STATE OF ECONOMY         RETURN ON STOCK a(%)        RETURN ON STOCK B(%)

a. Bear                   6.30                         -3.70
b. Normal                10.50                          6.40
c. Bull                  15.60                         25.30

1. Calculate the expected return on each stock
2  Calculate the standard deviation of returns on each stock
3. Calculate the covariance and correlation between the returns on the two stocks.

Hi!!


Let me start defining the variables:

E(RA) = expected return on Stock A
E(RB) = expected return on Stock B 
STDA = standard deviation of stock A
STDB = standard deviation of stock B
VarA = variance of stock A
VarB = variance of stock B
Cor(RA,RB) = correlation between RA and RB
Cov(RA,RB) = covariance between RA and RB


- Part a).

Since it is not stated we can consider the same probability for each
possible state of the economy, that is 1/3, then:

E(RA) = (1/3)*(0.063) + (1/3)*(0.105) + (1/3)*(0.156) =
      = 0.1080 =
      = 10.80%

So the expected return on Stock A is 10.80%.


E(RB) = (1/3)*(-0.037) + (1/3)*(0.064) + (1/3)*(0.253) =
      = 0.0933 =
      = 9.33%

The expected return on Stock B is 9.33%.



- Part b).

For definition and formulas of variance and standard deviation see
"Standard Deviation and Variance: Common Measures of Variability"
http://davidmlane.com/hyperstat/A16252.html


VarA = (1/3)*(0.063 - 0.108)^2 + (1/3)*(0.105 - 0.108)^2 +
       + (1/3)*(0.156 - 0.108)^2 =	
     = 0.001446

STDA = sqrt(0.001446) =
     = 0.0380 =
     = 3.80%

The standard deviation of returns of Stock A is 3.80%.


VarB = (1/3)*(-0.037 - 0.0933)^2 + (1/3)*(0.064 - 0.0933)^2 + 
       + (1/3)*(0.253 - 0.0933)^2 =
     = 0.014447 

STDB = sqrt(0.014447) =
     = 0.1202 =
     = 12.02%

The standard deviation of returns of Stock B is 12.02%.



- Part c).

For formulas and definitions see "Covariance and correlation"
http://www.ncbi.nlm.nih.gov/books/bv.fcgi?rid=mga.section.2685
or
"Statistical Review with formula of, and rules for the mean, variance,
covariance, correlation coefficient":
http://www.kaspercpa.com/statisticalreview.htm


Cov(RA,RB) = (1/3)*(0.063 - 0.108)*(-0.037 - 0.0933) + 
             + (1/3)*(0.105 - 0.108)*(0.064 - 0.0933) +
             + (1/3)*(0.156 - 0.108)*(0.253 - 0.0933) =
           = 0.004539

The covariance between the returns of the two stocks is 0.004539.

Cor(RA,RB) = Cov(RA,RB) / (STDA * STDB)
           = 0.004539 / (0.0380 * 0.1202) =
           = 0.9937

The correlation between the returns of the two stocks is 0.9937.



Search strategy:
variance formula
covariance correlation formula


I hope this helps you. Feel free to request foe a clarification if you need it.


Regards,
livioflores-ga