# Summary Statistics (Finance)

While total return, return relative, and wealth index are useful measures of return for a given period of time, in investment analysis we also need statistics that summarize a series of total returns. The two most popular summary statistics are arithmetic mean and geometric mean.

Arithmetic Mean: The most popular summary statistic is the arithmetic mean. Hence the word mean refers to the arithmetic mean, unless otherwise specified. The arithmetic mean of series of total returns is defined as:
n
R = Σ Ri
i=1
N
where
R= arithmetic mean
Ri = ith value of the total return( i= 1, …n)
N= number of total returns

To illustrate suppose the total returns from stock a over a five year period are as follows:

Year 1 Total return (percentage)

1 19.0
2 14.0
3 22.0
4 – 12.0
5 5.0

The arithmetic mean for stock A is

R = 19 + 14 + 22 – 12 + 5 / 5 = 9.6 percent.

Geometric Mean: When you want to know the central tendency of a series of returns, the arithmetic mean is the appropriate measure. It represents the typical performance for a single period. However, when you want to know the average compound rate of growth that has actually occurred over multiple periods, the arithmetic mean is not appropriate. This point may be illustrated with a simple example. Consider a stock whose price is 100 at the end of year 0. The price declines to 80 at the end of year 1 and recovers to 100 at the end of year 2. Assuming that there is no dividend payment during the two year period, the annual returns and their arithmetic mean are as follows:

Return for year 1 = 80 – 100 / 100 = — 0.20 pr – 20 percent

Return for year 2 = 100 – 80 / 80 = 0.25 or 25 percent

Arithmetic mean return= — 20 + 25 / 2 = 2.5 percent.

Thus we find that though the return over the two year period is nil, the arithmetic mean works out to 2.5 percent. So, this measure of average return can be misleading. In a multi-period context, the geometric mean describes accurately the “true” average return. The geometric mean is defined as follows:

GM = [1 + R1) (1+ R2 ) …. (1+ Rn)] 1/n – 1

Where

GM = geometric mean return

Ri = total return for period i

(i = 1,….., 2)

N= number of time periods

Notice that the geometric mean is the nth root of product resulting from multiplying a series of return minus one.

To illustrate, consider the total return relative from stock A over a 5 year period:

Year Total return (%) Return relative

1 19 1.19
2 14 1.14
3 22 1.22
4 — 12 0.88
5 5 1.05

The geometric mean of the returns over the 5 year period is:

GM = [(1.19) (1.14) (1.22) (0.88) (1.05)] 1/5 – 1

= 1.089 – 1 = 0.089 or 8.9 percent

The geometric mean reflects the compound rate of growth over time. In the above illustration, stock A has generated a compound rate of return of 8.9 percent over a period of 5 years. This means that an investment of one rupee produces a cumulative ending wealth of Rs 1.532 [1(1.089)5]. Notice that the geometric mean is lower than the arithmetic mean [9.6 percent].

The geometric mean is always less than the arithmetic mean, except when all the return values being considered are equal. The difference between the geometric mean and the arithmetic mean depends on the variability of the distribution. The greater the geometric mean and the arithmetic mean is approximated by the following formula:

(1 + Geometric mean) 2 ~ (1 + Arithmetic mean) 2 – (Standard deviation)2

In the above formula, you will find a new term viz., standard deviation, which is the most popular measure of variability.