Question

F: risk free  

The correlation between Asset A and Asset B is -0.2.

Asset F A B

Standard Deviation 0 0.35    0.6

Expected Return 0.02 0.12 0.26

Bob wants a portfolio with expected return of 22%. Investors can borrow and lend at the risk-free rate. Calculate the standard deviation of each. G is the global minimum variance combination of A and B. T is the tangency portfolio composed of A and B.

1) Invest only in assets A and B, standard deviation of Portfolio AB =

2) Invest only in assets F and A, standard deviation of Portfolio FA =

3) Invest only in assets F and B, standard deviation of Portfolio FB =

4) Invest only in assets F and G, standard deviation of Portfolio FG =

5) Invest only in assets F and T, standard deviation of Portfolio FT =

Answer 1

The return of a portfolio is the weighted return of the two stocks

So Return of this portfolio = 0.2163 * 16% +0.7837 *12% = 12.8653%

The standard deviation of a portfolio is given by

Where Wi is the weight of the security i,

is the standard deviation of returns of security i.

and is the correlation coefficient between returns of security i and security j

1. To achieve a return of 22% , suppose Bob invests proportion w of his wealth in A and (1-w) in B

then w*0.12+(1-w)*0.26 =0.22

=> w= 0.2857

So, 28.57% needs to be invested in A and 71.43% in B

Standard deviation of Portfolio =(0.2857^2*0.35^2+0.7143^2*0.6^2+2*0.2857*0.7143*0.35*0.6*(-0.2))^0.5

=0.4202 or 42.02%

2.

To achieve a return of 22% , suppose Bob invests proportion w of his wealth in F and (1-w) in A

then w*0.02+(1-w)*0.12 =0.22

=> w= -1

1-w=2

So, -100% needs to be invested in F and 200% in A (i.e. Bob must borrow an amount equal to his investment at risk free rate and invest both own as well as borrowed amount in A)

Standard deviation of Portfolio =2*0.35 =  70% (as standard deviation of risk free asset F is 0)

3.

To achieve a return of 22% , suppose Bob invests proportion w of his wealth in F and (1-w) in B

then w*0.02+(1-w)*0.26 =0.22

=> w= 0.1667

1-w=0.8333

So, 16.67% needs to be invested in F and 83.33% in B

Standard deviation of Portfolio =0.8333*0.6 = 50% (as standard deviation of risk free asset F is 0)

4. The Global Minimum variance portfolio weights for two stocks S and B  are given as

let S=A

So, weight of A = (0.6^2-0.35*0.6*(-0.2))/(0.35^2+0.6^2-2*0.35*0.6*(-0.2)) = 0.7096
Weight of B = 0.2904

So, Expected Return of G = 0.7096*0.12+0.2904*0.26 =0.1607

So, standard deviation of G =

=(0.7096^2*0.35^2+0.2904^2*0.6^2+2*0.7096*0.2904*0.35*0.6*(-0.2))^0.5

=0.2734 or 27.34%

To achieve a return of 22% , suppose Bob invests proportion w of his wealth in F and (1-w) in G

then w*0.02+(1-w)*0.1607 =0.22

=> w= -0.4219

1-w=1.4219

So, -42.19% needs to be invested in F and 142.19% in G (i.e. Bob must borrow an amount equal to 42.19% of his investment at risk free rate and invest both own as well as borrowed amount in G)

Standard deviation of Portfolio =1.4219*0.2734 = 38.87%    (as standard deviation of risk free asset F is 0)

5. The tangency portfolio T weights in two stock case is given by

and W_B= 1 - W_A

So, weight of A = ((0.12-0.02)*0.6^2-(0.26-0.02)*0.35*0.6*(-0.2))/(0.12-0.02)*0.6^2+(0.26-0.02)*0.35^2-(0.12-0.02+0.26-0.02)*0.35*0.6*(-0.2))

=0.5783

Weight of B = 1-0.5783 =0.4217

So, Expected Return of T = 0.5783*0.12+0.4217*0.26 =0.1790

So, standard deviation of T =

=(0.5783^2*0.35^2+0.4217^2*0.6^2+2*0.5783*0.4217*0.35*0.6*(-0.2))^0.5

=0.2907 or 29.07%

To achieve a return of 22% , suppose Bob invests proportion w of his wealth in F and (1-w) in T

then w*0.02+(1-w)*0.1790 =0.22

=> w= -0.2576

1-w=1.2576

So, -25.76% needs to be invested in F and 125.76% in T (i.e. Bob must borrow an amount equal to 25.76% of his investment at risk free rate and invest both own as well as borrowed amount in T)

Standard deviation of Portfolio =1.2576*0.2907 = 36.56%    (as standard deviation of risk free asset F is 0)