Compound interest
With compound interest, returns themselves go on to earn returns. The effect stays modest for a long time and only becomes large in the later years, which is why starting early beats starting big.
With simple interest, you get the same amount every year. With compound interest, you get interest on a growing amount every year, because last year's interest earns interest too.
The difference is tiny at first and becomes enormous later. €10,000 at 7 percent becomes roughly €19,700 after ten years. After twenty years, it's not double that, it's roughly €38,700. After thirty years, roughly €76,100.
That implies something most beginners don't like: the decisive factor is time, not cleverness. Someone who starts at twenty and contributes little often ends up ahead of someone who starts at forty and contributes a lot.
A useful rule of thumb is the number 72. Divide 72 by the return in percent, and you get the years until it doubles. At 6 percent, that's twelve years; at 9 percent, eight years.
The final value of a starting amount follows K_n = K_0 · (1 + i)^n. The final value of a regular contribution made at the end of each period follows E = R · ((1 + i)^n − 1) / i. Both expressions are exponential in n, which explains why growth is largest in the final periods: the absolute gain in any period equals K_{n-1} · i, and so it grows exponentially in its own right.
The doubling time follows exactly from n = ln(2) / ln(1 + i). The common rule using 72 exploits the fact that ln(2) ≈ 0.693 and ln(1+i) ≈ i for small i, with 72 chosen over 69.3 because it has many divisors and the approximation is slightly more accurate in the practically relevant range of 6 to 10 percent.
Practically significant is the sensitivity to i over long periods. The factor (1 + i)^n reacts strongly to small changes in i when n is large, which is why a one-percentage-point difference in cost can amount to roughly a quarter of your final wealth over thirty years. It's the exact same mechanism, just applied to fees instead of returns.
Summary
- Compound interest acts late, but powerfully.
- 72 divided by the return gives the years until it doubles.
- Because the effect is exponential, costs act exponentially too.
Did you get it?
How long does it take to double your money at 8 percent?
About nine years, since 72 divided by 8 is 9.
Why is the gain in the final years the largest?
Because the annual gain is calculated on capital that's already grown, and so it grows exponentially in its own right.
Why do fees have such a strong effect over long periods?
Because they use the same exponential mechanism, just against you.