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Contrast a stated annual rate with the effective annual rate

CFA® Level I · Quantitative Methods · The Time Value of Money in Finance · QM.1.d

The idea

A stated (nominal) annual rate says nothing until the compounding frequency is attached to it. The effective annual rate converts it into the return actually earned over a year: EAR = (1 + stated / m)^m - 1, where m is the number of compounding periods. More frequent compounding raises the EAR, approaching e raised to the stated rate, minus 1, as compounding becomes continuous. Comparing two quoted rates with different frequencies is meaningless until both are expressed as EARs, which is precisely why lenders are usually required to disclose one.

Formulas

Effective annual rate from a stated rate

EAR = (1 + stated ÷ m)ᵐ − 1

stated
the quoted, or nominal, annual rate
m
compounding periods a year

Comparing two quoted rates with different compounding frequencies. Quoted rates are not comparable until both are EARs.

Effective annual rate under continuous compounding

EAR = eʳ − 1

r
the stated annual rate
e
2.7183, the base of natural logarithms

The ceiling that more frequent compounding approaches. Daily compounding is already within a basis point of it.

A practice question

From the question bank, with the reason each wrong option is wrong.

A rate is quoted as 12% compounded monthly. Is the effective annual rate higher or lower than 12%?

  1. Higher — correct
  2. LowerWhy not: Backwards. Interest credited during the year goes on to earn interest itself, so the effective rate can only exceed the stated one.
  3. Exactly 12%Why not: They are equal only when compounding is annual. Twelve compounding periods lift the effective rate to 12.68%.

Interest earned early in the year itself earns interest, so the effective rate exceeds the stated rate whenever compounding happens more than once a year.

A harder one

A rate of 12% is compounded quarterly. What is the effective annual rate?

  1. 12.00%Why not: The stated rate. Four compounding periods make the effective rate higher.
  2. 12.55% — correct
  3. 12.68%Why not: Monthly compounding: (1 + 0.01)¹² − 1. Quarterly is four periods of 3%, which comes to 12.55%.

Each quarter earns 12% ÷ 4 = 3%, so the effective rate is 1.03⁴ − 1 ≈ 12.55%.

The hardest tier

Two loans quote the same stated annual rate, one compounded quarterly and one compounded daily. Which costs the borrower more, and why?

  1. They cost the same, since the stated rates matchWhy not: A stated rate means nothing without its compounding frequency — that is exactly why effective rates have to be disclosed.
  2. The quarterly loan, because fewer compounding periods means larger onesWhy not: The periodic rate is larger, but it is applied far less often. More frequent compounding wins: interest starts earning interest sooner.
  3. The daily loan, because more frequent compounding produces a higher effective rate — correct

The effective annual rate rises with compounding frequency, so daily compounding costs more than quarterly at the same stated rate. This is why the stated rate alone cannot be compared across products.

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