CFA Level I Exam · The Time Value of Money in Finance

How to Calculate Effective Annual Rate (EAR)

Updated 7 October 2026 · Fact-checked

The effective annual rate (EAR) is the actual return you earn in one year after compounding. Divide the stated annual rate by the number of periods m, add 1, raise to the power m, and subtract 1. For continuous compounding, use EAR = e^(stated rate) − 1.

Understand Effective Annual Rate and Compounding Frequency

A stated annual rate (also called the quoted or nominal rate) is the yearly rate a bank advertises. It does not tell you how much you really earn until you know how often interest is added. That frequency is the compounding frequency, m: 1 for annual, 2 for semiannual, 4 for quarterly, 12 for monthly, 365 for daily.

The periodic rate is the rate applied in each compounding period. You get it by dividing the stated annual rate by m. A 12% stated rate compounded monthly has a periodic rate of 1% per month.

Interest earns interest. With 12% compounded monthly, each month's 1% is added to the balance, and the next month's 1% is earned on the bigger balance. After one year the balance has grown by more than 12%. That actual one-year growth is the effective annual rate (EAR). The EAR is the only figure you can compare fairly across investments with different compounding frequencies.

For a given stated rate, a higher frequency gives a higher EAR, but the gain shrinks as m grows. The limit is continuous compounding, where m is infinite and the growth factor for one year is e^r. Here e ≈ 2.71828. The EAR is always at least the stated rate. It equals the stated rate only when compounding is annual.

The same logic works in reverse. If you know the EAR and want the periodic rate for a shorter period, take a root, not a division. This matters when you discount monthly or quarterly cash flows with an annual effective rate.

Key formulas to remember

Periodic rate: Periodic rate = stated annual rate ÷ m; m is the number of compounding periods per year.
Effective annual rate (discrete compounding): EAR = (1 + stated rate ÷ m)^m − 1; Equals the stated rate when m = 1. Rises with m.
EAR with continuous compounding: EAR = e^(stated rate) − 1; Use the stated rate as a decimal. Upper limit of EAR for a given stated rate.
Future value with m periods per year: FV = PV × (1 + stated rate ÷ m)^(m × N); N is years. Stated rate and m must match the periods in the exponent.
Future value with continuous compounding: FV = PV × e^(stated rate × N); N in years.
EAR to periodic rate: Periodic rate = (1 + EAR)^(1 ÷ m) − 1; Use a root, not EAR ÷ m.
EAR to continuously compounded stated rate: Stated rate = ln(1 + EAR); Inverse of e^r − 1.

How to solve Effective Annual Rate and Compounding Frequency questions

Use this order for any EAR or compounding-frequency question. It keeps the units straight.

1Identify the stated annual rate and the compounding frequency m. Words such as quarterly, monthly or daily give m. 'Continuously' means use e.
2Convert the stated rate to a decimal and compute the periodic rate: stated rate ÷ m.
3Compute the one-year growth factor: (1 + periodic rate)^m, or e^r for continuous compounding.
4Subtract 1 to get the EAR. Convert to a percentage.
5If the question asks for a future value over several years, raise the growth factor to the number of years, or use FV = PV × (1 + r ÷ m)^(mN).
6If comparing investments, rank them by EAR only, never by stated rate.
7If the question gives an EAR and asks for a periodic or stated rate, work backward with roots or ln, as in the formula list.
8Check the answer: EAR must be at least the stated rate and below the continuous-compounding value.

Quickest way: Rank-first shortcut for three-option questions

When to use it: Use this when the options differ by small amounts and you have about 90 seconds.

Before calculating, note the order: EAR with continuous compounding > daily > monthly > quarterly > annual, for the same stated rate.
Cross out any option below the stated rate, since EAR is never lower than the stated rate.
Estimate with EAR ≈ r + (m − 1) × r² ÷ (2m) for small rates. For 8% quarterly this is 8% + 3 × 0.0064 ÷ 8 = 8% + 0.24% = 8.24%. Use it only as a rough check, and rely on the calculator when options are close.
Confirm with the calculator. On the BA II Plus press 2nd, ICONV. Enter NOM, press the down arrow, enter C/Y, press the down arrow to EFF and press CPT. For continuous compounding, key r as a decimal, press 2nd, LN (e^x), then subtract 1.
On the HP 12C, compute 1 plus the periodic rate (for example 1 ENTER 0.02 +), then enter m (for example 4) and press y^x, then subtract 1 (1 −).

