Problem #9 Assume that you may trade 3 securities: a risk-free asset that retums

Question

Problem #9 Assume that you may trade 3 securities: a risk-free asset that retums r, = 5%, a bond fund with expected return E(,)-8% and standard deviation ?(-)-12% and a stock fund with expected return E()= 13% and standard deviation ?(r.)-20%. In addition the coefficient of correlation between the bond fund and the stock fund is pas 0.3. Recall the formula to compute the variance ofa portfolio ?:-(mo, ). (ws.). 2(w,o,)(ws?.0s . You perform some computations and determine the minimum variance portfolio consists a. of w 0.82 in the bond fund and w 0.18 in the stock fund. Calculate the expected return and standard deviation of the minimum variance portfolio. 82 You perform more computations and determine the optimal risky portfolio consists of w: = 0.40 in the bond fund and wg = 0.60 in the stock fund. Calculate the expected return and standard deviation of the optimal risky portfolio.

Explanation / Answer

Answer

Bond = Return 8% , SD = 12%

Stock = Return 13% , SD = 20%

rbs = 0.30

Ans (1) Wb = 0.82 Ws = 0.18

Expected Return of portfolio = Wb * ERb + Ws * ERs = 0.82(8%) + 0.18(13%) = 8.72%

variance of Portfolio = (Wb * SDb)2 + (Ws * SDs)2 + 2*Wb*Ws*SDb*SDs*rbs

= (0.82*0.12)2+ (0.18*0.20)2 + 2*0.82*0.18*0.12*0.20*0.30 = 0.00968+0.001296+0.00425 = 0.0130873

there fore SD is (0.0130873)1/2 = 0.1144 = 11.44%

Ans (2) In case of optimal risky portfolio

Wb = 0.40 Ws = 0.60

Expected Return of portfolio = Wb * ERb + Ws * ERs = 0.40(8%) + 0.60(13%) = 10.4%

variance of Portfolio = (Wb * SDb)2 + (Ws * SDs)2 + 2*Wb*Ws*SDb*SDs*rbs

= (0.40*0.12)2+ (0.60*0.20)2 + 2*0.40*0.60*0.12*0.20*0.30 = 0.002304+0.0144+0.003456 = 0.02016

there fore SD is (0.02016)1/2 = 0.1419 = 14.19%

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