Rate of Return - the Right Way to Determine

Also known as the average annualized return or geometric mean of return, a potentially confusing term in the mathematics of finance.

Compared to simple arithmetic mean, geometric mean takes into account the time period and the effect of compounding, and is a more accurate representation of investment return.

Let's take an example something we can all relate to - unit trust fund. A hypothetical fund - Fund LCF has the rate of return as below:

Year 1: 15%    Year 2: -15%

At a glance, the "too-casual observer" will conclude that over the 2 years period, the investment return is 0% of your capital. This is arithmetic mean of return calculation. You break even.

Not true actually. Say, if you invested RM 10,000, you investment will be at RM 11,500 by end of Year 1. By end of Year 2, your money would have reduced to RM 9,775 (0.85 x 11,500). So you actually lose RM 225, or 2.25% of your initial capital.

The more accurate method to know the rate of return within the 2 years is by using geometric mean. It's really no rocket science. The calculation is as such:

[ (1+ r1) x (1 + r2) ]^(1/n) - 1, whereby r1 = 0.15 and r2 = -0.15, n = number of years, 2

...and you will get, -1.13%. This is the annual averaged loss over the time frame of 2 years. "WTF?" you asked. Unfortunately, this is the number that represents the reality in this case.

Let's compute this again. By end of Year 1, your capital would have reduced to : (1-0.01314) x RM 10,000 = RM 9,886.86

By end of Year 2, your ending balance will be:  (1-0.01314) x RM 9886.86 = RM 9,775.00

To quote Investopedia,

...when it comes to annual investment returns, the numbers are not independent of each other. If you lose a ton of money one year, you have that much less capital to generate returns during the following years, and vice versa. Because of this reality, we need to calculate the geometric average of your investment returns in order to get an accurate measurement of what your actual average annual return over the five-year period is.

It may seem confusing as to why geometric average returns are more accurate than arithmetic average returns, but look at it this way: if you lose 100% of your capital in one year, you don't have any hope of making a return on it during the next year. In other words, investment returns are not independent of each other, so they require a geometric average to represent their mean.

The problem with calculating and reporting an arithmetic mean is that it always overstates the correct (geometric) annualized return. Interestingly, how much it overstates the actual geometric mean is highly related to the volatility of the returns of the asset. Hence the arithmetic average return is usually the one posted in ads for mutual funds and other investments. The only time when the arithmetic and geometric average will be the same is when the individual returns being averaged are the same for each period being analysed.

Now after knowing this, ask your financial planner or unit trust agent how the historical returns of the funds are computed when they show you their data. Don't let them paint a too rosy picture of the promised return (no one can guarantee your investment return anyway!). You might be misled by total returns published over a time period, which is not a true performance indicator especially if you are looking for capital preservation and consistency in investing your hard earned money. Look beyond the publicized investment returns being touted by the advertisers or investment managers of the products being solicited. Knowing the 'average' return of an investment is not very useful unless you are able to differentiate between the arithmetic average return and the geometric average return.