Compound vs Simple Interest: Where the Difference Comes From

Simple interest pays you on your original deposit. Compound interest pays you on your original deposit plus every payment you have already received. That is the entire conceptual difference, and it sounds too small to matter.

It is not small. Over a working lifetime it is the difference between a comfortable retirement and an uncomfortable one.

The two formulas

P is the principal, r the annual rate as a decimal, t the time in years, and n the number of compounding periods per year. In the simple case, time enters as multiplication. In the compound case, it enters as an exponent. Everything that follows is a consequence of that one structural difference.

What it looks like on $10,000 at 7%

Years	Simple	Compound (annual)	Gap
1	$10,700	$10,700	$0
5	$13,500	$14,026	$526
10	$17,000	$19,672	$2,672
20	$24,000	$38,697	$14,697
30	$31,000	$76,123	$45,123
40	$38,000	$149,745	$111,745

Read the gap column downwards. For the first year there is no difference at all. There is nothing yet to compound. By year five the gap is 4% of the balance and easy to dismiss. By year thirty the compound figure is nearly two and a half times the simple one, and by year forty the difference alone is eleven times the original deposit.

This shape is why compounding advice always sounds unconvincing to people in their twenties and obvious to people in their fifties. The curve is nearly flat exactly when you are being told about it.

The Rule of 72

Divide 72 by the interest rate and you get, approximately, the number of years to double your money.

• At 6%: 72 ÷ 6 = 12 years
• At 8%: 72 ÷ 8 = 9 years
• At 12%: 72 ÷ 12 = 6 years

It comes from taking logarithms of the compound formula. Doubling means (1 + r)^t = 2, so t = ln(2) / ln(1 + r). Since ln(2) ≈ 0.693 and ln(1 + r) ≈ r for small r, you get t ≈ 0.693 / r, or 69.3 divided by the percentage rate. 72 is used instead because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and it happens to be more accurate in the mid single digits where most real rates sit.

The same rule works against you on debt. A credit card at 24% doubles what you owe in three years if you never pay.

Compounding frequency, and its ceiling

More frequent compounding gives more growth, but with sharply diminishing returns. At 10% on $1,000 for one year:

Frequency	n	Balance	Effective rate
Annual	1	$1,100.00	10.0000%
Quarterly	4	$1,103.81	10.3813%
Monthly	12	$1,104.71	10.4713%
Daily	365	$1,105.16	10.5156%
Continuous	∞	$1,105.17	10.5171%

The jump from annual to quarterly is worth 38 basis points. Everything from monthly to infinity is worth 4.6 basis points combined. There is a hard limit, and it is P × e^rt: the continuous compounding case, where e is Euler’s number.

The practical lesson: when comparing two accounts, the rate is what matters. A product advertising "daily compounding" at 4.5% loses to one compounding annually at 4.7%.

Comparing like with like

To compare products with different frequencies, convert everything to an effective annual rate:

This is the number that answers "what would I actually earn over a year". In most jurisdictions lenders and savings providers are required to publish it, as APY for deposits, APR for loans, though APR conventions vary and sometimes include fees. Where both a nominal and an effective figure are given, use the effective one.

Where simple interest still shows up

It has not vanished. Simple interest is genuinely used for:

• Short-term instruments such as treasury bills, commercial paper and most money-market conventions under a year, where the difference is negligible and the arithmetic is easier.
• Some car and personal loans, particularly in the US, where interest accrues daily on the outstanding balance without being capitalised.
• Statutory interest on late payments and court judgments, where legislators chose predictability over accuracy.
• Flat-rate consumer lending, where it is used because it makes the headline rate look lower than it is: see the EMI article for the conversion.

The variable that matters most

Given the exponential form, it is worth asking which input rewards attention. Take $10,000 at 7% for 30 years, giving $76,123, and improve each input by a tenth:

• Principal $10,000 → $11,000: final balance $83,735. +$7,612
• Rate 7% → 7.7%: final balance $93,120. +$16,997
• Time 30 → 33 years: final balance $93,253. +$17,130

Principal scales linearly: 10% more in gives 10% more out. Rate and time sit in the exponent, and both deliver roughly double that. Three extra years at the end of a thirty-year horizon is worth more than a 10% larger starting deposit, which is an unintuitive result and the strongest argument for starting early rather than starting big.