This blog is all about math.  No, wait, don’t leave.  It’s really about making sure you understand how to properly calculate the returns you get from your investments.  Because if you’re not getting the return you expect, you might be putting your future goals at risk.

Most everybody is probably familiar with the concept of arithmetic average.  For example, if there are five students in a class, and their final exam test scores were 100%, 90%, 85%, 80%, and 70%, you can calculate the average test score by adding up the five scores and dividing by the number of scores (five in this case).  The result would be 85%.

Suppose we were to use the arithmetic average to determine the returns from a mutual fund investment over time.  Our example fund happens to be highly volatile.  Assume that at the end of year one its price doubles in value, but a year later it drops by half (which means the fund price is back to where it started).  By the end of year three it has doubled again, and by the time year four comes to a close its price has once again dropped in half.  If you had invested a thousand dollars in that fund, after four years you’d have found yourself still with only a thousand dollars.  What was the average return? Intuitively it’s 0%.  But let’s use some mathematical rigor to calculate it.  The return in year one was 100%, in year two it was -50%, it was 100% in year three, and again -50% in year four.  The arithmetic average tells you that your average annual return was 25%.  Oops!  What went wrong?

The problem is that arithmetic average only works when the numbers you are averaging are independent of each other.  In the case of an investment, each year’s performance will have an impact on the subsequent year’s performance.  If you’ve doubled your money in one year, you have a lot more to lose the second year, and vice versa.  Instead of using the arithmetic return, we utilize the geometric return instead to address this dependency.  For those of you with a penchant for math, here’s the formula: ⁿ√(A₁ × A₂ × … × A_n).

Let’s apply it to our example.  We add 100% to each year’s return to avoid having negative numbers.  You can think of it this way: the price at the end of year one is 200% of the price at the start, the price at the end of year two is 50% of the price at the beginning of the year, etc.  So our returns can be expressed as 200%, 50%, 200%, and 50% over the four years.  If you use your handheld calculator to multiply those four numbers, then take the 4^th root, you will find the answer to be 100%.  That means that on average each year’s price was 100% of the previous year’s price.  Subtracting the 100% we added in the beginning, the formula tells us the average return was 0%, which is correct.

Geometric returns are also known as time-weighted returns when applied to investments, because they measure the compounded rate of growth assuming the investment was made at a single time.  This is the way mutual funds report performance.  Since the fund manager has no control over when investors buy and sell shares in the fund, the time-weighted return most accurately reflects how well the fund performed ignoring the effects of inflows and outflows.

Unfortunately time-weighted returns do not really measure how well you yourself are doing with your investments.  That’s because your actual return does depend on the timing of your purchases and your sales.  For example, if you buy shares of a mutual fund at a time when prices are high and/or sell them when prices are low, your actual return could be lower than the fund’s reported returns for the period.  If you are working with an investment advisor you’ll probably have access to reports that provide a more accurate view.  For the record the internal rate of return (IRR) is the calculation that will tell you exactly how well you did with your investments.

Time-weighted returns are the clearest way to compare the performance of different mutual funds over similar time periods. But always remember that historical fund performance is by no means the best indicator of future fund performance.