Life Annuities and Assurances: Actuarial Basics Explained

A life annuity pays a regular income for as long as someone is alive, such as a monthly pension. A life assurance (life insurance) is the mirror image: it pays a single lump sum when someone dies. Both are payments that depend on whether a person is alive.

They transfer the risk of "living too long" or "dying too soon" from families to insurers. Mathematics prices these promises fairly: the present value of expected payments must equal what the customer pays.

Key Concepts

• Annuity — a series of payments made while a life survives. An annuity-due pays at the start of each period; an annuity-immediate pays at the end.
• Assurance — a benefit paid on death. A whole-life assurance pays whenever death occurs; a term assurance pays only if death occurs within a fixed number of years.
• a_x and ä_x — actuarial symbols for the expected present value of a life annuity to a life aged x.
• A_x — the expected present value of a whole-life assurance to a life aged x.
• Discount factor v — v = 1 / (1 + i), where i is the interest rate. It converts a future payment to its present value.

Formulas

• ä_x = Σ v^k · _k p_x for k = 0, 1, 2, … — annuity-due: each year's payment (at the start of year k) is multiplied by the probability of surviving to that payment and discounted. Assumes yearly payments of 1 unit.
• A_x = Σ v^{k+1} · _k p_x · q_{x+k} for k = 0, 1, 2, … — whole-life assurance: payment at the end of the year of death, weighted by the probability of dying in that year and discounted.
• Identity: A_x + d · ä_x = 1, where d = i / (1 + i) — always sums to 1 for whole-life contracts at constant interest.

Worked Example

A 2-year model: survival probabilities _1 p_x = 0.98, _2 p_x = 0.95; interest i = 5%, so v ≈ 0.95238. Price a 2-year annuity-due paying 1,000 yearly.

1. Step 1: Payment at time 0 (certain, since the person is alive now): 1,000 × 1 = 1,000.
2. Step 2: Payment at time 1, paid only if alive: expected discounted value = 1,000 × _1 p_x × v = 1,000 × 0.98 × 0.95238 ≈ 933.33.
3. Step 3: Total fair price = 1,000 + 933.33 = 1,933.33.
4. Step 4: The insurer charges 1,933.33 today; averaged over many customers, this exactly funds the yearly payments.

Common Misconceptions

• "An annuity is just a savings account." No. A savings account returns your money on demand. An annuity pools longevity risk: those who die early subsidize those who live long, which is why annuities pay more per year than safe savings.
• "Term assurance and whole-life assurance cost the same." Whole-life costs more because it is guaranteed to pay out one day; term assurance often pays nothing, so its premium is much lower.
• "The annuity factor ä_x is a guaranteed number of years." It is an expected present value, averaged over survival chances and discounted by interest — not a promised number of payments.

Real-World Applications

• Retirement pensions — retirees convert lump sums into life annuities for a guaranteed lifetime income.
• Family protection — term and whole-life assurances pay dependants when the breadwinner dies.
• Structured settlements — courts use annuities to pay injury compensation over a victim's lifetime.

Annuities pay while you live; assurances pay when you die. Both are priced with the equivalence principle — expected discounted payments equal the premium. Master these, and the rest of insurance mathematics follows.