Problem 1 ( BKM. Q3 of Chapter 7 ) ( 10 points1 ) What must be the beta of a portfolio with E ( rP ) = 20. 0 % . if the hazard free rate is 5. 0 % and the expected return of the market is E ( rM ) = 15. 0 % ? Answer: We use E ( rP ) = ? P * ( E ( rM ) – r degree Fahrenheit ) + r degree Fahrenheit. We so have: 0. 20 = ? P * ( 0. 15-0. 05 ) + 0. 05. Solving for the beta we get: ? P =1. 5.
Problem 2 ( BKM. Q4 of Chapter 7 ) ( 20 points ) The market monetary value of a security is $ 40. Its expected rate of return is 13 % . The riskless rate is 7 % . and the market hazard premium is 8 % . What will the market monetary value of the security be if its beta doubles ( and all other variables remain unchanged ) ? Assume that the stock is expected to pay a changeless dividend in sempiternity. Hint: Use zero-growth Dividend Discount Model to cipher the intrinsic value. which is the market monetary value. Answer: First. we need to cipher the original beta before it doubles from the CAPM. Note that: ? = ( the security’s hazard premium ) / ( the market’s hazard premium ) = 6/8 = 0. 75. Second. when its beta doubles to 2*0. 75 = 1. 5. so its expected return becomes: 7 % + 1. 5*8 % = 19 % . ( Alternatively. we can happen the expected return after the beta doubles in the undermentioned manner.
If the beta of the security doubles. so so will its hazard premium. The current hazard premium for the stock is: ( 13 % – 7 % ) = 6 % . so the new hazard premium would be 12 % . and the new price reduction rate for the security would be: 12 % + 7 % = 19 % . ) Third. we find out the implied changeless dividend payment from its current market monetary value of $ 40. If the stock pays a changeless dividend in sempiternity. so we know from the original informations that the dividend ( D ) must fulfill the equation for a sempiternity: Price = Dividend/Discount rate 40 = D/0. 13 ? D = 40 * 0. 13 = $ 5. 20 Last. at the new price reduction rate of 19 % . the stock would be deserving: $ 5. 20/0. 19 = $ 27. 37. The addition in stock hazard has lowered the value of the stock by 31. 58 % . Problem 3 ( BKM. Q16 of Chapter 7 ) ( 10 points )
A portion of stock is now selling for $ 100. It will pay a dividend of $ 9 per portion at the terminal of the twelvemonth. Its beta is 1. 0. What do investors anticipate the stock to sell for at the terminal of the twelvemonth if the market expected return is18 % and the hazard free rate for the twelvemonth is 8 % ? Answer: Since the stock’s beta is equal to 1. its expected rate of return should be equal to that of D + P1 ? P0. hence. we can work out for P1 as the market. that is. 18 % . Note that: Tocopherol ( R ) = P0 9 + P1 ? 100 the followers: 0. 18 = ? P1 = $ 109. 100 Problem 4 ( 15 points ) Assume two stocks. A and B. One has that E ( radium ) = 12 % and E ( rubidium ) = 15. % . The beta for stock A is 0. 8 and the beta for B is 1. 2. If the expected returns of both stocks lie in the SML line. what is the expected return of the market and what is the riskless rate? What is the beta of a portfolio made of these two assets with equal weights?
Answer: Since both stocks lie in the SML line. we can instantly happen its incline or the hazard premium of the market. Slope = ( E ( rM ) – releasing factor ) = ( E ( r2 ) – E ( r1 ) ) / ( ?2- ?1 ) = ( 0. 15-0. 12 ) / ( 1. 2-0. 8 ) = 0. 03/0. 4= 0. 075. Puting these values in E ( r2 ) = ?2* ( E ( rM ) – releasing factor ) + rF one gets: 0. 15 = 1. 2*0. 075 + releasing factor or releasing factor =0. 06=6. 0 % . The Expected return of the market is so given by ( E ( rM ) – 0. 06 ) = 0. 075 giving: Tocopherol ( rM ) = 13. 5 % . If you create a portfolio with these two assets seting peers sums of money in them ( every bit weighted ) . the beta will be ?P = w1*?1+w2*?2= 0. 5*1. 2+0. 5*0. 8 = 1. 0. Problem 5 ( 15 points ) You have an plus Angstrom with one-year expected return. beta. and volatility given by: Tocopherol ( radium ) = 20 % . ? A =1. 2. ? A =25 % . severally. If the one-year riskless rate is r f =2. 5 % and the expected one-year return and volatility of the market are E ( rM ) =10 % . ? A =15 % . what is the alpha of plus A? Answer: In order to happen the alpha. ? A. of plus A we need to happen out the difference between the expected return of the plus E ( radium ) and the expected return implied by the CAPM which is r f + ? A ( E ( rM ) – r degree Fahrenheit ) .
That is. show its expected return as: ? A = E ( radium ) – r f + ? A ( E ( rM ) – r degree Fahrenheit ) ) . Since we know the expected return of the market. the beta of the plus with regard to the market. and the riskless rate. alpha is given by: ? A = E ( radium ) – ? A ( E ( rM ) – r degree Fahrenheit ) – r f = 0. 20 – 1. 2 ( 0. 1 – 0. 025 ) – 0. 025
= 0. 085 = 8. 5 % .
Problem 6 ( BKM. Q23 of Chapter 7 ) ( 20 points ) Consider the followers informations for a one-factor economic system. All portfolios are good diversified. _______________________________________ Portfolio E ( R ) Beta ———————————————————-A 10 % 1. 0 F 4 % 0 ———————————————————-Suppose another portfolio Tocopherol is good diversified with a beta of 2/3 and expected return of 9 % . Would an arbitrage chance be? If so. what would the arbitrage scheme be? Answer: You can make a Portfolio G with beta equal to 1. 0 ( the same as the beta for Portfolio A ) by taking a long place in Portfolio E and a short place in Portfolio F ( that is. borrowing at the riskless rate and puting the returns in Portfolio E ) . For the beta of G to be 1. 0. the proportion ( tungsten ) of financess invested in E must be: 3/2 = 1. 5
The expected return of G is so: E ( roentgenium ) = [ ( ?0. 50 ) ? 4 % ] + ( 1. 5 ? 9 % ) = 11. 5 % ?G = 1. 5 ? ( 2/3 ) = 1. 0 Comparing Portfolio G to Portfolio A. G has the same beta and a higher expected return. This implies that an arbitrage chance exists. Now. see Portfolio H. which is a short place in Portfolio A with the returns invested in Portfolio G: ?H = 1?G + ( ?1 ) ?A = ( 1 ? 1 ) + [ ( ?1 ) ? 1 ] = 0 E ( rhesus factor ) = ( 1 ? roentgenium ) + [ ( ?1 ) ? rA ] = ( 1 ? 11. 5 % ) + [ ( ? 1 ) ? 10 % ] = 1. 5 % The consequence is a nothing investing portfolio ( all returns from the short sale of Portfolio A are invested in Portfolio G ) with nothing hazard ( because ? = 0 and the portfolios are good diversified ) . and a positive return of 1. 5 % . Portfolio H is an arbitrage portfolio.
Problem 7 ( 10 points ) Compare the CAPM theory with the APT theory. explicate the difference between these two theories? Answer: Apt applies to well-diversified portfolios and non needfully to single stocks. It is possible for some single stocks non to be on the SML. CAPM assumes rational behaviour for all investors ; APT merely requires some rational investors: APT is more general in that its factor does non hold to be the market portfolio. Both theoretical accounts give the expected return-beta relationship. 3