Nominal vs effective annual rate: 12% quoted is 12.68%
A bank quotes 12% compounded monthly. A candidate who reads that as a 12% return answers 12.00%, but the effective annual rate is 12.68%, because each month's interest starts earning interest of its own for the rest of the year. The gap is small. It is also the entire point of the question.
The stated rate is a label, not a return
A stated annual rate (also called the quoted or nominal rate) is a convention for quoting, nothing more. It tells you the periodic rate only after you divide it by the compounding frequency, and it tells you the actual annual return only after you compound that periodic rate back up. Two steps, and candidates who skip the second one lose the mark.
Periodic rate = Stated annual rate / m
EAR = (1 + Periodic rate)^m − 1
Here m is the number of compounding periods in a year: 2 for semiannual, 4 for quarterly, 12 for monthly. Notice that the stated rate never appears in a value calculation by itself. It is an input to the periodic rate, and the periodic rate is what actually does the work.
The one case where stated and effective agree is annual compounding. With m = 1 the exponent disappears and 12% stated is 12% effective. Every other frequency pushes the effective rate above the stated one, and a question that names a frequency has already told you the two numbers differ.
Working the 12 percent example
Take 12% compounded monthly. The periodic rate is one percent a month, and twelve of those compound:
Periodic rate = 0.12 / 12 = 0.01
EAR = (1.01)^12 − 1 = 1.126825 − 1 = 12.68%
In money: 1,000 invested at the quoted 12% grows to 1,126.83 after a year, not 1,120.00. The extra 6.83 is interest earned on interest, and it is the whole difference between the two answer choices.
Change only the frequency and the same 12% quote produces a ladder:
Semiannual: (1.06)^2 − 1 = 12.36%
Quarterly: (1.03)^4 − 1 = 12.55%
Daily: (1 + 0.12/365)^365 − 1 = 12.75%
The steps shrink as m grows. Going from annual to semiannual buys 36 basis points; going from daily to infinitely often buys less than a quarter of one. That ceiling is continuous compounding, where the discrete formula gives way to the exponential:
EAR = e^(Stated annual rate) − 1 = e^0.12 − 1 = 12.75%
Daily and continuous both round to 12.75% at two decimals, which is exactly why an exam question that wants to separate them will quote more decimals or ask which is larger rather than asking for a value.
The reverse direction is the harder version
Questions run the calculation backwards more often than candidates expect. Given a target effective annual rate of 12%, the stated rate under monthly compounding is not 12%: it is lower, because monthly compounding will do part of the work.
Stated annual rate = [(1.12)^(1/12) − 1] × 12 = 11.39%
Two quotes describing the same return look different on the page. That is what makes the effective annual rate the right comparison tool: it is the only one of these numbers that means the same thing across contracts with different compounding conventions.
Quick reference
| Question wording | What it establishes | Risk if missed |
|---|---|---|
| Stated annual rate, quoted rate, nominal rate | A quoting convention that must be divided by m | Using it directly as the annual return |
| Compounded monthly, quarterly, semiannually | The value of m, so the periodic rate and the exponent | Compounding the stated rate instead of the periodic rate |
| Effective annual rate | The one comparable figure across frequencies | Comparing two loans on their quoted rates |
| Continuously compounded | Use e raised to the stated rate, not a power of m | Guessing a large m instead of the exponential |
Read the frequency before the rate. That single habit converts most of these questions from a calculation into a lookup, because once you know m the formula has one shape and no decisions left in it.
The same compounding logic drives geometric versus arithmetic mean returns: in both, adding periodic figures together gives one answer and compounding them gives the true one.
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