Compound interest: little by little, then suddenly

Interest that earns interest

The previous lesson established what interest is, the price of time and risk, and previewed the difference this one is built on.

Simple interest pays you on what you put in. Compound interest pays you on what you put in plus everything the money has already earned, interest earning interest, which then earns interest of its own. In year one the difference is pennies. Put $100 in at 10% for two years: simple interest gives you $120, compound gives you $121. One dollar. But compounding is a snowball rolling downhill: the growth itself grows, and the curve that crawled for a decade eventually climbs faster than your contributions do. Every serious saving and investing decision for the rest of your life sits on top of this one piece of math.

Here is the whole thing as a formula, so you can rebuild any table on this site in a spreadsheet:

A = P(1 + r/n)^(nt)

where P is what you start with, r is the yearly rate as a decimal, n is how many times a year interest is added, t is years, and A is what you end with. The $121 above is 100 × (1 + 0.10/1)^(1×2).

How often it compounds matters (a little)

The n in that formula is the compounding frequency, and it is the detail most calculators hide. More frequent compounding means slightly more money. Take $1,000 at 6% for two years:

That last column is why banks quote two numbers. The APR is the plain yearly rate before compounding; the APY (or effective annual rate) is what you actually earn once compounding is counted. A 10% rate compounded twice a year is really 10.25% a year. Compare savings accounts on APY and loans on APR, and never compare one against the other.

The rule runs against you as well. A credit card advertising 24% APR, compounding monthly, charges an effective 26.8% a year.

The Rule of 72

You can estimate compounding in your head. Divide 72 by the yearly growth rate and you get roughly how many years the money takes to double:

It is an approximation, but a good one, and it cuts both ways. The same rule tells you how fast a debt doubles on the lender’s side of the counter, and how fast inflation halves what a hidden shoebox of cash can buy.

Starting early beats starting big

Here is the claim that sounds like a trick and is not. Two savers each put away $100 a month and earn 8% a year, compounded monthly. One starts at 18, one waits until 28. Both stop at 65:

Ten years of delay cost $12,000 in contributions and about $350,000 in outcome. The head start is not the extra deposits; it is the extra doublings. At 8%, money doubles about every 9 years; starting at 18 buys the snowball one more full doubling, and the last doubling is always the biggest.

Sharper still: if the early starter contributed only from 18 to 28, $12,000 total, and then never added another dollar, that abandoned account would still reach about $350,000 by 65, beating the late starter’s 37 years of faithful deposits. That figure is the same $350,000 as the gap above, and not by coincidence: the whole advantage of starting early is those first ten years of deposits, left to compound for the rest of the run. Time in is worth more than money in.

Honesty about the assumptions: 8% is a plausible long-run return on invested money, not a promise from anyone, and these figures ignore taxes, fees, and inflation. Real returns arrive lumpy, not smooth (Ecclesiastes 9:11, time and chance happen to all). Other lessons here run the same illustration at 7% (why start now does), and the gap between those two numbers is roughly the honest width of anyone’s knowledge about future returns. The shape of the math, though, is not in doubt: whatever the rate turns out to be, the person who started earlier gets more doublings out of it.

Gathering little by little

Scripture backed the slow snowball long before anyone charted it: “Wealth gotten by vanity shall be diminished: but he that gathereth by labour shall increase” (Proverbs 13:11, KJV), steady gathering, little by little, over windfalls and schemes. The same book points to the ant storing food in summer, season after small season (Proverbs 6:6-8). And in the parable of the talents, the servant condemned is not the one with the smallest account but the one who buried what he was given, while even bank interest is named as the obvious floor (Matthew 25:27). The parable’s first meaning is stewardship of everything God entrusts (gifts, time, the gospel), not an investment manual; but it plainly assumes that letting entrusted resources sit sterile is fear, not faithfulness. See investing and multiplication for where the line sits between patient growth and greed.

One caution the math never gives you: compounding is a mechanism, not a master. The rich fool of Luke 12:16-21 had excellent growth and a dead soul by morning. Compound toward purposes (generosity, provision, good work), not toward a bigger barn.

The practice

• Run your own numbers in the Third Servant Compounder: your age, your monthly amount, your rate. Watch what one more decade does.
• Use the Rule of 72 on one real number in your life this week: a savings rate, a card’s interest rate, the inflation rate.
• If you can start with only a small amount, start with the small amount. The table above is the argument: the calendar is doing most of the work, and it only runs one way. Two Khan Academy lectures, linked and not embedded, sit beside this page on the video shelf. The SEC’s definition is on further reading.

Try it yourself: open the interactive tool for this lesson.