Compound Interest, Explained With Real Numbers

• Compounding means returns earn returns. Over one year it is trivial; over thirty it dominates everything.
• Rule of 72: divide 72 by your rate to get the doubling time. Close enough for mental arithmetic between about 5% and 12%.
• Ten years of early saving can beat thirty years of late saving — a real worked example below.
• A 1% annual fee costs roughly 25% of your final pot over 30 years, because fees compound too.
• Inflation compounds against you as well. Nominal returns flatter what you can actually buy.

The mechanism

Simple interest pays a return on your original amount only. Put in 10,000 at 7% simple, and you get 700 every year forever: a straight line.

Compound interest adds each period's return to the balance, and the new balance earns next period's return. Year one pays 700. Year two pays 7% of 10,700, which is 749. Year three pays 7% of 11,449, which is 801.

The formula is:

A = P × (1 + r/n)^(n × t)

P = starting amount
r = annual rate (as a decimal)
n = compounding periods per year
t = years

Watch what those small annual differences accumulate into. Starting with 10,000 at 7%, compounded annually:

Years	Simple interest	Compound interest	Difference
5	13,500	14,026	526
10	17,000	19,672	2,672
20	24,000	38,697	14,697
30	31,000	76,123	45,123
40	38,000	149,745	111,745

At ten years the gap is unremarkable. At forty years the compound figure is nearly four times the simple one. Nothing changed about the rate — only the number of times the return got to act on itself.

This is the shape people consistently misjudge. Asked to guess the 40-year figure after being told the 10-year figure, most estimates land far too low, because the mind extrapolates the line it has already seen.

The Rule of 72

You rarely need a calculator to sanity-check a claim about returns. Divide 72 by the annual percentage rate and you get the approximate number of years for money to double.

Annual return	Rule of 72 estimate	Exact answer
2%	36 years	35.0 years
5%	14.4 years	14.2 years
7%	10.3 years	10.2 years
10%	7.2 years	7.3 years
15%	4.8 years	5.0 years

It is a genuinely useful filter. If someone offers you 3% and suggests your money will double in a decade, the rule says 24 years and the pitch is wrong. It also works in reverse on debt: a 22% credit card balance doubles in a little over three years if left alone.

Time versus money

Here is the example worth internalising, because it produces a result most people refuse to believe until they check it.

Two savers, both assuming a 7% annual return, both contributing 5,000 a year:

• Ava invests 5,000 a year from age 25 to 35 — ten years, 50,000 contributed — then never adds another penny and lets it sit until 65.
• Ben invests nothing until 35, then contributes 5,000 a year from 35 to 65 — thirty years, 150,000 contributed.

At 65:

	Contributed	Value at 65
Ava (10 years, early)	50,000	~526,000
Ben (30 years, late)	150,000	~472,000

Ava contributed one third as much and finished roughly 54,000 ahead.

The reason is that Ava's money got more doublings. Her first contribution compounds for 40 years, which at 7% is close to four doublings — each one larger than the last. Ben's contributions never get that runway, and the final doubling is always the biggest one in absolute terms.

The practical reading is not "if you are 40, you have lost." Ben still turned 150,000 into 472,000, which is an excellent outcome, and the second-best time to start is obviously now. The reading is that the earliest money you invest is the most valuable money you will ever invest, and small amounts in your twenties genuinely outrank larger amounts later.

Regular contributions

Most people do not invest a lump sum; they contribute monthly. That is the future value of an annuity:

FV = PMT × [((1 + r)^n − 1) / r]

PMT = payment per period
r   = rate per period
n   = number of periods

Contribute 500 a month for 30 years at 7% annual (0.5833% monthly, 360 payments) and you finish with roughly 610,000. You will have paid in 180,000. The remaining 430,000 — about 70% of the total — is growth.

That ratio is the thing to hold on to. In a long-horizon plan, most of the final number is not the money you saved. It is the return on the return.

What eats it: fees

Fees are usually quoted as a small annual percentage, which is exactly what makes them so easy to underrate. A fee is charged on your whole balance, every year — so it compounds against you with the same relentlessness that your returns compound for you.

Take 30 years:

Net return	Growth multiple	Value of 100,000
7.0% (0.05% fee)	7.61×	761,000
6.0% (1.05% fee)	5.74×	574,000

The 1% difference in annual fee costs about 187,000, or roughly 25% of the final pot. Extend to 40 years and the gap widens further.

This is the entire practical argument for keeping costs low, and it needs no forecasting skill to act on. Future returns are unknowable; the fee is printed on the document. It is one of the very few variables in investing you control with certainty.

What also eats it: inflation

The other compounding force running against you is the falling purchasing power of money. A 7% nominal return with 3% inflation is not 4% real — the correct calculation is a ratio:

real return = (1 + nominal) / (1 + inflation) − 1
            = 1.07 / 1.03 − 1
            = 3.88%

Over 30 years that distinction is severe. At 7% nominal your money grows 7.6×. At 3.88% real it grows 3.1×. Both statements describe the same investment; only the second tells you what you can actually buy.

Inflation is also the reason cash is not the safe option people assume. Money in a current account earning nothing loses purchasing power every year, compounding quietly downward. Over 30 years at 3% inflation, cash retains about 41% of its real value.

Where the 7% comes from, and its honest limits

Every figure above uses 7%. That is a conventional planning assumption, roughly reflecting the long-run real return of broad developed-market equities, or a nominal return net of a modest inflation assumption. Over very long periods, US equity markets have delivered something in the region of 10% nominal and 6.5–7% real, though the number moves depending on the start date, the index and the country.

Three caveats that matter:

Returns are not smooth. No market delivers 7% every year. It delivers −20%, +25%, +9%, −6%. The compounding maths still applies, but volatility means the path matters enormously if you need to withdraw at a bad moment — an issue that dominates in the years around retirement.

Past averages are not forecasts. They are evidence about a range of plausible outcomes, not a promise. Using 5% instead of 7% in every calculation above is a defensible act of prudence.

Survivorship bias is real. Long-run return series are dominated by markets that did well and kept operating. Markets that were closed, expropriated or destroyed do not appear in the average.

None of this undermines the core arithmetic. It just means compounding is a mechanism, not a guarantee — and the mechanism rewards the two things you actually control: starting early, and keeping costs down.

Further reading: The US SEC's compound interest calculator lets you check every figure here. For long-run return data, the Credit Suisse/UBS Global Investment Returns Yearbook (Dimson, Marsh and Staunton) is the standard reference across markets and over a century of history.

This article explains arithmetic, not investments. It is general information and not financial advice. Rates of return are assumptions used for illustration, and actual investments can lose money.