Question

2.

Mr. Brown would like to create a portfolio that is composed of Asset 1 and Asset 2. The correlation coefficient of Asset 1 and Asset 2 is .70. You are given the following information about Asset 1 and Asset 2.

E(R₁)   =   0.12                      E(s₁)   =   0.04

E(R₂)   =   0.16                      E(s₂)   =   0.06

            Mr. Brown is considering three possible combinations of Asset 1 and Asset 2.

            Option 1 :                   w₁ = 0.75                   w₂ = 0.25

            Option 2 :                   w₁ = 0.50                   w₂ = 0.50

            Option 3 :                   w₁ = 0.25                   w₂ = 0.75      

           

1. Calculate the expected return and the standard deviation for each option.

Show your calculations.

2. Mr. Brown has only Option 1, Option 2 and Option 3 available to him.

What will determine Mr. Brown’s choice among these three options? Explain.

Answer 1

Solution:

Option 1:

Expected return:

w1 = 0.75

w2 = 0.25

E(R1) =0.12

E(R2) = 0.16

Expected Return = sum( w1 * E(R1) + w2 * E(R2))

Standard Deviation:

= sqrt ( w₁²σ₁² + w₂²σ₂² + 2w₁w₂p_(1,2)σ₁σ₂)

w_{1 =} 0.75

w_{2 =} 0.25

σ_{1 =} 0.04

σ_{2 =} 0.06

p_{(1,2) =} 0.70

Applying the values,

= sqrt ( 0.75² * 0.04² + 0.25² * 0.06² + 2 * 0.75 * 0.25 * 0.04*0.06*0.70)

= sqrt( 0.5625 * 0.0016 + 0.0625 * 0.0036 + 0.00063 )

=sqrt( 0.0009 + 0.000225 + 0.00063)

=sqrt (0.001755)

=0.042

= 4.2%

Option 2:

w1 = 0.50

w2 = 0.50

E(R1) =0.12

E(R2) = 0.16

Expected Return = sum( w1 * E(R1) + w2 * E(R2))

Standard Deviation:

= sqrt ( w₁²σ₁² + w₂²σ₂² + 2w₁w₂p_(1,2)σ₁σ₂)

w_{1 =} 0.50

w_{2 =} 0.50

σ_{1 =} 0.04

σ_{2 =} 0.06

p_{(1,2) =} 0.70

Applying the values,

= sqrt ( 0.50² * 0.04² + 0.50² * 0.06² + 2 * 0.50 * 0.50 * 0.04*0.06*0.70)

= sqrt (0.00214)

= 0.0463

= 4.63%

Option 3:

Expected return:

w1 = 0.25

w2 = 0.75

E(R1) =0.12

E(R2) = 0.16

Expected Return = sum( w1 * E(R1) + w2 * E(R2))

Standard Deviation:

= sqrt ( w₁²σ₁² + w₂²σ₂² + 2w₁w₂p_(1,2)σ₁σ₂)

w_{1 =} 0.25

w_{2 =} 0.75

σ_{1 =} 0.04

σ_{2 =} 0.06

p_{(1,2) =} 0.70

Applying the values,

= sqrt ( 0.25² * 0.04² + 0.75² * 0.06² + 2 * 0.25 * 0.75 * 0.04*0.06*0.70)

= sqrt( 0.002755)

= 0.052

= 5.2%

Mr. Brown's choice can be determined using the mean variance theory. For a desired rate of return, Mr Brown may choose the option with the least risk, ie, lower standard deviation. Higher the risk , Higher the returns. Thus if Mr Brown is willing to assume more risk , he can choose Option 3 which gives a higher expected return of 15% . But if he wants to obtain returns with minimal risk, Mr. Brown may opt for Option 1 which gives a return of 13%.