Question

You want to invest in a portfolio that consists of two stocks: AAA and BBB. According to your analysis, you are expecting three states of nature: recession, steady and expansion, with probabilities of 0.2, 0.5 and 0.3 respectively. Current AAA stock price is $50, while BBB stock is $40. The following table show the analysts' prediction for the price in each state: State of Economy Probability AAA stock BBB STOCK Recession 0.2 $41 Steady 0.5 $54 $42 Expansion 0.3 $58 $47 $51 Suppose you decided to construct a portfolio, p, such that 60% of your wealth is allocated to AAA stock and 40% to BBB stock. Given this allocation, solve the following two questions: (1 mark) Compute the expected rate of return on p. b. (2 marks) Compute the portfolio's measure of risk. a.

Answer 1

first we Have to calculate rate of return for each state

AAA stock:

Recession = (51 - 50) / 50 = 2%

Steady = (54 - 50) / 50 = 8%

Expansion = (58 - 50) / 50 = 16%

BBB Stock :

Recession = (41 - 40) / 40 = 2.5%

Steady = (42 - 40) / 40 = 5%

Expansion = (47 - 40 ) / 40 = 17.50%

Probability AAA stock BBB stock AAA stock(A') BBB stock(B") P*(A-A')^2 P*(B-B')^2 P*(A-A')*(B-B') 0.2 2 2.5 0.4 0.5 15.488 12.0125 13.64 0.5 8 5.00 4 2.5 13.52 16.53125 14.95 0.3 16 17.5 4.8 5.25 5.808 2.7 3.96 Expected Return 9.2 8.25 34.816) 31.24375 32.55

a)

Port folio return = weighted average return

= (0.6*9.2%) + (0.4*8.25%)

= 8.82%

b)

Portfolio Risk = [(Wa*Stand.Dev.A)^2 + (Wb*Stand.dev.B)^2 + 2*Wa*Wb*stand. dev A * Stand.Dev.B * correlation]^(1/2)

Standard deviation of A = [P*(A-A')^2]^(1/2)

= (34.816)^(1/2) = 5.90%

Standard deviation of B = (31.2345)^(1/2) = 5.59%

Correlation = $\sum P*(A-A')*(B-B') / Stand.dev.A * Stand. Dev.B$

= 32.55 / (5.90*5.59)

= 1

so Portfolio risk = Wa*stand.dev A + Wb * Stand. dev. B (only when correlation is 1)

= (0.60*5.9) + (0.40*5.59)

= 5.78%