This variability can be measured. That is,
risk can be quantified! The most common measure
of risk is the standard deviation --
a numerical measure of the dispersion around the
expected value. Here is how the standard
deviation (designated by s ) is calculated for
our two dice probability distributions:
- compute the expected return;
- square the variance of each return from the
expected return;
- multiply this by the probability;
- sum these weighted values;
- calculate the square root of this sum (the
standard deviation).
This gives a numerical indication of how far
the returns are dispersed from the average.
There are various measures of risk: beta, standard
deviation, R-squared. (See a comparison
of these three measures of risk.) cFor an excellent
primer on risk in securities markets, see Vanguard's
Investing
Primer. |
| Return |
Square
of variance |
Probability |
Weighted
square |
$2 |
(7 -
2)2 = 25 |
1/36
= .028 |
0.694 |
$3 |
(7 -
3)2 = 16 |
2/36
= .056 |
0.889 |
$4 |
(7 -
4)2 = 9 |
3/36
= .083 |
0.750 |
$5 |
(7 -
5)2 = 4 |
4/36
= .111 |
0.444 |
$6 |
(7 -
6)2 = 1 |
5/36
= .139 |
0.139 |
$7 |
(7 -
7)2 = 0 |
6/36
= .167 |
0.000 |
$8 |
(7 -
8)2 = 1 |
5/36
= .139 |
0.139 |
$9 |
(7 -
9)2 = 4 |
4/36
= .111 |
0.444 |
$10 |
(7 -
10)2 = 9 |
3/36
= .083 |
0.750 |
$11 |
(7 -
11)2 = 16 |
2/36
= .056 |
0.889 |
$12 |
(7 -
12)2 = 25 |
1/36
= .028 |
0.694 |
| Statistics: |
|
| Sum
of weighted squares |
5.833 |
| Standard
deviation
(square root of sum) |
2.415 |
|
2.2.2 Expected Return