Insurance and Actuarial Analytics

Introduction

Insurance converts uncertain losses into relatively predictable premiums through pooling, classification and capital support. Financial analytics enables insurers to estimate how often claims occur, how costly they are, when they will be settled and whether sufficient premiums, reserves and capital exist to meet obligations.

Unlike many banking datasets, insurance data contain exposure differences, censoring, delayed reporting, extreme losses and contractual modifications such as deductibles and policy limits. Consequently, actuarial analytics must combine statistical modelling with accounting consistency, insurance law, governance and professional judgment.

Learning Objectives

After studying this chapter, students should be able to:

1. Model claim frequency, severity and aggregate loss.
2. Calculate incurred claims, loss ratios and combined ratios correctly.
3. Explain risk classification and estimate premiums using segmentation models.
4. Apply chain-ladder and Bornhuetter–Ferguson reserving methods.
5. Analyse mortality and survival using life-contingency models.
6. Distinguish net premium reserves from gross premium reserves.
7. Evaluate fraud indicators, catastrophe scenarios and solvency measures.
8. Assess the ethical, regulatory and managerial implications of insurance analytics.

1. The Insurance Analytics Framework

An insurer's analytical process connects exposure, claims, pricing, reserving and capital.

Figure: Insurance analytics is a feedback cycle linking experience, pricing, reserving and capital management.

The basic unit of analysis is exposure. One policy insured for a full year normally represents one policy-year; a six-month policy represents 0.5 policy-years. Claim counts should not be compared without adjusting for exposure.

2. Frequency, Severity and Aggregate Loss

2.1 Claim frequency

Let N denote the number of claims generated by an exposure during a specified period. A common starting model is the Poisson distribution:

Here, λ is the expected number of claims for the exposure period. Under this model, E[N]=Var(N)=λ. It is most appropriate when events are conditionally independent and occur at a stable rate.

Insurance counts often exhibit overdispersion, meaning that observed variance exceeds the mean. A negative binomial model can accommodate this. Under one common parameterisation,

where μ is expected frequency and α>0 is a dispersion parameter. The negative binomial model may represent unobserved differences among policyholders or contagion-like claim processes.

2.2 Claim severity

Claim severity is the monetary amount of an individual loss, conditional on a claim occurring. Let X denote the ground-up loss. Common models include:

Contract terms convert the ground-up loss into the insurer's payment. With deductible d, payment limit L and coinsurance proportion c, the insurer payment is

where (X-d)_+=max(X-d,0), 0≤c≤1, and d, L, X and Y are measured in the same currency. The formula assumes the limit applies to the amount above the deductible.

2.3 Aggregate loss

For claim count N and individual payments X₁,…,X_N, aggregate loss is

If severities are independent and identically distributed, independent of N, with mean μ_X and variance σ_X², then conditioning on N gives

For a Poisson count with parameter λ, this becomes

These relationships may be invalid when claim severities are dependent, claims arise from the same catastrophe, or frequency and severity share common drivers.

Worked Example: Pure Premium

A motor portfolio has 1,000 policy-years of exposure, expected frequency 0.08 claims per policy-year and expected severity CU2,500 per claim, where CU denotes currency units.

Expected claims are 1,000×0.08=80. Expected aggregate loss is

The pure premium per policy-year is therefore

The pure premium covers expected claims only; it does not cover expenses, profit, risk margins or capital costs.

3. Claims Measurement and Performance Ratios

3.1 Paid, outstanding and incurred claims

Claims may be reported and settled long after the policy period. The following quantities must therefore be distinguished:

• Payments during the period: claim cash paid between the opening and closing valuation dates.
• Cumulative paid claims: all claim payments from origin to the valuation date.
• Case reserves: estimates for reported but unpaid claims.
• IBNR: claims incurred but not reported, including development on known claims where applicable.

For a reporting period, incurred losses are

Outstanding reserve should be consistently defined at both dates, normally as case reserves plus IBNR. Equivalently, when cumulative data for the same group of claims are used:

Using an unspecified “paid loss” in the period formula is incorrect because it may be confused with cumulative payments.

3.2 Loss and combined ratios

The incurred loss ratio is

Earned premium is the portion of written premium attributable to expired insurance coverage. The expense ratio should be calculated on the basis specified by the insurer or regulator; a common management definition is underwriting expenses divided by earned premium.

