Compound Interest: The Maths of Time (and of Debt)

ℹ️ This is education, not financial advice. The maths is exact; the returns and rates used are illustrative, not predictions. Real investment returns vary and can be negative.

The most important idea in personal finance, and it's one line

There is a single mechanism that quietly decides whether money grows into wealth or debt grows into a trap. It's not complicated. It's this:

You earn interest on your interest.

That's the whole thing. Interest gets added to your money, and then next time, you earn interest on the bigger amount — including the interest you already earned. It compounds. And because it compounds, it doesn't grow in a straight line — it curves, gently at first and then startlingly.

Einstein probably never actually called it the eighth wonder of the world, but the quote stuck for a reason: almost nobody's intuition handles exponential growth. Your brain expects straight lines, so compound growth always looks slow right up until it looks impossible. Understanding it is the difference between "I'll start saving later" and "I need to start now" — and, on the other side, between "it's just a small balance" and "why do I still owe this."

Why this deserves a whole lesson

Because the intuition failure is expensive, in both directions.

On the growth side, misjudging it means starting late — and with compounding, starting late is the one mistake you can never fully undo, because you can't get the years back. On the debt side, the exact same mechanism runs in reverse: credit card interest compounds against you, which is why a balance that felt manageable becomes one that won't die.

Same maths. One force. It's either working for you or on you, and this lesson is about knowing which.

Mechanism 1 — Simple vs compound

Simple interest pays only on your original amount:

£1,000 at 10% simple, for 3 years:
  Year 1: +£100  → £1,100
  Year 2: +£100  → £1,200   (still 10% of the original £1,000)
  Year 3: +£100  → £1,300

Compound interest pays on the growing total:

£1,000 at 10% compound, for 3 years:
  Year 1: +£100.00  → £1,100.00
  Year 2: +£110.00  → £1,210.00   (10% of £1,100, not £1,000)
  Year 3: +£121.00  → £1,331.00

Over three years the gap is small — £31. That's the trap in the intuition: early on, compounding looks barely different from simple. The gap is invisible for years, and then it becomes the whole story. Give the same example thirty years and the compound version is more than seven times larger than where it started, while simple has merely quadrupled.

The formula, once, so the mechanism is exact:

Final = Principal × (1 + r)^n

  r = the rate per period (10% = 0.10)
  n = the number of periods
  the ^n — "to the power of n" — is where the curve comes from

That little exponent is the entire difference between a line and a curve. You don't need to compute it by hand — a spreadsheet or any calculator does it — but you need to feel what it means: each year multiplies, it doesn't add.

Mechanism 2 — Time matters more than amount

Here's the counterintuitive part that changes decisions.

With compounding, when you start matters more than how much you put in. Because the early money has the most years to compound, a small amount invested early can beat a large amount invested late.

The classic illustration (illustrative, 7% annual return):

ANA invests £2,000/year from age 18 to 28 (11 years), then STOPS.
   Total put in: £22,000.

BEN invests £2,000/year from age 28 to 65 (37 years).
   Total put in: £74,000.

At 65, assuming ~7% a year:
   Ana (£22k in, stopped at 28)  →  roughly £430,000
   Ben (£74k in, started at 28)  →  roughly £340,000

(Illustrative — not a promise; returns vary and can be negative.) Read that again. Ana put in a third as much and ended with more, purely because her money had a decade's extra head start, and in compounding the earliest years do the heaviest lifting.

This is the single most valuable thing a 16-year-old can understand about money: the years you have are your biggest asset, and they're the one thing you can't buy back later. Not being rich yet doesn't matter nearly as much as being early.

Mechanism 3 — The Rule of 72

A mental-maths shortcut worth memorising, because it lets you feel the curve without a spreadsheet:

Years to double ≈ 72 ÷ interest rate

  at 2%  → 72 ÷ 2  = 36 years to double
  at 6%  → 72 ÷ 6  = 12 years
  at 9%  → 72 ÷ 9  = 8 years
  at 24% (a credit card!) → 72 ÷ 24 = 3 years

It's approximate but close, and it does two jobs. On the growth side it shows why a couple of extra percent matters so much — the difference between doubling every 12 years and every 8 is enormous over a lifetime. On the debt side, that last line should stop you cold: at credit-card rates, a debt you ignore doubles in about three years.

Mechanism 4 — The same force, in reverse: debt

Everything above runs backwards on money you owe, and this is where it does real damage to real people.

