Simple vs Compound Interest: Why the Curve Bends

0:00Same 5%, two very different years ▶

Same $10,000, same 5%: $500 in year one, $2,058 in year thirty. Nothing changed — not the rate, not the rules. So where did the other fifteen hundred come from?

0:20The one idea you need ▶

All you need for this video is one idea: interest is a percentage of whatever is in the account right now. Keep that in mind and every number that follows will make sense.

0:31Simple interest: the straight line ▶

Let's start with the simple version. You put in $10,000, and say a bank pays you 5% of that every year, in cash. Five percent of $10,000 is $500, and with simple interest that is what you get — every single year. Your deposit plus the cash you collected grows in a straight line: $10,000, $15,000, $20,000, $25,000. Each year adds the same $500, so the line never bends.

1:03Compound interest: the interest stays in ▶

Now change one rule: the bank no longer pays you in cash, it leaves the $500 in the account. Interest is still paid once a year, on whatever is in there. In year two, you earn 5% of $10,500 — that's $525. Twenty-five dollars more than last year, because last year's interest is now earning interest too. It sounds like a rounding error — but watch what it does over thirty years.

1:31Year 10: the curve that looks like a line ▶

After ten years, simple interest has taken you to $15,000. Compound interest has taken you to $16,289. The difference is $1,289 — about the price of a laptop. Ten years in, the two lines are almost on top of each other. This is where most people stop believing in the curve.

1:58Year 20 and 30: the bend ▶

Keep going to year twenty: simple interest gives you $20,000, compound gives you $26,533. Now the gap is $6,533 — five times what it was at year ten. And at year thirty: $25,000 on the straight line, $43,219 on the curve. The gap has grown to $18,219 — nearly twice the money you put in. In the first ten years, interest on interest added about $1,300.

2:38Taking $43,219 apart ▶

In the last ten, almost $12,000. Let's take that $43,000 apart. Ten thousand is the money you put in. Fifteen thousand is simple interest — $500 a year, for thirty years. And the remaining $18,219 is interest earned by interest. After thirty years, more than half of the growth came not from your money,

3:05The formula, from the picture: A = P(1 + r)^n ▶

but from the interest your interest earned. The straight line was easy: $10,000, plus $500 times n, the number of years. The curve has a formula too, built on one small idea: adding 5% is the same as multiplying by 1.05. Here is why, writing P for the $10,000 you put in. P plus 5% of P is P + 0.05·P, which is 1.05 × P. So each year, the balance is multiplied by 1.05. After thirty years, that is 1.05 multiplied by itself thirty times. Mathematicians write that as 1.05 to the power of thirty, which is 4.3219. Ten thousand times that is $43,219 — the formula is just the repeated multiplication, written short. In that formula, r is the rate per year,

4:09Doubling time: n = log 2 / log 1.05 ▶

n is the number of years, and A is what you end with. At 5%, your money doubles roughly every fourteen years — 14.2, to be exact. Ten thousand becomes $20,000 just after year fourteen, and $40,000 around year twenty-eight. The second doubling is worth $20,000 — twice the first — and it took the same fourteen years. That is the bend, in one sentence: each doubling is worth as much as everything that came before it. And that 14.2 is not a rule of thumb, it comes from the formula. Doubling means the growth factor itself reaches two: 1.05ⁿ = 2. Finding that n is exactly what a logarithm does: log 2 / log 1.05 = 14.2.

4:58Starting one year later ▶

Suppose you start one year later, so your money has twenty-nine years instead of thirty. You do lose the first year's $500 — but not just that. Left on the pile for twenty-nine years, that $500 becomes $2,058. Which is exactly the last year of the curve: $41,161 instead of $43,219.

5:30The whole story in one picture ▶

One year at the start costs more than four times what it looks like — and that is the number we opened with. So, here is the whole story: simple interest is a straight line, $500 a year. Compound interest is a curve, because each year's interest joins the pile and earns interest of its own. By year thirty, the curve has added $18,219 that the straight line never will. After thirty years, more than half of the growth is interest on interest —

6:00What's next: the Rule of 72 ▶

and that's the math behind the money. Next time: the Rule of 72 — and why dividing 72 by 5 lands so close to 14.2. And a question for you: at what year did you expect the curve to pull away? Let me know in the comments below. I'm the MoneyMathician — subscribe, and the next one will find you. Until then: do the math!