Interest That Earns Its Own Interest
Compound interest is interest paid on both your original money and the interest it has already earned, so the balance grows by a larger amount each period and its total curves upward over time instead of rising in a straight line. · 12 min
Money left alone in the right place does something strange: it earns, and then its earnings earn. In the first year you get interest on what you saved. In the second year you get interest on the savings and on last year's interest. Each year the base is a little bigger, so each year's gain is a little bigger too. Over a few years the effect is mild. Over decades it is enormous — and almost everyone's first guess about it is far too low, because we picture a straight line.
Guess before you learn
You put $1,000 somewhere that earns about 8 percent a year and never touch it. Guess what it grows to after 40 years.
It becomes roughly $21,700 — more than twenty times the original, from a single $1,000 and time. Keep your guess in pencil: most people guess a few thousand, because we instinctively add interest in a straight line instead of letting it compound on itself.
9–12
3–5
When you save money, a bank can pay you a little extra for keeping it there. That extra is called interest. The clever part: next time, you earn interest on your savings and on the interest you already got. So the total grows faster and faster the longer you wait.
A small amount left alone for many years can grow into a surprisingly big one. Time does most of the work.
6–8
Interest is a percentage a bank or investment pays you for holding your money. With compound interest, each period's interest is added to your balance, and the next period pays interest on that larger balance. So you earn interest on your interest. At 10 percent, $1,000 earns $100 the first year, but $110 the second — because the second year is figured on $1,100, not $1,000.
The yearly gains keep rising, so the running total does not climb in a straight line — it bends upward, gently at first and then steeply. The two things that feed it most are the rate and, above all, the time you leave it alone.
9–12
Compound growth is multiplicative: a balance at rate r multiplies by (1 + r) each period, so after n periods it is the start times (1 + r) to the nth power. That exponent is why the curve accelerates — growth is proportional to the current balance, and the balance keeps rising. Simple interest, paid only on the original, would instead trace a straight line.
Time dominates because it sits in the exponent while the rate sits in the base. Doubling the years does far more than doubling the rate. A rough gauge is the rule of 72: dividing 72 by the percent rate estimates the years to double — about 9 years at 8 percent. This is the engine behind long-horizon saving, and, reversed, behind long-horizon debt.
K–2
You save 10 coins. The bank likes savers, so it adds 1 coin. Now you have 11. Next time it looks at all 11 and adds a bit more. Your pile grows a little faster each time.
Undergrad
Each period multiplies the balance by (1 + r), so a start of B₀ becomes B₀(1 + r)ⁿ after n periods — geometric growth. The continuous-compounding limit is the exponential B₀·eʳᵗ. What makes the accumulation feel counterintuitive is that the growth rate is proportional to the level: the larger the balance, the faster it grows, so linear intuition systematically underestimates terminal wealth over long horizons.
Sensitivity is asymmetric between the base and the exponent. Because the balance depends on the rate polynomially but on the number of periods exponentially, marginal years outrank marginal rate at long horizons — the case for starting early rather than chasing yield. The rule of 72 approximates the doubling time (ln 2 divided by ln of one-plus-r); it is accurate to a percent or two across ordinary rates and makes the exponential legible without a calculator.
Postgrad
Discrete compounding is the linear map sending this period's balance to (1 + r) times itself, with closed form B₀(1 + r)ⁿ; the continuous limit solves to B₀·eʳᵗ. The defining feature is self-proportional growth. Human forecasting anchors on affine models, so exponential quantities are predicted low, with error that grows in the horizon — the documented exponential-growth bias.
Terminal wealth scales like (1 + r) to the nth power — polynomial in the rate but exponential in time — so at long horizons a marginal year outweighs a marginal point of rate, and early contributions are worth disproportionately more than later ones. The same mechanism runs in reverse on compounding debt, where the borrower occupies the lender's exponential — the bridge to the folio that follows.
compound interest
Interest paid on both your original money and the interest already earned, so each period's gain is larger than the last and the total curves upward over time. Contrast with simple interest, paid only on the original.
Why is this true?
Why does a compounding balance grow in a curve rather than a straight line?
Because each period's interest is figured on a balance that already includes all the previous interest. The gain is proportional to a total that keeps rising, so the gains themselves keep rising — and a series of ever-larger yearly increases traces an upward-bending curve, not a constant slope.
Compound $1,000 at 10% for three years — the steps fade as you master them
1,000 × 0.10 = 100, so balance = 1,100
1,100 × 0.10 = 110, so balance = 1,210
1,210 × 0.10 = 121, so balance = 1,331
100, then 110, then 121 — each larger
This is the most hopeful arithmetic in the whole course — and the most dangerous, because it runs both directions. The same force that grows your savings grows a balance you owe. When you are the lender, compounding is a gift; when you are the borrower, you are standing inside someone else's exponential. The next unit turns to that side: what borrowing truly costs, and the traps built to keep the curve running against you.
Note
Struggling to feel why time beats rate? The Atelier of Mind has a short exercise on exponential-growth bias — the reason our minds draw compounding as a straight line.
Practice — new ink and old, interleaved
1.Without looking back: what are the three shares of 50/30/20, and what does each cover?
Fifty percent for needs like housing and food, thirty percent for wants like dining out and hobbies, and twenty percent for saving and extra debt payoff — all applied to take-home pay.
How close were you? Grade yourself honestly — it sets your review date.
2.Take-home pay is $1,800. Under 50/30/20, how many dollars go to needs?
3.Which of these expenses holds the same amount month after month?
4.From folio ten: you want $3,600 in 12 months and interest is negligible over that year. How many dollars per month?
5.From folio nine: given compounding, should the emergency fund be invested in stocks to earn that growth?
6.Without looking back: what is compound interest, and why does time matter more than the rate over long horizons?
Compound interest is interest paid on your original money plus the interest already earned, so gains grow each period; time matters most because it sits in the exponent while the rate sits in the base, so more years multiply the balance far more than a slightly higher rate does.
How close were you? Grade yourself honestly — it sets your review date.
7.Which statement about a compounding balance is true?
8.Without looking back: what is a budget, and which figure does it assign?
A budget is a plan written before you spend, and it assigns your take-home pay by giving every dollar a job in advance.
How close were you? Grade yourself honestly — it sets your review date.
9.A $2,000 balance grows at 10 percent, compounded yearly. What is the balance after the first year, in dollars?