Compound growth
See how growth on growth pulls away from plain growth, why the years you use matter more than the amounts you add, and how to estimate a doubling time in your head.
Compound growth
Growth calculated on the current total, not on the amount you started with. Last year's growth joins the pile and earns alongside it. In Persian and in English it is often called interest on interest, and the same machine runs under any growing amount: savings, investments and debts alike.
Two accounts, both at 10% a year. One pays 10% of the original 1,000 forever. The other pays 10% of whatever is there this year. In year one they are identical. By year thirty they are not remotely the same.
1,000 left alone for 10 years at 10% a year
(10% is for illustration only, not a promise)
Simple growth: 10% of the ORIGINAL, every year
1,000 + (100 x 10) = 2,000.00
Compound growth: 10% of the CURRENT total
year 1 1,000 x 1.10 = 1,100.00
year 2 1,100 x 1.10 = 1,210.00
year 3 1,210 x 1.10 = 1,331.00
...
year 10 1,000 x 1.10^10 = 2,593.74Simple 2,000.00
Compound 2,593.74
The extra 593.74 is growth earned by earlier growthIn year one the two are identical. The gap opens in year two, when the 10% is charged on 1,100 rather than on 1,000.
Check yourself
Two accounts both start at 1,000 and both grow at 10% a year. One always pays 10% of the original amount; the other pays 10% of the running total. After exactly one year, how do they compare?
- The compound one is already ahead by 100
- They are identical: both hold 1,100
- The compound one is ahead, but only by 10
- The simple one is ahead, because it pays out something every single year
Show the answer
They are identical: both hold 1,100
Yes. Compounding has nothing to work on in the first year. It needs last year's growth before it can grow anything, which is exactly why its advantage is invisible early and enormous late.
1,000 growing at 10% a year, nothing added, nothing taken out.
| After | It is worth |
|---|---|
| 0 years | 1,000 |
| 5 years | 1,611 |
| 10 years | 2,594 |
| 20 years | 6,727 |
| 30 years | 17,449 |
The first ten years add 1,594. The last ten add 10,722.
Check yourself
In that table, the 1,000 grows by more in its last ten years than it did in its first twenty.
Show the answer
True
True, and it is the point of the whole lesson. Years 20 to 30 add 10,722; years 0 to 20 add 5,727. Each year's 10% is charged on a bigger total than the year before, so the same rate moves more money every year. The end of the run does the heavy lifting.
THE LEVER
Time does more than the amount
Compound growth multiplies. Putting in more money changes the number being multiplied. Adding years changes how many times it is multiplied, and over a long stretch that wins by a distance. A year you did not use is the one input you can never buy back, at any price.
Doubling what you put in doubles the result. Doubling the years at 10% does much more than double it: ten years turns 1,000 into 2,594, and twenty years turns it into 6,727.
Two savers, both growing 10% a year (illustration only)
MINA pays in 1,200 a year for 10 years, then stops
and leaves it alone for 20 more years.
value at year 10 19,125
19,125 x 1.10^20 128,660
total she ever paid in 12,000
OMID waits 10 years, then pays in 1,200 a year
for the next 20 years.
value at year 30 68,730
total he ever paid in 24,000Mina paid in 12,000 and ends with about 128,700
Omid paid in 24,000 and ends with about 68,700Omid put in twice as much and finished with roughly half as much. Same rate, same discipline. The only difference is which ten years each of them used.
Check yourself
Mina paid in 12,000 over her first ten years and ended with about 128,700. Omid paid in 24,000 over his last twenty and ended with about 68,700. What did Mina's early start actually buy her?
- Twenty extra years of growth on money that was already in place
- A higher rate of return than the one Omid was getting
- Nothing real: the example only works because she happened to go first
- Lower costs, because she made half as many payments as he did
Show the answer
Twenty extra years of growth on money that was already in place
Right. They had exactly the same rate. Her first 1,200 had thirty years to multiply; his last 1,200 had one. Early money is multiplied more times.
Check yourself
Using 72 divided by the rate, match each yearly rate to roughly how long it takes to double
Show the answer
- 3% a year → About 24 years
- 6% a year → About 12 years
- 9% a year → About 8 years
- 12% a year → About 6 years
- 24% a year → About 3 years
Check yourself
Reza has a spare amount this month. His savings grow 30% a year where prices rise 40%, and he also has a card debt charging 60% a year. What is the strongest thing he can do with it?
- Add it to the savings, because 30% a year is still growth and growth compounds
- Split it between the savings and the card so that both positions improve at once
- Hold it in cash until inflation comes down and the picture is clearer
- Put it against the card: cutting a 60% cost beats a real return below zero
Show the answer
Put it against the card: cutting a 60% cost beats a real return below zero
Right. His savings are really compounding at about −7% a year, while the card compounds at 60% against him. Removing the most expensive compounding is the one move with a certain result.
Lesson recap
- Compound growth is charged on the current total, so last year's growth earns alongside your original money.
- In year one compound and simple are identical. The gap opens later and then widens fast.
- Years matter more than amounts: Mina paid in half of what Omid did and finished with roughly twice as much.
- Divide 72 by the rate for a rough doubling time. It is an approximation and drifts at high rates.
- The same machine runs on debt, against you, and it runs on the real rate, not the advertised one.