Today we’re going to calculate returns correctly

I’m sure you’re all off to a great new year as the markets have never been more promising and abundantly green! So let’s figure out your returns!

There are a few ways to calculate your returns. First the basics:

If you’re trying to see how much you made in one investment, the formula is simply:

[(Ending Balance / Initial Balance) - 1] x 100 = Your Percentage Return

Example: On February 1st, you invested $100 into stock XYZ. At the end of the month you closed out of your position and sold the stock for $110. Your return is:

(110/100 - 1) x 100 = 10%

Now what if you wanted to calculate your average monthly return, across multiple months?

There are two ways to do this:

- Arithmetic Average
- Geometric Average

## Using the Arithmetic Average for Calculating Returns

Everyone knows how to calculate the arithmetic average. You add up all the numbers and divide by how many numbers you had. If your monthly returns were:

- January: 6%
- February: 10%
- March: 13%
- April: -9%

Then your arithmetic average would be (6+10+13-9)/4 = 5% Monthly

However there is a major flaw with using the arithmetic average.

It is an oversimplification of the numbers and doesn’t take into account compounding before and after.

Example:

- May you made a 50% return
- June you made a -50% return

The arithmetic average return across this period would be [(50 – 50)/2] x 2 months = 0%

In reality, if you started off with $100 and made a 50% return, you’d have $150. Then when you lost 50%, you’d be down to $75, not $100, even though the arithmetic return said you’ve netted 0%!

This is why in finance, the Geometric Average is a more precise indicator of returns.

## Using the Geometric Average in Calculating Returns

The formula for the geometric average is this example would be the product of (1+ your monthly return) minus 1. So in order to correctly calculate the returns you made between May and June, you would enter

(1+50%) x (1-50%) - 1 = -0.25

This shows accurately that you made a return of -25% across those two time periods.

So if you wanted to calculate your returns for the entire year, the complete formula would be:

(1+Jan) x (1+Feb) x (1+Mar) x … x (1+Dec) – 1

### Geometric Return Example:

Say you’re monthly returns for last year were as follows:

Your geometric average for the year would be:

[ (1 + 2%) x (1 + 3%) … (1 + 2%) x (1 + 1%) ] – 1 = 14.08%

Let’s pretend you started off with $100 and see your result:

As you can see, you ended the year with $114.08, exactly a 14.08% increase over your initial $100.

Now if you had taken the arithmetic average of monthly returns, which is 1.33%, and multiplied by 12 months, it would’ve given you 16%. Slightly off but not the correct number we’d want to report.

If you want to figure out your annual return across *n* number of years, there’s one more step you will have to take to correctly calculate average: Taking the *nth* root before subtracting 1. The complete formula becomes:

[(1+Year 1) x (1+Year 2) x… x (1+Year *n*)]^(1/n) – 1

If you start with $100 and your annual returns are as follows:

- First Year: 22% ($100 -> $122)
- Second Year: 10% ($122 -> $134.20)
- Third Year: -3% ($134.20 -> $130.17)

Then your average annual return following the geometric formula would be:

[(1+22%) x (1+10%) x (1-3%)]^(1/3) – 1 = 9.19%

Double checking our work we have:

- First Year: $100 x (1+9.19%) = $109.19
- Second Year: $109.19 x (1+9.19%) = $119.22
- Third Year: $119.22 x (1+9.19%) = $130.17

*Numbers have been rounded off in each step of the calculation.*

### Conclusion

Hopefully this may have enlightened some of you readers. Returns are dependent on each other and so a simple average will *simply* not do.

Fortunately, our brokerage accounts calculate our returns for us. They also account for cash deposits and withdrawals. So while you’re investing your portfolio, you can simply pull up your brokerage account and check your returns.