Credit card interest compounds against you, typically quoted around 20–30% a year, often applied monthly. And the mechanism has a specific cruelty built into how repayment works:

The minimum payment trap. Card statements show a "minimum payment" — often around 1-3% of the balance. Paying only the minimum means most of your payment covers interest, and the balance barely moves. Because the remaining balance keeps compounding, a debt paid at the minimum can take years, sometimes decades, and cost far more in interest than the original amount.

Illustrative: £2,000 on a card at ~22%, paying only the ~2% minimum
each month → well over a decade to clear, and more paid in interest
than the original £2,000.

Same £2,000 at a fixed £100/month → cleared in about two years, a
fraction of the interest.

(Illustrative; exact figures depend on the card's rules.) The lesson isn't "never use credit" — it's that compounding debt is the same wonder working against you, and the minimum payment is designed around it, not against it. Paying more than the minimum, or clearing the balance monthly, is how you stay on the right side of the curve.

The order of operations this implies: high-interest debt compounding at 22% is a guaranteed 22% loss, and no investment reliably beats that. Which is why the common guidance is to clear high-interest debt before investing — you can't out-earn a debt growing faster than the market.

What this means for you

Not advice — the implications of the maths.

• Starting early beats starting big. The years are the asset. If you take one thing from this lesson, it's that time in the market matters more than the amount, because the early years compound the longest.
• Small percentages are not small. 1% more in fees (MN-03), or 1% more in return, is a large fraction of your final wealth over decades. The Rule of 72 is why.
• Credit card debt is the same mechanism pointed at you. At ~22%, ignoring it means it roughly doubles every three years, and the minimum payment keeps you in the curve rather than out of it.
• Compounding rewards patience and punishes impatience — symmetrically. It's slow enough early to make you doubt it, which is exactly why most people quit or delay before it pays off.

Try it: build the two curves (40 min)

The output. You'll model both directions in a spreadsheet (see TL-01 if formulas are new).

1. Build a savings model:

    A         B                          C
1   Year      Balance                    Interest earned
2   0         1000
3   =A2+1     =B2*(1+$E$1)               =B3-B2
    ...drag down 30 rows...

E1  = 0.07   (label it "annual return")

Drag it down 30 years. Watch the Interest earned column — notice it grows every year even though you add nothing. That column growing is compounding.

2. On the same sheet, model a debt:

    Balance                              Payment    Interest added
    2000                                 100        =bal*(0.22/12)
    =prev - payment + interest_added     100        ...
    ...drag down until the balance hits zero...

Count how many months it takes to clear. Then change the payment to only 2% of the current balance each month and watch how much longer it takes.

3. Chart both. Put years on the horizontal axis. See the savings curve bend upward and the debt take far longer to clear at the minimum.

4. Write three sentences: how much of your final savings balance was money you added vs interest; how much sooner the debt clears at £100/month vs the 2% minimum; and what the Rule of 72 predicts for each rate you used.

✅ Finish check: one spreadsheet showing savings compounding up over 30 years and a debt being cleared, both charted, with your three written observations.

Summary card

• Compound interest = you earn interest on your interest. It curves, not lines — slow, then sudden.
• Final = Principal × (1 + r)^n. The exponent is where the curve comes from: each period multiplies.
• Time beats amount. Starting early with a little can beat starting late with a lot — the early years compound the longest, and you can't buy them back.
• Rule of 72: years to double ≈ 72 ÷ rate. At 6%, 12 years. At a 24% card, ~3 years.
• Small percentages are huge over decades — in returns and in fees (MN-03).
• Debt is the same force in reverse. Credit cards at ~22% compound against you fast.
• The minimum payment is a trap — most of it is interest, so the balance barely moves and the debt can last a decade-plus.
• Clear high-interest debt before investing — you can't reliably out-earn a 22% guaranteed loss.
• Compounding rewards patience and punishes delay, symmetrically.

Sources

1. SEC — Investor.gov: compound interest guidance and calculator
2. US CFPB — guidance on credit card interest and minimum payments
3. Bernstein, W. — The Four Pillars of Investing, 2002

Next lesson: MN-05 — Risk, Volatility and Diversification (L2) Related: MN-03 Index Funds and ETFs · MN-11 Building a Budget · MN-13 Credit and Credit Scores · TL-01 Spreadsheets From Zero Path: Reading the Market — 3/6