1. Covariance and Correlation

2. Chap 5-2
Chapter Topics
 Covariance and Its Applications in Finance
 Correlation

3. Chap 5-3
Random Variable
 Random Variable
 Outcomes of an experiment expressed numerically
 E.g., Toss a die twice; count the number of times
the number 4 appears (0, 1 or 2 times)
 E.g., Toss a coin; assign $10 to head and -$30 to a
tail = $10
T = -$30

4. Chap 5-4
Discrete Random Variable
 Discrete Random Variable
 Obtained by counting (0, 1, 2, 3, etc.)
 Usually a finite number of different values
 E.g., Toss a coin 5 times; count the number of tails
(0, 1, 2, 3, 4, or 5 times)

5. Chap 5-5
Probability Distribution
Values Probability
0 1/4 = .25
1 2/4 = .50
2 1/4 = .25
Discrete Probability Distribution
Example
Event: Toss 2 Coins Count # Tails
T
T
T T This is using the A Priori Classical
Probability approach.

6. Chap 5-6
Discrete Probability Distribution
 List of All Possible [Xj , P(Xj) ] Pairs
 Xj = Value of random variable
 P(Xj) = Probability associated with value
 Mutually Exclusive (Nothing in Common)
 Collective Exhaustive (Nothing Left Out)
   0 1 1j jP X P X  

7. Chap 5-7
Summary Measures
 Expected Value (The Mean)
 Weighted average of the probability distribution

 E.g., Toss 2 coins, count the number of tails,
compute expected value:
   j j
j
E X X P X   
 
        0 .25 1 .5 2 .25 1
j j
j
X P X 
   


8. Chap 5-8
Summary Measures
 Variance
 Weighted average squared deviation about the mean

 E.g., Toss 2 coins, count number of tails, compute
variance:
(continued)
     
2 22
i iE X X P X      
  
   
           
22
2 2 2
0 1 .25 1 1 .5 2 1 .25
.5
i iX P X  
     



9. Chap 5-9
Covariance and Its Application
     
 
1
th
th
th
: discrete random variable
: outcome of
: discrete random variable
: outcome of
: probability of occurrence of the
outcome of an
N
XY i i i i
i
i
i
i i
X E X Y E Y P X Y
X
X i X
Y
Y i Y
P X Y i
X


        
th
d the outcome of Yi

10. Chap 5-10
Computing the Mean for
Investment Returns
Return per $1,000 for two types of investments
          100 .2 100 .5 250 .3 $105XE X      
          200 .2 50 .5 350 .3 $90YE Y      
P(Xi) P(Yi) Economic Condition Dow Jones Fund X Growth Stock Y
.2 .2 Recession -$100 -$200
.5 .5 Stable Economy + 100 + 50
.3 .3 Expanding Economy + 250 + 350
Investment

11. Chap 5-11
Computing the Variance for
Investment Returns
        
2 2 22
.2 100 105 .5 100 105 .3 250 105
14,725 121.35
X
X


      
 
        
2 2 22
.2 200 90 .5 50 90 .3 350 90
37,900 194.68
Y
Y


      
 
P(Xi) P(Yi) Economic Condition Dow Jones Fund X Growth Stock Y
.2 .2 Recession -$100 -$200
.5 .5 Stable Economy + 100 + 50
.3 .3 Expanding Economy + 250 + 350
Investment

12. Chap 5-12
Computing the Covariance for
Investment Returns
P(XiYi) Economic Condition Dow Jones Fund X Growth Stock Y
.2 Recession -$100 -$200
.5 Stable Economy + 100 + 50
.3 Expanding Economy + 250 + 350
Investment
       
   
100 105 200 90 .2 100 105 50 90 .5
250 105 350 90 .3 23,300
XY        
   
The covariance of 23,000 indicates that the two investments are
positively related and will vary together in the same direction.

13. Chap 5-13
Computing the Coefficient of
Variation for Investment Returns


 Investment X appears to have a lower risk
(variation) per unit of average payoff (return)
than investment Y
 Investment X appears to have a higher
average payoff (return) per unit of variation
(risk) than investment Y
 
121.35
1.16 116%
105
X
X
CV X


   
 
194.68
2.16 216%
90
Y
Y
CV Y


   

14. Chap 5-14
Sum of Two Random Variables
 The expected value of the sum is equal to the
sum of the expected values
 The variance of the sum is equal to the sum
of the variances plus twice the covariance
 The standard deviation is the square root of
the variance
     E X Y E X E Y  
  2 2 2
2X Y X Y XYVar X Y        
2
X Y X Y  

15. Correlation Coefficient
Chap 5-15

16. Correlation Coefficient
Correlation Coefficient =
23500
121.35 𝑥 194.68
=0.9947

17. Find the Covariance and correlation
coefficient of the data below
Temperature Number of Customers
98 15
87 12
90 10
85 10
95 16
75 7
Chap 5-17

18. Chap 5-18

19. Chap 5-19

20. Exercises
 Collect the PM2.5 values on hourly basis for
the past 24 hrs and Calculate the mean,
median, mode, variance, standard deviation,
covariance and correlation. Get data from
Air4Thai website
 Get the value of THB against USD for the past
one month and calculate the metrics.
 Get the values of SET index for the past 2
weeks and calculate the metrics
Chap 5-20