What is a Cumulative Return?
A cumulative return measures how much your portfolio grew over a specified period of time.
For example, let’s say a portfolio starts with $100. Two years later, it ends at $110. You didn’t add or take away money during the two year period. The cumulative return for the two year period is 10% (calculated as $10 gain / $100 beginning value).
Periodic Returns – The Building Blocks of the Cumulative Return
A cumulative return is typically made up of periodic returns. A periodic return is a return for a smaller time period within the cumulative time period.
For example, the two year time period is comprised of two annual returns. The annual returns are the periodic returns within the cumulative time period. Similarly, the two year time period is comprised of 24 monthly returns, which are also periodic returns.
Typically, you are given periodic returns and asked to calculate the cumulative return. Let’s take a look at how that is done.
Calculating Periodic Returns
The path from $100 to $110 could have occurred multiple ways, but the cumulative return is 10% regardless. For example, the portfolio could have dipped from $100 to $97, and then went back up to $110. Or it could have went up to $115 and then back down to $110. Let’s look at the scenario where the portfolio steadily climbed from $100 to $105 to $110.
Period 1 (Jan – June): The return is 3%, calculated as the $3 gain over the $100 beginning market value.
Period 2 (July – Dec): The return is 1.94%, calculated as the $2 gain over the $103 beginning market value. Remember, this period starts with the $103 we ended with in period 1, so the $2 gain in percentage terms is lower than 2% since it was earned on a higher amount. In other words, a $2 gain on $100 would be 2%, but it’s lower when it’s earned on a higher denominator.
First, we cannot simply add these, since 3% + 1.94% does not equal 5%.
So how do we combine these returns? We have to use what is called geometric linking. But don’t worry, it’s not as scary as it sounds. It’s a simple concept actually.
1. We started with $100 and grew to $103. Let’s use some simple algebra to figure out how we do that.
Well, we can’t simply put a 3% in there, since 3% of 100 is $3—not $103.
So, we need to include the $100. To do this, we multiply the $100 by 1 to get the $100, plus the 3% to get the $3. That is the same as multiplying by 1+ 3% or 1.03.
Now, to get from $103 to $105, we need to do the same thing, Multiply $103 by 1 + 1.94% or 1.0194.
So to get from the $100 to the $105, we would take $100 x 1.03 x 1.0194 = $105.
Most of the time, the returns are reported without the specific dollars. So to get just the returns you can multiply the 1.03 x 1.0194 = 1.05. That means the amount of the investment grew by 5%. To get rid of the “1” which represents the starting investment, and only reflect the growth of the investment, we subtract 1 from the geometric linking:
(1 + Return1) x (1 + Return2) – 1 = Return For The Entire Period*
(1.03) x (1.0194) – 1= 5%
* The fancy term for the entire period’s return is the “cumulative return,” since you are accumulating the returns. The return 1 (3%) and return 2 (1.94%) are called the periodic returns.