A combined ratio below 100% indicates an underwriting profit before investment income and certain non-underwriting items.

Worked Example: Calendar-Year Performance

An insurer records opening outstanding reserves of CU1.2 million, payments during the year of CU3.8 million and closing outstanding reserves of CU1.5 million. Earned premium is CU5.5 million and underwriting expenses are CU1.1 million.

The combined ratio is 94.55%. The portfolio generated an underwriting margin of 5.45% of earned premium, subject to reserve estimation uncertainty.

4. Segmentation, Underwriting and Premium Analytics

Risk segmentation groups policyholders according to characteristics associated with expected loss. Examples include vehicle type, location, age of building, occupation, coverage level and prior claims. A valid variable should be predictive, legally permissible, available at decision time and sufficiently stable.

Generalised linear models (GLMs) are widely used because they connect risk factors to expected outcomes while controlling for other variables. A Poisson frequency model with exposure e_i may use

The term log(e_i) is an offset with coefficient fixed at one. Exponentiating a coefficient gives a multiplicative relativity, holding other variables constant. Gamma GLMs with a log link are frequently used for positive claim severity.

Expected loss cost for segment i is

If F is fixed expense per policy, e is the proportion of premium consumed by variable expenses and r is the target profit-and-capital proportion, a simplified indicated gross premium is

This formula requires e+r<1 and consistent per-policy units. Actual pricing must also consider reinsurance, taxes, policyholder behaviour, competition, regulation and uncertainty.

Segmentation creates ethical and legal concerns. A statistically predictive variable may be unacceptable if it acts as a proxy for protected status, creates unjustified disparate impacts or cannot be explained. Insurers should test model stability, calibration and fairness across relevant groups.

5. Claims Reserving

A claims reserve estimates future payments for losses that have already occurred. It is not a savings account; it is a model-based liability estimate subject to revision.

5.1 Development triangles and chain-ladder

A development triangle organises cumulative claims by accident period and development age. The volume-weighted age-to-age factor is

where C_i,j is cumulative paid or incurred claims for accident period i at development age j. Only accident periods observed at both ages are included.

The cumulative development factor from age j to ultimate is

Estimated ultimate claims and reserves are

The chain-ladder method assumes that historical development patterns remain applicable and that the selected triangle uses a consistent claim measure. Changes in inflation, settlement practices, coverage or case-reserving policy can violate this assumption.

Worked Example: Chain-Ladder

Cumulative claims, in CU millions, are:

The development factors are

Assuming 36 months is ultimate, the 2023 ultimate estimate is 22.5×1.20=CU27.0 million, giving a reserve of CU4.5 million. The 2024 estimate is 18.0×1.50×1.20=CU32.4 million, giving a reserve of CU14.4 million. The total indicated reserve is CU18.9 million.

5.2 Bornhuetter–Ferguson method

The Bornhuetter–Ferguson (BF) method combines an independent prior estimate of ultimate claims with observed development. If p_i is the estimated proportion reported or paid to date, and U_i^prior is the a priori ultimate loss estimate, then

The observed amount and p_i must use the same basis—paid with paid, or reported with reported. For 2024 above, p=1÷(1.50×1.20)=0.5556. If the prior ultimate is CU30 million:

The BF ultimate is 18+13.33=CU31.33 million. BF is less responsive than chain-ladder to unusually high or low early claim emergence.

6. Mortality and Survival Analysis

Let T_x be the future lifetime of a person currently aged x. The survival function is

and the cumulative probability of death by time t is {}_tq_x=1−{}_tp_x. If a continuous lifetime distribution has density f_x(t), its force of mortality is

The force of mortality has units of deaths per unit of time, conditional on survival to that time.

For yearly models, let K_x=⌊T_x⌋ be the curtate future lifetime. For an n-year term insurance paying one unit at the end of the year of death, the present-value random variable is

where v=(1+i)^-1, i is the annual effective interest rate and the indicator ensures that no benefit is paid after the term. Without the indicator, the variable would represent whole-life insurance rather than term insurance.

Its expected present value is

assuming death benefits are paid at year-end and mortality and interest follow the valuation basis.

Worked Example: Three-Year Term Insurance

Suppose annual conditional death probabilities are 0.010, 0.012 and 0.015, and i=5%. The expected present value of a CU100,000 death benefit is

This is the single net premium under the stated assumptions. It excludes expenses, profit and risk margins.

