# Arithmetic Mean versus Geometric Mean in the world of investments

In the world of investments, the focus is mostly on knowing the central tendency of a series of returns. Hence arithmetic mean is commonly employed. Why should, you may ask, the arithmetic mean be preferred to the geometric mean?

To answer this question, let us consider an example. Suppose the equity share of Modern Pharma has an expected return of 15 percent in each year with a standard deviation of 20 percent. Assume that there are two equally possible outcomes each year, + 45 percent and – 15 percent (that is, the mean plus or minus one standard deviation). The arithmetic mean of these returns is 15 percent, (45 – 15) / 2, whereas the geometric mean of these returns is 11.0 percent, [(1.45) (0.85)]1/2 — 1.

An investment of one rupee in the equity share of Modern Pharma would grow over a two year as follows:
Probability

1.45 2.10 0.25

0.85 1.23 0.50

0.72 0.25

Notice that the median (middle outcome) and mode (most common outcome) are given by the geometric (11.0 percent), which over a two year period compounds to 23 percent (1.112 = 1.23). The expected value of all possible outcomes, however, is equal to:

(0.25 x 2.10) + (0.50 x 1.23) + (0.25 x 0.72) = 1.32

Now 1.32 is equal to (1.15)2. This means that the expected value of the terminal wealth is obtained by compounding up the arithmetic discount rate.

Put differently, the arithmetic mean is the appropriate mean because an investment that has uncertain returns will have a higher expected terminal value than an investment that earns its compound or geometric mean with certainty every year. In the above example, compounding at the rate of 11 percent for two years produces a terminal value of Rs 1.23, for an investment of Re 1.00. But holding the uncertain investment which yields high returns (45 percent per year for two years in a row) or middling returns (45 percent in year 1 followed by – 15 in year 2 or vice versa) or low returns ( — 15 percent per year for two years in a row), yields a higher expected terminal value, Re 1.32. This happens because the gains from higher than expected returns are greater than the losses from lower than expected returns.

Therefore in the investment markets, where returns are described by a probability distribution, the arithmetic mean is measure that accounts for uncertainty and is the appropriate ones for estimating discount rates and the cost of capital.

Real Returns:

The returns discussed so far are nominal returns, or money returns. To convert nominal returns into real returns, an adjustment has to be made for the factor of inflation:

Real return 1 + Nominal return / 1+ Inflation rate – 1

Example: The total return for the equity stock for a year during was 18.5 percent. The rate of inflation during that year was 5.5 percent. Thus the real (inflation-adjusted) total return was:

1.185 / 1.055 – 1 = 0.123 or 12.3 percent.