Compounding Forward
Future value asks what money becomes if left to grow. A single amount compounds by multiplication; a stream of payments compounds each deposit for the time it is invested:
FV = PV × (1 + r)n + PMT × [(1 + r)n − 1] / r
The first term handles the lump sum, the second the payments. Nothing else is required, which is why this single expression underlies retirement projections, savings goals and every compound growth question.
The Shape of Compounding
Growth is not a straight line and not intuitive. $25,000 at 7% with $500 a month added:
| Years | Paid in | Value | Growth | Growth as a share |
|---|---|---|---|---|
| 5 | $55,000 | $71,237 | $16,237 | 23% |
| 10 | $85,000 | $136,784 | $51,784 | 38% |
| 20 | $145,000 | $361,432 | $216,432 | 60% |
| 30 | $205,000 | $812,898 | $607,898 | 75% |
The last decade of that thirty-year run adds more value than the first twenty combined. This is the entire argument for starting early, and it is arithmetic rather than exhortation.
The Rule of 72
Divide 72 by the annual rate to approximate the years to double. At 6%, twelve years; at 9%, eight; at 12%, six. It is accurate within a few months for rates between 4% and 15%, and it is the fastest way to sanity-check any growth claim in your head.
Compounding Frequency
More frequent compounding produces more, but the effect is small and bounded. $10,000 at 6% for one year: $10,600 compounded annually, $10,616.78 monthly, $10,618.31 daily, $10,618.37 continuously. The gap between annual and monthly matters slightly; the gap between daily and continuous never does.
Beginning or End of Period
An annuity due — payments at the start of each period — is worth exactly one period's growth more than an ordinary annuity. At 7% annually that is 7% more; on a monthly schedule it is about 0.58% more. Small, but the correct setting matters for rent, insurance and some pension arrangements.
Frequently Asked Questions
What return should I assume?
The S&P 500's long-run nominal average is close to 10%, or roughly 7% after inflation. For planning, 6–7% nominal is a defensible conservative figure; anything above 10% in a long projection should be treated with suspicion.
Should I use nominal or real returns?
Either, as long as you are consistent. A nominal projection produces a large number in future dollars; a real projection produces a smaller number in today's buying power, which is usually easier to reason about.
What is the difference between future value and compound interest?
None, mathematically. Future value is the answer; compound interest is the mechanism producing it.