2.9 The Geometric Mean (OPTIONAL)
2.9 THE GEOMETRIC MEAN (OPTIONAL)
In Section 2.2, we defined the mean to be the average of a set of population or sample measurements. This mean is sometimes referred to as the arithmetic mean. While very useful, the arithmetic mean is not a good measure of the rate of change exhibited by a variable over time. To see this,
consider the rate at which the value of an investment changes—its rate of return. Suppose that an
initial investment of $10,000 increases in value to $20,000 at the end of one year and then decreases
in value to its original $10,000 after two years. The rate of return for the first year, R1, is
R1 5 a
20,000 2 10,000
b 3 100% 5 100%,
10,000
and the rate of return for the second year, R2, is
R2 5 a
10,000 2 20,000
b 3 100% 5 250%.
20,000
Although the value of the investment at the beginning and end of the two-year period is the same,
the arithmetic mean of the yearly rates of return is (R1 1 R2)y2 5 (100% 1 (250%))y2 5 25%.
This arithmetic mean does not communicate the fact that the value of the investment is
unchanged at the end of the two years.
To remedy this situation, we define the geometric mean of the returns to be the constant
return, Rg, that yields the same wealth at the end of the investment period as do the actual
returns. In our example, this says that if we express Rg, R1, and R2 as decimal fractions (here
R1 5 1 and R2 5 20.5), then
(1 1 Rg ) 2 3 10,000 5 (1 1 R1 )(1 1 R2 ) 3 10,000
Rg 5 2(1 1 R1 )(1 1 R2 ) 2 1
5 2(1 1 1)(1 1 (20.5)) 2 1
5 21 2 1 5 0.
or
Therefore, the geometric mean Rg expresses the fact that the value of the investment is
unchanged after two years.
In general, if R1, R2, . . . , Rn are returns (expressed in decimal form) over n time periods,
then we have the following:
The geometric mean of the returns R1, R2, . . . , Rn is
n
Rg 5 2(1 1 R1 )(1 1 R2 ) . . . (1 1 Rn ) 2 1,
and the ending value of an initial investment I experiencing returns R1, R2, . . . , Rn is I(1 1 Rg)n.
As another example, suppose that in year 3 our investment’s value increases to $25,000, which
says that the rate of return for year 3 (expressed as a percentage) is
R3 5 a
25,000 2 10,000
b 3 100%
10,000
5 150%.
Since R1 5 1, R2 5 20.5, and R3 5 1.5 (expressed as decimals), the geometric mean return
at the end of year 3 is
3
Rg 5 2
(1 1 1)(1 1 (20.5))(1 1 1.5) 2 1
5 1.3572 2 1
5 0.3572,
and the value of the investment after three years is
$10,000(1 1 0.3572) 3 5 $25,000.
1
2
Chapter 2 Descriptive Statistics
Exercises for Section 2.9
CONCEPTS
2.76 Explain the interpretation of the geometric mean return
for an investment.
2.77 If you know the initial value of an investment and its
geometric mean return over a period of years, can you
compute the ending value of the investment? If so,
how?
METHODS AND APPLICATIONS
Year
S&P/TSX Index
2005
2006
2007
2008
2009
2010
9,142.98
11,441.58
12,923.66
13,926.76
9,234.11
11,866.90
Source: http://tmx.quotemedia.com/pricehistory.php.
2.78 Suppose that a company’s sales were $5,000,000 three
years ago. Since that time sales have grown at annual
rates of 10 percent, 210 percent, and 25 percent.
a. Find the geometric mean growth rate of sales over
this three-year period.
b. Find the ending value of sales after this three-year
period.
2.79 Suppose that a company’s sales were $1,000,000 four
years ago and are $4,000,000 at the end of the four
years. Find the geometric mean growth rate of sales.
2.80 The S&P/TSX stock index is a commonly used measure
of stock market performance in Canada. In the following
table, we give the value of the S&P/TSX index on the
first day of market trading for each year from 2005 to
2010.
a. Show that the percentage changes (rates of return)
for the S&P/TSX index for the years from 2005 to
2006 and from 2006 to 2007 are, respectively, 25.1
percent and 13.0 percent (that is, 0.251 and 0.130
expressed as decimal fractions).
b. Find the rates of return for the S&P/TSX index for
each of the years from 2007 to 2008; from 2008 to
2009; from 2009 to 2010.
c. Calculate the geometric mean return for the
S&P/TSX index over the period from 2005 to 2010.
d. Suppose that an investment of $1,000,000 is made in
2005 and that the portfolio performs with returns
equal to those of the S&P/TSX index. What is the
investment portfolio worth in 2010?