Actuarial Present Values of Life Insurance Benefits

Introduction and context

Life insurance benefits are contingent cash flows: payment depends on whether and when an insured person dies, or whether the person survives to a specified date. Actuarial mathematics converts these uncertain payments into values at a common valuation date. The central quantity is the expected present value (EPV), which combines financial discounting with mortality probabilities.

The EPV is fundamental to premium calculation, reserving, solvency assessment, product design, embedded-value analysis and financial reporting. This chapter develops the principal life insurance benefit functions under both discrete and continuous models, then extends them to deferred, endowment and varying-benefit contracts. Expenses and payment-timing assumptions are also incorporated.

Learning objectives

By the end of this chapter, the reader should be able to:

1. Explain the relationship between survival probabilities, death probabilities, discount factors and EPVs.
2. Derive and calculate EPVs for term, whole-life, endowment and deferred insurance.
3. Distinguish discrete benefits payable at the end of the year of death from continuously payable benefits.
4. Value increasing, decreasing and otherwise varying death benefits.
5. Incorporate acquisition, maintenance, claim and settlement expenses.
6. Evaluate the effects of interest, mortality, payment timing and modelling assumptions.
7. Apply the framework to product pricing and professional actuarial decision-making.

1. Fundamental notation and assumptions

Let a life aged x be observed from the valuation date. The future lifetime is denoted by T_x, measured in years. The curtate future lifetime is K_x = ⌊T_x⌋, the number of complete future years lived. Let i be the effective annual interest rate and v = (1 + i)⁻¹ the annual discount factor. For a fractional time t, the discount factor is vᵗ = (1 + i)⁻ᵗ under an equivalent annual effective rate.

The principal mortality functions are:

The survival function and force of mortality satisfy

and, where a density exists,

These relationships assume a valid mortality model and a continuously defined force of mortality. In discrete calculations, annual death probabilities or a life table are sufficient.

The EPV of a benefit whose amount is B(T_x) and whose payment time is τ(T_x) is generally

The expectation is taken over the distribution of future lifetime. This formula is the unifying principle for all contracts in this chapter.

2. Discrete death-benefit models

2.1 n-year term insurance

An n-year term insurance pays 1 at the end of the year of death if death occurs within n years. Its EPV, usually denoted Ā_{x:n|} in actuarial notation, is

Here k = 0, …, n − 1 identifies the year of death. The term {}_k p_x q_{x+k} is the probability of surviving k years and then dying during the next year. The factor v^{k+1} discounts the payment made at the end of that year.

The formula assumes annual benefit payments at the end of the year of death, a level benefit of 1, and no death benefit after duration n.

For a benefit amount S, the EPV is S A_{x:n|}. If the benefit is payable at the beginning of the year of death rather than its end, the discount factor changes, and the result is not generally identical.

2.2 Whole-life insurance

A whole-life insurance pays 1 at the end of the year of death, regardless of when death occurs:

For practical work, the infinite sum is evaluated over the relevant life table ages. A whole-life EPV can also be related to the EPV of an annuity-due. The whole-life annuity-due value is

Under the same annual, end-of-year death-payment convention,

where d = i/(1+i) is the annual discount rate. This identity follows by comparing the first-year payment and the surviving annuity stream, and assumes the same interest and mortality basis.

2.3 Pure endowment and endowment insurance

A pure endowment pays 1 at time n if the insured survives n years. Its EPV is

An n-year endowment insurance combines an n-year term insurance with a pure endowment:

Thus, it pays 1 at the end of the year of death if death occurs during the term, or 1 at time n if the insured survives. For sum insured S, multiply the expression by S.

This decomposition is particularly useful in pricing because it separates the mortality-dependent component from the survival benefit.

