Compound Interest Explained: Why Starting Early Beats Saving More

Simple interest pays on the original amount. Compound interest pays on the original amount plus all interest already earned. That single difference turns a straight line into a curve, and over decades the curve dominates everything else.

Most people meet the idea as a cheerful graph and then make decisions as though interest were simple — underestimating both what a long horizon is worth and what a small ongoing fee costs. The arithmetic below is the whole of it.

The gap, concretely

£10,000 at 7% for 30 years:

• Simple: £10,000 + (£700 × 30) = £31,000
• Compound: £10,000 × 1.07³⁰ = £76,123

The £45,000 difference is interest earning interest. No additional money was contributed.

The formula

A = P × (1 + r/n)^(n×t)

• P — the starting amount
• r — the annual rate as a decimal (7% → 0.07)
• n — compounding periods per year
• t — years

For the example above: 10,000 × (1 + 0.07/1)^(1×30) = 10,000 × 7.6123 = £76,122.55. The whole formula is really just the growth factor 1.07 raised to the number of periods; everything else is bookkeeping about what a period is.

When you also add money each month, a second term is needed — the future value of a series of deposits:

A = C × ((1 + i)^m − 1) ÷ i

where C is the deposit, i the monthly rate and m the number of deposits. £100 a month at 7% for 30 years (i = 0.07/12, m = 360) comes to £121,997 from £36,000 of contributions. The investment calculator combines both terms.

Frequency: real, but overrated

£10,000 at 5% for 10 years:

Annual to monthly gains £181. Daily to continuous gains under a pound. Rate and time matter far more than frequency — a common misconception is that daily compounding is a meaningful advantage.

The Rule of 72

Divide 72 by the rate for approximate doubling time:

Accurate enough for mental checks between about 4% and 12%, and the fastest way to test whether a growth claim is plausible. The exact figure is ln(2) ÷ ln(1 + r); the rule drifts at very high rates, which is why 72 ÷ 36 = 2 years badly underestimates the true 2.2 years.

Run it in reverse to sanity-check a promise. "Doubles in three years" implies roughly 72 ÷ 3 = 24% a year, sustained.

Why ten years early beats three times the money

Two savers, both at 7% compounded monthly:

Ama saves £200/month from 25 to 35, then stops. Contributed: £24,000. Ben saves £200/month from 35 to 65. Contributed: £72,000.

Ama's pot is worth £34,617 when she stops at 35. Left alone for 30 years it multiplies by (1 + 0.07/12)³⁶⁰ = 8.12, reaching roughly £281,000. Ben finishes with about £244,000.

Ama contributed a third as much and finished ahead, because her money compounded for thirty extra years. This is the strongest possible argument for starting early, and it is arithmetic rather than motivation.

Note what the gap is *not*: it is not large. Ben is only about 13% behind on three times the contributions — late saving still works, it just costs more. The point is the price of delay, not the futility of starting late.

What quietly destroys it

Inflation

A 7% nominal return with 3% inflation is not 4% real — it is (1.07 ÷ 1.03) − 1 = 3.88%. Over 30 years that is the difference between multiplying your money by 7.61× and by 3.14× in purchasing power. Always reason in real terms over long horizons — see the inflation calculator.

Fees

A 1% annual fee sounds trivial. It turns a 7% return into 6%, and over 30 years the growth multiple falls from 7.61× to 5.74× — removing 24.6% of the final balance. On £10,000 that is £18,688 handed over for one percentage point a year. The fee compounds alongside the returns, which is why the damage is so much larger than "1% × 30 years". This is the strongest practical argument for low-cost index funds.

Interrupting it

Withdrawing and restarting resets the curve to its flat beginning. Compounding rewards being left alone more than it rewards being optimised.

Working backwards: what rate did I actually get?

If an investment went from £8,000 to £20,000 over nine years, the compound annual growth rate is:

CAGR = (20000 ÷ 8000)^(1/9) − 1 = 10.72%

This is the only honest way to compare investments held for different lengths of time, and it is always lower than the average of the yearly returns when those returns were volatile.

The same force in reverse

Credit card debt at 22% APR compounds against you. At monthly compounding, £5,000 left untouched becomes £5,000 × (1 + 0.22/12)¹² = £6,218 after a year and £11,959 after four — it more than doubles without a single new purchase. The credit card payoff calculator shows why clearing high-interest debt usually beats investing: paying off a 22% balance is a guaranteed 22% return, which no portfolio reliably offers.

This is not financial advice, and real returns are never the smooth percentage these models assume. Use the compound interest calculator to test your own numbers, and the retirement calculator to see what a given contribution rate actually buys over a working lifetime.