6.1 Censoring and empirical survival models

Life, health and disability observations are often right-censored because the event has not occurred when observation ends. The Kaplan–Meier estimator is

where d_j is the number of events at time t_j and n_j is the number at risk immediately before that time. The estimator assumes censoring is independent of the event process, conditional on relevant information.

The Cox proportional-hazards model relates covariates to event intensity:

It assumes covariate hazard ratios are proportional over time. Diagnostics should test this condition rather than treating it as automatic.

7. Life Insurance Premiums and Reserves

Under the equivalence principle, the expected present value (EPV) of net premiums equals the EPV of benefits on the pricing basis.

A prospective net premium reserve at duration t is

Future expenses are not included in a net premium reserve. A gross premium valuation instead uses actual or valuation gross premiums and explicitly includes future expenses:

Both calculations depend on mortality, lapse and interest assumptions. Gross premium valuations may also incorporate options, guarantees, reinsurance and risk adjustments where required by the relevant accounting or regulatory framework.

8. Fraud Detection Analytics

Insurance fraud may involve staged events, inflated losses, false identities, provider collusion or duplicate claims. Analytical approaches include:

• supervised classification using labelled historical investigations;
• anomaly detection for unusual claims without reliable labels;
• network analysis linking claimants, providers, addresses and bank accounts;
• text analytics applied to adjuster notes and claim narratives;
• image and metadata analysis, subject to authenticity and privacy controls.

A fraud model should support investigation rather than automatically declare guilt. Important evaluation measures include precision, recall, false-positive rate, expected investigation value and avoided loss. Accuracy alone is misleading when fraud is rare.

Models may learn historical investigator bias rather than fraud itself. Governance therefore requires human review, reason codes, access controls, appeal mechanisms, drift monitoring and restrictions on sensitive information.

9. Catastrophe Risk, Dependence and Solvency

Catastrophes violate the assumption that policyholder losses are independent. A single cyclone, earthquake, pandemic or cyber event may produce thousands of correlated claims.

A catastrophe model commonly combines:

1. a hazard module describing event frequency and intensity;
2. an exposure module locating insured assets or lives;
3. a vulnerability module translating intensity into damage;
4. a financial module applying deductibles, limits and reinsurance.

For loss variable L, value at risk at confidence level α is

Tail value at risk, for a continuous loss distribution, is

TVaR describes average loss beyond the selected quantile and therefore gives more information about tail severity. For distributions with atoms, a general expected-shortfall definition should be used rather than the simple conditional formula.

A simplified solvency ratio is

If eligible own funds are CU20 million and required capital is CU15 million, the ratio is 133.3%. Interpretation depends on local rules governing asset eligibility, liability valuation, diversification and capital tiers.

Scenario analysis complements probabilistic models. Scenarios should consider inflation, climate trends, litigation, cyber accumulation, reinsurance default and operational disruption. Reverse stress testing asks which combinations of events could make the insurer's business model unviable.

10. Illustrative Integrated Case: Meridian General Insurance

Background

Meridian General Insurance is a fictional motor insurer with 50,000 policy-years of annual exposure. Management is reviewing pricing adequacy, claim reserves, fraud controls and catastrophe resilience.

Problem

Recent claim severity has increased because of parts inflation. Urban segments show higher claim frequency, and management suspects organised repair-shop fraud. The board also wants to understand the effect of a major flood on solvency.

Analysis

The pricing model projects 4,250 claims with average severity of CU3,200, producing expected losses of CU13.6 million. Calendar-year data show payments during the year of CU10.8 million, opening outstanding reserves of CU4.6 million and closing outstanding reserves of CU5.9 million.

With earned premium of CU18.5 million and underwriting expenses of CU4.2 million:

• loss ratio = 12.1÷18.5=65.4%;
• expense ratio = 4.2÷18.5=22.7%;
• combined ratio = 88.1%.

A claim-triangle review indicates that recent accident periods are developing faster than older periods. Actuaries therefore supplement chain-ladder estimates with BF estimates based on current exposure and severity trends.

A fraud model refers 220 claims for investigation, of which 88 are confirmed problematic. Referral precision is 88÷220=40%. This is higher than the prior manual referral precision of 25%, but investigators review all referrals before action.

