FIN2101 BUSINESS FINANCE II MODULE 2-CAPITAL ASSET PRICING MODEL(CAPM) QUESTION 1 The best combination of expected value of return and standard deviation depends upon the investor's utility function. Explain this statement QUESTION 2 Explain how the existence of a risk-free asset affects optimal portfolio selection QUESTION 3 Explain briefly just what the Capital Asset Pricing Model tells us QUESTION 4 Using the CAPM, estimate the appropriate required rate of return for the three shares listed below, given that the risk-free rate is 5% and the expected return on the market portfolio 17% Share Beta B 0.90 QUESTION 5 (a) Determine the expected return and beta for the following portfolio Share Portfolio Beta Expected Return 40 11 (b) Given the information above, draw the security market line and show where the return on the market portfolio is 12%. How would you interpret these findings?cted securities fit on the graph. Assume that the risk-free rate is 8% and that the exp
July 2003 FIN2101 BUSINESS FINANCE II MODULE 2 – CAPITAL ASSET PRICING MODEL (CAPM) QUESTION 1 The best combination of expected value of return and standard deviation depends upon the investor's utility function. Explain this statement. QUESTION 2 Explain how the existence of a risk-free asset affects optimal portfolio selection. QUESTION 3 Explain briefly just what the Capital Asset Pricing Model tells us. QUESTION 4 Using the CAPM, estimate the appropriate required rate of return for the three shares listed below, given that the risk-free rate is 5% and the expected return on the market portfolio is 17%. Share Beta A 0.75 B 0.90 C 1.40 QUESTION 5 (a) Determine the expected return and beta for the following portfolio: Share Portfolio Weighting % Beta Expected Return % 1 40 1.00 12 2 25 0.75 11 3 35 1.30 15 (b) Given the information above, draw the security market line and show where the securities fit on the graph. Assume that the risk-free rate is 8% and that the expected return on the market portfolio is 12%. How would you interpret these findings?
QUESTION 6 In five successive time periods, the market portfolio had a return of 10%, 12%, 6%0,-4% and 1%. a given security had the following returns over the same five time periods: 15%, 13%, Calculate (a) the standard deviation of returns (b) the correlation of returns between the market portfolio and the security (c) the systematic risk of the security QUESTION 7 Johnson Manufacturing Ltd is considering several investments. The rate on Treasury notes currently 6.75% and the expected return for the market is 12%. What would be the required rates of return for each investment (using the CAPM)? Security Beta 1.50 B 0.60 QUESTION 8 CSB Ltd has a beta of 0. 765. If the expected market return is 11.5% and the risk-free rate is 7.5%, what is the appropriate required return of CSB (using the CAPm)? QUESTION 9 The expected return for the general market is 12. 8%, and the market risk premium is 4.3% Tasaco, LBM and Exxos have betas of 0.864, 0.693 and 0.575 respectively. What are the correspond ing required rates of return for the three securities? Questions 4, 5,7,8 and9 taken from Petty et al, Financial Management, 2nd edn, Pearson Education, Sydney, 2000
July 2003 QUESTION 6 In five successive time periods, the market portfolio had a return of 10%, 12%, 6%, -4% and 1%. A given security had the following returns over the same five time periods: 15%, 13%, 4%, -12% and –2%. Calculate: (a) the standard deviation of returns; (b) the correlation of returns between the market portfolio and the security; (c) the systematic risk of the security. QUESTION 7 Johnson Manufacturing Ltd is considering several investments. The rate on Treasury notes is currently 6.75% and the expected return for the market is 12%. What would be the required rates of return for each investment (using the CAPM)? Security Beta A 1.50 B 0.82 C 0.60 D 1.15 QUESTION 8 CSB Ltd has a beta of 0.765. If the expected market return is 11.5% and the risk-free rate is 7.5%, what is the appropriate required return of CSB (using the CAPM)? QUESTION 9 The expected return for the general market is 12.8%, and the market risk premium is 4.3%. Tasaco, LBM and Exxos have betas of 0.864, 0.693 and 0.575 respectively. What are the corresponding required rates of return for the three securities? Questions 4, 5, 7, 8 and 9 taken from Petty et al, Financial Management, 2nd edn, Pearson Education, Sydney, 2000
