Question

                Expected Return     Standard Deviation

Stock fund (S)             20%                          30%

Bond fund (B)             12%                          15%  

Correlation = .10

7. If you were to use only the two risky funds, and still require an expected return of 14%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio in Problem 9. What do you conclude?

Problem 9

Stock Expected Return         Standard Deviation

A            10%                                     5%

B            15                                        10

          Correlation = −1

expected return 14%

Answer 1

	ER	Sd
Stock fund (S)	20	30
Bond fund (B)	12	15

Let proportion of Stock fund be x, therefore proportion of Bond fund would be 1-x.

14 = 20x+12(1-x)

14 = 20x+12-12x

2 = 8x

x = 2/8

=0.25

1-x = 0.75

Proportion of Stock fund is 25% and of Bond fund is 75%.

Standard deviation of Stock fund(S) and Bond fund (B) = {(X²_s Sd²_s+ (X²_B Sd²_B)+(2 X_s X_B(Sd_s Sd_B r_sB))}^1/2

= {(0.25² *30²)+(0.75²*15²)+(2*0.25*0.75*30*15*0.10)}^1/2

= (0.0625*900)+ (0.5625*225)+(2*0.25*0.75*45)}^1/2

= (182.8125+16.875)^1/2

= (199.6875)^1/2

= 14.13%

	ER	Sd
A	10	5
B	15	10

Let proportion of Stock A be x, therefore proportion of Stock B would be 1-x.

14 = 10x+15(1-x)

14 = 10x+15-15x

5x = 1

x = 1/5

=0.20

1-x = 0.80

Proportion of Stock A is 20% and of Stock B is 80%.

Standard deviation of Stock A and Bond B = {(X²_A Sd²_A) + (X²_B Sd²_B) + (2 X_A X_B (Sd_A Sd_B r_AB))}^1/2

= {(0.20² *5²)+(0.80²*10²)+(2*0.20*0.80*5*10*(-1))}^1/2

= (0.04*25)+ (0.64*100)+(2*0.20*0.80*(-50))}^1/2

= (65-16)^1/2

= (49)^1/2

= 7%

Lower of standard deviation is always better to opt.

As per above calculations, Expected returns of both the portfolios are same, however Standard deviations are as follows:

Standard deviation:

Portfolio (SB) = 14.13%

Portfolio (AB) = 7%

Since standard deviation of Portfolio AB is lower therefore Portfolio AB should be Opt.