Question

Suppose a fund has a portfolio with two risky assets; stock and bond. Annual expected return of stock is 0.15 and standard deviation of 0.10 and expected return of bond is 0.08 and standard deviation of 0.07. The correlation-coefficient between stock and bond is 0.2. while t-bill has annual return of 0.03

Draw the opportunity set with 25% increment in bond fund. Also indicate the variance minimizing weight for bond and stock

Draw the optimal CAL line and calculate the sharp ratio

If the investor requires the complete portfolio standard deviation of 5%, how much of his fund to be invested in the risky portfolio (in terms of proportion, how big is y?)

Answer 1

Expected return on the portfolio = w(x)*E(x) + w(y)*E(y)

Standard deviation of portfolio =

where x and y are the securities

We use excel to calculate the opportunity set and excel solver for variance minimizing weight for bond and stock.

Weight in stock	Weight in bond	Expected return	Standard deviation of portfolio
0%	100%	8.0000%	0.4900%
25%	75%	9.7500%	0.3906%
75%	25%	13.2500%	0.6456%
100%	0%	15.0000%	1.0000%

For variance minimizing weights, we use an excel solver with the following constraints

Solving, we get

Minimum variance weights

	Weights	E[r]	Std. dev	Correlation
Stock	0.2893	0.15	0.1	0.2
Bond	0.7107	0.08	0.07
Portfolio	1	10.02%	6.24%

Draw the optimal CAL line and calculate the sharp ratio

CAL line equation =

We first need to find the optimal risky portfolio which maximises the sharpe's ratio. Using an excel solver,

Solving, we get the weights after maximising sharpe's ratio. This sharpe ratio = 1.293955 is the slope of the CAL equation

Standard deviation of the portfolio	CAL
0%	0.03
5%	0.09469775
10%	0.1593955
15%	0.22409325
20%	0.288791
25%	0.35348875
30%	0.4181865
35%	0.48288425
40%	0.547582
45%	0.61227975
50%	0.6769775
55%	0.74167525
60%	0.806373
65%	0.87107075
70%	0.9357685
75%	1.00046625
80%	1.065164
85%	1.12986175
90%	1.1945595
95%	1.25925725
100%	1.323955

If the investor requires the complete portfolio standard deviation of 5%, how much of his fund to be invested in the risky portfolio (in terms of proportion, how big is y?

We use an excel solver to set the portfolio standard deviation as 5% and calculate the weights

Weight in Stock	Weight in Bond	Weight at risk-free	Expected return	Standard deviation
37.24%	38.20%	24.56%	0.09379	5.000%