A flood stress test estimates gross losses of CU28 million, reinsurance recoveries of CU16 million and CU2 million of additional operational and reinstatement costs. The net stress impact is therefore CU14 million. With pre-stress eligible own funds of CU19 million, only CU5 million would remain in this simplified static scenario, demonstrating serious vulnerability even though the pre-stress solvency requirement is met.

Approach and Outcome

Meridian introduces quarterly severity monitoring, geographically refined but fairness-tested pricing, higher flood reinsurance limits and stricter repair-provider network controls. It also establishes reserve ranges rather than relying only on a point estimate. The board adopts risk appetite triggers tied to combined ratio, reserve deterioration, reinsurance exhaustion and solvency coverage.

Learning Points

• A favourable current combined ratio does not guarantee resilience to catastrophe losses.
• Calendar-year incurred claims and accident-year ultimate claims answer different questions.
• Reserving methods should reflect changing development conditions.
• Fraud analytics should prioritise investigations, not replace due process.
• Pricing, reserving, reinsurance and capital decisions must be evaluated jointly.

11. Governance, Regulation and Contemporary Developments

Insurance analytics operates within accounting, conduct and prudential requirements. IFRS 17, where applicable, measures groups of insurance contracts using fulfilment cash flows and a contractual service margin; it should not be confused with solvency capital regulation. Local regulators may impose separate reserving, capital and stress-testing rules.

Good model governance includes documented ownership, data lineage, independent validation, change control, performance thresholds and board-level reporting. Personally identifiable and health information require especially strong privacy safeguards.

Contemporary developments include climate-conditioned catastrophe models, telematics-based motor pricing, wearable health data, automated damage estimation, generative AI for claim-document processing and synthetic data for privacy-preserving development. These technologies can improve speed and granularity but create model, cybersecurity, explainability and discrimination risks.

Best practice is to combine statistical evidence with contractual knowledge, actuarial judgment and explicit uncertainty analysis. Estimates should be reproducible, monitored and communicated as ranges when point estimates would imply false precision.

Chapter Summary

Insurance analytics models uncertain claim frequency, severity and timing to support underwriting, pricing, reserving and capital management. Aggregate loss depends on both claim counts and individual severities, while policy terms transform ground-up losses into insurer payments. Incurred losses must use payments during the period and changes in outstanding reserves consistently.

Loss and combined ratios measure underwriting performance but remain sensitive to reserve estimates. GLMs enable exposure-adjusted segmentation, while chain-ladder and Bornhuetter–Ferguson methods estimate unpaid claims using different balances of experience and prior expectations. Mortality and survival models support life-contingent pricing, and a term-insurance present value must explicitly restrict payment to deaths within the term. Net premium reserves exclude expenses; gross premium valuations include future expenses and gross premiums.

Fraud detection, catastrophe modelling and solvency analytics broaden the analysis from expected losses to operational integrity and tail resilience. Effective practice requires model validation, ethical segmentation, regulatory compliance, human oversight and integrated management of pricing, reserves, reinsurance and capital.

Key Terms

Aggregate loss: Total payments generated by all claims in a specified portfolio and period.

Bornhuetter–Ferguson method: Reserving method combining observed emergence with an independent prior ultimate-loss estimate.

Case reserve: Estimate of the unpaid amount on a reported claim.

Claim frequency: Expected or observed number of claims per unit of exposure.

Claim severity: Monetary size of a claim, usually conditional on occurrence.

Combined ratio: Sum of the loss ratio and underwriting expense ratio.

Curtate future lifetime: Number of complete future years lived by an insured person.

Earned premium: Premium attributable to insurance coverage already provided.

Exposure: Measure of the amount of risk insured over a period.

Generalised linear model: Regression framework linking a response distribution to predictors through a link function.

IBNR: Reserve for claims incurred but not yet reported, including relevant development provisions.

Incurred losses: Period claim payments plus the change in outstanding claim reserves.

Loss ratio: Incurred losses divided by earned premium.

Net premium reserve: EPV of future benefits minus EPV of future net premiums.

Pure premium: Expected claim cost per exposure unit before expenses and margins.

Solvency ratio: Eligible own funds divided by required solvency capital.

Survival function: Probability that future lifetime or event time exceeds a specified duration.