FIN2101 BUSINESS FINANCE II SOLUTIONS TO TUTORIAL QUESTIONS MODULE 2-CAPITAL ASSET PRICING MODEL(CAPM)
July 2003 FIN2101 BUSINESS FINANCE II SOLUTIONS TO TUTORIAL QUESTIONS MODULE 2 – CAPITAL ASSET PRICING MODEL (CAPM)
QUESTION 1 The literature is a little deficient in clearly explaining the relevance of utility theory to portfolio selection and the notions of indifference curves and efficient frontiers. The students should understand not merely that a particular point on a diagram represents something but why it does and why an investor may choose to prefer that point as opposed to othe possibilities An individual investor's choice between different portfolios will depend on personal preference(util ity theory). The portfolio with the maximum util ity is the one at the point of tangency of the opportunity set with the highest indifference curve. a risk averse investor will choose a portfolio which is at a point of tangency between his indifference curve and the efficient frontier QUESTION 2 When risk-free assets do not exist, the optimal portfolio selection will be found on the efficient frontier With the introduction of a risk-free asset we get a new efficient frontier which is drawn as the Capital Market Line(CML), with all points(except point M)on the old"efficient QUESTION 3 CAPM attempts to provide a means of understanding the relationship between expected return and systematic or non-diversifiable or unavo idable risk and evaluation of securities in the following context. In market equilibrium, a security is expected to provide a return commensurate with its unavoidable risk. Put simply, the greater the unavoidable risk of a security, the greater the return that investors will expect from that security QUESTION 4 Rr+ kA=0.05+075×(017-0.05 0.14or14% kB=05+[090×(0.17005 =0.158or15.8% kc=005+[40×(017-005 =0.2l8or21.8%
July 2003 QUESTION 1 The literature is a little deficient in clearly explaining the relevance of utility theory to portfolio selection and the notions of indifference curves and efficient frontiers. The students should understand not merely that a particular point on a diagram represents something but why it does and why an investor may choose to prefer that point as opposed to other possibilities. An individual investor's choice between different portfolios will depend on personal preference (utility theory). The portfolio with the maximum utility is the one at the point of tangency of the opportunity set with the highest indifference curve. A risk averse investor will choose a portfolio which is at a point of tangency between his indifference curve and the efficient frontier. QUESTION 2 When risk-free assets do not exist, the optimal portfolio selection will be found on the efficient frontier. With the introduction of a risk-free asset, we get a new efficient frontier which is drawn as the Capital Market Line (CML), with all points (except point M) on the “old” efficient frontier now dominated by points on the "new" frontier. QUESTION 3 CAPM attempts to provide a means of understanding the relationship between expected return and systematic or non-diversifiable or unavoidable risk and evaluation of securities in the following context. In market equilibrium, a security is expected to provide a return commensurate with its unavoidable risk. Put simply, the greater the unavoidable risk of a security, the greater the return that investors will expect from that security. QUESTION 4 ( ) ( ) ( ) ( ) 0.218 or 21.8% k 0.05 1.40 0.17 - 0.05 0.158 or 15.8% k 0.05 0.90 0.17 - 0.05 0.14 or 14% k 0.05 0.75 0.17 - 0.05 k R b k - R C B A j F j m F = = + = = + = = + = +
QUESTION 5 =(040×0.12)+(0.25×0.1)+(035×0.15) 0.048+0.0275+0.0525 =0.128or12.8% =(040×100)+(025×075)+(035×130) 0.40+0.1875+0455 (b) Shares 1 and 2 seem to be right in line with the secur ity market line, which suggests that they are earning a fair return, given their systematic risk. Share 3, on the other hand, is earning more than a fair return(above the security market line). We might be tempted to conclude that security 3 is undervalued
