Financial Markets and Valuation - Tutorial 5: SOLUTIONS Capital sset Pricing Model, Weighted verage Cost o Capital & Practice Questions (*) denotes those problems to be covered in detail during the tutorial session Capital sset Pricing Model (CPM) Problem 1. (Ross, Westerield & Jae) Suppose the risk-ree rate is 6.3 % and the market portolio has an expected rate o return o 14.8 %. The market portolio has a variance o 0.011. Portolio Z has a correlation coeicient with the market o 0.45 and a variance o 0.0169. ccording to the CPM, what is the expected rate o return on portolio Z? r 6.3% [ ~ Ε r m] 14.8% σ 0. 011 ρ 0.45 σ 0. 0169 ρ z, m Cov ( Rz, Rm) ( 0.13)( 0.11) ( Rz, Rm) 0. 006435 z, m Cov β Z Cov z ( ) R Z R m σ, m 0.006435 0.011 Ε [ r ) Z ] 6.3% + 0.53 [.8% 6.3% ] m 0.53 14 10.8% (*) Problem. (Ross, Westerield & Jae) Johnson Paint stock has an expected return o 19 % with a beta o 1.7, while Williamson Tire stock has an expected return o 14 % with a beta o 1.. ssume the CPM is true. What is the expected return on the market? What is the risk-ree rate? Ε ( ~ rj) 19% r + 1.7 [ Ε ( ~ rm) r ] - Equation Ε ( ~ rw) 14% r + 1. [ Ε ( ~ rm) r ] - Equation B 0.5 Ε ~ rm [ r ] [ ] Equations () (B): 5% ( ) Risk Premium: ( rm) r ( r~ m ) Ε ~ 10% Ε r + 10% FMV/Tutorial 5 Solutions/Sept.-Oct. 006 1
Substituting into Equation () : r 1.7 [ r + 10% r ] 19% + r % Ε ( ~ r m) 1% Problem 3. (Ross, Westerield & Jae) Suppose you have invested $30,000 in the ollowing our stocks. Security mount invested Beta Stock $5,000 0.75 Stock B 10,000 1.10 Stock C 8,000 1.36 Stock D 7,000 1.88 The risk-ree rate is 4 % and the expected return on the market portolio is 15 %. Based on the CPM, what is the expected return on the above portolio? βρ ( 1 )( 0.75) + ( 1 )( 1.10) + ( 1.36) ( 4 ) + ( 7 )( 1.88) 6 3 1.93 4% + 1.93 15% 4% 18.% Ε ( r~ ρ ) [ ] 15 30 (*) Problem 4. (Ross, Westerield & Jae) There are two stocks in the market, stock and stock B. The price o stock today is $50. The price o stock next year will be $40 i the economy is in a recession, $55 i the economy is normal, and $60 i the economy is expanding. The attendant probabilities o recession, normal times and expansion are 0.1, 0.8 and 0.1, respectively. Stock pays no dividend. ssume the CPM is true. Other inormation about the market includes: σ (m) 10% σ (B) 1 % E(R b ) 9 % Correlation Coeicient (, m) 0.8 Correlation Coeicient (B, m) 0. Correlation Coeicient (, B) 0.6 a. I you are a typical risk averse investor, which stock would you preer? Why? b. What are the expected return and standard deviation o a portolio consisting o 70 % o stock and 30 % o stock B? c. What is the beta o the portolio in part (b)? FMV/Tutorial 5 Solutions/Sept.-Oct. 006
(a) Price o Security today $50 Expected Price o Security next year (0.1)($40) + (0.8)($55) + (0.1)($60) $54 ~ $54 $50 Ε $50 ( r ) 8% σ [ 0.0 0.0 8] + 0.8 [ 0.10 0.0 8] + 0.1 [ 0.0 0.0 ] 0.1 8 0.0096 σ σ 9.8% ρ σ σ, m m β σ m 0.8 0.1 0.098 0.1 0.1 0.784 The beta o stock B is: B,M B M B B,M M B M 0. 0.1 0.10 0. 4 The return on stock B is higher than the return on stock. The risk o stock B, as measured by its beta, is lower than the risk o. Thus, a typical risk-averse investor will preer stock B. b. E ( Rp ) 70% x E ( Ra) + 30% x E ( Rb) 70% x 8% + 30% x 9% 8.3% σ P ~ ~ x1 σ1 + ( 1 x1 ) σ + x1(1 x1 )cov( R1, R ) [(0.7) (0.098) + (0.3) (0.1) + () (0.7)(0.3)(0.6*0.098*0.1) ] 1/ (0.0089655) 1/ 9.47% c. β P 0.7*0.784+0.3*0.40.6 FMV/Tutorial 5 Solutions/Sept.-Oct. 006 3
