Question

Question 3

Consider the following information:

State of Economy Probability Recession 0.30 Normal 0.50 Boom 0.20

Required:

Rate of return

Stock A 2% 7% 12%

Stock B

5% 4% 3%

1. Calculate the expected return for the two Stocks A and B respectively.

2. Calculate the standard deviation for the two Stocks A and B respectively.

3. If you own a portfolio that has $1.3 million invested in Stock A and $2.2 million

   invested in Stock B. Calculate the expected return and standard deviation for the

   portfolio.

4. Compute the portfolio standard deviation using the weighted average of individual

   asset's standard deviation for the portfolio as mentioned in part (c) above.

5. Comment and explain the different results obtained in parts (c) and (d) above

   critically [within 200 words].

Answer 1

expected return =

where P = probability

standard deviation =

where X = return with respect to probability

X' = expected return

P = probability

covariance = P*(A-A')*(B-B')

correlation coefficient = covariane / stand. dev. A * Stand. dev .b

from the above image we can see that

a) expected return of Stock A = 6.5%

Stock B = 4.1%

b) standard deviation of stock A = (12.25)^1/2 = 3.5%

Stock B = (0.49)^1/2 = 0.7%

c)

first we have to calculate correlation coefficient

we have already calculated covariance = -2.45

so correlation coefficient = -2.45 / (3.5 * 0.7)

= -1

when correlation coefficient = -1 then

standard deviation of portfolio = Wa * stand. dev. A - Wb * Stand. dev. B

Expected return of port folio = Wa * Ra + Wb * Rb

where , Wa and Wb = weights of stock A and B

Ra and Rb = expected returns of stovk A and B

total amount invested in stock A and B = 1.3 + 2.2 = $3.5 million

weight of Stock A = 1.3 / 3.5 = 0.37

Stock B = 0.63

so expected return of port folio = 0.37*6.5 + 0.63*4.1

= 2.405 + 2.583

= 5% (rounded to nearest integer)

standard deviation of portfolio = 0.37*3.5 - 0.63*0.7

= 0.86%

d) if we use weighted average for standard deviation of portfolio then

standard deviation of portfolio = Wa * stand. dev. A + Wb * Stand. dev. B

= 0.37*3.5 + 0.63*0.7

= 1.74%

the standard deviation obtained from part c and d are not equal infact standard deviation from part d is almost double from what we have calculated in Part c. this is because there exist a correlation between the stocks which is -1(perfect negative).in part d we ignore the correlation between the stocks. how ever weighted average can be used only when correlation between stocks is +1(perfect positive).