Page 69 - Mathematics of Business and Finance

Chapter 1 | Review of Basic Arithmetic 49

th
The geometric mean is defined as the n root of the product of n number of items (terms).
In other words, to find the geometric mean for a set of n number of items, multiply the numbers
1
th
together and take the n root ( power ).
n
If there are n terms (x , x , x , x , x ... x ), then the geometric mean, G, is:
5
1
3
2
4
n
Formula 1.5(c) Geometric Mean
1
n
G = n x . x . x . x . x . ... . x or G = ex ∙ x ∙ x ∙ x ∙ x ∙ ... ∙ x o
1 2 3 4 5 n 1 2 3 4 5 n
Note: Geometric mean, G, of a set of values is the number G that satisfies

×
×
×
x ∙ x ∙ x ∙ x ∙ x ∙ ... ∙ x = G × G × G × G ... = G G=. x x x ........ x n n G G G G ..... = n n
.

3
1
2
1 4 2 3 5
n times
For most returns in finance including yields on bonds, stock returns, and other market risk investments,
the geometric average provides a more accurate measurement of the true return by considering
year-over-year compounding.
Suppose you invested in the financial markets for five years and if your portfolio returns each year
were 25%, 15%, –45%, 5%, and 10%:
(a) Using the arithmetic average, the average annual return is +2.00%, calculated as follows:
25% + 15% – 45% + 5% +10% = +2.00%
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(b) Using the geometric average, the average annual return is –1.80%, calculated as follows:
1
5
Geometric Average = [(1 + 0.25)(1 + 0.15)(1 – 0.45)(1 + 0.05)(1 + 0.10)] = 0.981997778
Average Annual Return = 1 – 0.981997778 = –0.018002221 = –1.80%

Therefore, the average annual return is –1.80%.

In this case, since the investment return is compounding, the annual return of –1.80% over the 5-year
period, as calculated by the geometric average, is the most appropriate measure.
Note: The following sequence of annual returns over a 5-year period

Year 1 Year 2 Year 3 Year 4 Year 5

+25% +15% –45% +5% +10%

is the same as a decrease by 1.8% each year

Year 1 Year 2 Year 3 Year 4 Year 5

–1.80% –1.80% –1.80% –1.80% –1.80%

That is,

5
(1.25)(1.15)(0.55)(1.05)(1.10) is equal to (1 – 0.018002221) = 0.913171876