Question

Calculate the expected return and standard deviation for a portfolio consisting of 30% in Ant Ltd and the remainder in Bell Ltd.

Please explain your calculations thoroughly, showing all the formulas and calculations themselves. Please do not use excel.

State of economy	Probability of state	Rate of return Ant Ltd	Rate of return Bell Ltd
Recession	15%	2%	-30%
Normal	55%	10%	18%
Boom	???	15%	31%

Answer 1

probability of Boom = 100 - 55 - 15

= 30%

Expected return = sum of (probability of state * return of state)

expected returrn of A = 0.15 * 0.02 + 0.55 * 0.1 + 0.3 * 0.15

= 10.3%

expected returrn of B = 0.15 * -0.3 + 0.55 * 0.18 + 0.3 * 0.31

= 14.7%

E(X^2) =  sum of (probability of state * return of state^2)

expected returrn of A = 0.15 * 0.02^2 + 0.55 * 0.1^2 + 0.3 * 0.15^2

= 0.01231

expected returrn of B = 0.15 * -0.3^2 + 0.55 * 0.18^2 + 0.3 * 0.31^2

= 0.06015

variance = E(X^2) - (E(X))^2

variance of A = 0.01231 - 0.103^2

= 0.001701

variance of B = 0.06015 - 0.147^2

= 0.038541

Standard deviation = sqrt(variance)

Standar deviation of A = sqrt(0.001701)

= 4.12%

standard deviation of B = sqrt(0.038541)

= 19.63%

expected return of portfolio = 0.3 * 0.103 + 0.7 * 0.147

= 13.38%

standard deviation of portfolio = 0.3 * 0.0412 + 0.7 * 0.1963

= 14.98%