Interest - Simple and Compound Explained
Interest is the cost of borrowing money or the reward for lending it. When you borrow, you pay interest. When you save or invest, you earn it. Understanding the two types of interest – simple and compound – is one of the most practically useful skills in financial mathematics.
Rules governing interest are among the oldest surviving pieces of written law: the Babylonian Code of Hammurabi, carved into stone around 1750 BCE, fixed maximum interest rates that lenders could charge on loans of grain and silver, protecting borrowers from exploitative rates. Compound interest is a much later refinement – the English mathematician Richard Witt published the first dedicated book of compound interest tables, Arithmeticall Questions, in 1613, giving merchants and bankers a practical way to calculate growth without laborious repeated multiplication. Remarkably, the mathematical constant e (Euler's number, ≈ 2.71828), which underlies continuous compounding, was first stumbled upon by the Swiss mathematician Jacob Bernoulli in 1683 while he was investigating exactly this question: what happens to compound interest as the compounding periods become infinitely frequent.
Simple Interest
Simple interest is calculated only on the original amount (the principal). It does not grow over time – the same amount of interest is added each period.
I = P × R × T
Where: I = interest, P = principal, R = annual interest rate (as a decimal), T = time in years.
Total amount = P + I = P(1 + RT)
Compound Interest
Compound interest is calculated on the principal plus the interest already earned. Each period, the interest is added to the balance, and the next period's interest is calculated on that larger amount – the snowball effect.
A = P(1 + r/n)nt
Where: A = final amount, P = principal, r = annual rate (decimal), n = number of compounding periods per year, t = years.
Compounding Frequency
| Frequency | n (per year) | Effect |
|---|---|---|
| Annual | 1 | Interest added once a year |
| Semi-annual | 2 | Every 6 months |
| Quarterly | 4 | Every 3 months |
| Monthly | 12 | Every month |
| Daily | 365 | Every day – highest return |
The more frequently interest compounds, the greater the final amount – though the difference between monthly and daily is usually small in practice.
Simple vs Compound: A Comparison
| Year | Simple interest (5%) | Compound interest (5%) |
|---|---|---|
| Start | £1 000 | £1 000 |
| 1 | £1 050 | £1 050.00 |
| 5 | £1 250 | £1 276.28 |
| 10 | £1 500 | £1 628.89 |
| 20 | £2 000 | £2 653.30 |
| 30 | £2 500 | £4 321.94 |
Annual Percentage Rate (APR)
APR is the standardised annual interest rate that includes fees and charges, used to compare the true cost of borrowing across different loans and credit products. A higher APR means a more expensive loan. By law, lenders must display the APR so consumers can make fair comparisons.
Worked Examples
I = 4 000 × 0.06 × 5 = £1 200.
Total = £4 000 + £1 200 = £5 200.
A = 4 000 × (1.06)5 = 4 000 × 1.3382 = £5 352.90.
Compound earns £152.90 more than simple interest over 5 years.
A = 2 500 × (1 + 0.04/12)12 × 3 = 2 500 × (1.003333)36 = 2 500 × 1.12749 = £2 818.73.
I = 1 040 − 800 = £240.
R = I / (P × T) = 240 / (800 × 3) = 240 / 2 400 = 0.10 = 10% per year.
Finding Principal, Rate, or Time
Rearranging I = PRT:
P = I / (RT) R = I / (PT) T = I / (PR)
Key Takeaways
- Simple interest: I = PRT. The same amount is added each period.
- Compound interest: A = P(1 + r/n)nt. Interest earns further interest.
- More frequent compounding = higher final amount.
- APR is the standardised rate for comparing borrowing costs.
Practice: Simple and Compound Interest
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