The compound interest formula turned around: solve for the payment, then see how horizon, return and fees move it.
To reach 1,000,000 in 30 years at an illustrative 7 per cent a year, compounded monthly, you need to invest about 820 a month. Only 295,089 of the million is your own money; the other 704,911 is compound growth. Give yourself 40 years and the figure falls to 381 a month; give yourself 20 and it rises to 1,920.
Worked in full in Stock Market Investing for Beginners by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The return is an assumption for a diversified, low-cost equity index fund held for decades, not a forecast. Real returns arrive unevenly, and some decades disappoint. The arithmetic only tells you what a given return would require.
| Input | Value |
|---|---|
| Target value | 1,000,000 |
| Years of investing | 30 |
| Number of monthly payments | 360 |
| Annual return, nominal, before fees (illustrative) | 7% |
| Monthly rate (7% ÷ 12) | 0.5833% |
| Payment timing | end of each month |
The usual question runs forward: what will a monthly amount grow to? This one runs backwards. Start from the future value of a regular payment and solve it for the payment.
Future value: FV = P × ((1 + i)n − 1) ÷ i
Solved for the payment: P = FV × i ÷ ((1 + i)n − 1)
i = 0.07 ÷ 12 = 0.005833, n = 360, so (1.005833)360 = 8.1165
(8.1165 − 1) ÷ 0.005833 = 1,219.97: each 1 invested monthly becomes 1,219.97 after 30 years
P = 1,000,000 ÷ 1,219.97 = 819.69 a month
Excel: =PMT(7%/12, 360, 0, -1000000) returns 819.69
The middle line is the useful one. At 7 per cent over 30 years, the factor 1,219.97 converts any monthly amount into a final value and back again. Divide the target by it and you have the payment; multiply the payment by it and you have the target.
If you invest at the start of each month instead, every payment gets one more month of growth. Add a final argument of 1, =PMT(7%/12, 360, 0, -1000000, 1), and the payment falls to 814.94, a saving of 4.75 a month. The timing matters far less than the horizon.
At 819.69 a month for 360 months you pay in 295,089. The pot is 3.39 times what went in, and 70.5 per cent of the final million is growth. For comparison, 500 a month over the same 30 years reaches 609,985, which is why the figure is not simply 500 scaled to a round number.
The horizon changes the split more than anything else:
| Years | Per month | Total paid in | Growth share of the million |
|---|---|---|---|
| 20 | 1,920 | 460,717 | 53.9% |
| 25 | 1,234 | 370,338 | 63.0% |
| 30 | 820 | 295,089 | 70.5% |
| 35 | 555 | 233,197 | 76.7% |
| 40 | 381 | 182,870 | 81.7% |
Waiting ten years, from a 30-year horizon to a 20-year one, raises the monthly figure by 1,100, to 2.34 times as much. That is the cost of delay stated as a payment rather than as a lost balance, and it is usually the more persuasive form, because the payment is what has to come out of a salary.
The return is the one input nobody controls, so read the answer as a range. The table below solves the same equation for each combination.
| Annual return | 20 years | 25 years | 30 years | 35 years | 40 years |
|---|---|---|---|---|---|
| 4% | 2,726 | 1,945 | 1,441 | 1,094 | 846 |
| 5% | 2,433 | 1,679 | 1,202 | 880 | 655 |
| 6% | 2,164 | 1,443 | 996 | 702 | 502 |
| 7% | 1,920 | 1,234 | 820 | 555 | 381 |
| 8% | 1,698 | 1,051 | 671 | 436 | 286 |
| 10% | 1,317 | 754 | 442 | 263 | 158 |
Two readings matter. First, the 6 per cent row is also the 7 per cent row after one percentage point of annual fees. On a 30-year plan that point lifts the required payment from 820 to 996, 176 more a month or 21.4 per cent. Put the other way, 820 a month invested at a net 6 per cent reaches 823,393, not a million. The fee is a cost you choose, which is why it deserves more attention than the return you cannot. The longer version of that argument is in what one percentage point of annual fees actually costs.
Second, the rows spread apart as the horizon lengthens. Over 20 years the 4 per cent saver needs about twice what the 10 per cent saver needs; over 40 years more than five times. A long horizon makes the target cheap, but it also makes the answer more sensitive to an assumption nobody can check in advance.
The first is forgetting inflation. A million in 30 years is not a million of today's money. At an illustrative 2.5 per cent inflation it buys what about 476,743 buys now. To hold a million of today's purchasing power you would need a nominal 2,097,568, and that takes about 1,719 a month at 7 per cent. Alternatively, run the whole calculation in real terms with a real return of about 4.4 per cent, (1.07 ÷ 1.025) − 1; either way, decide which million you mean before you solve for it.
The second is solving with annual compounding and then dividing by twelve. Paying 10,586 once a year at 7 per cent also reaches a million in 30 years, but that is 882 a month in equivalent effort, because each annual payment waits up to a year to start compounding. The third is ignoring growth altogether: a million divided by 360 months is 2,778 a month, more than three times the compounded answer.
The figure from the formula is a payment that has to be kept up through every bad year. A plan that needs 820 a month and gets it 300 months out of 360 is not a plan for a million.
=PMT(rate/12, months, 0, -target) in Excel.The blank calculator in the free workbook for this case answers the two questions worth making a habit of: what one more year of waiting costs, and what half a point of extra fee takes away. To run the forward version of this sum, see how much 500 a month grows to in 30 years.
At an illustrative 7 per cent a year, compounded monthly, about 1,920 a month for 240 months, or 460,717 paid in. At 10 per cent it would be 1,317 a month and at 4 per cent 2,726. Over 20 years growth supplies only 53.9 per cent of the million, so the saving itself has to do most of the work.
Use PMT with the monthly rate, the number of months, a present value of zero and the target as a negative future value: =PMT(7%/12, 360, 0, -1000000) returns 819.69. Add a fifth argument of 1 for payments at the start of each month, which gives 814.94. FV does the reverse calculation from a payment to a pot.
No. At an illustrative 2.5 per cent inflation, 1,000,000 in 30 years buys about what 476,743 buys today. To hold a million of today's purchasing power you would need a nominal 2,097,568, which takes about 1,719 a month at 7 per cent rather than 820.
Compounding and costs are the subject of the workbook for chapters 4 and 8 of Stock Market Investing for Beginners. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
Get the book on Amazon →Free companion files
Also on Amazon UK · Amazon Germany · Amazon France · Amazon Canada
Reading guide: personal finance and investing for beginners → · All 681 articles →
If this book helped, or didn’t, a few lines on Amazon are worth more than they look: they are what the next reader goes on. Write a review. The workbook stays free either way.