# Predicting Portfolio Growth: Average vs. Compound Returns

by on Aug 5, 2013

When evaluating investments, you’ll often run across “average” return values. As useful as these numbers seem, they can be misleading.

Why do I say this? Because they’re almost certainly arithmetic averages. Thus, they ignore the effects of volatility and overstate the returns that you can actually expect.

Not sure what I mean? Then read on… I’ll explain with a few simple examples.

## Introducing CAGR

Before we go any further, I want to introduce you to a concept known as the compound annual growth rate, or CAGR. The CAGR is essentially a “smoothed” rate of return that dampens the effect of volatility on portfolio growth.

In other words, CAGR provides a much more accurate prediction of expected returns vs. the average annual return. Technically speaking, CAGR is an average (mean) rate of return. But it’s the geometric mean vs. the arithmetic mean.

Let’s run some numbers to see why this matters.

## Constant returns over time

Let’s assume that you have a portfolio with a starting value of \$100k and that, over a period of three years, it returns 10%, 10%, and 10%. Here’s how your portfolio value would look over time:

• Start: \$100,000
• Year 1: \$110,000
• Year 2: \$121,000
• Year 3: \$133,100

It should be obvious from the above that your average annual return is 10%/year (i.e., [10 + 10 + 10]/3 = 10). To estimate the CAGR, divide the ending value by the starting value and then take the nth root, where n is the # of years, as follows:

`(\$133,100/\$100,000)^(1/3) = 1.10`

Note: Taking the nth root is the same as raising it to the 1/n power.

So the CAGR is likewise 10%. In fact, the CAGR will always be less than or equal to the average annual return. When the returns are identical over time, the two values are equal. When the returns vary, the CAGR is less than the average annual return.

## Variable returns over time

Now let’s assume that your portfolio instead returned 30%, 0%, and 0%. In this case, the average annual return is still 10%/year. But what about CAGR?

Well, in this case, your portfolio would look like this over time:

• Start: \$100,000
• Year 1: \$130,000
• Year 2: \$130,000
• Year 3: \$130,000

As you can see, the final value is a bit below that in the equal returns scenario, so it should come as no surprise that the CAGR will be a bit lower. Here’s the math:

`(\$130,000/\$100,000)^(1/3) = 1.0914`

So the CAGR is 9.14% — in other words, your portfolio has taken on the same value as if it had grown 9.14%/year.

And… One more example.

Let’s say that your portfolio grew 40% in year 1, was flat in year 2, and dropped 10% in year 3. Once again, we have an average annual return of 10%, but…

Here’s what your portfolio would look like over time:

• Start: \$100,000
• Year 1: \$140,000
• Year 2: \$140,000
• Year 3: \$126,000

In this case, your CAGR would be ca. 8.01%, as follows:

`(\$126,000/\$100,000)^(1/3) = 1.0801`

That’s nearly 2% below the average annual return. Volatility really matters.

## Real world data

Now, let’s consider some real world data. Looking at stock market performance from 1926-2012, the average annual return was 11.88%. Care to guess what the CAGR was over the same time period? Here’s the answer: 9.87%

That’s a shade over 2% less than the average annual return. Think about that for a minute. If you use the average annual return for planning purposes, you’ll wildly overstate your returns. Remember, small differences can have a huge impact.

Looking back over the time period in question, if you applied the average annual return to \$1k invested in 1926, you’d expect it to have grown to over \$17M by the end of 2012. In reality, that \$1k would have grown to ca. \$3.6M — nearly 80% less!

The bottom line: when making financial projections, you should always use the CAGR of an investment (as opposed to the average annual return) to predict how its value will change over time. If you don’t, you’ll likely be disappointed.

Comments on this entry are closed.

Previous post:

Next post: