Suppose that there are two independent economic factors, F1 and F2. The risk-fre

Question

Suppose that there are two independent economic factors, F1 and F2. The risk-free rate is 4%, and all stocks have independent firm. specific components with a standard deviation of 51%. Portfolios A and B are both well-diversified with the following properties: Expecte Return Portfolio Beta on F1 Beta on F2 1.3 3.2 2.1 29% 26% -0.21 What is the expected return-beta relationship in this economy? Calculate the risk-free rate, rs and the factor risk premiums, RP1 and RP2, to complete the equation below. (Do not round intermediate calculations. Round your answers to two decimal places.) rf RP1 RP2 7.351%

Explanation / Answer

We will state both expected return of portfolio A and B in terms of equation 1

29=4+ (1.3×RP1) +( 2.1×RP2)                          ……… 2 (Portfolio A)

26=4+ (3.2×RP1) +( -0.21×RP2)                       ………3 (Portfolio B)

The above is a quadratic equation where there are 2 equations with 2 unknowns. So multiplying equation 2 by (3.2/1.3)=2.461538 and subtracting equation 3 from 2 we get:

71.38462 =9.846154+ (3.2×RP1) + (5.169231×RP2)

-26=-4- (3.2×RP1) - ( -0.21×RP2)

=45.38462=5.846154+4.959231RP2

= RP2=7.9727=7.98%

   Now plugging in RP2 in equation 3 we get RP1

           26=4+ (3.2×RP1) +( -0.21×7.98)                 

          23.6758=3.2 RP1

               RP1=7.398688=7.40 %

rf

4%

RP1

7.40%

RP2

7.98%

         

2a)In a single index model the variance of security 2 can be found as:

Thus the variance of security is sum of beta adjusted variance relative to the market (2m ) plus unsystematic risk or variance of the security (ei2).Using the above equation we will find the variance of security A, B, and C.

              2A= (1.3)2 × (0.22)2+(0.27)2= 0.154696=15.47%

               2B= (1.5)2 × (0.22)2+(0.13)2= 0.1258=12.58%

               2C= (1.7)2 ×(0.22)2+(0.22)2= 0.188276=18.83%

b) Now if there is infinite number of stocks with similar characteristics then there will be no unsystematic risk as it will be diversified away. Hence equation 1 discussed in question 2a will become:

2=i2 2m….(2)

(Note: Since unsystematic risk is diversified away the component ei2 equals 0).

Hence using equation 2 the variance of security A, B and C will be:

                 2A= (1.3)2 × (0.22)2= 0.081796=9% (rounded to nearest whole number)

                 2B= (1.5)2 × (0.22)2= 0.1089=11%   (rounded to nearest whole number)

                 2C= (1.7)2 ×(0.22)2= 0.139876=14% (rounded to nearest whole number)

The mean return of the securities will simply be equal to the expected return of the securities which is already given:

Mean

Variance

A

14

9

B

16

11

C

18

14

     

rf

4%

RP1

7.40%

RP2

7.98%

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