Financial Mathematics

Key concepts

• Compound: A = P(1 + i)ⁿ
• Future value annuity: F = x · [(1+i)ⁿ − 1] / i
• Present value annuity: P = x · [1 − (1+i)⁻ⁿ] / i
• Effective rate: (1 + i/m)^m − 1

How it works

Read the question to identify: lump sum vs annuity, future vs present value, nominal vs effective rate, and the compounding period. Convert the interest rate to match the payment period before substituting. Draw a time line for every multi-step question, marking the deposits or payments, any withdrawals and any change in interest rate. Then decide whether you are moving money forward in time (future value) or back to the start (present value), and use the matching formula.

Step by step

1. Identify the scenario

Lump sum or annuity? Future or present value?

2. Convert the rate

Match i and n to the compounding period.

3. Substitute

Use the correct formula and a calculator.

4. Round properly

Round to two decimals (rand and cents) at the end.

Worked examples

R10 000 invested for 5 years at 9% p.a. compounded monthly. Find A.

1. i = 0.09/12 = 0.0075; n = 60
2. A = 10000(1.0075)^60 ≈ R15 657

Answer: ≈ R15 657

How many years does it take R5 000 to double at 8% p.a. compounded annually?

1. 10 000 = 5 000(1,08)ⁿ, so 1,08ⁿ = 2
2. n = log 2 / log 1,08 = 0,30103 / 0,03342

Answer: n ≈ 9,01, so just over 9 years

A R200 000 car loan is repaid monthly over 5 years at 12% p.a. compounded monthly. Find the monthly repayment.

1. Present value annuity with i = 0,01 and n = 60
2. 200 000 = x[1 − (1,01)⁻⁶⁰] / 0,01
3. [1 − (1,01)⁻⁶⁰] / 0,01 ≈ 44,955

Answer: x ≈ R4 448,89 per month

Convert 9% p.a. compounded monthly to an effective annual rate.

1. 1 + i_eff = (1 + 0.09/12)¹²
2. = 1.0075¹² ≈ 1.0938

Answer: ≈ 9.38% p.a.

A machine costing R80 000 depreciates at 12% p.a. on the straight-line method. Find its value after 4 years.

1. A = P(1 − in)
2. A = 80 000(1 − 0.12 × 4) = 80 000 × 0.52

Answer: R41 600

How much must be saved each month to have R100 000 in 5 years at 7% p.a. compounded monthly?

1. Use F = x[(1 + i)ⁿ − 1] / i with i = 0.07/12 and n = 60
2. 100 000 = x × 71.5929…
3. x = 100 000 / 71.5929…

Answer: ≈ R1 396.79 per month

Common mistakes

• Mixing annual and monthly rates
• Using future value formula when present value is required
• Rounding too early
• Mixing up straight-line depreciation A = P(1 − in) with reducing-balance depreciation A = P(1 − i)ⁿ.

Frequently asked questions

What's the difference between nominal and effective rates?

Nominal ignores compounding within the year; effective accounts for it.

What is a sinking fund?

A savings plan, usually with monthly payments, set up to pay for a known future cost such as replacing equipment. It is calculated with the future value annuity formula.

When do I use logarithms in finance?

To find n, the number of periods. Rearrange the formula so that (1 + i)ⁿ equals a number, then n = log(number) / log(1 + i).