Actuarial Foundations, Pricing and Reserving

Introduction

Insurance converts uncertain future losses into relatively predictable premiums. Actuarial work supports this process by combining probability, statistical data, financial mathematics and professional judgment. In life insurance, actuaries model survival, death, disability and policyholder behaviour over long periods. In general insurance, they commonly model how often claims occur, how severe they are and how long settlement takes.

Pricing and reserving address different questions. Pricing determines the premium required for future coverage. Reserving estimates the insurer’s present obligation for coverage already provided or claims already incurred. Both activities must recognize uncertainty, expenses, investment income, capital requirements, regulation and the quality of available data.

Learning Objectives

After studying this chapter, students should be able to:

1. apply basic probability and expected-value concepts to insurance losses;
2. interpret mortality and morbidity tables;
3. apply the time value of money to insurance cash flows;
4. calculate expected losses using frequency–severity models;
5. distinguish pure, net and gross premiums;
6. calculate and interpret loss and expense ratios;
7. explain basic life and general insurance reserves; and
8. evaluate modelling assumptions, uncertainty and governance requirements.

1. Insurance Risk as a Random Variable

A random variable assigns a numerical value to an uncertain outcome. Let X represent the amount of a claim in currency units. If X can take values x₁,x₂,…,x_n with probabilities p₁,p₂,…,p_n, its expected value is:

Here, E[X] is the long-run average loss per exposure, x_i is a possible loss amount and p_i is its probability. The probabilities must be non-negative and sum to one. The expected value is expressed in the same currency units as X.

Variance measures dispersion around the expected value:

The standard deviation is σ_X = √Var(X). It has the same units as the loss variable and therefore provides a more directly interpretable measure of volatility than variance.

Worked Example 1: Expected Loss

A policy produces no claim with probability 0.96, a claim of CU 10,000 with probability 0.03 and a claim of CU 50,000 with probability 0.01. The expected loss is:

This does not mean every policy will cost CU 800. It means that, across a sufficiently large and stable portfolio with comparable risks, the average claim cost is expected to approach CU 800.

Law of Large Numbers

If risks are sufficiently numerous, comparable and not excessively dependent, the average observed loss tends to become more stable as portfolio size increases. Insurance pooling relies on this principle. However, diversification is weakened by correlated losses such as pandemics, earthquakes, inflation shocks or cyber events affecting many policies simultaneously.

2. Mortality Models and Life Tables

Mortality measures the occurrence of death in a population. A mortality table summarizes death and survival probabilities by age, sex or other permitted risk classifications. Its principal symbols include:

• q_x: probability that a life aged exactly x dies before age x+1;
• p_x: probability that the life survives one year, where p_x = 1-q_x;
• l_x: expected number of survivors to exact age x from a chosen initial cohort;
• d_x: expected deaths between ages x and x+1.

The life-table relationships are:

These relationships assume that the table’s mortality probabilities apply to the cohort being modelled.

Illustrative Mortality Table

The decimal survivor counts are theoretical expected counts rather than literal numbers of people.

Multi-Year Survival and Death Probabilities

Under the age-specific probabilities in the table, the probability that a life aged 40 survives three complete years is:

The probability of dying within those three years is:

Worked Example 2: Three-Year Mortality

Using the illustrative table:

Therefore:

Thus, the probability of death before age 43 is approximately 3.5574%. This calculation uses successive conditional survival probabilities; adding annual death probabilities would be incorrect because each later probability applies only to survivors.

Selecting a Mortality Table

Mortality varies by calendar period, age, underwriting class, occupation, location and socioeconomic conditions. Common table types include population tables, insured-life tables, select tables that reflect recent underwriting, and ultimate tables applying after the selection effect diminishes. Actuaries may incorporate mortality improvement because future mortality can differ systematically from historical mortality.

3. Morbidity Tables and Health-Related Decrements

Morbidity refers to illness, injury, disability or other adverse health states rather than death. It is central to health, critical illness, disability income, long-term care and workers’ compensation insurance.

Important measures include:

• Incidence: rate at which new cases or claims begin among exposed healthy lives;
• Prevalence: proportion of a population in a specified health state at a point or during a period;
• Claim termination rate: probability that an active claim ends through recovery, death or expiry;
• Average duration: expected time spent on claim;
• Cost per episode: expected payment associated with one illness or disability event.

Incidence and prevalence are not interchangeable. Prevalence depends on both the rate of new cases and their duration.

Illustrative Annual Morbidity Table

The following fictional table represents one-year illness-claim experience. An exposed healthy life is a person eligible to experience a new claim during the year.