2.4 Deferred insurance

An m-year deferred whole-life insurance pays 1 at the end of the year of death, provided death occurs after the deferment period. Its EPV is

Using conditional valuation at age x + m,

The factor v^m {}_m p_x represents the value of reaching the beginning of the insurance period; A_{x+m} values the subsequent whole-life insurance. A deferred n-year term insurance, covering durations m through m+n−1, is

3. Continuous models

A continuously payable whole-life insurance pays 1 immediately on death. Its EPV is

For an n-year continuous term insurance,

A continuous pure endowment remains payable at a fixed time and therefore has the same EPV as under a discrete model:

The continuous n-year endowment EPV is

The distinction between discrete and continuous models is a payment-timing assumption, not necessarily a different mortality law. Continuous payment generally produces a larger EPV than end-of-year payment because death claims are paid earlier. The difference can be material for high interest rates, short contracts or large claim volumes.

Constant-force illustrative result

Suppose the force of mortality is constant at μ and the force of interest is δ, where v^t = e^{−δt}. Then {}_t p_x = e^{−μt}, and the continuous whole-life EPV is

This is a simplified model. Real mortality varies significantly with age, so the result should not be used as a substitute for a calibrated life table.

4. Worked example: annual term and endowment benefits

Assume a five-year contract on a life aged 40. Let i = 5%, v = 1/1.05 = 0.952381. Suppose the relevant probabilities are:

The five-year term EPV is

Therefore,

For a death benefit of 100,000 monetary units, the EPV is approximately 528.70 monetary units. The pure endowment component is

Thus the five-year endowment EPV per unit is approximately 0.7839, subject to the supplied rounded probabilities. The large difference between the term and endowment values reflects the substantial survival benefit.

5. Varying benefits

Many products have benefits that change by duration. Let b_{k+1} be the benefit paid at the end of year k + 1 if death occurs in that year. The EPV of an n-year varying term insurance is

For an increasing benefit b_{k+1} = k + 1,

For a decreasing benefit b_{k+1} = n − k,

A benefit that changes continuously with death time, b(t), has EPV

These models are useful for mortgage protection, inflation-linked cover, family-income replacement and contracts with loyalty or persistency bonuses. The benefit specification must be separated from expenses and from the mortality assumption; otherwise product comparisons become misleading.

6. Expenses and payment timing

A benefit EPV is not the same as the EPV of insurer outgo. Expenses may include:

• initial acquisition expense at issue;
• renewal or maintenance expenses at policy anniversaries;
• claim administration expense at death;
• settlement expense at benefit payment;
• premium-related expenses proportional to premium or benefit.

If an initial expense e₀ is payable immediately, its EPV is e₀. If renewal expense e_k is payable at time k provided the policy is in force, its EPV is

If a claim expense c_{k+1} is paid with a death claim at the end of year k + 1, its EPV is

Therefore, total expected outgo is the sum of benefit and expense EPVs, provided the cash flows are additive and valued on the same basis. If expenses are a proportion r of the death benefit, c_{k+1} = r b_{k+1}, and the claim-expense EPV is r times the benefit EPV.

Timing matters. An expense payable at the beginning of a year uses v^k, while one payable at the end uses v^{k+1}. A maintenance expense payable only while alive at time k uses {}_k p_x; a death expense uses {}_k p_x q_{x+k}. Confusing these probabilities is a common modelling error.

7. Integrated applied case: pricing a family-protection product

Background

An insurer is designing a five-year family-protection policy for a life aged 40. The death benefit is 100,000 monetary units, paid at the end of the year of death. An initial acquisition expense of 150 is paid at issue, annual maintenance expense is 20 at each year-end while the policy remains in force, and claim administration expense is 500 when a death claim is paid.

Problem

Management requires the expected present value of insurer outgo on a mortality and interest basis of 5% interest, using the probabilities in the earlier worked example.

Analysis and approach

The death-benefit EPV is

The maintenance-expense EPV is approximately

Using the supplied probabilities, this is approximately 20 × 4.38 = 87.60. The claim-expense EPV is 500 × 0.005287 = 2.64. Total expected outgo is therefore approximately

This is not automatically the required premium. A premium calculation must also specify premium timing, premium expenses, profit or risk margins, capital costs, lapse assumptions, taxes and the distinction between best-estimate and pricing bases.

