Annuities & Loan Repayments (Grade 12)

Future value for savings and present value for loans, working out a monthly instalment, and finding the outstanding balance part-way through a loan.

1The two formulae

Future value (saving up)
F = x[(1 + i)ⁿ − 1]i

Present value (paying off)
P = x[1 − (1 + i)⁻ⁿ]i

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Choose by asking when the lump sum exists. Saving towards an amount in the future ⇒ F. Borrowing money you have now ⇒ P. The only difference in the formula is the sign of the exponent.

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i is the rate per payment period, not per year. For 9% p.a. compounded monthly, i = 0,0912 = 0,0075 and n counts months.

2Future value: saving

Worked ExampleR1 500 is deposited monthly for 20 years at 9% p.a. compounded monthly. Determine the future value.

1
i = 0,0912 = 0,0075 and n = 20 × 12 = 240.
Convert to the payment period first, always.
2
F = 1500[(1,0075)²⁴⁰ − 1]0,0075.
3
= R1001830,30.
Total deposited was only 240 × 1 500 = R360 000 — the rest is interest.

3Present value: a loan instalment

Worked ExampleA bond of R800 000 is repaid monthly over 20 years at 9% p.a. compounded monthly. Determine the instalment.

1
i = 0,0075, n = 240, P = 800 000.
2
800 000 = x[1 − (1,0075)⁻²⁴⁰]0,0075.
Present value — the money exists now.
3
x = 800 000 × 0,00751 − (1,0075)⁻²⁴⁰.
Rearrange for x.
4
= R7197,81 per month.
Over 240 months that is about R1,73 million repaid on an R800 000 loan.

4Outstanding balance

The balance still owing is the present value of the payments still to come — not the amount already paid subtracted from the loan.

Worked ExampleHow much is still owing on that bond after 5 years?

1
Payments made: 5 × 12 = 60. Payments remaining: 240 − 60 = 180.
2
Balance = ${R(inst)}[1 − (1,0075)⁻¹⁸⁰]0,0075.
Present value of the 180 payments still due.
3
= R709656,39.
After a quarter of the term, barely R60 000 of the capital has been repaid — the early payments are almost all interest.

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If a question says payments start one month after the loan, the formula works as it stands. If there is a delay, adjust n and be explicit about it.

5Using logarithms to find n

When the unknown is the number of periods, it sits in the exponent — and the only way down from an exponent is a logarithm.

A = P(1 + i)ⁿ ⇒ n = log(A ÷ P)log(1 + i)

Worked ExampleHow long will R50 000 take to grow to R80 000 at 9% p.a. compounded annually?

1
80 000 = 50 000(1,09)ⁿ.
2
1,6 = (1,09)ⁿ.
Divide both sides by 50 000.
3
n = log 1,6log 1,09.
Take logs of both sides and use the power law.
4
n = 5,45 years.
Interest is only paid at year end, so in practice you would wait 6 full years.

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Any base works, as long as you use the same base top and bottom. The calculator's log and ln keys both give the same n.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
R1 500 is deposited monthly for 20 years at 9% p.a. compounded monthly. Determine the future value.
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i = 0,0075, n = 240 → F = R1001830,30
2
How much was actually deposited in total in that annuity?
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240 × 1 500 = R360 000
3
A bond of R800 000 is repaid monthly over 20 years at 9% p.a. compounded monthly. Determine the monthly instalment.
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x = 800 000 × 0,00751 − (1,0075)⁻²⁴⁰ = R7197,81
4
Determine the outstanding balance on that bond after 5 years.
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PV of the remaining 180 payments = R709656,39
5
Write down the value of i for 9% p.a. compounded monthly.
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0,0912 = 0,0075
6
Which formula do you use when saving towards a future amount?
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Future value, F = x[(1+i)ⁿ − 1]i
7
Which formula do you use for a loan taken out today?
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Present value, P = x[1 − (1+i)⁻ⁿ]i
8
How is the outstanding balance on a loan calculated?
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As the present value of the payments still outstanding.