For ages 40–49, annual incidence is:

Thus, about 2.1% of comparable healthy lives are expected to generate a new claim during one year. If each person can have at most one episode and the benefit equals the listed average episode cost, the expected annual claim cost per exposed life is:

This simplified estimate is inappropriate when recurrent claims are material. In that case, claim frequency per exposure—not merely first-onset incidence—must be modelled. Disability income valuation may also require a multi-state model covering healthy, disabled, recovered and dead states.

4. Time Value of Money

Insurance premiums may be received years before or after claim payments. Cash flows at different dates are therefore not directly comparable.

For an effective annual interest rate i, the one-year discount factor is:

The present value of an amount C_t payable at the end of year t is:

The units of present value are currency at the valuation date. These formulas assume annual compounding and a deterministic interest rate. More advanced models may use yield curves or stochastic interest rates.

Expected Present Value

If a benefit B_t is paid at time t only when an event occurs with probability π_t, its expected present value is:

The event probabilities must be defined consistently. For death benefits, π_t is normally the probability that the insured survives to the start of year t and dies during that year.

Worked Example 3: Three-Year Term Insurance

A three-year policy issued at age 40 pays CU 100,000 at the end of the year of death. Annual death probabilities are 0.010, 0.012 and 0.014, and the effective annual interest rate is 5%.

The expected present value of benefits is:

With v=1÷1.05=0.95238095, this gives approximately:

Suppose equal annual net premiums are payable at the beginning of each year while the insured is alive. The expected present value of premium payments of CU 1 each is:

Under the equivalence principle, expected present values of net premiums and benefits are equal. Therefore, the annual net premium is:

This net premium excludes expenses, taxes, profit margins, capital costs and risk margins.

5. Frequency–Severity Models

General insurance claim costs are commonly decomposed into claim frequency and claim severity. Let:

• N be the number of claims during a period;
• X_j be the amount of claim j;
• S be aggregate loss.

Then:

If N is independent of identically distributed severities and both expectations are finite, then:

The expected aggregate loss therefore equals expected frequency multiplied by expected severity.

Worked Example 4: Pure Risk Cost

A motor portfolio has expected claim frequency of 0.12 claims per vehicle-year and expected severity of CU 18,000 per claim. Its expected annual loss per vehicle is:

The unit calculation is consistent: claims per vehicle-year multiplied by currency per claim gives currency per vehicle-year.

Aggregate Loss Variance

Under the same independence assumptions:

If claim count follows a Poisson distribution with parameter λ, then E[N]=Var(N)=λ, giving the special-case result:

This Poisson formula should not be generalized to overdispersed or dependent claim counts. Real portfolios may exhibit catastrophe dependence, inflation, deductibles, policy limits and reporting delays.

6. Premium Construction

Pure, Net and Gross Premiums

A pure premium is the expected claim cost per exposure. In general insurance it is often frequency multiplied by severity. A net premium in life insurance commonly equates the expected present value of premiums with the expected present value of benefits. A gross premium additionally allows for expenses, taxes, commissions, reinsurance, risk, capital and profit.

A simplified gross-premium formula is:

where:

• G = gross premium per policy;
• L = expected claim cost per policy;
• F = fixed expense per policy;
• e = variable expense proportion of premium;
• r = risk and profit margin proportion;
• t = premium tax or levy proportion.

This formula applies only when the proportional components are defined as percentages of gross premium and e+r+t<1.

Worked Example 5: Gross Premium

Using the expected motor loss of CU 2,160, suppose fixed expenses are CU 300, variable expenses are 20% of premium, the risk and profit margin is 8%, and premium tax is 2%.

The expected loss ratio is 2,160÷3,514.29=61.46%. Pricing should also examine sensitivity to claim inflation, adverse selection, catastrophe accumulation and uncertain parameter estimates.

Figure: Data and actuarial models support both premium determination and reserve estimation.

7. Performance Ratios

The loss ratio measures claims relative to earned premium:

Incurred claims normally equal claims paid plus the increase in outstanding claim reserves over the period. Earned premium is the portion of premium attributable to expired coverage.

The expense ratio is:

Analysts must identify whether written or earned premium is used because accounting conventions vary. A simplified combined ratio is:

A combined ratio below 100% indicates an underwriting profit before investment income; a ratio above 100% indicates an underwriting loss. The ratio alone does not measure total profitability because it excludes investment returns and the cost of capital.

8. Life Insurance Reserving

A life insurance reserve is a valuation-date measure of the insurer’s obligation under in-force contracts. Under a prospective approach:

Here, V_t is the reserve at time t, conditional on the policy being in force. Assumptions must be consistent regarding mortality, morbidity, lapses, expenses, interest rates, guarantees and policyholder options.

A retrospective reserve accumulates past premiums and investment income and subtracts benefits and expenses already incurred. Under idealized consistent assumptions, prospective and retrospective approaches can produce equivalent results. In practice, updated assumptions, acquisition expenses and contractual features may lead to differences.