Tail value at risk: Expected loss in the tail beyond a selected confidence threshold, subject to its precise distributional definition.

Review Questions

Short Answer Questions

1. Distinguish claim frequency from claim severity.
2. Why must exposure be included in insurance frequency models?
3. Define incurred losses for a reporting period.
4. What is the role of the term indicator in a term-insurance present value?
5. Why can classification accuracy be misleading in fraud detection?

Long Answer Questions

1. Explain the compound loss model and derive its mean and variance under the stated independence assumptions.
2. Compare chain-ladder and Bornhuetter–Ferguson reserving, including their assumptions and limitations.
3. Explain how GLMs support insurance segmentation and discuss associated fairness risks.
4. Distinguish net premium and gross premium reserves and identify the cash flows included in each.
5. Discuss how catastrophe models, reinsurance and solvency capital interact in portfolio management.

Analytical / Application Questions

1. An insurer has payments during the year of CU7 million, opening outstanding reserves of CU3 million and closing outstanding reserves of CU4.5 million. Calculate incurred losses and the loss ratio if earned premium is CU12 million.
2. A development factor to ultimate is 1.40 and cumulative reported claims are CU10 million. Calculate the chain-ladder reserve. Then calculate the BF reserve if the prior ultimate is CU13 million.
3. Design a governance framework for an AI fraud model, addressing performance, fairness, privacy, human review and model drift.

Multiple Choice Questions with Answers

Question 1

A portfolio has expected frequency 0.05 claims per policy-year and expected severity CU4,000. What is the pure premium per policy-year?

A. CU80

B. CU200

C. CU800

D. CU4,050

Correct Answer: B. CU200

Question 2

An insurer pays CU6 million during the year. Outstanding reserves are CU2 million at the beginning and CU3.5 million at the end. What are incurred losses for the year?

A. CU4.5 million

B. CU6.0 million

C. CU7.5 million

D. CU11.5 million

Correct Answer: C. CU7.5 million

Question 3

Which random variable correctly represents a unit benefit payable at the end of the year of death under an n-year term insurance?

A. v^K_x+11\{K_x<n\}

B. v^K_x+1

C. vⁿ1\{K_x≥n\}

D. (1+i)^K_x+11\{K_x<n\}

Correct Answer: A. v^K_x+11\{K_x<n\}

Question 4

Cumulative reported claims are CU12 million and the estimated proportion reported is 60%. If the a priori ultimate loss is CU20 million, what is the Bornhuetter–Ferguson reserve?

A. CU4 million

B. CU8 million

C. CU12 million

D. CU20 million

Correct Answer: B. CU8 million

Question 5

Which item is excluded from a conventional prospective net premium reserve?

A. Expected present value of future benefits

B. Expected present value of future net premiums

C. Expected present value of future expenses

D. Valuation interest assumptions

Correct Answer: C. Expected present value of future expenses

Question 6

An insurer has incurred losses of CU9 million, underwriting expenses of CU3 million and earned premium of CU15 million. What is its combined ratio?

A. 60%

B. 70%

C. 80%

D. 120%

Correct Answer: C. 80%

Question 7

Why is a negative binomial model often preferred to a Poisson model for insurance claim counts?

A. It requires claim severity to be normally distributed.

B. It can represent count variance greater than the mean.

C. It guarantees that claims are independent.

D. It eliminates the need to measure exposure.

Correct Answer: B. It can represent count variance greater than the mean.

Question 8

Eligible own funds are CU24 million and required solvency capital is CU18 million. What is the solvency ratio?

A. 75.0%

B. 100.0%

C. 133.3%

D. 142.0%

Correct Answer: C. 133.3%

Question 9

Which statement best describes tail value at risk for a continuous loss distribution?

A. It is the most likely annual loss.

B. It is the average loss conditional on exceeding the selected VaR threshold.

C. It is always lower than value at risk at the same confidence level.

D. It measures underwriting expenses relative to premium.

Correct Answer: B. It is the average loss conditional on exceeding the selected VaR threshold.

Question 10

What is the most appropriate operational use of a fraud-detection score?

A. Automatically establish criminal liability

B. Replace all claim investigations

C. Prioritise claims for controlled human investigation

D. Reject every claim above the score threshold

Correct Answer: C. Prioritise claims for controlled human investigation