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CFA Level I Exam · Returns of Financial Assets and Instruments

Annualizing Returns and Compounding Conventions for CFA Level 1

Updated 7 October 2026

To annualize a return, compound it to a 1-year horizon: annualized return = (1 + periodic return)^(periods per year) − 1. For a continuously compounded return, use EAR = e^r − 1 and r = ln(1 + EAR). Always match the period to the rate before you compute.

Understand Annualizing Returns and Compounding Conventions

A return only means something when you know its time period. A 2% return in one month is very different from a 2% return in one year. To compare returns, you put them on the same basis, usually one year. This is annualizing.

There are two ways to annualize. With compounding, you assume each period's gain earns the same return again. A monthly return of 1% becomes (1.01)^12 − 1 = 12.68% a year. With simple annualizing, you multiply by the number of periods: 1% × 12 = 12%. The CFA curriculum treats compounding as the right method for returns, because gains are reinvested.

The effective annual rate (EAR) is the annual return after compounding. A stated annual rate with m compounding periods per year gives EAR = (1 + stated rate ÷ m)^m − 1. More frequent compounding gives a higher EAR for the same stated rate.

Take compounding to its limit and you get continuous compounding. Here the stated rate is a continuously compounded return, r, and EAR = e^r − 1. The reverse is r = ln(1 + EAR). This r is also called the log return or ln(1 + HPR) for one period.

Log returns have a useful property: they add across periods. Simple holding period returns do not add; you must multiply growth factors. Also, a continuously compounded return is always lower than the matching effective annual rate, for any positive rate.

Key formulas to remember

Annualizing a periodic return (compounded)
Annualized return = (1 + R_period)^c − 1
c = number of periods in one year, e.g. 12 for monthly, 4 for quarterly, 52 for weekly.
Annualizing a multi-period holding period return
Annualized return = (1 + HPR)^(365 ÷ days) − 1
Use the year-length convention given in the question. Use 1 ÷ years if the horizon is in years.
Simple annualizing (not compounded)
Annualized return ≈ R_period × c
Use only if the question says so. It ignores reinvestment.
Effective annual rate
EAR = (1 + stated rate ÷ m)^m − 1
m = compounding periods per year. The stated rate is the annual quoted rate.
EAR with continuous compounding
EAR = e^r − 1
r is the continuously compounded stated annual rate, as a decimal.
Continuously compounded return
r = ln(1 + HPR) = ln(P_end ÷ P_start)
Also written ln(1 + EAR) when HPR is the annual return.
Adding log returns
r(0 to T) = r1 + r2 + … + rT
Holds for continuously compounded returns over consecutive periods.

How to solve Annualizing Returns and Compounding Conventions questions

Use this method for any annualizing or compounding question. The key is to match the rate, the period and the formula.

  1. 1Identify what you are given: a periodic return, a stated annual rate with a compounding frequency, or a continuously compounded rate.
  2. 2Identify what is asked: an annualized return, an EAR, a continuously compounded rate, or a return over a different period.
  3. 3Write the number of periods per year (c or m) or the horizon as a fraction of a year.
  4. 4Convert percentages to decimals before you start.
  5. 5Choose the formula: power of (1 + R) for compounding, e^r − 1 for continuous to EAR, ln(1 + EAR) for EAR to continuous.
  6. 6Compute on the calculator, then subtract 1 if you are finding a return.
  7. 7Check the answer: if a stated rate is compounded more than once a year, the EAR is above the stated rate. For continuous compounding, e^r − 1 > r when r is positive, so the continuous rate is below the EAR.
  8. 8Pick the option nearest your answer; options go from smallest to largest.

Quickest way: Calculator shortcuts for powers, e^x and ln

When to use it: Use this when a question needs a power, an exponential or a log and you have about 90 seconds.

