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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Time Value of Money: Compound Interest and Discounting

Time value of money means a rupee today is worth more than a rupee later, because it can earn interest. You move money through time using accumulation factors (1 + i)^t and discount factors v^t, where v = 1 ÷ (1 + i). Convert every rate to one effective basis before you compare or combine cash flows.

What this chapter covers

This chapter builds the base for all of CM1. You learn how money grows under simple and compound interest, how to bring future payments back to today using discount factors, and how interest rates can be quoted in different ways: effective rates, discount rates, nominal rates and the force of interest.

The four topics link together. Simple and compound interest give you the growth rule. Present values reverse that rule. The effective rate of discount gives the same idea from the other side, where interest is paid at the start of the period. Nominal rates and the force of interest extend it to any compounding frequency, and to the continuous limit.

Everything later in CM1 depends on this. Annuities, equation of value, loans, bonds, and pricing and reserving for life contracts all use v, i, d, δ and i^(m) constantly. In CM2 the same ideas return in asset valuation and interest rate models. If this chapter is weak, later chapters will feel slow and error-prone.

CM1 Theory of interest rates carries 25% of the 2026 syllabus, and Equation of value (20%) and Pricing and reserving (35%) both rely on these conversions. Recent CM1 Paper A papers open with multiple-choice questions of 2 marks each, and many of these test quick rate conversions. Written questions then expect you to state your assumptions and show clean working. The Paper B computer-based exam also needs you to code or enter these same formulas correctly. The chapter is short, so effort here repays you across the whole paper.

Time value of money: compound interest and discounting: topics in the order to study them

  1. 1Simple and Compound InterestStart here because it defines accumulation, and the contrast between linear and exponential growth is the base for all later formulas.
  2. 2Present Value and Discount FactorsOnce you can accumulate, you reverse the process with v = 1 ÷ (1 + i), which you will use in every later chapter.
  3. 3Effective Rate of DiscountIt builds on v, since d = i ÷ (1 + i) = 1 − v, and it prepares you for annuities-due and discount-based instruments.
  4. 4Nominal Rates and Force of InterestStudy this last because it needs all the earlier ideas: it generalises them to any compounding frequency and to continuous time.

How to prepare Time value of money: compound interest and discounting

Aim to make the conversions automatic, then practise using them inside longer questions. Work with a calculator and keep a one-page list of relations.

  1. Read each topic once and write the core relations in your own words: 1 + i = v⁻¹, d = iv, d = 1 − v, and (1 + i^(m)/m)^m = 1 + i.
  2. Derive the formulas yourself at least once, for example why d = i ÷ (1 + i), so you can rebuild them if you forget.
  3. Practise converting between i, d, i^(m), d^(m) and δ until you can move between any two without hesitation. Always find the effective annual rate first.
  4. Solve short questions on accumulating and discounting single payments, including time periods that are not whole years.
  5. Do questions where the rate changes over time. Use the product of accumulation factors for varying rates, and ∫ of δ(s) for a varying force of interest.
  6. Practise timed multiple-choice questions, then write full answers to past-style written questions, stating the rate basis and the time point you use.
  7. If you prepare for Paper B, repeat key calculations in R or Excel and check them against your hand answers.

Common mistakes in Time value of money: compound interest and discounting

  • Treating a nominal rate as an effective annual rate

    Fix: Divide a nominal rate by m to get the per-period rate, then convert to an effective annual rate with (1 + i^(m) ÷ m)^m − 1 if you need it.

  • Mixing up i and d

    Fix: Remember d = i ÷ (1 + i), so d is always smaller than i for positive rates. Check your answer against that.

  • Using simple interest when compound interest applies, or the reverse

    Fix: Look for words such as compounded, effective, or convertible. Use simple interest only when the question states it.

  • Discounting for the wrong number of periods

    Fix: Draw a timeline, mark the valuation date, and count the exact time from each payment to that date.

  • Rounding rates too early

    Fix: Store full calculator values and round only the final answer. In multiple-choice questions, rounding errors can push you to a wrong option.

  • Forgetting to state assumptions in written answers

    Fix: Begin the working with a line such as 'Let i be the effective annual rate', and define each symbol you introduce.

Last-day revision: Time value of money: compound interest and discounting

  • Simple interest: A(t) = P(1 + it). Compound interest: A(t) = P(1 + i)^t.
  • Discount factor: v = 1 ÷ (1 + i), and PV = payment × v^t.
  • Effective rate of discount: d = i ÷ (1 + i) = iv = 1 − v.
  • Link: i − d = id, so 1 − d = v.
  • Nominal rate: (1 + i^(m) ÷ m)^m = 1 + i, where i^(m) is paid m times a year.
  • Nominal discount: (1 − d^(m) ÷ m)^(−m) = 1 + i.
  • Force of interest: δ = ln(1 + i), so e^δ = 1 + i and v = e^(−δ).
  • Generally δ(t) = (dA/dt) ÷ A(t), and A(t) = exp(∫ δ(s) ds) from 0 to t.
  • For a constant force of interest, accumulation over t years is e^(δt).
  • As m → ∞, both i^(m) and d^(m) tend to δ, and for m > 1, d < d^(m) < δ < i^(m) < i.
  • Compare rates only after converting them to the same effective basis.
  • State the time unit, rate basis and valuation date in written answers.

Time value of money: compound interest and discounting practice questions

Time value of money: compound interest and discounting in other exams

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

Time value of money: compound interest and discounting: frequently asked questions

Do I need to memorise all the interest rate conversion formulas for CM1?

You should know the main ones well: v, d, i^(m), d^(m) and δ in terms of i. Understanding how they are derived helps you rebuild any you forget. Check the formula sheet rules for your session, as IAI may change what is supplied.

What is the difference between the effective rate of discount and the effective rate of interest?

Interest is paid at the end of the period and is measured against the opening amount. Discount is paid at the start and is measured against the closing amount. They are linked by d = i ÷ (1 + i).

Why is the force of interest important?

It describes interest as a continuous rate at each instant, δ(t). It lets you handle varying rates through integration and gives clean formulas such as v = e^(−δ) for a constant force of interest. It also links to continuous models used later in the course.

How much of this chapter appears in the Paper B computer-based exam?

Paper B tests the subject through R or Excel, and these conversions often sit inside larger calculations. You should be able to enter the formulas correctly and check results against hand calculations. The exact content varies by session, so practise both formats.