Standard Deviation
Example 1: Without Probabilities
The return of investments in two companies, Kaya Pa and Suko Na, are given. Calculate the
standard deviation of these companies and compare their risks.
𝑆𝐷 = √
∑(𝑥 − 𝑥̅ )2
𝑛−1
SUKO NA COMPANY
MONTH
RETURN (%)
KAYA PA COMPANY
MONTH
RETURN (%)
AUG
SEP
6
7
AUG
SEPT
4
6
OCT
NOV
8
9
OCT
NOV
8
10
DEC
10
DEC
12
Solution:
Where:
= return observed in a period
= the arithmetic mean of the returns observed
= number of observations in the data set
Kaya Pa Company
RETURN
(𝒙)
6
7
8
9
10
=𝟖
𝑛=5
∑(𝑥−𝑥̅ )2
𝑆𝐷 = √
𝑛−1
̅
𝒙−𝒙
(𝒙 − 𝒙
̅) 𝟐
-2
-1
0
1
2
4
1
0
1
4
∑(𝑥 − 𝑥̅ )2 = 10
10
𝑆𝐷 = √5−1 = 1.58
Suko Na Company
RETURN
(𝒙)
4
6
8
10
12
=𝟖
̅
𝒙−𝒙
(𝒙 − 𝒙
̅) 𝟐
-4
-2
0
2
4
16
4
0
4
16
∑(𝑥 − 𝑥̅ )2 = 40
𝑛=5
∑(𝑥−𝑥̅ )2
𝑆𝐷 = √
𝑛−1
40
𝑆𝐷 = √5−1 = 3.16
⁘Kaya Pa Company has lower risk since it has lower standard deviation while Suko Na
Company has higher risk since it has higher standard deviation
Example 2: With Probabilities
An investor wants to calculate the standard deviation experience by his investment portfolio in the last
five months. Below are the return figures and probabilities:
MONTH
AUGUST
SEPTEMBER
OCTOBER
NOVEMBER
DECEMBER
RETURN
0.20
0.40
0.50
0.75
0.80
PROBABILITY
O.1
0.2
0.4
0.2
0.1
Formula for Standard Deviation:
𝑆𝐷 = √∑ 𝑃(𝑅 − 𝐸(𝑅))2
Where:
P = probability
R = rate of return
E(R)= expected return
𝑷
0.1
0.2
0.4
0.2
0.1
𝑹
0.20
0.40
0.50
0.75
0.80
𝑬(𝑹)
0.02
0.08
0.2
0.15
0.08
=0.53
(𝑹 − 𝑬(𝑹))
-0.33
-0.13
-0.03
0.22
0.27
(𝑹 − 𝑬(𝑹))𝟐
0.1089
0.0169
0.0009
0.0484
0.0729
𝑷 ∗ (𝑹 − 𝑬(𝑹))𝟐
0.01089
0.00338
0.00036
0.00968
0.00729
∑ 𝑷 ∗ (𝑅 − 𝐸(𝑅))2 = 0.0316
𝑆𝐷 = √0.0316 = 0.17776
𝑆𝐷 = 17.78%
We, expect a return of 53% based on the Summation of E(R) section, but the standard of our
expectation is 17.78%.
Above – 17.78% + 53% = 70.78%
Below – 17.78% - 53% = - 35.22%
Return can be observed or measured between -35.22% and 70.78%