Outcome and learning points

The case shows that non-claim expenses can be material even when mortality claims are rare. It also demonstrates that expense timing and policy-continuance probabilities must be modelled separately. A professional pricing model would test sensitivity to mortality deterioration, interest-rate changes, expense inflation, lapses and alternative claim-payment timing.

8. Assumptions, limitations and governance

EPVs are model-based quantities, not guaranteed cash outcomes. Important assumptions include:

1. Mortality: Life-table rates may not reflect underwriting selection, medical improvement or portfolio composition.
2. Interest: A single deterministic rate ignores yield-curve shape, reinvestment risk and market volatility.
3. Timing: Annual, monthly and continuous payment conventions produce different values.
4. Dependants: Lapses, underwriting, policy alterations and premium holidays may affect exposure.
5. Expenses: Fixed and variable expenses behave differently under scale changes and inflation.
6. Model risk: Coding, data, rounding and aggregation errors can materially distort results.

Actuaries should document data sources, valuation bases, materiality thresholds, validation tests and model changes. Governance should include independent review, reproducible calculations and sensitivity analysis. Under applicable insurance regulation and financial-reporting standards, assumptions may need to reflect prescribed contract boundaries, discounting rules, risk adjustments and disclosure requirements. The exact treatment depends on jurisdiction and reporting framework.

9. Contemporary practice and technology

Modern actuarial systems calculate EPVs using life tables, projection models and scenario-based cash-flow engines. Spreadsheet models remain useful for transparency, but production systems increasingly use controlled code, automated testing and version management. Machine learning may assist mortality or lapse modelling, but it does not remove the need for actuarial judgement. Models should be explainable enough for review, monitored for drift and assessed for discriminatory effects, especially where underwriting data include sensitive characteristics.

Best practice is to reconcile aggregate results to closed-form formulas where possible, compare discrete and continuous approximations, test extreme but plausible scenarios, and maintain a clear audit trail from assumption to reported liability.

Chapter Summary

Actuarial present values convert contingent life insurance payments into values at a common date. The general principle is the expected discounted benefit, with the expectation taken over future lifetime. For discrete end-of-year claims, the key building block is {}_k p_x q_{x+k}v^{k+1}; for continuous claims, it is {}_t p_x μ_{x+t}v^t dt. Term, whole-life, pure endowment, endowment and deferred insurance follow directly from this framework. Varying benefits are handled by multiplying each death probability by its duration-specific benefit. Expense valuation requires the correct payment-time discount factor and the correct survival or death probability. Reliable professional work depends on explicit assumptions, validated data, sensitivity testing and sound governance.

Key Terms

EPV: Expected present value of a contingent cash flow.

Future lifetime T_x: Number of years lived after age x.

Curtate future lifetime K_x: Completed future years lived, ⌊T_x⌋.

Survival probability: Probability of surviving a specified future period.

Death probability: Probability of dying during a specified period.

Force of mortality: Instantaneous mortality rate μ_{x+t}.

Discount factor: Present-value factor applied to a future payment.

Term insurance: Insurance payable only if death occurs within a stated term.

Whole-life insurance: Insurance payable whenever death occurs.

Pure endowment: Benefit payable if the insured survives to a specified date.

Endowment insurance: Combination of term insurance and pure endowment.

Deferred insurance: Insurance whose cover begins after a deferment period.

Varying benefit: Benefit amount dependent on duration or death time.

Annuity-due: Payments made at the beginning of successive periods.

Acquisition expense: Expense incurred when a policy is issued.

Maintenance expense: Expense incurred while a policy remains in force.

Claim expense: Expense associated with processing or settling a claim.

Force of interest: Continuously compounded interest rate corresponding to discounting by e^{−δt}.

Model risk: Risk arising from inappropriate assumptions, design, implementation or use of a model.

Review Questions

Short Answer Questions

1. Define the expected present value of a life insurance benefit.
2. Distinguish {}_t p_x from q_{x+t}.
3. Why does a continuous death benefit usually have a different EPV from an end-of-year benefit?
4. What is the relationship between an endowment insurance and its term and pure-endowment components?
5. Why must expense timing be specified in an actuarial valuation?

