Question

uppose that there are two independent economic factors, F1 and F2. The risk-free rate is 7%, and all stocks have independent firm-specific components with a standard deviation of 37%. Portfolios A and B are both well-diversified with the following properties: Portfolio Beta on F1 Beta on F2 Expected Return A 1.3 1.7 27 % B 2.2 –0.17 24 % What is the expected return-beta relationship in this economy? Calculate the risk-free rate, rf, 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.) E(rP) = rf + (βP1 × RP1) + (βP2 × RP2)

Answer 1

The expected return-beta relationship is given by:
E(rP)=Rf + βP1*RP1 + βP2*RP2

Where,
E(rP) is the expected return on the portfolio
Rf refers to risk free rate
RP1 is the risk premium on factor 1
RP2 is the risk premium on factor 2
BP1 and BP2 refers to beta on factors F1 and F2 respectively

Given that, the risk-free rate=7%
Beta on factor F1=1.3
Beta on factor F2=1.7
Expected return= 27%
Now, putting these values in the equation E(rP)=Rf + βP1*RP1 + βP2*RP2, we get
27%=7% + 1.3*RP1 + 1.7*RP2
This is the equation for portfolio A
=>27%-7%- 1.3*RP1=1.7*RP2
=>(20%-1.3*RP1)/1.7=RP2

Equation for portfolio B will be:
24%=7%+2.2*RP1+(-0.17)*RP2
=>24%-7%=2.2RP1-0.17RP2
=>17%=2.2RP1-0.17RP2
Substituting the value of RP2, we get
17%=2.2RP1-0.17*(20%-1.3*RP1)/1.7
=>0.17*1.7=1.7*(2.2RP1)-0.17*(0.2-1.3RP1)

=>0.289=3.74RP1-0.17*0.2+0.17*1.3RP1
=>0.289=3.74RP1-0.034+0.221RP1
=>0.289+0.034=3.74RP1+0.221RP1
=>0.289+0.034=3.74RP1+0.221RP1
=>0.323=3.961RP1
=>0.323/3.961=RP1
=>RP1=0.081545064 or 8.15% (rounded up to two decimal places)

Now, (20%-1.3*RP1)/1.7=RP2
Substituting the values of RP1 in (20%-1.3*RP1)/1.7=RP2, we get
RP2=(20%-1.3*0.081545064)/1.7
=(.2-0.106008583)/1.7
=0.093991417/1.7
=>RP2=0.055289069 or 5.53% (rounded up to two decimal places)

As per the question asked on completing the equation, we will have the following equation after substituting the required values:
E(rP)=Rf + βP1*RP1 + βP2*RP2 will be E(rP)=7% + βP1*8.15% + βP2*5.53%

Note:
Further, we can also substitute the values of BP1 and BP2, to get the value of expected portfolio return.
=7% + 1.3*8.15% + 1.7*5.53%
=0.07 + 0.10595 + 0.09401
E(rP)=0.26996 or 26.996%