Why average annual return of investment is geometric mean and not arithmetic mean
When you invest in anything other than the savings accounts and fixed deposits, the annual returns usually changes from positive to negative from year to year. To measure the success of any investment strategy over the years or some other time periods chosen, one will need to measure the average annual return (or other time periods), and see of it is a significant positive number to decide whether the strategy is working well as expected. However, due to how the annual return is calculate in the first place, one cannot simply find out what is the average annual return from end of year 1 to year n by just adding up the annual return for each year and dividing by n, where n is usually an integer.
How annual return from year to year is measured?
Assume now is end of year 1, and you want to measure the annual return of stock portfolio from its beginning value one year ago to now. The worth of portfolio one year ago is $20 000 and it worth $26 000 now.
Rate of return for year 1 = 26 000 – 20 000 X 100% / 20 000
= 30%
Assume that now is end of year 2, and you want to measure the annual return of stock portfolio from its beginning value one year ago to now. The worth of portfolio one year ago is $26 000 and it worth $30 000 now.
Rate of return for year 2 = 30 000 – 26 000 X 100% / 26 000
= 15.4% (3 significant figures)
Assume that now is end of year 3, and you want to measure the annual return of stock portfolio from its beginning value one year ago to now. The worth of portfolio one year ago is $30 000 and it worth $21 000 now.
Rate of return for year 3 = 21 000 – 30 000 X 100% / 30 000
= -30%
Assume that now is end of year 4, and you want to measure the annual return of stock portfolio from its beginning value one year ago to now. The worth of portfolio one year ago is $21 000 and it worth $40 000 now.
Rate of return for year 4 = 40 000 – 21 000 X 100% / 21 000
= 90.5% (3 significant figures)
Take note that the value of portfolio one year ago is used for calculating rate of return now and not all the way back by using year 0. The average return for these four years is NOT 30% + 15.4% + (-30%) + 90.5% and then divided by four.
The correct way to accurately measure the average return over these four years is by using the geometric mean.
Why using geometric mean to calculate the average return over these four years is correct and not otherwise as mentioned above?
Detailed mathematical explanation will be beyond the scope of my intelligence this blog post. The simple explanation is that investment returns are not independent of each other. If you lose 100% of your capital in one year, you will have not capital to generate return for the next year or the annual return of each year depends on amount of capital in preceding year. As a result, the average return cannot be 0% if you gained 100% at the end of year 1 and lost everything by the end of year 2, the average return will be -100% because you will have lost everything.
Now is the 21st century and there is Internet, Microsoft Excel and a program similar to Microsoft Excel from OpenOffice such that you don’t need to manually calculate the geometric mean yourself by using the multiplication tables or pressing the calculator keys. In fact, there a few websites and a statistical function in Excel from both Microsoft Office and OpenOffice that can return the geometric mean by simply entering the annual return for each year.
Anyway, the equation for calculating geometric mean may looks bombastic to non-science, engineering and maths students.
Where n is the number of time periods, usually in years and y is the percentage return for each year (or return for that year).
1. Microsoft Excel and OpenOffice
Both Microsoft Office and OpenOffice called it GEOMEAN function in their respective spreadsheet program. To find out the geometric mean is incredibly easy by entering all the annual returns in a row or column of cells, then enter =GEOMEAN(number1, number2 . . . .) in a cell and select the row or column of annual returns.
As you can see from the equation above, if you got negative returns, there will be errors as square root of a negative number is undefined. As a result, you need to add 100 to each of the annual return, then the return value will need to subtract 100 to get the geometric mean.
2. Various Online Websites
There are quite a number of websites providing web applications for calculating geometric mean whereby one just need to enter the annual return for each year. The following are two of them.
a. Horton’s Geometric Mean Calculator
http://www.graftacs.com/geomean.php3
Same as spreadsheet programs above, need to add 100 to each year percentage return, then the geometric mean need to subtract 100 from the return value.
b. Hugh’s Mortgage and Financial Calculators
http://www.hughchou.org/calc/areturn.php
For this particular website, it is programmed in such a way that you can enter negative percentage return if there is any and still get the correct geometric mean without the need to add 100. As you can see, if you enter -100% for any given year, the geometric mean or average annual return will be -100%.
Related posts:
