What is Geometric Linking?

Geometric linking is a fancy term used to describe how returns are combined over time.  For example, let’s say you are given 36 monthly returns for your investment portfolio and want to know what your combined 3 year return is.  That is where geometric linking comes in.
The formula for geometric linking is:

The blue rectangles represent the inputs, which are the returns you are trying to combine.  
But don’t worry if this looks scary to you.  Once you understand why returns are combined this way it will be easy to remember!
In the rest of the post, I will provide you with:
• An example of geometric linking
• An Excel template so you can see how it’s calculated in practice
• The rationale for why the geometric linking formula works and plain addition doesn’t

An Example of Geometric Linking

Here are the steps to use the formula above:
1) Gather your series of returns.  In the below example I use 12 monthly returns which we will geometrically link, or combine, into a 1 year return.
2) Add 1 to each return
3) Multiply to the series of 1 + return’s
4) Deduct 1 from the product of step #3

 

Excel Template with Geometric Linking

Many investment firms use software to calculate a geometrically linked return, but sometimes you need to calculate it in Excel.
Excel has a function called PRODUCT, which actually allows you to highlight the series of returns and it will multiply them together (step 3 above).  This is what Performance Analysts typically use to geometrically link.  But BE CAREFUL, this function is considered an “array formula” and you need to press Shift + Control + Enter when you leave the cell.  If you go into the cell and then leave it by pressing Enter only, it will give you the wrong result.  In the below template I show how the formula is used.  

Geometric Linking Excel Template

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

 

Why Geometric Linking Works and Plain Addition Doesn’t

For illustrative purposes, we will use a simplified example of 2 monthly returns and calculate the cumulative 2 month return.  The same principle can be applied to any series of daily, monthly or annual returns.
• You start with $100, make $5 in month 1, and an additional $5 in month 2.  This leads to the following monthly returns:

• I put the question mark in the 2 month cumulative return.  However, we know this should be 10% – we made $10 on $100 and didn’t add or take away any money.
• We are going to prove how to calculate the 10% cumulative return using the monthly returns of 5% and 4.76% instead of the $10 dollar gain/loss

Step 1

In dollar terms, we see the investment started with $100 and grew to $105. 
However, if you are given a 5% return instead of the $5 gain, how do you show that?
We need to carry over both the $100 and the 5% gain into the next period, since we are investing both.  To do this, we multiply the $100 by 1+ 5%, or 1.05.  Multiplying by 1 carries over the $100 initial investment, and multiplying by 5% carries over the $5 profit.  
 

Step 2

In order to get from $105 to $110, we need to do the same thing,  Multiply $105 by 1 + 4.76% or 1.0476.
 
 

Step 3

Combining the steps in 1 and 2 together, we can get from $100 to $110:

Step 4

In the above example, we included the initial $100 in the formula.  Most of the time, returns are reported without the specific dollars.  So how can we do that?
To utilize only the returns in the formula, you can easily remove the dollar values by diving by the $100 and you end up with:
The 1.10 means the initial investment grew by 10%.  To isolate the return of 10% (get rid of the “1” on the right side which represents the initial value), we subtract 1 from the right side:

And that is the geometric linking formula!!

 

So what would happen if we just add the 2 monthly returns together to get the 10% return?

Well, 5% + 4.76% = 9.76%, which is not 10%.
But why does adding returns not work?  9.76% on $100 would mean that you made $9.76 on $100.  But you made $10. 
By adding the 4.76% in month 2, you are implying that you made $4.76 in profits, instead of the $5 you actually made.  Why is 4.76% smaller than the 5%?  Because you are investing $105 in month 2 (the initial $100 + $5 profit)

In other words, with returns you assume that you are reinvesting profits and losses from the prior period.  So any future gains and losses are earned on the latest value.  Adding does not do that.  Adding actually assumes that the return was earned on the initial $100, which is not correct!
In summary, I hope you now understand how and why geometric linking works!  Please let me know your thoughts below or send me a direct message at info@learninvestmentperformance.com.
For an explanation on how to calculate the average annual return from the compounded return, please read my post on cumulative and annualized returns.

 

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