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2.2 Calculating returns
Our first question was: what is the mean expected total return for
next year? We calculate this by taking each of the possible returns
and weighting it by its relative probability. As our table is so simple
and symmetrical, it is not difficult to see that the weighted mean
return is 7% per annum.
Our second question was: what is the degree of risk or uncertainty
in this mean figure? In other words, how widely dispersed are the
possible outcomes around the mean expected outcome? The most
commonly used statistical measure of dispersion is the standard
deviation. We need to be aware of some limitations in its use, but
first here is how the calculation would look in the case of returns
on Company X's shares. We shall use the notation
E (R)
to denote
the mean expected return, which is
7%
per annum in this case.
Table 2 Calculation of standard deviation of actual total returns on
a oneyear investment in shares of Company X in each of the last
50 years
A
n
n
u
al
re
tu
r
n
R
I
Dis
pe
rsi
on
E
(R
) −
R
i
Sq
ua
re
of
dis
pe
rsi
on
[
E(
R) −
R
I
]
2
Pr
ob
abi
lity
P
I
P
i
[
E(R
) −
R
i
]
2
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Investment risk
4
+3
9
0.2
1.8
6
+1
1
0.3
0.3
8
−1
1
0.3
0.3
10
−3
9
0.2
1.8
Sum of the squares of the probabilityweighted dispersions (=
variance)
V
= 4.2
Square root of variance
(
V)
= standard deviation
The standard deviation is normally expressed in the same units as
the expected return (in this case, per cent per annum) and is
intuitively easier to understand than the variance, especially with a
normal distribution where the possible outcomes are symmetrically
dispersed around the mean.
The particular usefulness of the standard deviation of normally
distributed data is the way it divides up the data so that:
68.3% of the data points lie within one standard
deviation on either side of the mean;
95.4% of the data points lie within two standard
deviations on either side of the mean.
In the case of an investment in shares of Company X, the
calculation in
Table 2
tells us that there is:
a 68.3% probability that the return will lie between
4.95% (i.e. 7 – 2.05) and 9.05% (i.e. 7 + 2.05), and
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a 95.4% probability that it will lie between 2.9% (i.e. 7
– 2 x 2.05) and 11.1% (i.e. 7 + 2 x 2.05).
The principal limitation on the use of standard deviation as a
measure of dispersion in investment returns is that in the real
world actual returns are not as neatly or symmetrically dispersed
as they were in our Company X example. However, the shape of
many longrun statistical series for investment returns is in fact
quite close to the familiar bellcurve of the normal distribution. So
for most practical purposes the standard deviation is as useful a
measure as we are likely to find.
 Winter '14