Life reserves often increase as insured persons age or as future premiums become insufficient relative to future benefits. Negative reserves may arise for some contracts under particular valuation rules, especially where substantial future premiums are expected.

9. General Insurance Reserving

General insurance liabilities are complicated by reporting and settlement delays. Major reserve categories include:

• Unearned premium reserve: premium relating to future coverage periods;
• Case reserve: claim-specific estimate for a reported claim;
• IBNR reserve: liability for claims incurred but not yet reported;
• IBNER reserve: additional development on reported claims whose current case reserves are inadequate;
• Claims-handling expense reserve: future cost of settling outstanding claims.

Unearned Premium Reserve

Under a pro rata assumption, a 12-month policy with premium CU 1,200 and nine months of coverage remaining has:

This assumes risk emerges uniformly. Seasonal or catastrophe-exposed contracts may require a non-linear earning pattern.

Basic Chain-Ladder Method

The chain-ladder method projects cumulative paid or incurred claims using historical development factors. Consider the following cumulative paid-claims triangle in CU thousands:

Volume-weighted age-to-age factors are:

Projected ultimate claims are:

Amounts are in CU thousands. The estimate assumes that historical development patterns remain relevant, no development occurs after 36 months and the triangle is sufficiently homogeneous. Changes in claims handling, inflation, legal decisions or portfolio mix can violate these assumptions.

10. Illustrative Case: Pricing and Monitoring a Motor Portfolio

Background

This fictional case concerns Horizon Mutual, which plans to price 10,000 annual motor policies. Its analysis estimates claim frequency at 0.10 claims per vehicle-year and average severity at CU 15,000.

Problem

Management must establish a sustainable premium and determine whether emerging claims experience remains consistent with pricing assumptions.

Analysis and Approach

Expected claim cost per policy is:

Fixed expenses are CU 250 per policy. Variable expenses equal 18% of gross premium, the capital and profit margin is 7%, and taxes equal 2%. Therefore:

Expected portfolio premium is approximately CU 23.973 million, while expected claims are CU 15 million. The expected loss ratio is about 62.57%.

After one year, incurred claims reach CU 16.2 million and all premium is earned. The actual loss ratio is:

Outcome

The actual ratio is 5.01 percentage points above expectation. Management should not automatically increase every premium. It should investigate whether the deterioration resulted from frequency, severity, inflation, a few large losses, reserve strengthening or changes in business mix. Credibility and statistical uncertainty are especially important when analysing small segments.

Learning Points

Pricing is not a one-time calculation. Insurers must monitor assumptions, reconcile expected and actual experience, maintain adequate reserves and document any corrective action. Segment-level analysis may support targeted underwriting or pricing changes, but classifications must be lawful and ethically defensible.

11. Model Risk, Governance and Contemporary Practice

Actuarial estimates are conditional on data and assumptions rather than certain predictions. Key limitations include sparse data, reporting delays, structural change, parameter uncertainty, model misspecification and extreme events. Inflation can affect both claim severity and settlement duration, while climate change, medical advances and emerging technologies may make historical experience less representative.

Professional practice therefore requires:

• validation and reconciliation of source data;
• explicit documentation of assumptions and methods;
• sensitivity and scenario analysis;
• independent review and governance approval;
• monitoring of actual versus expected outcomes;
• compliance with solvency, accounting and consumer-protection requirements; and
• communication of uncertainty rather than false precision.

Machine-learning systems can improve risk segmentation, fraud detection and severity prediction. However, opaque models may create discrimination, instability or weak explainability. Human oversight remains necessary, particularly where model outputs affect access to insurance or materially influence reported liabilities. Personal health and behavioural data must be collected and used lawfully, securely and proportionately.

Ethical pricing balances risk classification with fairness and social access. Actuaries must avoid prohibited discrimination, test for proxy bias and disclose material uncertainty. Reserving judgment should remain independent of pressure to manipulate reported profit or solvency.

Chapter Summary

Actuarial insurance analysis integrates probability, mortality, morbidity, time value of money and statistical loss modelling. Mortality tables describe survival and death, while morbidity tables measure illness or disability incidence, duration and cost. Expected present value allows contingent cash flows occurring at different dates to be compared consistently.

General insurance losses can be decomposed into frequency and severity. Premiums must cover expected losses as well as expenses, taxes, risk, capital and profit requirements. Loss and combined ratios support performance monitoring but must be interpreted using consistent premium and claim definitions.

Life reserves generally compare expected present values of future obligations and premiums. General insurance reserves include unearned premium, case, IBNR and claims-handling components. Methods such as chain ladder rely on historical development patterns and require professional judgment, validation and sensitivity analysis. Sound actuarial work combines quantitative methods with governance, ethical conduct and transparent communication.