Weighted verage Cost o Capital (WCC) Problem 5. (Ross, Westerield & Jae) The equity beta o dobe Online Company is 1.9. dobe online has a debt-to-equity ratio o 1. The expected return on the market is 13%. The risk ree rate is 7%. The cost o debt capital is 7%. The corporate tax rate is 35%. (a) What is dobe Online s cost o equity? (b) What is dobe Online s weighted average cost o capital? (a) Cost o Equity: 7 + 1.9 (13 7) 14.74% (b) B/(S+B) S/(S+B) 0.5 WCC 0.5(7)(0.65) +0.5(14.74) 9.645% (*) Problem 6. (Ross, Westerield & Jae) Calculate the WCC o the Luxury Porcelain Company. The book value o Luxury s outstanding debt is $60 million. Currently debt is trading at 10 percent o book value and is priced to yield 1 percent. The 5 million outstanding shares o Luxury stocks are selling or $0 per share. The required return on Luxury stock is 18 percent. The tax rate is 5 percent. B $60million * 1. $7million S $0*5million $100 million B/(S+B) 7 / 17 0.4186 S/(S+B) 100/17 0.5814 WCC 0.4186(1%)(0.75) + 0.5814(18%) 14.3% (*) Problem 7. (Ross, Westerield & Jae) Calgary Industries, Inc., is considering a new project that costs $5 million. The project will generate ater-tax (year-end) cash lows o $7 million or 5 years. The irm has a debt-to-equity ratio o 0.75. The cost o equity is 15% and the cost o debt is 9%. The corporate tax rate is 35%. It appears that the project has the same risk as that o the overall irm. Should Calgary take on the project? B/S 0.75 B/(S+B) 3/7 S/(S+B) 4/7 FMV/Tutorial 5 Solutions/Sept.-Oct. 006 4
WCC (4/7)15% + (3/7)(9%)(1-0.35) 11.08% NPV $5million + $819,99.04 5 t ( 1 0. ) t + $7million 1 1108 Thereore, the project should be undertaken. (*) Problem 8. (Ross, Westerield & Jae) National Electricity Company (NEC) is considering a $0 million modernization expansion project in the power station division. Tom Edison, the CFO, has evaluated the project; he determined that the project s ater-tax cash low will be $8 million, in perpetuity. In addition, Mr. Edison has devised two possibilities or raising the necessary $0 million: Issue 10-year, 10% debt; Issue common stock. NEC s cost o debt is 10% and its cost o equity is 0%. The irm s target debt-equity ratio is 00%. The expansion project has the same risk as the existing business, and it will support the same amount o debt. NEC is in the 34% tax bracket. Mr. Edison has advised the company to undertake the expansion. He suggests they use debt to inance the project because it is cheaper and its issuance costs are lower. (a) Should NEC accept the project? Support your answer with the appropriate calculations? (b) Do you agree with Mr. Edison s opinion o the expense o the debt? Why or why not? (a) NEC s debt equity ratio is. That means or every dollar o equity the irm has, it has two dollars o debt. The debt to value ratio o the irm is B/(S+B) which is equal to /(+1) /3. The equity to value ratio is 1/3. Thus, the WCC is: WCC (/3)(0.10)(1-0.34) + (1/3)(0.0) 0.1107 Thus, NPV -$0million + ($8million/0.1107) $5,67,389.34 Thereore, NEC should accept the project. FMV/Tutorial 5 Solutions/Sept.-Oct. 006 5
(b) Mr. Edison s conclusion is incorrect. Even though the issuing costs o debt are ar lower than those o equity, the irm must try and maintain its optimal capital structure. Thus, anytime all debt inancing is used, the irm will have to issue more equity in the uture to bring the capital structure back to the optimal. Revision Problems Problem 9. (Ross, Westerield & Jae) You are saving or the college education o your two children. They are two years apart in age; one will begin college in 15 years, the other in 17 years. You estimate your children s college expenses to be $1,000 per year per child. The annual interest rate is 15%, and it will remain 15% throughout the next 5 years. How much money must you place in an account each year to und your children s educations? You will begin payments one year rom today. You will discontinue payments when your oldest child enters college. ssume that the duration o the college degree is 4 years or each child and that the ees are due and are paid at the beginning o the college year. Solution : You have to pay $1,000 each year or 4 years, or each o your two children. The irst child will enter college in 15 years time while the second child will enter college in 17 years time. You are to assume that the ees are due and paid at the beginning o the college year. Given an annual interest rate o 15% which is expected to remain constant through the next 5 years, you are required to calculate the yearly payments (the irst starting one year rom now) that you need to invest so as to be able to pay or both your children s education when it comes due. You will stop making these payments when the oldest child enters college (i.e. 15 years rom now) The irst thing you need to do is to calculate the present value o your total inancial obligation, as ollows : PV (irst child s ees) $ 1,000* PVnnuityFactor(4yrs,15%) 14 (1.15) $8,473.36 PV (second child s ees) $ 1,000* PVnnuityFactor(4yrs,15%) 16 (1.15) $6,407.08 PV o total obligation $8,473.36 + $6,407.08 $14,880.44 FMV/Tutorial 5 Solutions/Sept.-Oct. 006 6
