Investing
Compound Interest: The Maths Behind Long-Term Growth
Compound interest is what happens when returns earn returns. It is the reason a modest amount invested in your twenties can outweigh a much larger amount invested in your forties, and the reason a one percentage point fee is far more expensive than it sounds.
Data snapshot
- Year over year
- +14.3%
- 12-month range
- 6,344 – 7,799
The S&P 500 index. It tracks the largest US listed companies and is the benchmark most broad US equity index funds are built to replicate.
The formula, and what each part does
For a lump sum, the future value is the principal multiplied by one plus the periodic rate, raised to the number of periods. For regular contributions, each payment compounds for however many periods remain, which is why the earliest contributions dominate the final total.
The exponent is the time variable, and it is the powerful one. Doubling the rate helps; doubling the time does far more, because growth is multiplicative rather than additive.
A worked example
Contribute $500 a month from age 25 to 65 at a 7% annual return and you finish with roughly $1.3 million, of which $240,000 is contributions and the rest is growth. Start the identical plan at 35 instead and the total falls to around $610,000 — less than half, for three-quarters of the contributions.
That ten-year gap is not a rounding difference. It is the single strongest argument for starting with a small amount now rather than a larger amount later.
What compounds against you
The same mathematics applies to every recurring drag on the portfolio.
- Fees: a 1% annual charge instead of 0.05% can consume a fifth or more of a forty-year balance.
- Inflation: at 3%, purchasing power roughly halves over 24 years, so plan in real terms.
- Taxes on interest and realised gains in a taxable account, which reduce the amount left to compound.
- Credit card interest, which compounds monthly at a rate no portfolio reliably matches.
Using the rule of 72
Divide 72 by the annual return to approximate the years needed to double. At 7% a balance doubles in roughly ten years; at 3% it takes about 24. The same rule tells you how fast a debt doubles at 20%: under four years.
It is an approximation, but it is accurate enough for mental arithmetic and useful for judging whether a projection you have been shown is plausible before you rely on it.
Try the compounding maths
Estimated balance after 20 years
$176,472
- Total contributed
- $77,000
- Interest earned
- $99,472
- If returns average 5.0%
- $136,873
- If returns average 9.0%
- $230,412
Year-by-year projection
Contributions and compound growth split out for every year, so you can see when growth starts outpacing what you put in.
| Year | Contributions | Growth | End balance |
|---|---|---|---|
| Year 1 | $8,600 | $479 | $9,079 |
| Year 2 | $12,200 | $1,253 | $13,453 |
| Year 3 | $15,800 | $2,344 | $18,144 |
| Year 4 | $19,400 | $3,773 | $23,173 |
| Year 5 | $23,000 | $5,566 | $28,566 |
| Year 6 | $26,600 | $7,749 | $34,349 |
| Year 7 | $30,200 | $10,350 | $40,550 |
| Year 8 | $33,800 | $13,399 | $47,199 |
| Year 9 | $37,400 | $16,929 | $54,329 |
| Year 10 | $41,000 | $20,974 | $61,974 |
| Year 11 | $44,600 | $25,572 | $70,172 |
| Year 12 | $48,200 | $30,762 | $78,962 |
| Year 13 | $51,800 | $36,588 | $88,388 |
| Year 14 | $55,400 | $43,095 | $98,495 |
| Year 15 | $59,000 | $50,333 | $109,333 |
| Year 16 | $62,600 | $58,355 | $120,955 |
| Year 17 | $66,200 | $67,217 | $133,417 |
| Year 18 | $69,800 | $76,979 | $146,779 |
| Year 19 | $73,400 | $87,707 | $161,107 |
| Year 20 | $77,000 | $99,472 | $176,472 |
Compound interest formula
FV = P(1 + r/n)^(nt) + PMT × [(1 + r/n)^(nt) − 1] / (r/n)
Where:
- FV = Future value
- P = Principal (initial investment)
- r = Annual interest rate (decimal)
- n = Compounding periods per year
- t = Time in years
- PMT = Contribution per period
What this result is based on
This tool uses only the figures you enter and standard arithmetic. No external dataset feeds the result, so there is nothing to cite beyond the formula shown on the page.
This compound interest calculator compounds your starting balance and ongoing contributions at the rate of return you choose, then splits the ending figure between what you contributed and what the market added.
Compounding rewards time far more than timing, which the year-by-year table makes obvious.
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