FRM Exam Part I · Insurance Companies and Pension Plans

Life Insurance Products and Mortality Tables for FRM Part I

Updated 11 October 2026 · Fact-checked

Life insurance products pay a benefit on death (term, whole, universal, variable) or pay income while you live (annuities). Mortality tables give the probability of death at each age. To price a policy, multiply each year's death probability by the benefit, discount to today, and sum. Annuities use survival probabilities instead.

Understand Life Insurance Products and Mortality Tables

A life insurance contract pays a benefit if the insured dies. The insurer collects premiums and invests them. Pricing rests on one idea: the premium must cover the expected present value of the benefits, plus costs and profit.

Term life covers a fixed period, such as 10 or 20 years. It pays only if death occurs within the term. It has no savings element, so it is the cheapest cover. Whole life (permanent insurance) covers you until death, whenever it happens. The premium is level, so in early years you pay more than the pure cost of cover. The excess builds a cash value that you can borrow against or take if you surrender the policy. An endowment policy pays on death or at the end of the term, whichever comes first.

Two flexible products add choice. Universal life lets the policyholder change the premium or the death benefit, and the cash value earns interest set by the insurer. Variable life lets the policyholder choose how the cash value is invested, for example in equity or bond funds, so the investment risk moves to the policyholder. Variable-universal life combines both features. The more control you get, the more investment risk you carry.

An annuity reverses the logic. You pay a lump sum or regular amounts, and the insurer pays you a regular income, often for life. A life annuity pays while the annuitant is alive. It may be immediate or deferred, and fixed or variable. A life insurer is hurt if people die earlier than expected (mortality risk). An annuity provider is hurt if people live longer than expected (longevity risk). Holding both books gives a partial natural hedge.

A mortality table shows, for each age, the probability of dying within the next year (often written q). From these numbers you build survival probabilities and the probability of dying in any given future year. Discount each benefit at the stated rate and you get its expected present value. The tables are estimates, so insurers also face the risk that real mortality differs from the table.

Key formulas to remember

One-year survival probability: p(x) = 1 − q(x); q(x) is the table probability that a person aged x dies within one year.
Survival to the end of year t: S(t) = (1 − q₀) × (1 − q₁) × ... × (1 − q(t−1)); Multiply the one-year survival probabilities. Here q₀ is for the starting age, q₁ for the next age, and so on.
Probability of dying in year t (t = 1, 2, ...): P(death in year t) = S(t−1) × q(t−1); You must survive to the start of the year, then die during it.
Expected PV of a death benefit (benefit paid at end of year of death): EPV = Σ B × P(death in year t) ÷ (1 + r)^t; This is the net single premium for a term policy, ignoring costs and profit.
Level annual premium, paid at start of each year while alive: P = EPV of benefits ÷ Σ [S(t−1) ÷ (1 + r)^(t−1)]; The denominator is the expected PV of $1 paid at the start of each year, starting with 1 at time 0.
Expected PV of an annuity-immediate: EPV = Σ C × S(t) ÷ (1 + r)^t; Payment C is made at the end of year t only if the annuitant is alive then.

How to solve Life Insurance Products and Mortality Tables questions

Use this method for any pricing or product question. Product-type questions need only steps 1 and 2.

1Identify the product. Decide whether it pays on death (term, whole, endowment) or on survival (annuity), and who bears the investment risk (insurer or policyholder).
2For concept questions, match the feature to the product: no cash value means term, flexible premium or benefit means universal, policyholder-chosen investments means variable.
3For pricing questions, list the table probabilities q for each year and the discount rate. Note when the benefit or payment is made (start or end of year).
4Build the survival probabilities S(t) by multiplying (1 − q) values.
5For death benefits, compute P(death in year t) = S(t−1) × q(t−1). For annuities, use S(t) at each payment date.
6Discount each amount by (1 + r)^t, using the correct t for the timing, then multiply by the benefit or payment.
7Sum the present values. For a level premium, divide by the expected PV of the premium stream (premiums paid only while alive).
8Check reasonableness: the single premium should be less than the benefit, and an annuity value should be less than payment × number of payments.

Quickest way: Table-column method

When to use it: Use it for any multi-year pricing question with a short mortality table or a list of q values.

Draw four columns: year, S(start of year), P(death in year), discount factor.
Fill S down the page by multiplying by (1 − q) each row.
Get P(death) by multiplying each row's S(start) by that row's q.
Multiply P(death) by the discount factor and sum the column. Multiply by the benefit.
For annuities, replace P(death) with S(end of year). For premiums at the start of the year, use S(start) and discount by (1 + r)^(t−1).
Use your calculator's memory to add the terms. Eliminate options that are above the benefit, or on the wrong side of an obvious bound.

