Now that we have an understanding of how compound interest works, let’s take a look at another important topic, Compound Annual Growth Rate (CAGR). CAGR is not the actual growth rate of the investment, but a rate that is used because it models that growth over time assuming a steady, smooth trend (removing the daily / periodic fluctuations – peaks and valleys).
To begin, let’s define what the rate of return on an investment is:
Or more simply stated:
For example, if we started off with an investment valued at $100 and it grew to a final value of $300 we would have:
Now, the Rate of Return (ROR) tells us nothing about the number of years it took to grow the investment. Furthermore, the growth rate per year is not evident. All we know is that the final return on the initial investment was 200%.
Let’s continue with this example and disperse values in between the start and stop points:
[table id=1 /]
The percentage change between one year and the next is also displayed. The Arithmetic Mean can be calculated by summing each year’s individual return and dividing it by the total number of years:
Based off of this calculation, we know that the average return on our investment was 46.51%. So, plugging this number in each year as our growth percentage should allow us to connect the starting point and the end point, right? Let’s see:
[table id=2 /]
For Year 1, we start with $100 as before. For Year 2, we take the initial starting value of $100 and add 46.51% (average annual growth rate). This gives us:
Clearly, using the Arithmetic Mean approach does not allow us to finish with the same value we initially assumed. The starting value is still $100. However, the end value is now $314.473 which overshoots from its defined value of $300.
A better approach to using the Arithmetic Mean would be to use Compound Annual Growth Rate:
Plugging in the numbers from above, we get:
Now, let’s go back to the chart and see if we can get the numbers to match:
[table id=3 /]
For clarity, here is the math behind the table:
Ok, the numbers are matching. Is there yet another tool we can use to validate this approach? What about the Compound Interest Formula? Let’s see…
First set n = 1 (compounding once per year)
This simple example does not allow us to see the “smoothing” effect that CAGR provides. To better illustrate, let’s track the S&P 500 from January 1990 to January 2012 (only looking at closing prices on the first trading day of January for each year). We calculate CACG using the same approach above, t= 22 years, Principal = $50,000.
The corresponding Compound Interest equation:
With this equation in hand, we are now able to replicate the performance of the S&P 500 over 22 years without the peaks and valleys. The curve would look like this: