Updated for the 2026-2027 CFA® Level I curriculum.
Continuous compounding models growth as the limit reached when the number of compounding periods increases without bound. A continuously compounded return, also called a log return, measures the change between two investment values using the natural logarithm.
For CFA Level I, you should know how to calculate this return, annualize it, and convert between continuously compounded and standard discrete returns.
Quick Answer
A continuously compounded return is the natural logarithm of the ending value divided by the beginning value. You can also calculate it as the natural logarithm of one plus the discrete holding-period return.
Continuously compounded returns are useful because returns from consecutive periods can be added together. Convert the result back to a discrete return with the exponential function.
Key Takeaways
Continuous compounding is the limiting case of increasingly frequent compounding.
A continuously compounded return is often called a log return.
The period return equals the natural logarithm of the ending value divided by the beginning value.
A discrete return converts to a continuous return using the natural logarithm.
A continuous return converts to a discrete return using the exponential function.
Continuously compounded returns can be added across consecutive periods.
Divide the period return by the holding period in years to calculate an annualized continuous rate.
What You Need to Know for CFA Level I
Focus on choosing the right formula and keeping the time period consistent:
Use the natural logarithm, written as inline equation code .
Identify whether the question provides beginning and ending values, a discrete return, or a continuous rate.
Separate the return for the holding period from the annualized return.
Express the holding period as a fraction of a year before annualizing.
Convert a continuous rate back to a discrete return when the question asks for an effective return.
Add continuously compounded returns across consecutive periods.
Treat the continuous rate as a return convention that summarizes observed growth. The investment price can still move unevenly within the period.
What Is Continuous Compounding?
Finite compounding credits interest a fixed number of times each year. Annual compounding applies interest once, quarterly compounding applies it four times, and monthly compounding applies it 12 times.
As the number of compounding periods increases, the investment value approaches the continuous-compounding limit. The relationship between beginning value and ending value is:
Where:
Inline equation code = beginning value
Inline equation code = ending value after time
Inline equation code = annual continuously compounded rate
Inline equation code = time in years
Inline equation code = base of the natural logarithm, approximately 2.71828
This equation says that the investment grows from inline equation code to inline equation code at the continuously compounded annual rate inline equation code over inline equation code years.
For example, an 8% annual continuously compounded rate produces a one-year growth factor of:
The equivalent effective one-year return is approximately 8.33%.
Continuously Compounded Return Formulas
The correct formula depends on the information provided and whether the question asks for a period return or an annualized rate.
Formula From Beginning and Ending Values
For a single holding period, calculate the continuously compounded return as:
Where:
Inline equation code = beginning value
Inline equation code = ending value
Inline equation code = continuously compounded return for the full holding period
If the holding period covers one year, the result is already an annual continuously compounded return.
Annualized Formula From Beginning and Ending Values
For a holding period of inline equation code years:
Dividing by inline equation code converts the holding-period return into an annualized continuous rate.
Formula From a Discrete Holding-Period Return
When the question gives you a discrete return inline equation code , use:
The relationship works because the value ratio equals one plus the discrete return:
Both continuous-return formulas therefore describe the same change in investment value.
How to Convert Between Continuous and Discrete Returns
Continuous and discrete returns use different conventions to express the same change in value.
Discrete return to continuously compounded return
Continuously compounded return to discrete return
Convert a Discrete Return to a Continuous Return
Suppose an investment earns a 12% discrete return.
The 12% discrete return and the 11.33% continuously compounded return describe the same change in wealth.
Convert a Continuous Return to a Discrete Return
Starting with the 11.33% continuous return:
The exponential function reverses the natural logarithm and returns the result to the standard discrete-return convention.
How Do You Annualize a Continuously Compounded Return?
Continuously compounded returns scale directly with time. Divide the holding-period return by the fraction of a year represented by the investment period.
Where inline equation code is measured in years.
Common time conversions include:
Holding Period | Value of |
|---|---|
Three months | 0.25 |
Six months | 0.50 |
Nine months | 0.75 |
One year | 1.00 |
Eighteen months | 1.50 |
Suppose an investment earns a 6% continuously compounded return over six months. Its annualized continuous rate is:
When the question asks for the equivalent effective annual return, convert the annualized continuous rate with:
For a 12% annual continuous rate:
The 12.00% figure is the annual continuously compounded rate. The 12.75% figure is its equivalent effective annual return.
