Growth on growth
Simple interest pays on the original sum. Compound interest pays on the sum plus everything already earned, so each period starts from a larger base than the one before. Over a long horizon that difference stops being a detail and becomes the entire result.
The formula
P is the opening amount, M each regular contribution, i the rate per period and n the number of periods. The first term grows the starting sum; the second grows each contribution for the time it remains invested.
Worked example: $1,000 plus $100 a month at 5% for 10 years
- Paid in: $13,000
- Final balance: $17,175
- Interest earned: $4,175, or 32% of what you contributed
Why the horizon matters more than the rate
| Years | Paid in | Balance | Interest share |
|---|---|---|---|
| 10 | $13,000 | $17,175 | 24% |
| 20 | $25,000 | $43,800 | 43% |
| 30 | $37,000 | $88,600 | 58% |
| 40 | $49,000 | $161,900 | 70% |
Contributions rise in a straight line; the balance does not. By year thirty most of the money was never paid in by you, and by year forty more than two-thirds of it was not.
Compounding frequency, and its ceiling
On the same $1,000 at 5% for ten years with no contributions:
- Annually: $1,628.89
- Quarterly: $1,643.62
- Monthly: $1,647.01
- Daily: $1,648.66
Moving from annual to monthly gains $18. Moving from monthly to daily gains $1.65. Beyond monthly the frequency stops mattering, which is why "compounded daily" in an advertisement is worth less than it sounds.
The rule of 72
Divide 72 by the rate for a quick doubling time. At 5% that is 14.4 years; at 8%, nine years; at 2%, thirty-six. The approximation is close enough for mental arithmetic up to about 15%.
It also works in reverse on costs. A 1% annual fee does not sound like much until you notice it removes roughly a fifth of a portfolio's growth over thirty years.
What this calculator leaves out
Inflation and tax, both of which apply to the result rather than to the contributions. A 5% nominal return with 2.5% inflation is 2.4% real, and the $17,175 above buys what about $13,400 buys today — barely more than you paid in.
It also assumes a constant rate. No investment delivers that, and the order in which returns arrive changes the outcome once withdrawals begin.
The same force, working against you
Compounding is symmetrical. A credit card balance at 22% left untouched grows the same way a portfolio does: $5,000 becomes $6,100 after a year, $9,079 after three and $11,076 after four, without a single new purchase.
This is why paying down high-interest debt beats investing at ordinary market rates. A guaranteed 22% return is not available anywhere else, and it is exactly what clearing that balance earns.
Where the money should sit
The 5% used here is a plausible blended rate, not a product. A savings account paying 4% and an index fund averaging 8% both compound by the same formula and behave very differently along the way.
The horizon decides which is appropriate. Money needed within three years belongs where it cannot fall, even at a lower rate; money not needed for twenty years loses more to inflation in cash than it risks in the market. The formula is indifferent — you are not.
Related calculators
- Investment calculator — the same maths at market rates and horizons
- Rule of 72 calculator — doubling time from any rate
- Real return calculator — what remains after inflation