Question

Suppose there are only two possible future states of nature: Good and Bad. There is a 25% probability that the future will be Good. Suppose also that there are two stocks: A and B. Stock A will return 7% if the future is Good and will return 1% if the future is Bad. Stock B will return 3% if the future is Good and -5% if the future is Bad. If you have a portfolio that contains 60% of Stock A and 40% of Stock B, what will be the standard deviation of the portfolio? (hint: the portfolio expected return is 0.3%)Select one:

a. 8.67%

b. 2.94%

c. 12.35%

d. 15.2%

e. None of the above

Answer 1

Exoected return on a portfolio can be calculated as weighted average of individual returns= (w1*r1)+(w2*r2)

If Future is good, expected return on Portfolio= (0.6*7%)+(0.4*3%)= 4.2%+1.2%= 5.4%

If Future is bad, expected return on Portfolio= (0.6*1%)+(0.4*-5%)= 0.6%-2%= -1.4%

The respective probabilities for Future being Good and Bad is 0.25 and 0.75. So, X1= 5.4%, X2= -1.4%, P(X1)= 0.25 and P(X2)= 0.75

Standard deviation for the probability distribution is calculated as sqrt(E(X^2)-(E(X))^2)

E(X^2)= X1^2*P(X1)+X2^2*P(X2)

= 5.4%^2*0.25+(-1.4%^2*0.75)

= 0.0009

Given E(X)= 0.3%= 0.003

So, Standard deviation= sqrt(E(X^2)-(E(X))^2)

= sqrt(0.0009-0.003^2)

= sqrt(0.0009-0.000009)

= sqrt(0.000891)

= 0.0294

= 2.94% (Option b).