Why Are Continuously Compounded Returns Additive?
Continuously compounded returns are additive because logarithms convert multiplied growth factors into sums.
Suppose an investment moves from inline equation code to inline equation code , and then from inline equation code to inline equation code .
The total continuously compounded return is:
The value ratio can be separated into two consecutive periods:
Taking the natural logarithm gives:
The continuously compounded returns for the two periods can therefore be added to find the total continuously compounded return.
Discrete returns combine through their growth factors:
This is the practical distinction to remember:
Add continuously compounded returns across consecutive periods.
Multiply discrete growth factors across consecutive periods.
Worked Example
An investment grows from 100 to 112 over nine months.
Calculate:
The nine-month discrete return
The nine-month continuously compounded return
The annualized continuously compounded return
The equivalent effective annual return
Step 1: Calculate the Nine-Month Discrete Return
The investment gained 12% during the nine-month holding period.
Step 2: Calculate the Nine-Month Continuous Return
The nine-month continuously compounded return is 11.33%.
Step 3: Annualize the Continuous Return
Nine months equals 0.75 years.
The annualized continuously compounded rate is 15.11%.
Step 4: Calculate the Equivalent Effective Annual Return
The results can be summarized as follows:
Measure | Result | What It Represents |
|---|---|---|
Nine-month discrete return | 12.00% | Standard return over the actual holding period |
Nine-month continuous return | 11.33% | Log return over the actual holding period |
Annualized continuous rate | 15.11% | Continuous return expressed on a one-year basis |
Effective annual return | 16.31% | Equivalent one-year discrete return |
Each result answers a different question. Check the requested return convention and time period before selecting your final answer.
How to Recognize a Continuous-Compounding Question
Look for wording such as:
Continuously compounded return
Log return
Natural logarithm
Annualized continuously compounded rate
Convert a discrete return to a continuous return
Convert a continuous rate to an effective return
Add returns across consecutive periods
The notation inline equation code signals the natural logarithm. Most financial calculators provide a dedicated key for this calculation.
Common Exam Traps
Using inline equation code instead of inline equation code .
Using the common logarithm instead of the natural logarithm.
Treating a period continuous return as an annualized rate.
Dividing by the number of months instead of expressing the holding period in years.
Reporting the continuous rate when the question asks for an effective discrete return.
Adding discrete returns across time instead of multiplying their growth factors.
Applying the exponential conversion before annualizing when the question asks for an annual continuous rate.
Confusing the annual continuously compounded rate with its equivalent effective annual return.
Practice Question
A security rises from 80 to 88 over six months. What is its annualized continuously compounded return?
9.53%
19.06%
20.00%
Correct Answer: B
First, calculate the six-month continuously compounded return:
Six months equals inline equation code . Annualize the result:
Option A gives the six-month continuous return before annualization.
Option C doubles the 10% discrete holding-period return and uses a different return convention.
The annualized continuously compounded return is 19.06%.
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FAQs About Continuously Compounded Returns
What Is a Continuously Compounded Return?
A continuously compounded return is the natural logarithm of an investment’s ending value divided by its beginning value. It is also called a log return.
What Is the Continuously Compounded Return Formula?
For one holding period, use:
When a discrete return is provided, use:
How Do You Convert a Continuously Compounded Return to a Standard Return?
Use the exponential function:
This gives the equivalent discrete return for the same holding period.
Why Are Continuously Compounded Returns Additive?
Investment growth factors multiply across consecutive periods. The natural logarithm converts those multiplied growth factors into a sum, allowing the period log returns to be added together.
Is a Continuously Compounded Return Lower Than a Discrete Return?
For the same positive holding-period gain, the continuously compounded return is numerically lower than the discrete return. Both describe the same change in investment value under different return conventions.
For example, a 10% discrete return is equivalent to a continuously compounded return of approximately 9.53%.
What Is the Difference Between a Continuous Rate and an Effective Annual Return?
The continuous rate uses logarithmic compounding. The effective annual return expresses the same annual growth using the standard discrete convention.
Convert an annual continuous rate to an effective annual return with: