WHAT
You may have heard the term CAGR. It’s an acronym for Compound Annual Growth Rate. Said another way, it provides investors or borrowers a way to compare returns of various time periods by converting them to an annualized rate that takes into account the compounding of interest or returns.
WHY
Practically, CAGR allows for an apples-to-apples comparison among investments of varying duration. For example, if investor A was up 50% in three years and investor B was up 30% in two years, which had a better average return? I also use CAGR to calculate my average annual returns over multiyear periods including fractional years.
HOW
Calculating CAGR is pretty simple. I’ll break it up into several steps.
- Step 1: Determine the return. First, you’ll want to determine the total return. For example, suppose that an investor had a return of 10% in the first year, 25% in the second year, and -7% in the first nine months of the third year. The total return would be calculated by multiplying 1.1, 1.25, and 0.93 and then subtracting 1. Note: a negative return for any of the period results in multiplying by a number less than 1. It’s important to subtract the 1 at the end. In this example, the answer is 0.27875 or 27.875%. Also, if one of the periods has a return of 100% or greater, then we need to use a number of 2 or greater. For example, if in one year the investor managed a return of 105%, then we would use 2.05 as the number for that period.
- Step 2: Determine the duration of the investment. It’s easy for whole years, but for partial years, we need to convert that partial year to a fraction. In the above example, the last period was only nine months so we will use 9/12 or 0.75. If the last period falls in the middle of a month, we can determine the number of elapsed days since the start of the year and divide that number by 365. For our example, the duration is 2 years, 9 months which is the same as 2.75.
- Step 3: Calculate the CAGR. Now we have an investment that yielded 27.875% in 2.75 years. I like to use a calculator for this last step. We will want to find the function that takes the “y-root of “x”. The symbol is circled on the image of the Apple iPhone’s scientific calculator (note: the regular calculator can be turned into a scientific calculator by flipping the iPhone 90 degrees; if you have a different scientific calculator then it should also have this function as a button):
“X” is the cumulative percentage gain plus 1. In our example, the return was 27.875% or 0.27875; we add 1 to this number and enter it into the calculator as 1.27875. Next, we hit the button “y-root of x” button on the calculator. “Y” is the number of years in fractional form. In the example, 2 years and 9 months is 2.75. After hitting the “y-root of x” button, we enter 2.75 and hit the equals button. The result for this example should be 1.09353111. The last thing we need to do is subtract 1 which results in 0.0935 or 9.35%. This is the CAGR for our example.
- Step 4: Check the calculation. This step is optional just to make sure that the calculation was done correctly. I like to check my work because I’ve messed up the calculation on occasion. We use the “x to the power of y” function which is the inverse of the “y-root of x” function. The image below shows how the “x to the power of y” function appears on the scientific calculator. Note that “X” is the CAGR and “Y” is the duration in years. We type in the CAGR as a decimal and add one. This will be 1.0935. Next, we hit the function “x to the power of y”. Then we enter “Y” which is the duration, 2.75 in our example followed by the equal button. The result is 1.27865, but we still need to subtract the 1 which then gives us 0.27865 or 27.865%. You will notice that the number is ever so slightly off; the reason for the discrepancy is a rounding error because we used 9.35% as the CAGR and not 9.35311%. In other words, at a CAGR of 9.35311% over a period of 2.75 years, we get a cumulative return of 27.875%. The two calculations will match if we’ve done the calculation correctly.
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