CALIFORNIA STATE UNIVERSITY, SACRAMENTO
College of Business Administration
PORTFOLIO THEORY
R E V I E W O F S TA T I S T I C S
Typical Formulas
Average: X = ( i n 

1
Xi )/n
Variance: Var = ( n  i

1
( Xi - X ) 2
) /
(n-1)
Standard Deviation:

= Var
Where X i
is the i th observation and n is the number of observations.
Expected Value = n  i

1
X i
P i
Where X i
is the i th outcome and P i
is the probability of that outcome
Example:
Economy
Good
Normal
Bad
Probability
30%
50%
20%
IBM
$150
$100
$ 80
Merck
$100
$ 65
$ 40
For IBM: X = (30%)(150) + (50%)(100) + (20%)(80) = $111.00
For Merck: X = (30%)(100) + (50%)(65) + (20%)(40) = $70.50
Variance (IBM) = (150-111)2 x .3 + (100-111)2 x .5 + (80-111)2 x .2 = 709
Standard Deviation (IBM):

=
709
= 26.63
Cov (x,y) =

(Xi – X ) (Y – Y ) n – 1

Cov (x,y) =

(Xi – X ) (Y – Y ) Pi
Example:
Cov(IBM, Merck)=(150-111)*(100-70.5)*.3 + (100-111)*(65-70.5)*.5 + (80-111)*(40-70.5)*.2= 564.50
TO STANDARDIZE COVARIANCE AND MAKE IT MORE MANAGEABLE, WE COMPUTE
 xy = Rho.
Cov (x, y)/
( 
(x) *

(y)
)
=
 xy = Rho
 xy = Rho is between 1 and –1
 xy = + 1 => (X and Y go same direction with the same magnitude)
 xy = - 1 => (X and Y go opposite direction with the same magnitude)
 xy = +.5 => (both X and Y go in the same direction but one only moves half as much)
 xy = 0 => (there is no relationship between X and Y)
The relationship between a stock and the market is depicted by the “Characteristic Line”.
Stock
C.L.
Slope =

Stock Market
If beta is large it means the steeper is the line. That is, that stock is very sensitive to the changes in the stock market
Beta =

= shows the sensitivity of one stock with respect to the stock market
 =  (Xi – X ) (Yi – Y ) /  (Xi – X ) 2