Compounding is the only exponential most people meet personally. Population growth, epidemics, Moore's law: those happen to other people, or to everyone, which is the same thing. A credit card happens to you. It's the one place where the math of exponentials shows up in a mailbox, addressed by name, and the tragedy is that the mailbox is the worst possible teacher.

The exchange rate

Start with what a loan is. You're moving consumption from the future into the present. The money you spend today was going to be earned next year, and a lender has agreed to hand it to you early. Interest is the price of that early delivery. It's an exchange rate between now-dollars and later-dollars.

That rate is roughly the sum of three things: how much the lender could have earned doing something else with the money, how much they expect inflation to erode it, and how likely they think you are to not pay. The first is the time value of money. The second is a hedge. The third is the reason your rate is different from your neighbor's.

Framed this way, "I have debt" means "I have sold some of my future to buy some of my present, and the price was set by what someone thought of my odds." That's a legitimate transaction. Most useful things, houses, educations, businesses, get bought that way. The problem is not debt. The problem is that the price compounds, and compounding isn't something the human brain estimates well.

Seventy-two

The one tool everyone should carry is the rule of 72. Divide 72 by the interest rate and you get the number of years it takes for a balance to double.

At 4%, about 18 years. At 8%, 9 years. At 24%, which is an unremarkable credit card rate, 3 years. Thirty-six percent, which some cards charge, doubles in two.

This is a linearization of the true formula, ln(2) / ln(1 + r), and it's accurate to within a few months for rates under about 20%. Above that it gets slightly optimistic. But the point isn't precision. The point is that the doubling time is something you can compute in your head, and once you can compute it you can feel it. A balance that doubles in three years is a balance that, left alone, quadruples in six and is eight times its size in nine. A $5,000 card at 24%, ignored, is $40,000 nine years later. Nobody intends that. The mailbox shows a minimum payment, not a doubling time.

Why the minimum payment is a trap by design

The minimum payment on a card is typically set around 1 to 2% of the balance plus that month's interest. This is engineered so that you're barely covering the interest and touching almost none of the principal. On a $5,000 balance at 24%, the monthly interest is about $100. A minimum payment of $150 retires $50 of principal. At that rate, the balance takes decades to clear, and you pay several times the original amount in interest.

The 2009 CARD Act in the US forced statements to show how long the minimum would take, and how much it would cost. The numbers are on the statement. They are in small type, in a box, and behavioral economists have found that most people still anchor on the minimum because it's the only number formatted as an instruction.

Amortization: where the payment goes

Fixed-payment loans, mortgages and car loans, have a different shape and a different surprise.

The payment is the same every month, but its composition changes. Each month's interest is the current balance times the monthly rate. Early on, the balance is large, so most of the payment is interest and only a little goes to principal. As the balance shrinks, the interest shrinks, and more of the same payment goes to principal. The curve is exponential in reverse.

The consequence people find shocking: on a thirty-year mortgage at 6%, after ten years of payments, a third of the term, you have paid off about 16% of the principal. After fifteen years, halfway through, a bit under 30%. The back half of the mortgage pays off most of the house. The front half mostly pays for the privilege of the back half.

This is also why extra principal payments early are so much more valuable than the same payments late. A dollar of principal removed in year one saves you thirty years of interest on that dollar. In year twenty-eight it saves you two.

The other direction

Compounding runs both ways, and this is the part that is supposed to be encouraging.

The same rule of 72 applied to an investment at 7% doubles every ten years. A thousand dollars at 25 is roughly eight thousand at 55, and roughly sixteen at 65, without adding anything. The whole retirement-savings industry is built on the observation that time is the input that matters, not the amount, and that most people get this backwards, saving aggressively in their forties and fifties when the doubling clock has mostly run out.

Debt is negative saving. Every dollar of debt at 24% is a dollar you would need to earn 24% on to break even, which no honest investment does. Paying off a card is the highest-yielding, zero-risk, tax-free investment available to almost anyone, and it doesn't feel like investing, because the balance just goes down instead of something going up.

A note on rates

Interest rates are the price of time and they move. For most of the 2010s, central bank rates in the rich world were near zero, and in a few places actually negative: banks were paying to lend to governments. Time was, briefly, free or better. Borrowing was cheap and the compounding argument ran weakly.

That ended. Rates rose sharply through 2022 and 2023, and every rule in this post got sharper along with them. A card that was 18% became 28%. The doubling time on the same balance dropped from four years to under three. The same debt got faster.

Which is the last thing to internalize: interest is a rate, and a rate is a speed. Debt is an amount and a velocity, and the velocity is set by someone else.

The quote that is not from Einstein

Compound interest is frequently called the eighth wonder of the world, attributed to Einstein. He never said it. The line first appears in the 1980s, in advertising copy for a financial services company, which is fitting, because the only people who fully understand compounding are the ones being paid by it.

You can be one of those people, in a small way, by being on the right side of the exponential. That's the whole trick. Time is the input. Interest is the exchange rate. Debt is just time you have already spent, with the meter running.