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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.
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:
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.
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.
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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Click box for answer.
False. It is best suited for daily or hourly periods.
$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.
Our trained humans found other terms in the category return math you may find helpful.
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