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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Equation of Value for IAI CM1: Chapter Guide

The **equation of value** sets the present value of money coming in equal to the present value of money going out, at a stated interest rate. Write every cashflow against time, bring each to one date using discount factors, then solve for the unknown rate, payment or time.

What this chapter covers

This chapter is the base of CM1 Actuarial Mathematics for Modelling. It starts with how money grows with time, using effective rates, nominal rates, force of interest and discount rates. It then builds accumulation and present value factors, and uses them to value any set of cashflows at any date.

The core idea is the equation of value: value of receipts = value of payments, at the same date and the same interest basis. From this you solve for an unknown payment, a time, or an interest rate (the yield). Level annuities, deferred annuities and increasing annuities are just special cases where the cashflow pattern gives a neat formula.

The chapter feeds the rest of the paper. Theory of interest rates (25% of the 2026 syllabus weighting) and Equation of value (20%) together make up a large part of CM1. Pricing and reserving later use the same logic, with survival probabilities added to the discounting. If your interest maths is shaky, life contingencies will feel much harder than it should.

Interest and equation of value questions are some of the most predictable marks in CM1. Almost every written question uses discounting somewhere, so an error here also costs marks in later parts. The methods are mechanical once you practise them, which makes this chapter a good place to gain secure marks in both the multiple-choice section and the written questions. It also helps in Paper B, where you may be asked to solve for a yield or build a loan schedule on the computer.

Equation of value: topics in the order to study them

  1. 1Time Value of Money and Interest RatesEverything else depends on being fluent with effective, nominal and force of interest rates and converting between them.
  2. 2Accumulation and Present Value FactorsYou need v, (1 + i)ⁿ and discount factors, including with a varying rate, before you can value any cashflow.
  3. 3Equation of Value PrincipleWith factors in hand, you can learn to choose a valuation date and set value of inflows equal to value of outflows.
  4. 4Annuities and Level Payment CashflowsAnnuity formulas are shortcuts for the equation of value, so they come once the general principle is clear.
  5. 5Solving for Unknown Interest Rate and PaymentsThis uses the principle and annuity formulas together, and adds interpolation and numerical solving for the rate.
  6. 6Applications: Project Appraisal and LoansIt comes last because it combines everything: net present value, internal rate of return, loan schedules and capital repaid.

How to prepare Equation of value

Spend your time on working through problems, not rereading formulas. This chapter rewards repetition and careful setup.

  1. Write the rate conversions from memory: (1 + i) = (1 + i⁽ᵐ⁾/m)ᵐ = e^δ and d = i ÷ (1 + i) = iv. Test yourself until they are automatic.
  2. Practise valuing simple cashflows on a timeline. Draw the line, mark each payment, pick a valuation date, then discount or accumulate.
  3. Learn the annuity formulas: a⌉ₙ = (1 − vⁿ) ÷ i, ä⌉ₙ = (1 − vⁿ) ÷ d, and s⌉ₙ = ((1 + i)ⁿ − 1) ÷ i. Know what timing each one assumes, payments at end or start of period.
  4. Solve unknown-payment and unknown-time questions first, since they need only algebra, then move to unknown-rate questions using linear interpolation and checks.
  5. Do loan and project questions in full: schedule of interest, capital repaid and outstanding balance, then NPV and IRR with a stated decision rule.
  6. Finish with timed mixed questions. For written answers, state the valuation date and rate used, show the equation, then the result. Redo each error a few days later.

Common mistakes in Equation of value

  • Mixing up annuity-immediate and annuity-due formulas.

    Fix: Draw the timeline first. If the first payment is at time 0, use ä; if at time 1, use a. Check by converting: ä⌉ₙ = (1 + i) × a⌉ₙ.

  • Using a nominal rate directly as the effective rate per period.

    Fix: Convert first. The effective quarterly rate is 8% ÷ 4 = 2%, and the effective annual rate is 1.02⁴ − 1.

  • Discounting for the wrong number of periods.

    Fix: Mark every payment on a line and count periods from the valuation date to each payment before writing the equation.

  • Not stating the valuation date or rate in a written answer.

    Fix: Write a line such as 'Equation of value at time 0, at 6% pa effective' before the equation. Method marks depend on this.

  • Accepting an interpolated rate without checking it.

    Fix: Substitute your rate back into the equation to confirm it is close, and say that the result is approximate.

  • Treating IRR as always unique and always meaningful.

    Fix: Look for more than one change of sign. If so, there may be several IRRs, so compare using NPV at the given rate.

Last-day revision: Equation of value

  • Effective annual rate i: (1 + i) = (1 + i⁽ᵐ⁾/m)ᵐ for nominal rate i⁽ᵐ⁾ convertible m times a year.
  • Force of interest δ = ln(1 + i), so 1 + i = e^δ.
  • Discount rate d = i ÷ (1 + i) = 1 − v, where v = 1 ÷ (1 + i).
  • Equation of value: PV of inflows = PV of outflows, at the same date and rate.
  • Any valuation date gives the same answer if the rate is constant; pick the one that simplifies the algebra.
  • a⌉ₙ = (1 − vⁿ) ÷ i for payments of 1 at the end of each year for n years.
  • ä⌉ₙ = (1 − vⁿ) ÷ d for payments at the start of each year, and ä⌉ₙ = (1 + i) × a⌉ₙ.
  • s⌉ₙ = ((1 + i)ⁿ − 1) ÷ i, and s⌉ₙ = (1 + i)ⁿ × a⌉ₙ.
  • A deferred annuity starting after k years has value vᵏ × a⌉ₙ for payments at end of years k+1 to k+n.
  • IRR is the rate that makes NPV equal to zero; check for sign changes in the cashflows.
  • Loan balance after t payments equals the PV of the remaining payments at the loan rate.
  • Interest in a loan payment = rate × opening balance; capital repaid = payment − interest.

Equation of value practice questions

Equation of value in other exams

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

Equation of value: frequently asked questions

What is the equation of value in CM1?

It states that the present value of all receipts equals the present value of all payments, at a given interest rate and valuation date. You use it to find an unknown payment, time or interest rate. Most annuity and loan results come from it.

Which valuation date should I choose?

Any date gives the same answer when the interest rate is constant. Pick the date that removes the unknown from awkward factors, often the date of an unknown payment or time 0. This keeps the algebra short and lowers the chance of errors.

How do I find an unknown interest rate in an exam?

If the equation reduces to a simple form, solve it directly. Otherwise, try two rates that give values on either side of zero, then use linear interpolation. Check the result by substituting it back.

Is this chapter useful for Paper B?

Yes. Paper B is computer-based, and tasks like finding a yield, building a loan schedule or computing NPV for many rates are well suited to R or Excel. Learn the hand method first so you can check the computer output.