Long Answer Questions

1. Derive the EPV of an n-year discrete term insurance and explain every term in the summation.
2. Explain and derive the decomposition of an endowment insurance into term insurance and pure endowment.
3. Compare discrete and continuous whole-life insurance models, including their assumptions and limitations.
4. Develop the EPV formula for a deferred whole-life insurance and interpret each factor.
5. Discuss the effect of mortality, interest, lapses and expenses on life insurance pricing.

Analytical / Application Questions

1. Using a supplied life table and interest rate, calculate the EPV of a five-year increasing term insurance and compare it with a level term insurance.
2. A product pays a death benefit at the end of the year of death and a maintenance expense at each policy anniversary while the policy is active. Construct the complete EPV of outgo and explain the probability attached to each cash flow.
3. Evaluate whether a continuous-payment approximation is acceptable for a high-value short-term policy. Identify the quantitative tests and governance evidence you would require before implementation.

Multiple Choice Questions with Answers

Question 1

For an n-year discrete term insurance paying 1 at the end of the year of death, which expression is correct?

A. ∑_{k=0}^{n−1} v^k {}_k p_x

B. ∑_{k=0}^{n−1} v^{k+1} {}_k p_x q_{x+k}

C. v^n {}_n p_x

D. ∫_0^n v^t {}_t p_x dt

Correct Answer: B. ∑_{k=0}^{n−1} v^{k+1} {}_k p_x q_{x+k}

Question 2

The EPV of a pure endowment of 1 payable at time n is:

A. A_{x:n|}

B. ∑_{k=0}^{n−1} v^{k+1}q_{x+k}

C. v^n {}_n p_x

D. A_x − A_{x:n|}

Correct Answer: C. v^n {}_n p_x

Question 3

A whole-life insurance benefit is paid:

A. Only if death occurs within five years

B. Only if the insured survives to age x+n

C. Whenever death occurs, subject to the contract terms

D. At issue regardless of survival

Correct Answer: C. Whenever death occurs, subject to the contract terms

Question 4

In a discrete death-benefit model, {}_k p_x q_{x+k} represents:

A. Survival for k years and death during the next year

B. Death at exactly time k with certainty

C. Survival for k+1 years

D. The annual discount rate

Correct Answer: A. Survival for k years and death during the next year

Question 5

Under a constant force of mortality μ and force of interest δ, the continuous whole-life EPV of a unit benefit is:

A. δ/(μ+δ)

B. μ/(μ+δ)

C. μδ

D. 1/(μδ)

Correct Answer: B. μ/(μ+δ)

Question 6

An m-year deferred whole-life insurance has EPV:

A. v^m {}_m p_x A_{x+m}

B. v^m {}_m q_x A_x

C. A_x + v^m {}_m p_x

D. {}_m p_x / v^m

Correct Answer: A. v^m {}_m p_x A_{x+m}

Question 7

If a death benefit increases with duration, the EPV calculation must:

A. Ignore the benefit amount and use only mortality

B. Multiply each duration-specific death probability by its corresponding benefit

C. Use survival probabilities only

D. Replace discounting with inflation

Correct Answer: B. Multiply each duration-specific death probability by its corresponding benefit

Question 8

A renewal expense payable at time k only if the policy is in force should generally use:

A. q_{x+k}

B. {}_k p_x

C. {}_k q_x v^{k+1} only

D. μ_{x+k} without discounting

Correct Answer: B. {}_k p_x

Question 9

A claim expense paid with a death claim should be valued using:

A. Survival probability alone

B. The probability of death in the relevant period and the claim-payment discount factor

C. The pure-endowment factor

D. No discount factor

Correct Answer: B. The probability of death in the relevant period and the claim-payment discount factor

Question 10

Which change generally increases the EPV of a continuously payable death benefit, holding mortality constant?

A. Lower interest rate

B. Later claim payment

C. Shorter benefit amount

D. Higher discount rate

Correct Answer: A. Lower interest rate