This note covers compound interest, depreciation, loan repayments and annuities, using the formula for single-sum problems and the GDC finance solver for repeated payments. These models describe savings, mortgages, car loans and pensions, so they appear often in real-world questions.
Key ideas
Compound interest. Interest is added to the balance and then earns interest itself. If \(PV\) is invested at a nominal annual rate of \(r\%\), compounded \(k\) times per year for \(n\) years, the future value is
\[ FV = PV\left(1+\frac{r}{100k}\right)^{kn} \]Here \(k=1\) for annual, \(2\) for half-yearly, \(4\) for quarterly and \(12\) for monthly compounding. The interest earned is \(FV-PV\).
Depreciation. If a value falls by \(r\%\) each year, multiply by \(\left(1-\frac{r}{100}\right)\) each year. After \(n\) years, \(FV = PV\left(1-\frac{r}{100}\right)^{n}\). This is the same idea as compound interest with a negative rate.
Loans and annuities. A loan is repaid by equal payments at regular intervals, and an annuity is a series of equal payments into (or out of) an account. These need the finance (TVM) solver on your GDC, which has these inputs:
- \(N\): the total number of payments.
- \(I\%\): the annual interest rate.
- \(PV\): present value, the amount at the start.
- \(PMT\): the payment each period.
- \(FV\): future value, the amount at the end. For a loan that is fully repaid this is 0.
- \(P/Y\) and \(C/Y\): payments per year and compounding periods per year. For monthly payments with monthly compounding both are 12.
Cash flowing out of your pocket is entered as negative and cash flowing in as positive. For a loan, \(PV\) is positive (you receive the money) and \(PMT\) comes out negative (you pay). Unless stated otherwise, payments are made at the end of each period. Enter four of the five quantities \(N\), \(I\%\), \(PV\), \(PMT\), \(FV\) and ask the solver to find the fifth.
Total paid and interest. Total paid \(= PMT\times N\). Total interest on a loan \(=\) total paid \(-\) amount borrowed.
Money amounts are normally given to 2 decimal places. Use the unrounded payment from the GDC for later calculations, and round only at the end.
Comparing options. Two investments with the same nominal rate but different compounding frequencies do not give the same return. More frequent compounding gives a slightly larger final value, because interest starts earning interest sooner. To compare options fairly, calculate the final value of each over the same time period.
Reading the situation. Before using the formula, decide whether the problem is a single sum (no extra payments, so use the formula) or has regular payments (use the solver). Then identify the direction of the cash. In a savings plan you pay in regularly and receive the balance at the end. In a loan you receive the money at the start and pay regularly.
Time to reach a target. To find how long a balance takes to reach a target, you can solve for \(N\) in the solver, or you can use the table or graph feature on your GDC to find where \(PV(1+\frac{r}{100k})^{x}\) first reaches the target. Remember that \(N\) counts payments, so convert it to years by dividing by \(P/Y\).
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| \(FV = PV\left(1+\dfrac{r}{100k}\right)^{kn}\) | Compound interest on a single sum with no extra payments. |
| \(FV = PV\left(1-\dfrac{r}{100}\right)^{n}\) | Depreciation at \(r\%\) per year. |
| Nominal rate | The stated annual rate \(r\%\), before compounding is taken into account. |
| Annuity | Equal payments at equal intervals, solved with the TVM solver. |
| Loan amortisation | Repaying a loan by equal payments so that the balance reaches 0 at the end. |
| Total interest | Total repaid minus amount borrowed (or final balance minus total deposited). |
Worked examples
Example 1 A student invests \(£8000\) at a nominal annual rate of \(3.6\%\), compounded monthly, for 5 years. Find the final value and the interest earned.
Here \(PV = 8000\), \(r=3.6\), \(k=12\), \(n=5\), so \(kn = 60\) and \(\dfrac{r}{100k} = \dfrac{3.6}{1200} = 0.003\).
\(FV = 8000\times(1.003)^{60} = 9575.158\ldots\)
Interest \(= 9575.16 - 8000 = 1575.16\).
Answer: final value \(£9575.16\); interest earned \(£1575.16\).
Example 2 A business borrows \(£15\,000\) at \(6\%\) per year, compounded monthly, and repays it with equal monthly payments over 4 years. Find the monthly payment and the total interest paid.
Use the TVM solver on the GDC. Enter \(N = 4\times12 = 48\), \(I\% = 6\), \(PV = 15000\), \(FV = 0\), \(P/Y = 12\), \(C/Y = 12\). Solve for \(PMT\).
The solver gives \(PMT = -352.2754\ldots\) The negative sign shows money paid out.
Total paid \(= 352.2754\ldots\times48 = 16\,909.22\).
Total interest \(= 16\,909.22 - 15\,000 = 1909.22\).
Answer: monthly payment \(£352.28\); total interest \(£1909.22\).
Example 3 Priya pays \(£200\) at the end of every month into a savings account paying \(4.8\%\) per year, compounded monthly. She starts with nothing. Find the balance after 10 years and the interest earned.
Use the TVM solver. Enter \(N = 120\), \(I\% = 4.8\), \(PV = 0\), \(PMT = -200\) (she pays money in), \(P/Y = 12\), \(C/Y = 12\). Solve for \(FV\).
The solver gives \(FV = 30\,726.39\ldots\)
Total deposited \(= 200\times120 = 24\,000\).
Interest \(= 30\,726.39 - 24\,000 = 6726.39\).
Answer: balance \(£30\,726.39\); interest earned \(£6726.39\).
Common errors
- Common error: using \(n\) instead of \(kn\) as the power, or forgetting to divide the rate by \(k\), when interest is compounded more than once a year.
- Common error: typing the rate as a decimal (0.06) in the solver. The solver expects the percentage (6).
- Common error: entering \(PV\) and \(PMT\) with the same sign, which gives a nonsense or error result. One must be positive and the other negative.
- Common error: setting \(P/Y\) and \(C/Y\) to 1 for a monthly problem, so that the solver treats \(N\) as years.
- Common error: rounding the payment and then multiplying by \(N\), which shifts the total by a few cents.
Exam tip: write down every value you enter in the solver (\(N\), \(I\%\), \(PV\), \(PMT\), \(FV\), \(P/Y\), \(C/Y\)) so that your method is clear even if a single entry is mistyped.
Check yourself
- Find the value of \(£5000\) invested for 8 years at \(2.5\%\) compounded annually.
Answer
\(FV = 5000(1.025)^{8} = 6092.01\ldots\) So \(£6092.01\). - A machine worth \(£24\,000\) loses \(12\%\) of its value each year. Find its value after 6 years.
Answer
\(24000\times0.88^{6} = 11\,145.70\ldots\) So \(£11\,100\) to 3 s.f. (\(£11\,145.70\) to the nearest cent). - How many whole years does it take for money invested at \(4\%\) compounded annually to double?
Answer
Solve \(1.04^{n}\ge2\) on the GDC (or by solving \(1.04^{n}=2\)): \(n = 17.67\ldots\) The balance first exceeds double after the end of year 18, so 18 years. - In the TVM solver, which inputs would you enter to find the monthly payment on a 25-year loan of \(£120\,000\) at \(5\%\) per year compounded monthly?
Answer
\(N = 300\), \(I\% = 5\), \(PV = 120000\), \(FV = 0\), \(P/Y = 12\), \(C/Y = 12\). Solve for \(PMT\) (it will be negative).