Most of us have seen the marketing brochures distributed by mutual fund companies and investment products salespeople as an inducement to place our money in a fund. At the very least, we’ve encountered historical returns posted within a 401k plan that some people use to help select where they place their money.

But have you ever noticed that when you look at the historical return data, your rate of return seems to magically be different? What gives? Some might explain this as entry and exit variances that alter the yield vs. the calendar year assumptions put in place on the historical data and to some degree this is true. But the real answer to this lies a little deeper and has to do with a little slight of hand, and some simple mathematics.

## Calculating Compound Annual Growth Rate, Timing Matters

Compound Annual Growth Rate (or CAGR) is another name for the **geometric mean** of a data set. We’ve noted before that some people on or around Wall St. seem to have a difficult time differentiating between the **arithmetic mean** (what we generally learn about in grade school) and the geometric mean (or maybe their confusion is intentional).

But further still we may run into problems when people ignore the important fact that geometric means are affected by timing. Keep in mind that the geometric mean is used to calculated rate of return over time, so it makes perfect sense that actual moment the investment takes place is of crucial consideration for this calculation.

**Two Methods of CAGR**

There are two methods to hand calculate compound annual growth rate, one is used when there is a lump sum amount we want to calculate the growth of over a period of time. The other is used when we have systematic cash flows and we wish to calculate the return we’ve received at some future date on those cash flows. The former calculation is ** much** simpler than the latter. This divergence is the answer to our question.

## How Marketing Brochures Calculate Return

All marketing brochures that I’m aware of, and all 401k stated returns calculate CAGR as if the investment was made as a single lump sum. Admittedly, this is the far easier way to discuss returns—one could calculate this return with nothing more complex than a desktop calculator.

But, I’m aware of no one whose entire investment process for their life commences with one single deposit into a brokerage account and never again receives new money—especially among the 401k crowd. Instead the investments take place over time and this fact is the key driver behind the difference in stated vs. experienced return.

## An Example for Clarity

If I take a $100,000 deposit and it grows at 8% per year that deposit should be worth roughly $216,000 ten years from now. If instead I only have $10,000 per year to invest, then I’ll need to achieve an annual return of about 13.64% in order to achieve the same return I got with the lump sum investment.

Now, mutual fund company employees and investment salespeople, wouldn’t it be awesome if you were allowed to take this data and state that the 10-year yield on a fund was 13.64%? Totally. But you can’t, because it’s not true.

Instead we have to calculate this a bit differently and come to much more sobering conclusion. If I instead calculate CAGR on the $10,000 annual investment at an assumed 8% per annum I end up with roughly $156,000, and if I go back and plug in an initial $100,000 lump sum and calculate CAGR on that amount growing to $156,000 my return is a not nearly as impressive 4.58% and now we see why they choose to state rate of return the way they do.

You see, when it comes to assets that vary in terms of value (like stocks) CAGR is a linear interpolation, and we can use that data to estimate values for other scenarios, and event to predict investment performances. But we have to keep in mind that periodic investment at a given rate of return is not the same thing as lump sum investing at a given rate of return (i.e. periodic investing with a return of 8% per annum is not equal to lump sum investing with an 8% return per annum).

And with assets like stocks it gets even more complicated because the price of the asset can move up and down.

## Example #2 Stock Fluctuations

Let’s say the 10 year yield on an asset is 8% when we look at a lump sum investment, but this was caused by a large upward swing in asset price in year 2 followed by virtually no movement at all in the asset price in years 3-10.

Our lump sum investment benefits by being all in before the swing, while our periodic investment gets dragged down by having us buy in mostly after the swing.

I bring this up merely to point out that merely looking at yield is a dangerous move if you wish to evaluate fund performance and make decisions about where you want to place your money—and also to note that there are some well regarded mutual fund rating services that completely ignore this sort of thing when they post top fund performers for a given period.

## Your Mileage will Vary

We all know that past performance does not guaranteed future results. But another disclosure that might help is that your approach to investing in a given security will probably vary dramatically from the assumptions use to calculated stated returns.

As much as we all might hate doing a little extra work. It’s crucial to spend a little time cutting through the fluff and figuring out if your investments are performing to your expectations, and what you might want to alter to take a more prudent approach building wealth.

This is another great explanation: you take an apparently obvious measure and prove it to be at best misused and misleading, and at worst meaningless!

Is there a single metric or group or metrics YOU would recommend for analyzing and comparing investments? I’d guess it would depend on the goals of the investor, but perhaps you can suggest some options for various goals or classes of assets.

Keep up the great work. I’m still laughing at your in-jokes and comments in the podcasts!

Hi Jeff,

In terms of a general metrics I would use, the most basic would be rate of return (or internal rate of return CAGR) for the systematic investment, which is more closely matched to the way in which people tend to invest. So it’s the example from the post that is the much more complicated calculation, and the one that looks much less attractive for the hypothetical investment results, but also places a much more realistic on assumed results.

On a lumps sum investment, I would suggest it’s prudent to look at a few what-if scenarios that knock the rate of return down a bit to determine where that leaves you and consider if that’s a result you’re willing to live with. You could also look at it from a probability density function point of view, which simply means looking at a lump sum investment at percentage likelihoods of various returns. Here’s where they fall if you crunch the numbers in Monte Carlo by assuming a normal distribution.

Thanks for the response. I re-read the January 7, 2012 blog post you referred to, and while I still do not quite “get” the math, I think I have a better feel for it.

I read the wikipedia page on Monet Carlo Method to see if it would help me better understand what you do. In the MC simulation, what exactly are you randomizing? Is it the order of yearly (historical) rates of return? I understand from a previous blog post about the timing risk – you never know when you’ll have a good year or a bad year, just that they do happen – so it seems to me randomizing the yearly “return” however you measure it would make sense. Am I far off here?

Nope not far off at all. The Monte Carlo simulation is merely taking the distribution we know about the market given it’s mean and standard deviation and using several randomized data points to predict investment scenario results to certain percentages of likelihood.

What we get are numerous randomized trials over investment periods with varying CAGR’s for the entire investing period. We can than take those end results to infer likelihood of achieving various investment returns. And as we see here, just because the average over a certain span of time is X, it doesn’t mean that our actual results if we are investing periodically will match the calculated average.