Lesson 04: Compound interest: how money grows over time

Lesson objectives:

• Define principal and compound interest, and separate compound from simple interest.
• Read the compound-interest formula and explain why compounding frequency changes the result.
• Recognize that a savings APY and a debt APR are the same force pointed in opposite directions.

Prerequisites: Lessons 01-03; comfort multiplying a number by a percentage | Previous << 03 | Next 05 >>

The savings account that grows faster every year

Your emergency fund is sitting in a savings account earning a little interest. Interest on a small balance sounds trivial, and for the first year it nearly is. But something odd happens if you leave it: the amount it earns each year keeps rising, even though you added nothing. Money that has been growing for twenty years grows faster this year than money that started last month. The mechanism behind that is compound interest, and once you can see it, two things click at once — why starting early matters more than starting big, and why a credit-card balance is so hard to escape. This lesson makes the mechanism concrete with your own arithmetic.

Explanation

Principal, and interest on top of it

Start with the plain words. The principal is the amount you begin with — the money you put in. Interest is a percentage the bank pays you for keeping money there (or that you pay a lender for borrowing). If you deposit $1,000 and the rate is 5% a year, after one year you have earned $50 of interest. So far this is simple.

The question that separates the two is: what happens to that $50 next year? Two answers exist, and they diverge more every year.

Simple interest versus compound interest

Under simple interest, the $50 sits aside and next year you again earn 5% of only the original $1,000 — another $50. The interest never joins the pile that earns.

Under compound interest, the $50 is added to the principal, so next year you earn 5% of $1,050, which is $52.50 — slightly more, because the interest is now itself earning interest. Formally, "compound interest is interest accumulated from a principal sum and previously accumulated interest," and it "is contrasted with simple interest, where previously accumulated interest is not added to the principal"¹. The gap between the two lines starts tiny and widens forever; that widening is the whole story of long-term saving.

Walk it forward yourself. Take a round 10% a year (an illustrative rate for clean arithmetic, not a promised return) on a $1,000 principal, compounded once a year. Fill in each year's balance — each is the previous balance plus 10% of the previous balance.

The formula, and why frequency matters

The pattern you just traced has a formula: A = P(1 + r/n)^(tn), where P is the principal, r the annual rate, t the number of years, and n how many times a year interest is added — the compounding frequency¹. You do not need to compute this by hand often; the point is what n does. "The compounding frequency is the number of times per given unit of time the accumulated interest is capitalized"¹, and it can be yearly, monthly, or daily.

More frequent compounding helps a little, because interest starts earning sooner. The same $1,000 at 5% for a year earns $50.00 compounded annually, but about $51.16 compounded monthly — the interest from January is itself earning by December. The effect is real but modest; frequency is a minor dial next to the two big ones, rate and time.

APY and APR: the same force, both directions

Two everyday abbreviations are just this mechanism, made honest for comparison. On savings, APY (annual percentage yield) is the one-year rate that already includes compounding: it "measures the total amount of interest paid on an account based on the interest rate and the frequency of compounding"². Because it bakes in n, APY is the number to compare between savings accounts — it already did the frequency math for you.

Pointed the other way, on debt, APR (annual percentage rate) is the yearly interest rate you are charged; for credit cards, "the interest rates are typically stated as a yearly rate ... called the annual percentage rate (APR)"³. The same interest-on-interest that grows your savings also grows an unpaid card balance — and cards commonly run 18% APR or more⁴. Compounding is not on your side or against you by nature; it amplifies whichever side the balance is on. That is the bridge to the next lesson.

Worked example (follow along)

Compare two savers to see time beat amount, using a round 7% annual return for illustration (not a guarantee):

• Early Ana puts in $2,000 once at age 25 and never adds more. At 7% compounded annually, after 40 years (age 65) it is $2,000 x 1.07^40 ≈ $29,900.
• Later Leo waits and puts in $2,000 once at age 45. After 20 years (age 65) it is $2,000 x 1.07^20 ≈ $7,740.

Same $2,000, same rate, same ending age. Ana ends with almost four times as much — not because she added more, but because her money compounded for twice as long, and the back half of a compounding curve is where the largest gains happen. The interest Ana earned in her 30s spent the next thirty years earning its own interest. This is why Lesson 05 treats when you start as a lever at least as strong as how much.

Want to run your own numbers without the algebra? The U.S. SEC's free compound-interest calculator takes a starting amount, a monthly contribution, a rate, a number of years, and a compounding frequency, and shows the result⁵ — you will use it in Lesson 06.

Your turn (faded example)

A $1,000 principal earns 8% a year, compounded annually. Reason about it without a calculator where you can.

• After year 1 the balance is ______ (principal plus 8% of it).
• The year-2 interest is 8% of ______ (which balance?), so it is ______ (more / less) than the year-1 interest.
• Over many years, this balance grows ______ (faster / at the same rate / slower) each year, with no new deposits.
• If instead this were an 8% APR credit-card balance you never paid down, the debt would grow by the ______ (same / opposite) mechanism.

Answer: after year 1, $1,000 + $80 = $1,080. Year-2 interest is 8% of $1,080 (the new balance, not the original $1,000), which is $86.40 — more than year 1's $80, because interest is now earning interest¹. The balance grows faster each year, since each year's base is larger. An unpaid 8% APR debt grows by the same mechanism — compounding does not care which side you are on; it amplifies the balance either way⁴³.

Summary + what's next

You can now define principal and compound interest, tell compound from simple by asking whether last period's interest joins the pile that earns, read what the formula's rate/time/frequency each do, and see that a savings APY and a debt APR are one mechanism facing opposite directions. Time, you saw, is the strongest lever.

That single fact — compounding cuts both ways, and hardest at high rates — sets the priority order for your money. The next lesson puts the moves in sequence: what to fund first, why high-interest debt jumps the queue, and what it costs to wait.