LOS M requires us to:
compare the use of arithmetic and geometric means when analyzing investment returns.
a. The purpose of this heading is to find out, which of the two ways of calculating mean, i.e. arithmetic mean or geometric mean is more appropriate.
b. While making the decision regarding the performance of the past, calculating the geometric mean is considered more appropriate. Whereas, for the forward-looking contexts such as making investment decisions, the arithmetic mean is considered more appropriate.
c. Consider the following example:
Suppose an investment of $ 100 earned a return of 40% at the end of year 1 and a loss of 40% during year 2.
Thus, the investment was worth $ 140 ($ 100*1.40) at the end of year 1 and $ 84 (i.e. $ 140*0.60) at the end of year 2.
If we average the returns using the arithmetic mean, we get the average returns as:
This is not the true reflection of the situation because the investor has actually lost $ 16 by the end of year 2; thus, there actually was a negative return.
And, if we calculate the geometric mean, we get the average returns as:
Thus geometric mean provides a constant annualized change in wealth that would have actually occurred if wealth grew at a constant rate of return.
Thus, we can say that for evaluating the past performance, the geometric mean was a more appropriate measure.
d. Consider another example:
Suppose an investor has $ 100 to invest. He can forecast that there is an equal possibility of the price of the investment going 40% up and down in each of the next three year’s investment horizons.
Thus, the possibilities of different prices in the next three years would be:
The arithmetic and geometric mean of the returns, at different possible price levels, are:
Thus, to conclude we can say that geometric mean provides a constant annualized change in wealth that would have actually occurred, but while looking forward we do not know what the return would be; we only work on the expected values. Therefore, for future analysis, the geometric mean does not work well; it is the arithmetic mean that gives more accurate results.