ANALYSIS OF FIRE INSURANCE CLAIMS USING STATISTICAL MODELS

CHAPTER ONE

INTRODUCTION

1.1 Background to the Study

Fire insurance indemnifies the insured against loss or damage to property caused by fire, lightning and, under extended cover, allied perils including explosion, flooding, riot and malicious damage. It is one of the oldest classes of general insurance and remains among the most significant: in Nigeria, fire business accounted for 20.4 per cent of non-life gross written premium in 2025, second only to oil and gas, and generated ₦246.3 billion in the first half of 2025, an increase of 53.3 per cent year on year (National Insurance Commission [NAICOM], 2026).

The statistical characteristics of fire claims present a distinctive modelling problem. Fire claim severity is extremely right-skewed: the great majority of claims are small, but the class periodically produces losses of a magnitude capable of threatening the solvency of the insurer a market fire, a factory total loss, a building collapse. This combination of low frequency and extreme severity in the upper tail means that the ordinary measures of central tendency convey almost nothing about the risk the insurer carries. The quantity that matters is the behaviour of the tail.

The actuarial literature has responded by developing and testing heavy-tailed and composite severity models. Brazauskas and Kleefeld (2016) fitted generalised Pareto, two versions of the lognormal–Pareto, two versions of the Weibull–Pareto and the folded-t distributions to Norwegian fire claims for 1981 to 1992, measuring tail risk under each. Cooray and Ananda (2005) introduced the composite lognormal–Pareto model precisely because a single distribution rarely fits both the body and the tail of loss data adequately. The extreme value approach, developed for insurance by McNeil (1997), models exceedances over a threshold by the generalised Pareto distribution under the Pickands–Balkema–de Haan theorem, and has been applied to fire portfolios to determine economic risk capital and optimal reinsurance retentions. The Danish fire insurance dataset has become a standard benchmark for this literature.

Nigeria's fire risk profile makes this analysis urgent rather than academic. Market fires, industrial and warehouse fires, and fires arising from electrical faults, generator use and fuel storage are frequent and often catastrophic in loss. Building regulation enforcement and fire service response capacity are limited in many areas. Under-insurance is widespread, since sums insured fixed in naira are rapidly eroded by inflation above thirty per cent and by naira depreciation that raises the replacement cost of imported building materials. Meanwhile, NIIRA 2025 expanded the enforcement of compulsory insurance of public buildings and introduced risk-based capital requiring insurers to quantify their underwriting risk (Federal Republic of Nigeria, 2025). Despite this, there is little Nigerian empirical work fitting

statistical models to fire claims data. This study addresses that gap.

1.2 Statement of the Problem

Nigerian insurers underwrite a fire portfolio whose loss distribution they have not characterised statistically. Four problems follow.

First, tail mis-specification. If a light-tailed distribution such as the gamma or exponential is fitted to data that are in fact heavy-tailed, the probability of extreme loss is understated, the required capital is understated, and the reinsurance retention is set too high. The Nigerian fire portfolio, with its exposure to market and industrial conflagrations, is precisely the case where this error is most costly.

Second, rating without risk differentiation. Fire premiums in Nigeria are commonly set by broad occupancy class and sum insured, with limited reference to construction type, fire protection, location, exposure to neighbouring risks or claims history factors the international literature identifies as significant determinants of both frequency and severity.

Third, reinsurance retention is not optimised. Determining the optimal excess of loss retention requires an estimate of the severity distribution above the threshold. Without a fitted tail model, retentions are set by market convention and capital availability rather than by reference to the insurer's own risk appetite.

Fourth, inflation and currency distortion of the severity series. Historical fire claim amounts denominated in naira are not comparable across years when inflation exceeds thirty per cent and the currency has depreciated by a factor of three. Fitting a distribution to unadjusted nominal data will confound the loss distribution with the price series.

The problem, therefore, is the absence of empirically fitted statistical models for Nigerian fire insurance claims, and the pricing, reserving, capital and reinsurance errors that follow from it.

1.3 Aim and Objectives of the Study

The aim of this study is to analyse fire insurance claims in Nigeria using appropriate statistical models.

The specific objectives are to:

1. examine the pattern, trend and descriptive characteristics of fire insurance claims in the study portfolio;

2. fit and compare candidate probability distributions to fire claim severity, including the lognormal, gamma, Weibull, Pareto, generalised Pareto and composite lognormal–Pareto models;

3. fit and compare candidate count distributions to fire claim frequency;

4. model the tail of the fire claim severity distribution using extreme value methods;

5. estimate the aggregate fire loss distribution and the associated risk measures; and

6. determine the effect of risk characteristics occupancy, construction, location, sum insured and fire protection on fire claim frequency and severity.

1.4 Research Questions

1. What are the pattern, trend and descriptive characteristics of fire insurance claims in the study portfolio?

2. Which probability distribution best describes fire claim severity in Nigeria?

3. Which count distribution best describes fire claim frequency in Nigeria?

4. What does extreme value analysis reveal about the tail of the Nigerian fire claim severity distribution?

5. What aggregate fire loss distribution and risk measures result from the fitted models?

6. What effect do risk characteristics have on fire claim frequency and severity?

1.5 Research Hypotheses

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

H₀₁: Fire insurance claim severity in Nigeria does not follow a lognormal distribution.

