Simple vs compound interest

Everything in module 4 rests on one idea: money has a time price. Rent a truck, pay by the day; rent money, pay by the year — and the rate is the rental fee. The only real question is whether the fee is charged on the original amount alone, or on the growing pile. That's the entire difference between simple and compound interest.

Simple: interest on the principal only

Put $10,000 to work at 6% simple interest. Each year adds the same $600 — interest is always computed on the original principal:

$A = P(1 + rt) = \$10{,}000 \times (1 + 0.06 \times 20) = \$22{,}000$

A straight line: predictable, walkable, linear.

Compound: interest on the interest

Now let each year's interest JOIN the principal before the next year is computed. Year 2 earns 6% on $10,600, not $10,000. Year 3 on $11,236. The formula swaps multiplication for an exponent:

$A = P(1 + r)^t = \$10{,}000 \times 1.06^{20} = \$32{,}071$

Same rate, same money, same twenty years — and the curve beats the line by $10,071, more than the original principal. Notice WHERE the gap grows: barely visible for five years, unmistakable at twenty. Compounding is not a fast force; it is a patient one, and it doesn't care which side of it you're on.

Both sides of the curve

• When you save or invest, the curve works for you — and starting early matters more than starting big, because the exponent feeds on years.
• When you borrow, the curve works against you. Carry a credit-card balance at 24% compounding monthly and the effective year costs (1 + 0.24/12)¹² − 1 = 26.8% — the compounding itself added 2.8 points. That gap between the sticker rate (APR) and the compounded truth (APY) widens with the rate; at high rates, always ask which one you're being quoted.

More frequent compounding uses the same machine — divide the rate by the periods, multiply the years by them:

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

Next lesson, the curve meets the shop: what a $40,000 truck loan actually costs, payment by payment.