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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Key Assurance and Annuity Contracts in Actuarial Mathematics

Key assurance and annuity contracts are the basic life insurance and annuity products that CM1 values. You write each benefit as a present value random variable, such as v^K, then take its expectation. That gives the actuarial present value, like Ax or äx. Learn the notation, the cash flows and the links between them.

What this chapter covers

This chapter covers the building blocks of life contingent valuation. You meet the main contracts: whole life, term, endowment, pure endowment and deferred versions, plus annuities paid immediately, in advance, deferred or for a limited term. For each, you learn the benefit, when it is paid and how to value it.

The method is the same every time. Define the time of death or the future lifetime as a random variable. Write the present value of the payments as a function of it. Take the expected value. That expected value is the actuarial present value, written in standard notation such as Ax, äx, ax:n̄| and so on. You also learn how assurance and annuity values connect, for example Ax = 1 − d × äx for a whole life assurance with benefit paid at the end of the year of death. This relationship lets you move between the two without recalculating.

The chapter links to the whole paper. It relies on interest rate theory and the equation of value, and it uses survival probabilities from life tables. It then feeds directly into premium calculation and reserving, which is the largest part of the CM1 syllabus, and into multiple life and multiple decrement models. If you are shaky here, those later chapters will be hard.

Pricing and reserving carries the biggest syllabus weighting in CM1, and every premium or reserve question starts from an assurance or annuity value. The multiple-choice section tests notation, relationships and quick numerical values, while written questions ask you to define random variables, derive expectations and sometimes variances. Marks are lost mainly through notation slips and wrong timing of payments, both of which you can fix with practice. Paper B, the computer-based exam, also builds these values in R or Excel, so a clear grasp of the formulas helps there too. A solid grasp of this chapter pays back across the entire paper.

Key assurance and annuity contracts: topics in the order to study them

  1. 1Life Insurance Contract Types and BenefitsYou need to know what each contract pays, and when, before you can write any formula for it.
  2. 2Annuity Contracts: Immediate, Due, Deferred and TemporaryAnnuities are the second family of cash flows, and the due versus immediate distinction is used in everything that follows.
  3. 3Present Value Random Variables and Actuarial NotationThis turns the contract descriptions into formulas and expected values, the core skill of the chapter.
  4. 4Relationships Between Assurance and Annuity ValuesOnce you have both families in notation, you can link them and cut your working in exams.
  5. 5Premium Payment Patterns and Special ContractsLevel, single and limited premiums and special benefit designs build on all the earlier topics, so they come last.

How to prepare Key assurance and annuity contracts

Build this chapter in layers. First the cash flows, then the random variables, then the formulas. Do not memorise symbols before you understand the payment each one stands for.

  1. For each contract, draw a timeline showing when the benefit is paid and on what condition: death, survival or both. Do this until you can do it without notes.
  2. Write the present value random variable for every contract, for example v^(K+1) for a whole life assurance paid at the end of the year of death, where K is the curtate future lifetime. Then write its expected value in standard notation.
  3. Practise the annuity types side by side: immediate versus due, temporary versus whole life, deferred versus immediate. Check each time whether the first payment is at time 0 or time 1.
  4. Derive the key relationships yourself at least once, such as Ax = 1 − d × äx, and the recursive links between values at successive ages. Derivation helps you recall and adapt them.
  5. Do numerical questions using a life table and a fixed interest rate. Check each answer with a sense test: an assurance value must be below the sum assured, and a due annuity must exceed the matching immediate annuity.
  6. Practise the written style: state assumptions, define the random variable, show the expectation, then the number. Finally rebuild a few values in R or Excel for Paper B.

Common mistakes in Key assurance and annuity contracts

  • Mixing up annuity-due and annuity-immediate

    Fix: Ask whether a payment is made at time 0. If yes, it is a due annuity (ä). Use äx = 1 + ax to convert.

  • Using the wrong exponent for the discount factor

    Fix: For end of year of death use v^(K+1). Write the payment time on your timeline before choosing the exponent.

  • Applying the relationship Ax = 1 − d × äx to the wrong contract

    Fix: Remember it holds for whole life assurance with the sum assured paid at the end of the year of death. Derive adjusted versions for other cases.

  • Errors in the deferred and temporary annuity formulas

    Fix: For deferral, include v^n × npx and value the annuity at age x + n. For temporary annuities, subtract the deferred part from the whole life value.

  • Dropping notation detail in written answers

    Fix: Define the variable, state the mortality basis and interest rate, then write the expectation. Examiners award method marks for this.

  • Not sense-checking numerical results

    Fix: Compare your answer to bounds: assurance below the sum assured, due above immediate, whole life above temporary.

Last-day revision: Key assurance and annuity contracts

  • Assurance pays on death, an endowment on death or survival to the term, a pure endowment only on survival.
  • Annuity-due pays at the start of each period; annuity-immediate pays at the end.
  • Whole life assurance, end of year of death: Z = v^(K+1), with APV Ax.
  • Whole life annuity-due: Y = ä(K+1)| and its APV is äx = Σ v^t × tpx, summed from t = 0.
  • Ax = 1 − d × äx, where d = i ÷ (1 + i), for whole life assurance paid at the end of the year of death.
  • äx = 1 + ax, since the due annuity includes the payment at time 0.
  • Term assurance plus pure endowment gives endowment assurance.
  • Deferred annuity: n|äx = v^n × npx × ä(x+n).
  • Temporary annuity: äx:n̄| = äx − n|äx.
  • Pure endowment APV: nEx = v^n × npx.
  • Always check timing and the condition for payment before writing the formula.
  • Sense-check values: assurance APV is below the sum assured.

Key assurance and annuity contracts practice questions

Key assurance and annuity contracts in other exams

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

Key assurance and annuity contracts: frequently asked questions

How are assurance and annuity values linked in CM1?

They are linked through the discount rate. For a whole life assurance paid at the end of the year of death, Ax = 1 − d × äx, with d = i ÷ (1 + i). This lets you find one value from the other.

What is the difference between an annuity-due and an annuity-immediate?

An annuity-due pays at the start of each period, so the first payment is at time 0. An annuity-immediate pays at the end of each period, so the first payment is at time 1. Because of this, äx = 1 + ax.

Do I need to derive formulas for the exam?

Yes, written questions often ask you to derive or justify a result, not just use it. Practise deriving the main relationships from the present value random variables so you can do it under time pressure.

Is this chapter useful for the computer-based paper?

Yes. Paper B asks you to calculate these values in R or Excel, and you need to know the formulas to build the calculations correctly. Practise coding a life table and computing Ax and äx.