# Why Your Rate of Return is always Different

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.