  1. Annualizing: on the TI BA II Plus, enter 1.01, press yˣ, enter 12, press = then − 1 = to get 0.1268.
  2. e^r − 1: on the TI BA II Plus, enter r as a decimal (e.g. 0.08), press 2ND then eˣ (the second function of the LN key), then − 1 =. For r = 0.08 this gives about 0.0833, an EAR of 8.33%.
  3. ln(1 + EAR): on the TI BA II Plus, for an EAR of 8.33% enter 1.0833, press LN. The result is about 0.08, the continuous rate of 8%.
  4. On the HP 12C, 1.01 ENTER 12 yˣ gives the factor; then 1 − gives the return. The HP 12C has no direct ln or eˣ keys, and it uses yˣ for powers. For continuous compounding, the TI BA II Plus is the easier choice.
  5. Estimate first: continuous rate is slightly below EAR, so you can often eliminate two options without full calculation.

Common mistakes in Annualizing Returns and Compounding Conventions

  • Multiplying a monthly return by 12 and calling it the annualized return.

    Simple annualizing is faster and feels natural.

    Fix: Use (1 + R)^12 − 1 unless the question asks for a simple annual figure.

  • Using the stated rate as if it were the EAR.

    The rate is quoted on an annual basis, so it looks effective.

    Fix: Divide the stated rate by m, raise (1 + that) to the power m, then subtract 1.

  • Forgetting to subtract 1 at the end.

    The calculator shows a growth factor like 1.1268.

    Fix: Always subtract 1 when the question asks for a return or a rate.

  • Entering the rate as 8 instead of 0.08 in e^r or ln.

    Percent figures are in the stem.

    Fix: Convert to decimals first. e^8 is a huge number, which signals the error.

  • Thinking a continuous rate is higher than the EAR.

    Continuous compounding sounds like the maximum growth.

    Fix: The EAR is the higher figure: e^r − 1 > r. The continuous rate is the lower input rate.

  • Using the wrong exponent when the horizon is shorter than a year.

    Mixing up 365 ÷ days with days ÷ 365.

    Fix: For a 90-day return, the exponent is 365 ÷ 90, which is greater than 1.

Worked examples

Example 1

A fund earns 1.5% in a quarter. What is its annualized return with compounding? A) 6.00% B) 6.14% C) 6.18%

Show the solution
  1. Quarterly return = 0.015, c = 4.
  2. Annualized = (1.015)^4 − 1.
  3. 1.015^2 = 1.030225.
  4. 1.030225^2 = 1.061364.
  5. Subtract 1: 0.061364, about 6.14%.
  6. Check: the simple figure 1.5% × 4 = 6.00% ignores compounding, so A is out.

Answer: B) 6.14%

Example 2

A bank quotes a stated annual rate of 6% compounded continuously. What is the effective annual rate? A) 5.83% B) 6.00% C) 6.18%

Show the solution
  1. r = 0.06.
  2. EAR = e^0.06 − 1.
  3. e^0.06 = 1.061837.
  4. Subtract 1: 0.061837, about 6.18%.
  5. Check: EAR must be above 6%, so A and B are out.

Answer: C) 6.18%

Exam tips

  • Read whether the question gives a stated rate with a frequency or a periodic return. This decides the formula.
  • Use the EAR above the stated rate to eliminate options quickly. A continuous rate below its EAR is a free check.
  • Log returns add across periods, and simple returns do not. If a question sums returns over time, it is asking about log returns.
  • Keep at least six decimals in intermediate steps. Rounding early can push you to a wrong neighbouring option.
  • If the horizon is in days, check the day count the question uses (365 is standard for annualizing).

Practice questions from Returns of Financial Assets and Instruments

Annualizing Returns and Compounding Conventions: frequently asked questions

How do I annualize a return in CFA Level I?

Compound the periodic return to a one-year basis: (1 + R)^c − 1, where c is the number of periods per year. For a multi-period holding period return, use an exponent of 1 ÷ years or 365 ÷ days.

What is the difference between EAR and holding period return?

A holding period return covers any length of time, such as 3 months or 5 years. The EAR is a return restated on a one-year basis with compounding. Convert a holding period return to an EAR by annualizing it.

How do I convert a continuously compounded return to an EAR?

Use EAR = e^r − 1, with r as a decimal. To go back, use r = ln(1 + EAR). The continuous rate is always a little below the EAR for positive rates.

Why do log returns matter more than simple returns?

Log returns add across time periods, which makes multi-period sums easy. Simple returns need compounding of growth factors. The curriculum uses log returns when it models prices as lognormal.