In this post, I will explain:

1) What is a cumulative return?
2) How do you calculate a cumulative return using geometric linking?
3) What is an annualized return?
4) How do you calculate an annualized return?

What Is A Cumulative Return For An Investment Portfolio?

A cumulative return measures, in percentage terms, how much your investments grew or declined over a specified period of time.  In other words, how much wealth you accumulated or lost due to your investments over time.
For example, let’s say a portfolio starts with $100.  Due to the investments held, five years later it is worth $130.  The 5 year cumulative return is 30% (calculated as $30 gain / $100 beginning value, since there are no flows).
A cumulative return can be calculated for any period of time (e.g. a 5 day cumulative return is the return earned over 5 days, a 5 year cumulative return is the return over 5 years, a since inception cumulative return is the return since the portfolio incepted). 

How Do You Calculate A Cumulative Return? 

Step 1: Gather the “Periodic Returns”

In the above example, I showed how you can calculate a cumulative return by taking the gain/loss over the beginning market value.  We took the $30 gained over five years and divided it by the $100 starting value to get a 30% return.
However, using cumulative gains/losses can only be used to calculate a cumulative return when there are no flows (money added or withdrawn from the account).  In the real world, people add or withdraw money from their accounts and so we have to use what are called “periodic returns” to calculate the “cumulative return.”
Periodic returns is a fancy term used to refer to the returns calculated for smaller time periods. For example, to calculate a 5 year cumulative return, you may have 5 annual returns, 60 monthly returns, or 1825 daily returns. The annual, monthly, and daily returns in these examples would be referred to as “periodic returns.” 
The following is an example where we have 5 annual returns. These will be used to calculate cumulative and annualized returns.

Step 2: Geometrically Link the “Periodic Returns” to Calculate the “Cumulative Return”

Now that we have our periodic returns, how do we combine them to get the 5 year cumulative return? Can we just add the 5 annual returns together?
No. You need to use something called “geometric linking.” But don’t worry, it’s not as scary as it sounds.
You essentially add a 1 to each return, and then multiply them together, using the following formula:
This formula is used because money, including profits and losses, from the current period is reinvested into the next period. If you want to know all the details behind geometric linking and why you need to use it, please read this post on geometric linking.
In practice, many investment firms use software to calculate a geometrically linked return, but you can also calculate it in Excel using a function called PRODUCT. The PRODUCT function allows you to highlight the series of returns and it will multiply them together.  This is what Performance Analysts typically use to geometrically link. 
If we apply the product formula to our series of 5 returns we get 30%.

Note: the Excel document is locked but you can save a copy and then edit as you like.

What Is An Annualized Return For An Investment Portfolio?

An annualized return shows you what the average annual return was over a certain time period. 
In our example, we earned 30% over a 5 year period, the annualized return will tell you what your average return is for each year.
But, wait!  Don’t break out your calculators and divide by 5!  We explained in the geometric linking post that returns are not added, they are linked using the below formula:
This is due to the fact that money from the current period, including profits and losses, is reinvested in the next period. So to calculate the annualized (aka “annual average”) return, we need to reflect that principle of reinvestment. 

So How Do You Calculate An Annualized Return?

Well, since we know returns are linked, we need to answer the following question:
What annual return, when linked over the years, will give you the correct cumulative return?
The Annual Avg Return is the same thing as the Annualized Return
We can simplify the above by combining the “1+ Annual Avg Returns.” Then we will have the 1+ Annual Avg Return raised to the number of years:
To solve for this annual average return (aka “annualized return”), you have to go back to high school algebra:
1) Add 1 to both sides of the formula, eliminating it on the left side and keeping it on the right side of the formula
2) Raise both sides to the power of (1 / number of years), eliminating the power on the left side and adding a power to the right
3) Subtract 1 from both sides, eliminating the 1 on the left side and putting it on the right
OK, that looks scary! Let’s break it down:
1) The “1 + Cumulative Return” represents the initial investment (the number 1) and the total aggregated return for the entire period (the cumulative return).
2) The raising to”1/number of years” is similar to dividing by the number of years, but since we are linking the returns together this essentially “unlinks” them.
3) The subtracting of 1 at the end removes the 1 that was used to represent the initial investment, so the return is isolated.
Let’s apply this to our example:
Ex) You earned 30% over 5 years.  What is the annualized return and what does that mean in plain English? 
If we divided the 30% by 5, it would give you 6%, but you can see the correct answer is 5.39%.
The difference between the arithmetic division (incorrect approach) and the geometric division (correct approach) will increase as the absolute returns get larger and the number of periods decrease.
What does the 5.39% represent in plain English? This is the same as saying if you earn 5.39% each year, and compound it, that will give you your 30% return (the cumulative return).
I hope you found this helpful. Please let me know your thoughts below or send me a direct message at info@learninvestmentperformance.com.

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