Put $100,000 away at 10% a year for 20 years. Simple interest gets you to $300,000. Compounding gets you to about $672,750. Same money, same rate, same years. Where did the other $372,750 come from?
Learning objectives. After this week you can (1) explain why a dollar today is worth more than a dollar later; (2) compute future value with compound interest; and (3) use the Rule of 72 to estimate how long money takes to double.
Remember this: compounding earns interest on your interest, and time is its most powerful input.
1) Why a dollar today beats a dollar tomorrow
A dollar you have today can be invested and earn a return, so it grows into more than a dollar by next year. That is the time value of money. Present value (PV) is what money is worth today; future value (FV) is what it grows into. Going from PV to FV is compounding; going from FV back to PV is discounting.
2) Compound interest
FV = PV × (1 + r)n
r is the rate per period, written as a decimal (10% = 0.10), and n is the number of periods. $100 at 10% for 2 years grows to 100 × 1.102 = 100 × 1.21 = $121. Simple interest would give $120. The extra $1 is interest earned on the first year's interest, and that gap widens every year you wait.
| PV | Rate | Years | FV |
|---|---|---|---|
| $100 | 10% | 1 | 100 × 1.10 = $110 |
| $100 | 10% | 2 | 100 × 1.21 = $121 |
| $1,000 | 10% | 3 | 1,000 × 1.331 = $1,331 |
Discounting runs the same formula backward: $121 received in 2 years at 10% is worth 121 ÷ 1.21 = $100 today.
3) The Rule of 72
Years to Double ≈ 72 ÷ Interest Rate (%)
At 6% a year, money doubles in about 72 ÷ 6 = 12 years; at 8%, about 9 years. It works backward too: to double in 6 years you need about 72 ÷ 6 = 12% a year. It is an estimate, close enough to check a plan in your head.
Quick self-check
(a) At 9% a year, about how long does money take to double? (b) What does $200 at 10% grow to in 2 years? (c) To double in 8 years, roughly what annual return do you need? (Answers: 72 ÷ 9 = 8 years; 200 × 1.21 = $242; 72 ÷ 8 = 9%.)
Common mistakes
- Using simple interest. Multiplying one year's gain by the years undercounts.
- Plugging in 10 instead of 0.10. The rate goes into the formula as a decimal.
- Treating the Rule of 72 as exact. It is a quick estimate, not the formula.
This week in the arena
Take your Total value from the Account panel as PV. Choose an annual rate you think is realistic for a portfolio like yours, and compute what it would grow to in 10 years. Then use the Rule of 72 to estimate how long that rate takes to double your money. Notice how much the answer moves if you change the rate by just two points.