Common mistakes in Effective Annual Rate and Compounding Frequency

Using the stated rate as if it were the effective rate.
The advertised rate looks like the yearly return.
Fix: Always ask how often interest compounds. Convert to EAR before comparing or discounting.
Forgetting to divide the stated rate by m, or dividing but forgetting to raise to the power m.
Students memorise half the formula.
Fix: Write (1 + r ÷ m)^m − 1 in full each time. Check that both steps appear.
Converting an EAR to a monthly rate by dividing by 12.
It feels like the reverse of multiplying.
Fix: Use (1 + EAR)^(1÷12) − 1. Dividing gives a periodic rate that compounds to more than the EAR.
Using the wrong exponent in a multi-year future value.
Students use N years instead of m × N periods, or use m × N with the annual rate.
Fix: Match the rate and the exponent: periodic rate with number of periods, or EAR with years.
Forgetting to subtract 1 and reporting the growth factor (such as 1.0824) as the EAR.
The calculator shows the growth factor before the final step.
Fix: Finish with −1. An EAR near 8% should look like 0.08, not 1.08.
Entering the rate as a whole number in e^x, such as e^6 instead of e^0.06.
The calculator accepts the key sequence without a warning.
Fix: Convert the percentage to a decimal before pressing e^x.

Worked examples

Example 1

A bank quotes a stated annual rate of 8% compounded quarterly. What is the effective annual rate? A. 8.00% B. 8.24% C. 8.50%

Show the solution

Stated rate = 0.08 and m = 4.
Periodic rate = 0.08 ÷ 4 = 0.02.
Growth factor = (1.02)^4 = 1.08243216.
EAR = 1.08243216 − 1 = 0.08243, or 8.24%.
Check: A equals the stated rate, which only holds for annual compounding. C is too high for quarterly compounding, since even continuous compounding gives e^0.08 − 1 = 8.33%.

Answer: B. 8.24%

Example 2

You invest €10,000 at a stated annual rate of 4% compounded continuously for 3 years. What is the future value? A. €11,200.00 B. €11,248.64 C. €11,274.97

Show the solution

Use FV = PV × e^(r × N).
r × N = 0.04 × 3 = 0.12.
e^0.12 = 1.127497.
FV = 10,000 × 1.127497 = €11,274.97.
Check the traps: €11,200.00 is simple interest (10,000 × 1.12). €11,248.64 is annual compounding (1.04^3). Continuous compounding must give the highest value.

Answer: C. €11,274.97

Exam tips

Questions often give the same stated rate with different frequencies and ask which has the highest EAR. Rank by frequency first, then confirm by calculation only if needed.
Watch for 'effective' versus 'stated' in the stem. When a question gives an EAR and a monthly cash flow, convert with a root, not a division.
Expect traps built from common errors: the stated rate, simple interest, and annual compounding all appear as wrong options. Compute the correct value and eliminate the other two.
Be fast with the e^x key on the BA II Plus (press 2nd, LN (e^x)). Practise continuous-compounding questions until the keystrokes are automatic.
With no penalty for wrong answers, always answer every question. On EAR comparison questions, use the frequency ordering to eliminate options before you guess.

Practice questions from The Time Value of Money in Finance

Effective Annual Rate and Compounding Frequency in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Effective Annual Rate and Compounding Frequency: frequently asked questions

What is the difference between stated annual rate and effective annual rate?

The stated annual rate is the quoted yearly rate before compounding is taken into account. The effective annual rate is the actual one-year return after interest is compounded. The two are equal only when interest compounds once a year.

What is the continuous compounding formula in CFA Level I?

For one year, EAR = e^r − 1, where r is the stated annual rate as a decimal. For several years, FV = PV × e^(rN). It gives the highest possible EAR for a given stated rate.

How do I calculate EAR on the BA II Plus?

Press 2nd, ICONV. Enter the stated rate at NOM, press the down arrow, and enter the compounding periods per year at C/Y. Press the down arrow to EFF and press CPT. The answer is shown as a percentage.

Why does more frequent compounding raise the EAR?

Interest is added to the balance sooner, so later interest is earned on a larger base. The increase gets smaller as frequency rises and approaches the continuous-compounding limit.