ARITHMETIC vs. GEOMETRIC AVERAGES
An arithmetic average is calculated as
n
 Xt
Arithmetic Average  t 1
n
Xt is the rate of return that you earned in year t
Suppose you invested $1,000 two years ago and your portfolio had the following values:
Year 0 $1,000
Year 1 $2,000
Year 2 $1,000
Your rate of return in the first year was +100% (from $1,000 to $2,000). In the second
year, you earned a rate of return of -50% (from $2,000 to $1,000). An arithmetic
average just sums these values and divides by the number of periods: (+100%-50%)/2
= 50%/2 = 25%
n
 Xt
1.0 - 0.5 0.5
Arithmetic Average  t 1 

 0.25 or 25%
n
2
2
It is clear that you did NOT average a 25% rate of return each year since you ended up
with the same dollar amount after two years that you started with.
A geometric average is calculated as
n
1  Geometric Average  n  (1  X t )
t 1
The  sign means to multiply together similar to how  means to add together. Thus,
the geometric average is
n
1  Geometric Average  n  (1  X t )
t 1
 2 (1  1.0) * (1  0.5)
 2 2 * 0.5  2 1.0  1.0 or 0%
As you can see, this gives you the ACTUAL average rate of return of 0% for the twoyear period.
In finance, the only meaningful average with respect to rates of return over time is the
geometric average. The geometric average can be calculated using Present Value and
Future Value calculations. For instance, in our example we could have used a future
value calculation such as
PV * (1  i) n  FV
or
$1,000 * (1  i) 2  $1,000
(1  i) 2  1.00
1  i  1.00
i  0.00
or 0%
In general, a growth rate or a rate of return can be calculated by the following
it
CFt
-1
CF0
This can be applied to any series of numbers over any period of time. For example,
following are the Consumer Price Indices over the past several years:
Year
CPI
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
109.6
113.6
118.3
124.0
130.7
136.2
140.3
144.5
148.2
152.4
156.9
160.5
163.0
166.6
172.2
177.1
179.9
184.0
188.9
195.3
201.6
207.3
215.3
214.5
218.0
Suppose we wanted to calculate the average rate of inflation between, for example,
1995 and 2010. We would use the previous equation in this manner:
it
CFt
-1
CF0
 (2010 -1995)
218.0
-1
152.4
 15 1.430446 - 1
 1.024153 - 1
 0.024153
or 2.4153%
To do this on your calculator, simply remember that
n
X
1
 Xn
Similarly, we can convert dollar amounts into constant dollar terms by using indexes
created from the CPI. This is important when estimating relationships using regression
analysis (time series) in order to abstract from inflationary consequences, particularly
when we’re trying to separate the fixed costs from the variable costs (as we’ll do later).
The CPI series is currently based in terms of 1982-84 dollars. That is, the 1983 CPI
average is approximately 100 and all other CPI are based upon 1983 as the starting
point. Suppose that we wanted to put a cash flow in the year 2000 into 1995 dollar
terms. In order to do this, we would first have to construct a new CPI where 1995 is the
base year by dividing each year’s 1982-84 CPI by the 1995 CPI. (Actually, we’ve
already done this for the year 2010.)
1995 Base Year CPI 
CPIt
CPI1995
Thus, the 1995 Base Year CPI for the year 2010 would be
218.0
 1.430446
152.4
Then, year 2010 cash flows could be converted into 1995 constant dollars by dividing
the year 2010 cash flows by the factor 1.430446 or, conversely, we could convert year
1995 cash flows into 2010 constant dollars by multiplying the 1995 cash flow by the
1.430446 factor.
Ex Ante Projections
If you are using historical averages for projecting into the future, the arithmetic
mean is the appropriate measure. To see why, reconsider our first example where we
earned a +100% return in the first year and –50% return in the second year. Based on
our experience, there was a 50% chance of earning a +100% return and a 50% chance
of earning a –50% return. If we look at this as a probability sequence, we get an
expected rate of return of 25%.
Possible
Return
Probability
+100%
-50%
*
*
0.5
0.5
=
=
+50%
-25%
+25%
This is also true in the 2-period case:
Time 0
Time 1
Time 2
4,000
2-period
Return
+300%
Joint
Probability
*
.25 =
75%
50%
2,000
50%
1,000
0%
*
.25 =
0%
1,000
0%
*
.25 =
0%
250
-75%
*
.25 = -18.75%
50%
1,000
50%
50%
500
50%
Expected 2-period Return =
56.25%
The 2-period return of 56.25% is equivalent to 25% per year. Thus, the arithmetic mean
represents all possible combinations of events and their associated probabilities of
occurrence and should be used for forecasting expected returns.