Compound Interest
Compound interest occurs when interest earned on an investment or deposit is added to the principal, so that future interest calculations include both the original amount and previously earned interest. This compounding effect creates exponential growth rather than linear growth. The formula A = P(1 + r)^n calculates the final amount A when principal P grows at rate r for n years.
Why it matters
Compound interest forms the foundation of long-term financial planning and appears throughout GCSE mathematics in Year 10. A £10,000 investment at 7% annual interest grows to £19,672 after 10 years, but £38,697 after 20 years — demonstrating how time amplifies the effect. Index funds, ISAs, and pension contributions all rely on compound growth, where starting early creates dramatic advantages. Someone investing £200 monthly from age 25 at 6% annual return accumulates £328,095 by age 65, whilst someone starting at age 35 with the same contributions reaches only £167,603. This mathematical principle explains why financial advisers emphasise early saving and why compound interest is often called the eighth wonder of the world.
How to solve compound interest
Compound Interest
- Compound interest earns interest on both the original principal AND on previously earned interest — that's why the curve bends upward over time.
- Annual: A = P(1 + r)n, where P is the principal, r the rate as a decimal, n the number of years.
- Monthly compounding: A = P(1 + r/12)12n.
- With monthly contributions PMT: future value = PMT × [((1 + i)n − 1) / i], where i = r/12 and n is the number of months.
- Index funds and savings accounts both rely on this — small early differences in rate, time, or starting age compound to outsized differences at the end.
Example: £10,000 at 7% for 20 years: A = 10000 · 1.07²⁰ ≈ £38,697.
Worked examples
You deposit £4,000.00 in a savings account paying 6% interest per year. How much is in the account after one year?
Answer: 4240
- Calculate the interest → 4000 × 6100 = 240 — Interest = principal × rate. £4,000.00 × 6% = £240.00.
- Add the interest to the principal → 4000 + 240 = 4240 — After one year: £4,000.00 + £240.00 = £4,240.00.
You invest £5,000.00 at 6% compound interest per year. What is the value after 3 years?
Answer: 5955
- Use the compound interest formula → A = P(1 + r)n — P is the principal, r the rate as a decimal, and n the number of years.
- Plug in the values → A = 5000 × (1 + 0.06)3 — P = £5,000.00, r = 0.06, n = 3.
- Compute the growth factor → (1 + 0.06)3 = 1.1910 — Raise 1 + r to the power n.
- Multiply by the principal → A ≈ £5,955.00 — £5,000.00 × 1.1910 ≈ £5,955.00 after rounding.
You invest £50,000.00 in an index fund returning 5% per year. What is the value after 12 years, assuming returns are reinvested?
Answer: 89793
- Apply A = P(1 + r)n → A = 50000(1 + 0.05)12 — Reinvested returns compound — the formula treats each year's gain as next year's principal.
- Compute the result → A ≈ £89,793.00 — After 12 years, £50,000.00 grows to approximately £89,793.00 — a gain of £39,793.00.
Common mistakes
- Confusing simple and compound interest leads to calculating £5,000 at 4% for 3 years as £5,600 (simple interest: 5000 + 3×200) instead of £5,624.32 using the compound formula.
- Applying the wrong time period results in monthly compounding errors, such as using n = 2 instead of n = 24 months when calculating 2 years of monthly compounding.
- Forgetting to convert percentages to decimals produces incorrect calculations like using r = 6 instead of r = 0.06, yielding vastly inflated results.