The next step is to ind the annuity payments starting one year rom today. Let the unknown annuity payment be $X, then $14,880.44 $X * PV nnuity Factor (15 years, 15%) $X * 5.8474 $14,880.44 Re-arranging, $X $,544.80 5.8474 (*) Problem 10. (Ross, Westerield & Jae) ter extensive medical and marketing research Pill, Inc. believes it can penetrate the pain reliever market. It can ollow one o two strategies. The irst is to manuacture a medication aimed at relieving headache pain. The second strategy is to make a pill designed to cure headache and arthritis pain. Both products would be introduced at a price o 4 $ per package in real terms. The broader remedy would probably sell 10 million packages a year. This is twice the sales rate or the headache-only medication. Cash costs o production in the irst year are expected to be 1.50$ per package in real terms or the headache-only brand. Production costs are expected to be 1.70$ in real terms or the more general pill. ll prices and costs are expected to rise at the general inlation rate o 5%. Either strategy will require urther investment in plant. The headache-only pill could be produced using equipment that would cost 10. $ million, last three years and have no resale value. The machinery required to produce the cure-all would cost 1$ million and last three years. t this time the irm would be able to sell it or $1 million in real terms. The production machinery would need to be replaced every three years at constant real costs. Suppose that or both projects the irm will use straight-line depreciation. The irm aces a corporate tax rate o 34%. The irm believes the appropriate real discount rate is 13%. Capital gains are taxed at the ordinary corporate tax rate o 34%. Which pain reliever should the irm produce? We will solve the question using nominal quantities. The nominal discount rate is (1 + inlation rate) x (1 + real rate) 18.65% Headache only ter tax operating income (using nominal prices and costs) is 1 3 qty 5000000 5000000 5000000 price 4. 4.41 4.6305 revenue 1000000 050000 315500 costs/unit 1.575 1.65375 1.736438 total cost 7875000 868750 868188 op. income 1315000 1378150 14470313 tax 446500 468565 4919906 NOPLT 866500 909565 9550406 FMV/Tutorial 5 Solutions/Sept.-Oct. 006 7
The total PV comes out to be: PV 866500/1.1865 + 909565/(1.1865)^ + 9550406/(1.1865)^3 $19,479,508.93 Depreciation tax shield: In nominal terms, it is expected to be (10. million/3)*(0.34) $1,156,000 Its total PV is $,487,51.65 The NPV o the project, thereore, is NPV - $10,00,000 + $19,479,508.93 + $,487,51.65 $11,767,030 Headache & rthritis only ter tax operating income (using nominal prices and costs) is 1 3 qty 10000000 10000000 10000000 price 4. 4.41 4.6305 revenue 4000000 44100000 46305000 costs/unit 1.785 1.8745 1.967963 total cost 17850000 1874500 1967965 op. income 4150000 5357500 665375 tax 811000 861550 90568 NOPLT 15939000 16735950 1757748 Total PV comes out to be $35,84,96.44 Depreciation tax shield: In nominal terms, it is expected to be (1 million/3)*(0.34) $1,360,000 Its total PV is $,96,495.488 Revenue rom the machine sale: The machine would sell or $1,000,000 (1.05) 3 $1,157,65. ter tax value $1,157,65(0.66) $764,03.5 PV $764,03.5/(1.1865^3) $457,413.1071 NPV o this option is thereore: -$1,000,000 + $35,84,96.44 + $,96,495.488 + $457,413.1071 $7,6,05.03 The irm should choose to manuacture the Headache & rthritis drug. FMV/Tutorial 5 Solutions/Sept.-Oct. 006 8