Arithmetic Return Definition and Tutorial for Investment Modeling

Arithmetic Return for investment performance analysis

Beginner

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.

How it is Calculated?

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.

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

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.

In a Sentence

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.

Video

This video can be accessed in a new window or App here , at our YouTube Channel or within this page below.

Arithmetic Return definition for investment modeling (4:24)

Video Script

The script includes two sections where we visualize and demonstrate the calculation of arithmetic return.

Visualize

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.

Demonstrate

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 =AVERAGE() function.

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.

Quiz

Click box for answer.

When positive, the average arithmetic return is always equal to or greater than the average geometric return. | True or False?

True. Geometric returns compound at a lower rate to reach the same place.

What is the average arithmetic return of 10%, -10% and 0% | 0% or -0.3%?

0%.

Questions or Comments?

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.

Related Terms

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

Arithmetic Return Definition and Tutorial