Actuarial Mathematics for Modelling · Key assurance and annuity contracts

Premium Payment Patterns and Special Contracts in Actuarial Mathematics

Updated 11 October 2026 · Fact-checked

Premium payment patterns say when a policyholder pays: once (single), for life, or for a limited term. You find the premium by the equivalence principle: expected present value (EPV) of premiums equals EPV of benefits plus expenses. Increasing or decreasing benefits change the benefit EPV, not the method.

Understand Premium Payment Patterns and Special Contracts

A life contract has two sides: benefits the insurer pays and premiums the policyholder pays. The premium payment pattern is the timing of the premiums. A single premium is paid once at the start. A level premium is the same amount each year while the policy is in force. A limited-term premium is paid only for a fixed number of years, say 10, even if cover lasts longer.

Every premium is paid only while the life is alive, so premiums form a life annuity-due. The EPV of level premiums P paid annually for the whole of life is P × ä_x. For premiums limited to n years, use the temporary annuity-due ä_x:n. For a single premium, there is no annuity: the premium is simply the EPV of benefits.

The equivalence principle sets EPV of premiums = EPV of benefits (plus expenses in a gross premium calculation). Solve for P. The net premium ignores expenses. The pattern only changes the annuity factor on the premium side.

Increasing and decreasing benefits change the sum assured over time. An increasing whole life assurance might pay a sum of k + 1 if death occurs in year k + 1. Its EPV is written (IA)_x. A decreasing term assurance, such as one that pays n − k if death occurs in year k + 1, has EPV (DA)^1_x:n. You value these by summing benefit × discount factor × probability of death in each year, or by splitting the benefit into level layers.

With-profits contracts add bonuses to the guaranteed sum assured, so the benefit grows. At this level you value them by treating declared bonuses as an increasing benefit, often at a compound or simple rate on the sum assured. Unit-linked contracts invest premiums in a fund, and the benefit depends on the unit fund value, often with a guaranteed minimum death benefit. The insurer's own cash flows (charges less expenses and claims) are the non-unit cash flows. A with-profits policyholder shares in profits through bonuses. A unit-linked policyholder bears the investment risk directly.

Key rules to remember

Equivalence principle (net): P × ä = EPV of benefits; Use ä_x for whole life premiums and ä_x:n for premiums limited to n years. Gross premium adds expenses to the right side.
Single premium: Single premium = EPV of benefits (+ EPV of expenses if gross); No premium annuity is needed.
Whole life assurance, level premium: P_x = A_x ÷ ä_x; Premiums payable annually in advance for life, benefit paid at end of year of death.
Limited-pay whole life: _nP_x = A_x ÷ ä_x:n; Premiums for n years only; cover for life.
Increasing whole life assurance (discrete): (IA)_x = Σ (k+1) × v^(k+1) × _k p_x × q_(x+k), summed over k = 0, 1, 2, …; Pays k+1 if death occurs in year k+1, paid at end of the year.
Increasing term assurance: (IA)^1_x:n = Σ (k+1) × v^(k+1) × _k p_x × q_(x+k), k = 0 to n−1; Same sum, cut off after n years.
Decreasing term assurance: (DA)^1_x:n = Σ (n−k) × v^(k+1) × _k p_x × q_(x+k), k = 0 to n−1; Pays n−k if death occurs in year k+1.
Link between increasing and decreasing term: (IA)^1_x:n + (DA)^1_x:n = (n+1) × A^1_x:n; Useful check: benefits (k+1) and (n−k) add to n+1 in every year.
Increasing annuity-due (discrete): (Iä)_x = Σ (k+1) × v^k × _k p_x, summed over k = 0, 1, 2, …; Payment k+1 at the start of year k+1, if alive.

How to solve Premium Payment Patterns and Special Contracts questions

Use this method for any premium or special-contract question. It keeps the premium pattern and benefit pattern separate.

1Write down the benefit: amount, when it is paid (end of year of death or immediate), and whether it is level, increasing or decreasing.
2Write down the premium pattern: single, level for life, or limited to a given term, and how often it is paid.
3Choose the annuity factor for premiums: ä_x for whole life, ä_x:n for n years, or none for a single premium.
4Compute the EPV of benefits. For level benefits use A_x or A^1_x:n. For varying benefits, sum year by year or split into level layers.
5Add the EPV of expenses if the question asks for a gross premium.
6Apply the equivalence principle: EPV premiums = EPV benefits (+ expenses). Solve for the premium.
7Check the answer: the limited-term premium should exceed the whole-life premium for the same benefit, and single premium should be smaller than the total of the level premiums paid.

Quickest way: Layer and annuity shortcut

When to use it: Use when the benefit increases or decreases by a fixed amount each year and you are given standard values such as A_x, ä_x or an (IA) value.