Key Terms

Actuarial reserve: Estimated value of an insurer’s obligation at a valuation date.

Case reserve: Claim-specific estimate established after a claim is reported.

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

Claim severity: Average monetary amount per claim.

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

Discount factor: Multiplier used to convert a future amount to present value.

Earned premium: Premium attributable to coverage already provided.

Expected present value: Probability-weighted present value of contingent cash flows.

Gross premium: Premium including claims, expenses, taxes, margins and other loadings.

IBNR: Reserve for claims incurred but not yet reported.

Incidence: Rate at which new morbidity cases arise among exposed lives.

Life table: Table showing mortality and survival experience by age.

Loss ratio: Incurred claims divided by earned premium.

Morbidity: Incidence and consequences of illness, injury or disability.

Mortality: Frequency or probability of death in a population.

Pure premium: Expected claim cost per exposure.

Survival probability: Probability that a life survives for a specified period.

Unearned premium reserve: Premium liability relating to future insurance coverage.

Review Questions

Short Answer Questions

1. Distinguish claim frequency from claim severity.
2. Explain the relationship between q_x and p_x.
3. Distinguish morbidity incidence from prevalence.
4. Define IBNR and explain why it arises.
5. What does a combined ratio above 100% indicate?

Long Answer Questions

1. Explain how mortality tables are constructed and used in life insurance pricing.
2. Discuss the role of the time value of money in pricing long-term insurance contracts.
3. Compare prospective and retrospective approaches to life insurance reserving.
4. Explain the principal components of a gross insurance premium.
5. Evaluate the assumptions, advantages and limitations of the chain-ladder method.

Analytical / Application Questions

1. A portfolio has annual claim frequency of 0.08 and average severity of CU 25,000. Calculate the pure premium and explain how dependence between claims could affect risk assessment.
2. Using the illustrative morbidity table, estimate the expected number and expected aggregate cost of new illness claims among 20,000 insured persons aged 50–59, assuming one episode at most per person.
3. An insurer reports earned premium of CU 40 million, paid claims of CU 22 million, an increase in outstanding reserves of CU 3 million and underwriting expenses of CU 10 million. Calculate the loss, expense and combined ratios and interpret the result.

Multiple Choice Questions with Answers

Question 1

Using the illustrative mortality table, what is the probability that a life aged 40 survives to age 43?

A. 0.03557368

B. 0.96442632

C. 0.98600000

D. 0.98800000

Correct Answer: B. 0.96442632

Question 2

In the illustrative morbidity table, what is the annual incidence for ages 40–49?

A. 0.009

B. 0.013

C. 0.021

D. 0.036

Correct Answer: C. 0.021

Question 3

What is the present value of CU 1,100 payable in one year at an effective annual interest rate of 10%?

A. CU 900

B. CU 990

C. CU 1,000

D. CU 1,210

Correct Answer: C. CU 1,000

Question 4

A portfolio has expected frequency of 0.12 claims per policy-year and expected severity of CU 18,000. What is its pure premium?

A. CU 1,500

B. CU 2,160

C. CU 18,000

D. CU 21,600

Correct Answer: B. CU 2,160

Question 5

Expected losses are CU 2,160 and fixed expenses are CU 300. If proportional premium loadings total 30%, what is the gross premium?

A. CU 2,460.00

B. CU 3,085.71

C. CU 3,514.29

D. CU 8,200.00

Correct Answer: C. CU 3,514.29

Question 6

An insurer has incurred claims of CU 600,000 and earned premium of CU 1,000,000. What is its loss ratio?

A. 40%

B. 60%

C. 100%

D. 166.67%

Correct Answer: B. 60%

Question 7

Under the prospective method, a life insurance reserve is principally calculated as:

A. Future premiums minus past expenses

B. Paid claims plus written premium

C. EPV of future benefits and expenses minus EPV of future premiums

D. Earned premium minus investment income

Correct Answer: C. EPV of future benefits and expenses minus EPV of future premiums

Question 8

Which reserve relates primarily to the unexpired portion of existing insurance coverage?

A. Case reserve

B. Unearned premium reserve

C. IBNR reserve

D. Claims-handling expense reserve

Correct Answer: B. Unearned premium reserve

Question 9

In the chapter’s claims triangle, what is the volume-weighted 24-to-36-month development factor?

A. 1.011111

B. 1.111111

C. 1.473684

D. 1.637427

Correct Answer: B. 1.111111

Question 10

A portfolio has expected claim frequency of 0.20 and expected severity of CU 1,000. What is its expected aggregate loss per exposure?

A. CU 50

B. CU 200

C. CU 800

D. CU 1,200

Correct Answer: B. CU 200