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Arithmetic Return is one of three methods for calculating return over multiple time periods. It is commonly used in forecasting applications. It is the 'total arithmetic return' when you add the returns together. And when you divide that total by the number of observations it becomes 'average arithmetic return'. This is time-uncertain, meaning, the order of returns does not matter.
Synonyms: arithmetic average return, arithmetic mean return
For context, arithmetic returns are used for forecasting because when looking into the future the investment modeler doesn't know in which order returns will come, so compounding isn't necessary.
As an example, let's say the ending value of an investment was $11 and
the beginning value was $10. The Excel formula would read
=(11/10)-1 for a result of 10%.
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.
So arithmetic return is a non-compounded version where you subtract one from the ending value divided by beginning value. That's it.
To calculate an average arithmetic return over multiple periods add up the returns and divide by the number of periods.
Because of its simple calculation and because we don't know in advance what order returns in the future will occur, arithmetic returns are often used in forward-looking applications like risk forecasting.
Doc: Is the arithmetic return of
100% followed by -50% equal to 0% or 25%?
Mia: I know! I know! It's 25% Doc, because arithmetic ignores compounding.
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Arithmetic Return definition for investment modeling (4:24)
The script includes two sections where we visualize and demonstrate the calculation of arithmetic return.
We're sitting here in Excel, and this is a snippet from our boot camp course.
Let's simplify this to two periods, but it applies to 3 and beyond. Here we have two versions of arithmetic return.
The arithmetic total return, sometimes just called arithmetic return involves adding the returns together.
And for the average, often called the arithmetic mean, take that total and divide by the number of observations. It is just the simple average we use every day, like two t-shirts selling for 24, is 12 per shirt. I put the total here in parentheses because that's similiar to what you'd do in Excel.
Let's peek at a chart, because often the eye picks up something you might not see in a table. Ok, looks good.
Let's walk through the total and average using two hypothetical stocks, over three periods.
ABC had returns of 10%, followed by -11%, then another positive 10%. XYZ had returns of 100%, followed by -50%, then it was flat.
The total arithmetic return calculation is logical. You could add each
one manually, like this, or use the
=SUM() function, with the range of
returns in parentheses.
We have observations here. And for the arithmetic average you could
do the division manually, like this. Or you oculd use the
You might ask, when would you choose the arithmetic method? It is the preferred method for risk studies and forward-looking estimates. You can't predict the order of returns in the future, instead you predict averages over long periods of time, typically.
The sentence earlier about time-uncertain is meaningful now. Let me show you why. Geometric returns, are more backward-looking, when you know the order of returns, like we did here. It is more reflective of the true client experience over a historical period.
Using the same data, but using geometric returns, look at the differences. 3% versus 2.5%, and 16.67% versus 0%. Hmmm. To see why, follow the link at the end.
Click box for answer.
True. Geometric returns compound at a lower rate to reach the same place.
Still unclear on Arithmetic Aeturn? Leave a question in the comments section on YouTube. Also, see a tutorial page and video on Stock return calculation methods from the Quant 101 Series. There we go over when to use arithmetic, geometric and log returns with examples in Excel.
Our trained humans found other terms in the category return math you may find helpful.
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