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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Annuity and Accumulation Functions Explained for IAI Actuarial

Annuity and accumulation functions measure how money grows with interest and how a series of payments is valued. The accumulation function A(t) gives growth; discount functions go backwards; annuity functions add up level or varying payments. You solve questions by drawing a timeline and valuing every payment at one date.

What this chapter covers

This chapter is the base of CM1 Actuarial Mathematics for Modelling. It starts with the accumulation function A(t), which tells you how much one unit invested at time 0 is worth at time t. The discount function v(t) is its reverse. From there you meet simple interest, compound interest, the force of interest δ(t) and varying rates.

The second half moves to streams of payments. You value annuity-immediate (payments at the end of each period) and annuity-due (payments at the start), then perpetuities, deferred annuities and m-thly annuities. The last topic covers increasing and decreasing annuities, where payments change in a pattern such as 1, 2, 3, ... or n, n−1, ..., 1.

The chapter feeds directly into the rest of the paper. Theory of interest rates and equation of value use the same tools. Loans, bonds, and cashflow valuation rely on annuity formulas. Later, life annuities, premiums and reserves are annuity functions weighted by survival probabilities. If this chapter is shaky, pricing and reserving will be hard.

Theory of interest rates and equation of value are two of the four CM1 syllabus areas, and together they carry a large share of the 2026 weighting (25% and 20%). Annuity functions also appear inside pricing and reserving questions, so one skill earns marks across the paper. In Paper A you need clean written working. In the multiple-choice questions and Paper B (computer-based) you need speed and accuracy. Students who master the timeline method lose few marks here.

Annuity and accumulation functions: topics in the order to study them

  1. 1Accumulation and Discount FunctionsEvery later formula is built on A(t) and v(t), so learn the definitions and the link A(t) × v(t) = 1 first.
  2. 2Simple and Compound Interest AccumulationThese are the two standard forms of A(t); you need them before handling rates that change.
  3. 3Force of Interest and Varying RatesThe force of interest generalises both forms and lets you handle rates that vary with time using δ(t) and an integral.
  4. 4Annuity-Immediate and Annuity-Due FunctionsWith the interest tools in place, you can now sum a series of level payments and see why a-due = (1 + i) × a-immediate.
  5. 5Perpetuities, Deferred and m-thly AnnuitiesThese are variations on the basic annuity, so they are easier once the immediate and due cases are secure.
  6. 6Increasing and Decreasing AnnuitiesThese are the hardest because payments change; they combine annuity formulas with a pattern, so study them last.

How to prepare Annuity and accumulation functions

Treat this chapter as one skill: put every payment on a timeline and value it at a single date. Practise that skill until it is automatic.

  1. Write the definitions of A(t), v(t), i, d, δ and the links between them (for example v = 1 ÷ (1 + i), d = i ÷ (1 + i)) on one page, and learn them cold.
  2. Derive the main annuity formulas yourself: a-angle-n = (1 − vⁿ) ÷ i and ä-angle-n = (1 − vⁿ) ÷ d. Deriving once makes them easier to recall and adapt.
  3. For each question, draw a timeline first. Mark payment dates and the valuation date, then decide what each payment is worth at that date.
  4. Do mixed practice on varying rates: use δ(t) with the integral exp(∫δ(s) ds) for accumulation, and check the units of time.
  5. Practise deferred, perpetuity and m-thly cases by converting them back to a basic annuity plus a shift in time.
  6. Work past exam questions under time. Write full steps for Paper A style answers, and repeat the key calculations on a calculator or in the computer-based format for Paper B.
  7. Finish each session by checking one answer using a different route, such as accumulating instead of discounting.

Common mistakes in Annuity and accumulation functions

  • Mixing up annuity-immediate and annuity-due

    Fix: Mark the first payment on the timeline. If it is at time 0 it is due; if at time 1 it is immediate. Use ä = (1 + i) × a to convert.

  • Using the wrong valuation date for deferred annuities

    Fix: Value the annuity at the date just before its first payment (immediate case), then discount that value back to the required date.

  • Treating nominal and effective rates as the same

    Fix: Convert first: (1 + i⁽ᵐ⁾ ÷ m)ᵐ = 1 + i. Then use the effective rate in all formulas.

  • Errors with the force of interest integral

    Fix: Write A(t₁, t₂) = exp(∫ from t₁ to t₂ of δ(s) ds) and state the limits before integrating.

  • Applying increasing annuity formulas to the wrong pattern

    Fix: Check the pattern against n = 2 or 3 by direct summation. If it does not match, rebuild the annuity from level pieces.

  • Dropping steps in written answers

    Fix: State the formula in standard notation, show the substitution, then the result. Method marks are often available even when arithmetic slips.

Last-day revision: Annuity and accumulation functions

  • A(t) is the value at time t of 1 invested at time 0; v(t) = 1 ÷ A(t) when A(t) is the only growth path.
  • Simple interest: A(t) = 1 + it. Compound interest: A(t) = (1 + i)ᵗ.
  • Force of interest: δ(t) = A′(t) ÷ A(t), and A(t) = exp(∫₀ᵗ δ(s) ds).
  • With constant δ, (1 + i) = e^δ, so δ = ln(1 + i).
  • a-angle-n = (1 − vⁿ) ÷ i (payments at end of each period); s-angle-n = ((1 + i)ⁿ − 1) ÷ i.
  • ä-angle-n = (1 − vⁿ) ÷ d; ä-angle-n = (1 + i) × a-angle-n.
  • Perpetuity-immediate = 1 ÷ i; perpetuity-due = 1 ÷ d.
  • Deferred annuity: value = vᵐ × a-angle-n for payments starting after m periods (immediate type).
  • m-thly annuity: payments of 1/m at each m-th of a year; use the nominal rate i⁽ᵐ⁾ and d⁽ᵐ⁾ carefully.
  • Increasing annuity (Ia)-angle-n = (ä-angle-n − n·vⁿ) ÷ i.
  • Decreasing annuity (Da)-angle-n = (n − a-angle-n) ÷ i.
  • Always check the valuation date and whether payments are at the start or end of each period.

Annuity and accumulation functions practice questions

Annuity and accumulation functions in other exams

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

Annuity and accumulation functions: frequently asked questions