Understanding Compound Interest: Nominal vs Effective Rate

Two savings accounts pay 5.9% and 5.75%. Two loans charge 7.4% and 7.5%. In both pairs the lower-looking number is the more expensive or the less generous one, and nothing in the headline rate says so. The missing information is how often interest is credited.

A rate quoted per year that is actually applied monthly is not a rate at all until you say what “applied monthly” does to it. That translation is the difference between nominal and effective, and it is a single worksheet on the BA II Plus.

What a nominal rate leaves out

A nominal annual rate is a periodic rate multiplied by the number of periods in a year. It is a quoting convention, chosen because multiplication is easier to explain than compounding.

1. EFF = (1 + NOM ÷ m)^m − 1 m = compounding periods per year

Multiplication builds the nominal rate; exponentiation builds the effective one. The gap between them is interest earning interest inside the year.

At 6% nominal compounded monthly you receive 0.5% twelve times. Because each month’s interest joins the balance, twelve applications of 0.5% multiply the money by 1.005¹² = 1.06167781, not by 1.06.

Compounding	Periods	Effective annual rate	After 1 year	After 10 years
Annual	1	6.000000%	£10,600.00	£17,908.48
Semiannual	2	6.090000%	£10,609.00	£18,061.11
Quarterly	4	6.136355%	£10,613.64	£18,140.18
Monthly	12	6.167781%	£10,616.78	£18,193.97
Weekly	52	6.179982%	£10,618.00	£18,214.89
Daily	365	6.183131%	£10,618.31	£18,220.29
Continuous	∞	6.183655%	£10,618.37	£18,221.19

Two features of that table are worth naming. The first split is the big one — going from annual to semiannual adds 9 basis points, while going from daily to continuous adds half of one thousandth. And the sequence converges rather than growing without limit: no amount of compounding frequency turns 6% into 7%. The ceiling is e⁰·⁰⁶ − 1 = 6.183655%.

Both conversions, in three keystrokes each

1. 2ND
2. · ICONV
3. · ↓
4. · 6
5. · ENTER
6. · ↓
7. · ↓
8. · 12
9. · ENTER
10. · ↑
11. · CPT

The worksheet order is NOM, EFF, C/Y. Enter NOM, drop past EFF to set C/Y, then come back up to EFF and press CPT.

EFF = 6.1678. Reverse it by filling in the field you know and computing the other:

1. 2ND
2. · ICONV
3. · ↓
4. · ↓
5. · 5
6. · ENTER
7. · ↓
8. · 12
9. · ENTER
10. · ↑
11. · ↑
12. · CPT

NOM = 4.888949, a periodic rate of 0.407412% a month. To earn a true 5% a year with monthly crediting, a bank needs to quote only 4.888949% nominal — which is exactly why it will advertise the 5% instead.

The comparison AER was invented to fix

Two savings accounts, £20,000 for one year.

	Account A	Account B
Quoted rate	5.9% annual	5.75% monthly
Compounding	Yearly	Monthly
Effective annual rate	5.9000%	5.9040%
Interest on £20,000	£1,180.00	£1,180.80

Account B pays 15 basis points less on paper and 0.4 basis points more in reality. UK rules require savings products to quote AER, the effective annual rate, so that this comparison can be made on one number — an account crediting 4.5% nominal monthly must advertise 4.59% AER, and on £20,000 that is £918.80 of interest against £920.00 for a flat 4.6% AER account.

The equivalent discipline does not exist on the other side of the Atlantic, where APR is a nominal rate by statute. A US credit card at 22.9% APR compounding monthly costs 25.4632% effectively; a UK card at 22.9% APR is already effective and works out at 1.7332% a month, a nominal 20.7983%. The same headline number, two different products.

Reading of “22.9%”	Effective annual rate	Interest for the year
Effective (UK APR)	22.9000%	£732.80
Nominal, compounded monthly	25.4632%	£814.82
Nominal, compounded daily	25.7252%	£823.21

Eighty-two pounds on the same balance at the same advertised rate, decided entirely by a convention the advertisement does not mention.

Loans: the shorter compounding period costs more

Same arithmetic, opposite sign. Borrowing £180,000 for one year.

	Loan A	Loan B
Quoted nominal rate	7.4%	7.5%
Compounding	Quarterly	Annual
Effective annual rate	7.6079%	7.5000%
Cost of £180,000 for a year	£13,694.21	£13,500.00

Loan A costs £194.21 more. Push the same 7.4% to monthly compounding and the effective rate rises to 7.6562%. For a borrower, more frequent compounding is always worse, and the quoted rate hides it.

The mistake that costs the most money

Wanting a monthly rate and dividing the effective annual rate by 12 is the single most common compounding error, because it looks like the reverse of the operation that produced the number.

Method	Monthly rate	Value after 10 years
Correct: nominal ÷ 12	0.500000%	£18,193.97
Wrong: effective ÷ 12	0.513982%	£18,500.24

The gap is £306.27, from a rate error of 1.4 basis points a month. Compounding an effective rate as though it were nominal double-counts the compounding you already added.

Going the other way needs a root, not a division: the monthly rate matching a 6.167781% EAR is 1.06167781^(1/12) − 1 = 0.5%. On the calculator, put 6.167781 in EFF, 12 in C/Y, and CPT on NOM.

Doubling time, without the rule of 72

The rule of 72 says 6% doubles money in twelve years. It is a nominal-rate approximation, and the real answer depends on the compounding.

Compounding	Effective rate	Doubling time
Rule of 72 estimate	—	12.00 years
Annual	6.000000%	11.90 years
Monthly	6.167781%	11.58 years
Continuous	6.183655%	11.55 years

The approximation is out by a month at annual compounding and five months at monthly. Fine for a conversation, not for a projection.

Where the distinction changes an answer, not just a number

• Comparing quoted products. Convert both to effective and compare once. Any other approach compares two different quantities.
• Bond yields. A semiannual-pay bond quoted at 8% has a bond-equivalent yield of 8% and an effective yield of 8.16%. Bond markets quote the nominal figure by convention, so a bond and a monthly-compounding loan are never directly comparable as quoted. Bond pricing on the BA II Plus uses the semiannual convention throughout.
• Discounting cash flows. An NPV with annual flows needs an effective annual discount rate. Feeding it a nominal rate that compounds monthly understates every discount factor and overstates the NPV.
• Credit card and overdraft costs. Daily compounding on 18% nominal gives 19.7164% against 19.5618% for monthly. Small on a month’s balance, material on a year of revolving debt.

Four things people get wrong

1. Assuming APR means the same thing everywhere. In the UK it is effective; in the US it is nominal. The same 22.9% describes two products that differ by £82 a year on a £3,200 balance.
2. Comparing a nominal rate with an effective one. The most common version is setting a savings AER beside a loan APR. Convert one of them first, always.
3. Believing more frequent compounding keeps helping. It converges. Beyond daily there is nothing left to win — the entire gap from daily to continuous on 6% is 0.0005 percentage points.
4. Dividing an effective rate to get a periodic one. Use the root, or the NOM field. Dividing 6.167781% by 12 is worth £306.27 of error on £10,000 over a decade.

Every time value of money answer inherits whichever convention you fed it, and the calculator cannot tell which one you meant. When a rate arrives from outside — a quote, a term sheet, an exam stem — decide whether it is nominal or effective before it goes anywhere near I/Y. The interest conversion calculator shows both readings side by side, and loan payment calculations are where getting it wrong shows up as real money.