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# Log Return Definition, Quiz and Calculation in Excel

The least known of the three return calculation methods is used for continuously compounded returns over very short timeframes like minutes or hours.
1. Define - Define and understand log return for investments.
2. Calculation - Learn how to calculate log returns.
3. Context - Use log return in a sentence.
4. Quiz - Test yourself.
Updated: February 17, 2021
The continuously compounded log return isn't as practical and as common as the geometric and arithmetic versions because interest typically doesn't compound over infinitely short time periods.

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## Understanding Log Return and Continuous Compounding

Intermediate

Log Return is one of three methods for calculating return and it assumes returns are compounded continuously rather than across sub-periods. It is calculated by taking the natural log of the ending value divided by the beginning value. (Using the LN on most calculators, or the `=LN()` function in Excel)

Synonyms: logarithmic return, continuously compounded return

For context, it's important to recall from high school math that the letter e is a constant value rounded to 2.71828. The unrounded number e can be found in Excel using `=EXP(1)`.

Much like pi which describes the ratio of circumference to the diameter of all circles, e can be used to quantify all continuously compounding processes like inflation, interest rates, population growth and even radioactive decay.

For practicality in Finance however, banks don't pay interest on a continuously compounding basis, so keep that in mind.

### How to Calculate Log Returns

As an example to find a continuously compounding rate of return, let's say the ending value of an investment was \$11 and the beginning value was \$10. The Excel function would read `=LN(11/10)` for a result of 9.53%.

To calculate a continuously compounded ending value on a \$100 investment at 3% per year for 5 years would result in an ending value of \$116.18, using the formula:

• Ending Value = Beginning Value * (e)^(rate * time)
• \$116.18 = \$100 * 2.71828^(0.03 * 5)

In math and statistics, a distinction is often made between discrete and continuous data. Log return is the more theoretical continuous version. In the practical world, however, most people think of returns broken into discrete periods instead.

• Arithmetic return - One period, not-compounded, discrete.
• Geometric return - Multi-period, compounded, discrete.
• Logarithmic return - Infinite-periods, compounded, continuous.

The corresponding ending value using non-compounded returns (arithmetic) would yield \$115.00. If compounded five times (geometric) the ending value would be \$115.93.

So log return is the non-discrete version, so continuous, meaning it answers what the ending value would be if the whole period was broken into an infinite number of sub-periods. See, it's theoretical.

Other than for very short periods of time (typically less than 1 day), and for theoretical applications involving calculus and precisely measuring curves and tests for the normality of investment return distributions, log returns are not as common as arithmetic and geometric returns.

### In a Sentence

Tom:  If log return compounds most frequently, why is the rate the lowest?
Doc:  Chew on this. Let's say \$1 grew to \$1.10 in a year; the non-compounding return is 10%.

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### Quiz

The log return is best suited for long periods of time. | True or False

False. It is best suited for daily or hourly periods.

What is the ending value of an investment of \$1,000 continuously compounded for 10 years at 7% annually? | \$1,700.00 or \$1,967.15 or \$2,013.75

\$2,013.75. The others represent arithmetic and geometric total returns, respectively. To confirm type =LN(2013.75/1000) in Excel to get 7%.

Still unclear on log returns? Check out the tutorial page and video on Stock return calculation methods from the Quant 101 Series on YouTube. There we go over when to use arithmetic, geometric and log returns.

### Related Terms

Our trained humans found other terms in the category return math you may find helpful.

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