Common mistakes in Life Insurance Products and Mortality Tables

Using q for year 2 or 3 directly as the probability of dying in that year.
The table shows probabilities conditional on being alive at that age. Students forget the person must first survive earlier years.
Fix: Always use P(death in year t) = S(t−1) × q(t−1).
Discounting a benefit paid at the end of the year of death by the wrong number of years.
Students discount by the age rather than by the number of years from today, or mix start-of-year and end-of-year timing.
Fix: Write the timing next to each cash flow. A death benefit in year t is discounted t years. A premium paid at the start of year t is discounted (t − 1) years.
Treating the premium stream as certain when finding the level premium.
Premiums stop on death, but students discount them as if all were paid.
Fix: Weight each premium by the probability the insured is alive at that date, S(t−1).
Saying variable life guarantees the cash value.
Students link the word life insurance with guarantees and overlook who chooses and bears the investment risk.
Fix: In variable life, the policyholder bears the investment risk. Universal life credits interest set by the insurer. Whole life has the most predictable cash value.
Mixing up mortality risk and longevity risk.
Both involve life expectancy, so the direction of loss is easy to reverse.
Fix: Life insurer loses if people die sooner than expected (mortality risk). Annuity provider loses if people live longer (longevity risk).
Believing term life builds cash value.
Students blur term and whole life because both are called life insurance.
Fix: Term is pure protection for a fixed period. Cash value belongs to permanent products such as whole life and universal life.

Worked examples

Example 1

A 3-year term policy pays $100,000 at the end of the year of death. The insured is alive today. The probabilities of dying in each successive year, given alive at the start of that year, are 1.0%, 1.5% and 2.0%. The discount rate is 5% per year. Find the net single premium and, if premiums are paid at the start of each year while alive, the level annual premium.

Show the solution

Survival probabilities: S(1) = 0.99 and S(2) = 0.99 × 0.985 = 0.97515.
Probability of death in year 1 = 0.01. Year 2 = 0.99 × 0.015 = 0.01485. Year 3 = 0.97515 × 0.02 = 0.019503.
Discount: year 1 ÷ 1.05 = 0.009524. Year 2 ÷ 1.1025 = 0.013469. Year 3 ÷ 1.157625 = 0.016848.
Sum = 0.039840. Multiply by $100,000 to get the net single premium of about $3,984.
Premium annuity factor: 1 + 0.99 ÷ 1.05 + 0.97515 ÷ 1.1025 = 1 + 0.942857 + 0.884490 = 2.827347.
Level premium = 3,984.07 ÷ 2.827347 ≈ 1,409.1.

Answer: Net single premium is about $3,984. The level annual premium paid at the start of each year is about $1,409.

Example 2

An insurer sells a 2-year life annuity-immediate paying $10,000 at the end of each year if the annuitant is alive. The annuitant is alive today. The probability of dying in year 1 is 2%, and the probability of dying in year 2, given alive at the start of year 2, is 3%. The discount rate is 4% per year. Find the expected present value of the payments.

Show the solution

Survival to end of year 1: S(1) = 1 − 0.02 = 0.98.
Survival to end of year 2: S(2) = 0.98 × (1 − 0.03) = 0.98 × 0.97 = 0.9506.
Discount factors: 1 ÷ 1.04 = 0.961538 and 1 ÷ 1.0816 = 0.924556.
PV factor for year 1 = 0.98 × 0.961538 = 0.942308. Year 2 = 0.9506 × 0.924556 = 0.878882.
Sum = 1.821190. Multiply by $10,000.

Answer: The expected present value is about $18,212.

Exam tips

Read the timing words closely: start of year or end of year, and benefit on death or on survival. They change the discount exponent and the probability used.
Most questions give q values, not survival values. Convert them yourself, and watch for questions that give cumulative survival instead.
For product questions, link each product to who bears investment risk: insurer (whole life, fixed annuity) or policyholder (variable life, variable annuity).
Sanity check numbers: a death-benefit premium must be below the benefit, and an annuity value below payment times the number of payments.
Remember the hedge: life insurance has mortality risk, annuities have longevity risk, and the two partly offset each other.

Practice questions from Insurance Companies and Pension Plans

Life Insurance Products and Mortality Tables in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Life Insurance Products and Mortality Tables: frequently asked questions

What is the difference between term life and whole life insurance?

Term life pays only if the insured dies within a fixed period and has no cash value. Whole life covers you until death, has a level premium and builds a cash value you can borrow against or take on surrender. Term is cheaper for the same death benefit.

How is universal life different from variable life?

Universal life gives flexibility over premiums and the death benefit, and the cash value earns interest set by the insurer. Variable life lets you choose the investments for the cash value, so you carry the investment risk. Variable-universal life combines both features.

How do you use a mortality table to price life insurance?

Convert the table's one-year death probabilities into the probability of dying in each future year. Multiply each by the benefit, discount to today and add them up. That gives the net single premium, which you can spread into a level premium using survival probabilities.

What is the difference between mortality risk and longevity risk?

Mortality risk is the risk that policyholders die sooner than expected, which hurts life insurers. Longevity risk is the risk that people live longer than expected, which hurts annuity providers and pension plans. A firm writing both has a partial natural hedge.

How does an annuity work?

You pay the insurer a lump sum or regular amounts and receive regular income, often for life. It can start at once (immediate) or later (deferred). In a fixed annuity the insurer sets the payments, and in a variable annuity the payments depend on investment performance.