IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Means and Variances of Assurance and Annuity Payments

Define the present value random variable Z for a benefit, then find E[Z] and Var(Z) = E[Z²] − (E[Z])². E[Z²] is the same expected value computed at a doubled force of interest (2δ). For annuities, link them to assurances with ä_x = (1 − A_x) ÷ d, so Var(Y) = Var(Z) ÷ d².

What this chapter covers

This chapter treats a life insurance or annuity payment as a random variable. The payment amount and its timing depend on how long a life survives, so its present value is random. You learn to write that present value as a function of the future lifetime, then find its mean and variance.

The mean gives the expected present value, which is the net premium or reserve basis. The variance measures the risk the insurer carries. You cover whole life, term, endowment and deferred assurances, and whole life, temporary and deferred annuities, paid annually, in advance or in arrears, or continuously.

This chapter is the base for the rest of CM1. Premium calculation, reserves, multiple decrement models and profit testing all use these present values. The mortality models come from earlier work on survival and life tables. If you are weak here, the Pricing and reserving part of the syllabus will feel hard. That part carries the largest topic weighting in the 2026 CM1 syllabus.

Pricing and reserving is the largest CM1 topic by syllabus weighting, and every question in it starts from these present values. Paper A written questions often ask for a mean and a variance in the same part, and Paper B needs you to build the same quantities in R or Excel from a life table. The method is mechanical once learned, so it is a reliable source of marks. It also pays off in the MCQs, where a recursion or a standard relationship can save several minutes.

Means and variances of assurance and annuity payments: topics in the order to study them

1Present Value Random Variables for Life ContingenciesEverything else is built on writing the present value as a function of the future lifetime, so learn this first.
2Mean and Variance of Assurance BenefitsAssurances are the simplest case: one payment, with a single present value to take expectations of, and the 2δ rule for the second moment.
3Mean and Variance of Annuity PaymentsAnnuities have a stream of payments, so the algebra is heavier, and you use the assurance results to keep it manageable.
4Relationships and Recursions Between Assurances and AnnuitiesOnce you know both, you can link them, for example ä_x = (1 − A_x) ÷ d, and use recursions to compute values and check answers.
5Assurances and Annuities with Varying BenefitsIncreasing and decreasing benefits extend the basic cases, so you need the fixed-benefit results solid first.

How to prepare Means and variances of assurance and annuity payments

Treat this chapter as a method to drill, not facts to memorise. Aim to build every result from the present value random variable.

Write the present value random variable for each product in terms of the future lifetime (K or T). Do this from memory until it is automatic, and state the assumptions, such as the interest rate and the mortality basis.
Derive E[Z] and E[Z²] for the whole life, term and endowment assurances. Show that E[Z²] is the mean at force of interest 2δ, or at interest rate 2i + i².
For endowment assurances and combinations of benefits, work out Var(Z) carefully, because the benefit pays on both death and survival and you cannot just add variances. Use Z² directly.
Learn the links between annuities and assurances: ä_x = (1 − A_x) ÷ d and Var(Y) = Var(Z) ÷ d² for the whole life case. Then practise term and deferred versions.
Practise recursions such as A_x = v q_x + v p_x A_(x+1). Use them to roll values back from a life table, as you would in Paper B.
Do varying benefits last. Split an increasing benefit into a sum of level benefits, and check each answer for reasonableness.
Finish with timed past paper questions. Write the formula, the working and the result in full. Then redo the same numerical questions in R or Excel.

Common mistakes in Means and variances of assurance and annuity payments

Using Var(Z) = E[Z]² instead of E[Z²] − (E[Z])².
Fix: Write Var(Z) = E[Z²] − (E[Z])² in every answer, and compute E[Z²] separately each time.
Computing the second moment at the original interest rate.
Fix: Remember that the second moment uses force 2δ, equivalently rate 2i + i². Check you have switched the rate before reading values.
Adding the variances of the death and survival parts of an endowment assurance.
Fix: Work with Z as one variable and find E[Z²] directly. The two indicator parts can never both be non-zero, so the cross-term is zero in E[Z²].
Mixing annuity-due and annuity-immediate, or the wrong discount factor in the relationship with assurances.
Fix: Check the timing of the first payment. Use d for the annuity-due link, and i for the annuity-immediate link with a_x = ä_x − 1.
Mismatching the timing of the death benefit, such as year-end against moment of death.
Fix: Read the payment timing first. Write the present value as v^(K+1) or v^T before any calculation.
Leaving out the assumptions and the working in written answers.
Fix: State the basis, give the formula in standard notation, show the substitution, then the result.

Last-day revision: Means and variances of assurance and annuity payments

Z is the present value random variable. E[Z] is its mean, and Var(Z) = E[Z²] − (E[Z])².
For a whole life assurance paid at the end of the year of death, Z = v^(K+1). With payment at the moment of death, Z = v^T.
E[Z²] is the mean of the same benefit at force of interest 2δ, so that v becomes v².
For an annuity-due, Y = ä_(K+1) with K the curtate future lifetime.
ä_x = (1 − A_x) ÷ d, where d = i ÷ (1 + i) = 1 − v. This holds for whole life with the same mortality basis.
Var(Y) = Var(Z) ÷ d² for the whole life annuity-due, where Var(Z) = ²A_x − (A_x)².
Term and endowment assurances: split Z into a death part and a survival part, and be careful with the cross-term.
Endowment assurance: A_(x:n) = A¹_(x:n) + A_(x:n)¹, where the second term is the pure endowment.
Recursion: A_x = v q_x + v p_x A_(x+1), and ä_x = 1 + v p_x ä_(x+1).
Varying benefits: write an increasing benefit as a sum of level benefits, or use the standard (IA) notation.
Always state your interest rate and mortality table at the start of an answer.
Check the sign and size of your answer: A_x lies between 0 and 1 for a unit benefit.

Means and variances of assurance and annuity payments practice questions

Means and variances of assurance and annuity payments in other exams

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

Means and variances of assurance and annuity payments: frequently asked questions

How do I find the variance of an assurance benefit?

Find E[Z] and E[Z²], then use Var(Z) = E[Z²] − (E[Z])². E[Z²] is the expected present value at the doubled force of interest, so you can use the same method with v replaced by v². Keep the benefit amount in mind, since a benefit of S multiplies the variance by S².

Why is the variance of an annuity linked to an assurance variance?

For a whole life annuity-due, Y = (1 − Z) ÷ d, where Z is the present value of a whole life assurance. So Var(Y) = Var(Z) ÷ d². This avoids summing a long series of payments directly.

Do I need to memorise all the formulas?

Learn the present value definitions and the main relationships, such as ä_x = (1 − A_x) ÷ d and the recursions. You can derive most other results from them. Check which formulas the Actuarial Formulae and Tables book gives you, and practise using it.

How does this chapter appear in Paper B?

Paper B is a computer-based exam in R or Excel. You may need to compute assurance and annuity values from a life table, then find means and variances using recursions. Practise building the table calculation yourself so that you can explain each step.