Split the benefit into level layers where possible, for example a decreasing benefit n, n−1, …, 1 is n × A^1_x:n minus an increasing amount.
Use the identity (IA)^1_x:n + (DA)^1_x:n = (n+1) × A^1_x:n to get one from the other.
Write the premium annuity with the correct term before computing anything.
Divide the EPV of benefits by the annuity factor last, and sanity-check size against a level-benefit case.

Common mistakes in Premium Payment Patterns and Special Contracts

Using ä_x for a limited-term premium.
Students focus on the whole life benefit and forget premiums stop early.
Fix: Match the annuity term to the premium term. Premium for n years means ä_x:n.
Using a_x (annuity-immediate) for premiums.
Both annuity symbols look similar.
Fix: Premiums are paid at the start of each period, so use annuity-due ä.
Paying the increasing benefit k instead of k+1 in year k+1.
The index k starts at 0, so the year number is k+1.
Fix: Write the benefit for the first year first. If year 1 pays 1, then year k+1 pays k+1.
Treating decreasing benefit n−k as n−k−1 or as n.
Mixing up where the sum starts or ends.
Fix: Check with year 1, which should pay n, and year n, which should pay 1.
Discounting the death benefit by v^k instead of v^(k+1).
Premium payments use v^k, so students reuse it.
Fix: Death benefit paid at end of year k+1 uses v^(k+1). Premiums at start of year k+1 use v^k.
Assuming unit-linked and with-profits policyholders bear the same risk.
Both are described as investment-linked.
Fix: Unit-linked: policyholder bears investment risk through unit values. With-profits: insurer smooths returns and declares bonuses on a guaranteed sum assured.

Worked examples

Example 1

A whole life assurance on (x) pays ₹10,00,000 at the end of the year of death. Premiums are payable annually in advance for 10 years only. Given A_x = 0.40 and ä_x:10 = 7.5, calculate the annual net premium.

Show the solution

EPV of benefits = 10,00,000 × A_x = 10,00,000 × 0.40 = ₹4,00,000.
EPV of premiums = P × ä_x:10 = 7.5P.
Equivalence principle: 7.5P = 4,00,000.
P = 4,00,000 ÷ 7.5 = ₹53,333.33.

Answer: Annual net premium = ₹53,333 (to the nearest rupee).

Example 2

A 3-year term assurance on (x) pays 3, 2, 1 (in ₹ lakh) if death occurs in years 1, 2, 3 respectively, at the end of the year of death. Given v = 0.95, q_x = 0.01, q_(x+1) = 0.02, q_(x+2) = 0.03, find the single premium (net) in rupees.

Show the solution

Survival: _0 p_x = 1, _1 p_x = 0.99, _2 p_x = 0.99 × 0.98 = 0.9702.
Year 1: benefit 3 lakh × v × 1 × 0.01 = 3 × 0.95 × 0.01 = 0.0285.
Year 2: benefit 2 × v² × 0.99 × 0.02. v² = 0.9025. So 2 × 0.9025 × 0.0198 = 0.03573900 (approx 0.035739).
Year 3: benefit 1 × v³ × 0.9702 × 0.03. v³ = 0.857375. 0.9702 × 0.03 = 0.029106. 0.857375 × 0.029106 = 0.024955 (approx).
Sum = 0.0285 + 0.035739 + 0.024955 = 0.089194 lakh.
Convert: 0.089194 × 1,00,000 = ₹8,919 (approx).

Answer: The single net premium is about ₹8,919.

Exam tips

Write the benefit and premium patterns as two separate lines before computing. Most lost marks come from mixing them up.
Show the equivalence equation explicitly. Examiners award marks for the correct setup even if arithmetic slips.
For increasing or decreasing benefits, check year 1 and the last year of your benefit formula before summing.
In short-answer parts on with-profits and unit-linked, state who bears investment risk and how benefits are determined. Keep it to a few clear sentences.
In the computer-based paper, build the sum in a table with columns for year, survival, death probability, benefit and discount factor so each step can be checked.

Practice questions from Key assurance and annuity contracts

Premium Payment Patterns and Special Contracts: frequently asked questions

What is the difference between single and level premiums?

A single premium is paid once at the start and equals the EPV of benefits. Level premiums are paid regularly while the policyholder is alive, and are found by dividing the EPV of benefits by the premium annuity factor.

How do I value an increasing whole life assurance?

Sum the benefit × v^(k+1) × _k p_x × q_(x+k) over all years, with benefit k+1 in year k+1. This sum is written (IA)_x. If given tables or standard values, use them directly.

How do with-profits and unit-linked contracts differ?

With-profits policies have a guaranteed sum assured plus bonuses declared by the insurer, who smooths returns. Unit-linked policies pay benefits tied to the value of an invested unit fund, so the policyholder bears the investment risk.

Why is a limited-term premium higher than a whole life premium?

The same benefits must be paid for from fewer premium payments. The premium annuity ä_x:n is smaller than ä_x, so the premium is larger.