insolvency and of the arbitrary premium increases that follow mis-priced portfolios.

1.7 Scope of the Study

The study covers health insurance claims data obtained from a selected Health Maintenance Organisation (or set of HMOs) operating in Nigeria over a period of not less than five consecutive years. The variables of interest are the number of claims per enrollee per period, the amount paid per claim, and the enrollee and plan characteristics available in the administrative records. Methodologically, the study is limited to parametric and semi-parametric statistical models of frequency, severity and aggregate loss; it does not extend to network pricing, provider contracting design, or health outcome evaluation.

1.8 Limitations of the Study

The study is constrained by the following. (i) Data access and confidentiality claims data are commercially sensitive, and insurers may release only anonymised or partially aggregated extracts, limiting the covariates available for modelling. (ii) Data quality Nigerian administrative claims records are known to contain coding inconsistencies, missing entries and duplicate submissions, which require cleaning that may itself alter distributional properties. (iii) Reporting delay claims incurred but not reported (IBNR) at the data cut-off are unobserved, so estimates of the most recent period may be understated. (iv) Generalisability results obtained from one or a few HMOs may not describe the whole Nigerian market. (v) Inflationary distortion the high and volatile medical-cost inflation experienced in Nigeria during the study period complicates comparison of nominal claim amounts across years, and adjustment assumptions will be required.

1.9 Operational Definition of Terms

Claim frequency: The number of claims arising from a policy or enrollee within a defined period of exposure.

Claim severity: The monetary amount of an individual claim.

Aggregate claim: The total of all claim amounts arising from a portfolio within a period, modelled as a compound distribution of frequency and severity.

Pure premium: The expected claim cost per unit of exposure, exclusive of expenses, profit and contingency loadings.

Capitation: A fixed periodic payment made to a health-care provider per enrolled person, irrespective of the volume of services rendered.

Health Maintenance Organisation (HMO): An organisation accredited by the NHIA to manage the provision of health-care services to enrollees under a health insurance plan.

Zero-inflation: The presence in count data of more zero observations than the assumed count distribution predicts.

Goodness of fit: The degree of agreement between a fitted theoretical distribution and the observed empirical distribution, assessed here by chi-square, Kolmogorov Smirnov, Anderson Darling and information-criterion statistics.

References

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