H₀₂: There is no significant difference in goodness of fit among the candidate severity distributions for Nigerian fire claims.

H₀₃: The fire claim severity distribution in Nigeria is not heavy-tailed.

H₀₄: Fire claim frequency in Nigeria does not follow a Poisson distribution.

H₀₅: Risk characteristics have no significant effect on fire claim frequency and severity in Nigeria.

1.6 Significance of the Study

For insurance companies, the fitted models supply the basis for fire premium rating, technical provision estimation, capital allocation to the fire account and the determination of optimal reinsurance retentions. For NAICOM, the study bears on the calibration of underwriting risk for fire business within the risk-based capital framework, and on the supervision of insurers underwriting concentrated fire exposures. For reinsurers, the estimated tail parameters are direct inputs to excess of loss pricing. For property owners, businesses and public authorities, the evidence on the distribution of fire loss severity supports investment in fire prevention and strengthens the case for realistic sum insured indexation. For the Federal and State Fire Services and building regulators, quantified fire loss severity translates the safety case into financial terms. For scholarship, the study supplies Nigerian evidence to a fire loss modelling literature built almost entirely on Danish, Norwegian and other European datasets, testing whether the distributional findings of that literature hold in a market with different construction standards, fire response capacity and enforcement.

1.7 Scope of the Study

The study covers fire insurance claims data obtained from one or more insurance companies licensed by NAICOM in Nigeria, over a period of not less than five consecutive years, covering fire and special perils business including extended perils cover where written. The variables examined are individual claim amounts, claim counts per period, and, where recorded, risk characteristics including occupancy class, construction type, location, sum insured, fire protection measures and policy deductible. Methodologically the study covers descriptive analysis, parametric distribution fitting by maximum likelihood, composite distribution fitting, goodness-of-fit testing by Kolmogorov–Smirnov, Anderson–Darling and chi-square statistics, model comparison by information criteria, extreme value analysis by both block maxima and peaks over threshold approaches, and regression modelling of frequency and severity on risk characteristics. Claim amounts are deflated to a common price basis. The study does not cover motor, marine, oil and gas or life business.

1.8 Limitations of the Study

(i) Sparse tail data the extreme losses that dominate the fire risk profile are by definition rare, so a five-year single-company dataset may contain too few large claims for stable tail parameter estimation. (ii) Threshold selection peaks over threshold results are sensitive to the choice of threshold, and the standard diagnostic tools for that choice are themselves subjective. (iii) Inflation and currency adjustment deflating a naira claim series over a period of thirty per cent inflation and threefold depreciation requires assumptions about the appropriate index, and results are sensitive to that choice. (iv) Under-insurance where sums insured are below replacement value, settled claim amounts understate actual losses and the fitted severity distribution is biased downward. (v) Risk characteristic recording Nigerian policy records frequently omit construction, protection and exposure details, limiting the covariate analysis. (vi) Reporting and settlement delay large fire claims often take years to settle, so recent periods are incomplete.

1.9 Operational Definition of Terms

Fire insurance: A contract of indemnity covering loss or damage to property caused by fire, lightning and, under extended cover, allied perils.

Claim severity: The monetary amount of an individual claim.

Heavy-tailed distribution: A probability distribution whose tail decays more slowly than exponentially, so that extreme values occur with non-negligible probability.

Generalised Pareto distribution (GPD): The limiting distribution of exceedances over a high threshold, used in the peaks over threshold approach to extreme value modelling.

Composite lognormal–Pareto model: A severity model using a lognormal density below a threshold and a Pareto density above it, joined to be continuous and differentiable.

Peaks over threshold (POT): An extreme value method that models the distribution of the excesses of observations above a chosen high threshold.

Block maxima: An extreme value method that models the distribution of the maximum observation within each of a sequence of blocks, using the generalised extreme value distribution.

Retention: The amount of loss an insurer retains on its own account before reinsurance responds.

Sum insured: The maximum amount payable under a policy, nominally representing the value of the insured property.

Under-insurance (average condition): The situation in which the sum insured is less than the value at risk, triggering a proportionate reduction in claim settlement.

References

Brazauskas, V., & Kleefeld, A. (2016). Modeling severity and measuring tail risk of Norwegian fire claims. North American Actuarial Journal, 20(1), 1–16. https://doi.org/10.1080/10920277.2015.1062784

Cooray, K., & Ananda, M. M. A. (2005). Modeling actuarial data with a composite lognormal-Pareto model. Scandinavian Actuarial Journal, 2005(5), 321–334. https://doi.org/10.1080/03461230510009763

Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling extremal events for insurance and finance. Springer.

Federal Republic of Nigeria. (2025). Nigerian Insurance Industry Reform Act, 2025. Federal Government Press.

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

McNeil, A. J. (1997). Estimating the tails of loss severity distributions using extreme value theory. ASTIN Bulletin, 27(1), 117–137. https://doi.org/10.2143/AST.27.1.563210

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.

Pickands, J. (1975). Statistical inference using extreme order statistics. Annals of Statistics, 3(1), 119–131. https://doi.org/10.1214/aos/1176343003