July 2003 QUESTION 5 (a) ( ) ( ) ( ) 0.128 or 12.8% 0.048 0.0275 0.0525 0.40 0.12 0.25 0.11 0.35 0.15 k w k n j 1 p j j = = + + = + + = = ( ) ( ) ( ) 1.0425 0.40 0.1875 0.455 0.40 1.00 0.25 0.75 0.35 1.30 b w b n j 1 p j j = = + + = + + = = (b) Shares 1 and 2 seem to be right in line with the security market line, which suggests that they are earning a fair return, given their systematic risk. Share 3, on the other hand, is earning more than a fair return (above the security market line). We might be tempted to conclude that security 3 is undervalued
QUESTION 6 lote that the question involves historical data and ex post analysis is therefore required Step 1-Calculate the arithmetic mean returns for the security and the market portfolio k k ks0.15+013+0.04-0.12-0.02 =0.036 kn=010+012+006-04+00 Step 2-Calculate the standard deviation of the returns for the security and the market portfolio 015-0036)+1013-006+00-006+(012-0069+(002003 ks 0.012996+0008836+0.000016+0024336+0.003136 =√001233 =0.11104 010-005+(0120053+(00605+(00400+01-05 5-1 0025+00049+0.0001+0.0081+00016 4 =√00043 =0.06557
July 2003 QUESTION 6 Note that the question involves historical data and ex post analysis is therefore required. Step 1 – Calculate the arithmetic mean returns for the security and the market portfolio 0.05 5 0.10 0.12 0.06 - 0.04 0.01 k 0.036 5 0.15 0.13 0.04 - 0.12 - 0.02 k n k k m S n i 1 i = + + + = = + + = = = Step 2 – Calculate the standard deviation of the returns for the security and the market portfolio ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0.06557 0.0043 4 0.0025 0.0049 0.0001 0.0081 0.0016 5 -1 0.10 - 0.05 0.12 - 0.05 0.06 - 0.05 - 0.04 - 0.05 0.01- 0.05 0.11104 0.01233 4 0.012996 0.008836 0.000016 0.024336 0.003136 5 -1 0.15 - 0.036 0.13 - 0.036 0.04 - 0.036 - 0.12 - 0.036 - 0.02 - 0.036 n -1 k - k 2 2 2 2 2 k 2 2 2 2 2 k n i 1 2 i k m S = = + + + + = + + + + = = = + + + + = + + + + = = =
QUESTION 6 (Continued) Step 3-Calculate the covariance of returns between the security and the market portfolio ∑[-kl2 n (0.15-00360.10-05)+(013-0036)0.12-005)+(0.04-0036006-005) Cov(ks, km)= +(-0.12-0.036-004-0.05)+(-0.02-0036001-005) 5-1 (0.114x005)+(094x07)+(0004x001)+(-0.56x-09)+(-0.056x-004) 0.0057+0.00658+0.000004+001404+0.00224 0.0286 0.00715
July 2003 QUESTION 6 (Continued) Step 3 – Calculate the covariance of returns between the security and the market portfolio ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 0.00715 4 0.0286 4 0.0057 0.00658 0.000004 0.01404 0.00224 4 0.114 x 0.05 0.094 x 0.07 0.004 x 0.01 - 0.156 x - 0.09 - 0.056 x - 0.04 5 -1 - 0.12 - 0.036 0.04 0.05 - 0.02 - 0.036 0.01 0.05 0.15 - 0.036 0.10 0.05 0.13 - 0.036 0.12 0.05 0.04 - 0.036 0.06 0.05 Cov k , k n -1 k - k k - k Cov k , k S m n i 1 2 2 ,i 1 1 ,i 1 2 = = + + + + = + + + + = + − − + − − + − + − = = =
QUESTION 6 (Contin Step 4- Calculate the correlation coefficient of returns 0. g 0.007 0.11104×0.06557 0.00715 0.0072808928 =0.982 Step 5-Calculate the systematic risk(beta) of the security Cov(k;, km) 0.0043
July 2003 QUESTION 6 (Continued) Step 4 – Calculate the correlation coefficient of returns ( ) ( ) 0.982 0.0072808928 0.00715 0.11104 0.06557 0.00715 Cov k , k r Cov k , k r S m 1 2 k k S m S,m k k 1 2 1,2 = = = = = Step 5 – Calculate the systematic risk (beta) of the security ( ) ( ) 1.663 0.0043 0.00715 Cov k , k b Cov k , k b 2 k S m S 2 k j m j m m = = = =
QUESTION 7 =R+b,×(kaR) kA=00675+[50×(012-0675 0.14625or14625% kB=0675+082x(0.12-00675 =0.11055or11055% kc=0675+060×(0.12-00675 =0.099or99% kp=00675+[15×(0.2-00675) =0.127875or127875% QUESTION 8 k =R+b x R =0075+0765(0115-075) =0.1056or10.56% QUESTION 9 If the market return is 12. 8% and the market risk premium is 4.3%, the risk-free rate of return must be8.5%(12.8-43) k,=RF+b,×(km-R) kr=085+[0864×(0043 =0.122152or12.2152% k1=085+[0693×(0043 =0.114799or114799% k:=085+0575×(043) 0.109725or109725%
July 2003 QUESTION 7 ( ) ( ) ( ) ( ) ( ) 0.127875 or 12.7875% k 0.0675 1.15 0.12 - 0.0675 0.099 or 9.9% k 0.0675 0.60 0.12 - 0.0675 0.11055 or 11.055% k 0.0675 0.82 0.12 - 0.0675 0.14625 or 14.625% k 0.0675 1.50 0.12 - 0.0675 k R b k - R D C B A j F j m F = = + = = + = = + = = + = + QUESTION 8 ( ) ( ) 0.1056 or 10.56% 0.075 0.765 0.115 - 0.075 k j RF b j k m - RF = = + = + QUESTION 9 If the market return is 12.8% and the market risk premium is 4.3%, the risk-free rate of return must be 8.5% (12.8 – 4.3). ( ) ( ) ( ) ( ) 0.109725 or 10.9725% k 0.085 0.575 0.043 0.114799 or 11.4799% k 0.085 0.693 0.043 0.122152 or 12.2152% k 0.085 0.864 0.043 k R b k - R E L T j F j m F = = + = = + = = + = +
