Core answer: compound future value = principal × (1 + annual return)ⁿ. ¥100k at 8% for 30 years → ¥1.006M, ten times the principal. Rule of 72: 72 ÷ 8 = 9 years to double. Compounding's power is not the rate — it is TIME: starting 10 years earlier beats earning 2% more.
The formula, derived
- Simple interest: interest never joins the principal. FV = P × (1 + r×n)
- Compound interest: each period's interest joins the principal and earns more. FV = P × (1 + r)ⁿ
The intuition: year 1 ends at P(1+r); year 2 grows that by (1+r) → P(1+r)²... year n gives P(1+r)ⁿ. The exponential curve is flat early and steep late — that bend is the "compounding inflection point."
Simple vs compound, measured
¥100k at 8%:
| Years | Simple | Compound | Gap |
|---|---|---|---|
| 5 | ¥140k | ¥147k | ¥7k |
| 10 | ¥180k | ¥216k | ¥36k |
| 20 | ¥260k | ¥466k | ¥206k |
| 30 | ¥340k | ¥1,006k | ¥666k |
The first 5 years feel identical; by year 20 it is a rout. Most people lose by not waiting for the bend.
Rule of 72 cheat table
Years to double ≈ 72 ÷ annual return:
| Annual return | Doubling time | 30-year multiple |
|---|---|---|
| 3% (deposits/bonds) | 24 years | ~2.4× |
| 5% | 14.4 years | ~4.3× |
| 8% (broad index long-run mean) | 9 years | ~10× |
| 10% | 7.2 years | ~17.4× |
| 15% | 4.8 years | ~66× |
Monthly investing, compounded
¥2,000/month at 8%:
| Years | Total contributed | Account value | Gain |
|---|---|---|---|
| 10 | ¥240k | ¥366k | ¥126k |
| 20 | ¥480k | ¥1,178k | ¥698k |
| 30 | ¥720k | ¥2,981k | ¥2,261k |
At year 30, 76% of the account is gains, not contributions.
Inflation's bite
Nominal compounding minus inflation is the real purchasing-power growth:
- 8% nominal − 2.5% inflation → ~5.5% real
- 3% nominal (deposits) − 2.5% inflation → ~0.5% real
That is the math behind "money in the bank quietly shrinks."
Common mistakes
- "Compounding is a scam/exaggeration": the formula is fine; the lie is products marketing "15% annual, stable, for 30 years." 8-10% long-run is already broad-index ceiling territory.
- "Double the return = double the wealth": forgetting inflation and taxes. A nominal double taking 14 years (at 5%) is maybe +60% in real purchasing power.
- "It is too late for me": the best time to plant a tree was ten years ago; the second-best is now. Starting ¥2,000/month at 30 still builds ¥2.98M by 60 — late beats never.