Financial Markets and Valuation - Tutorial 5: SOLUTIONS Capital Asset Pricing Model, Weighted Average Cost of Capital & Practice Questions (*) denotes those problems to be covered in detail during the tutorial session
Capital Asset Pricing Model (CAPM) Problem 1. (Ross, Westerfield & Jaffe) Suppose the risk-free rate is 6.3 % and the market portfolio has an expected rate of return of 14.8 %. The market portfolio has a variance of 0.0121. Portfolio Z has a correlation coefficient with the market of 0.45 and a variance of 0.0169. According to the CAPM, what is the expected rate of return on portfolio Z? Solution:
rf
Ε [~ r m] = 14.8%
= 6.3%
ρ z, m
= 0.45
ρ z, m
=
βZ
=
σ
2 z
σ m2
= 0.0121
= 0.0169
Cov (Rz , Rm ) (0.13)(0.11) Cov (Rz , Rm ) = 0.006435
) Ε [rZ ] =
Cov (R Z , Rm )
σ
2 m
0.006435 = 0.0121
=
0.532
6.3% + 0.532 [14.8% − 6.3%] = 10.82%
(*) Problem 2. (Ross, Westerfield & Jaffe) Johnson Paint stock has an expected return of 19 % with a beta of 1.7, while Williamson Tire stock has an expected return of 14 % with a beta of 1.2. Assume the CAPM is true. What is the expected return on the market? What is the risk-free rate?
Solution: Ε (~ r j ) = 19% = r f + 1.7 Ε (~ r m) − r f ~ Ε (r w) = 14% = r f + 1.2 Ε (~ r m) − r f r m) − r Equations (A) – (B): 5% = 0.5 Ε (~
[
Risk Premium:
[
]
[
]
[ Ε (~r m) − r ]
Ε (~ r m)
f
=
FMV/Tutorial 5 – Solutions/Sept.-Oct. 2006
f
]
-
Equation A
-
Equation B
= 10% r f + 10% 1
[
Substituting into Equation (A) : r f + 1.7 r f + 10% − r f Ε (~ r m ) = 12% ∴r = 2%
]
= 19%
f
Problem 3. (Ross, Westerfield & Jaffe) Suppose following four stocks. Security Amount invested Stock A $5,000 Stock B 10,000 Stock C 8,000 Stock D 7,000
you have invested $30,000 in the Beta 0.75 1.10 1.36 1.88
The risk-free rate is 4 % and the expected return on the market portfolio is 15 %. Based on the CAPM, what is the expected return on the above portfolio? Solution: βρ =
(16 ) (0.75) + (13 ) (1.10) + (1.36) (415) + (7 30) (1.88)
= ~ Ε (r ρ ) = =
1.293 4% + 1.293 [15% − 4%] 18.22%
(*) Problem 4. (Ross, Westerfield & Jaffe) There are two stocks in the market, stock A and stock B. The price of stock A today is $50. The price of stock A next year will be $40 if the economy is in a recession, $55 if the economy is normal, and $60 if the economy is expanding. The attendant probabilities of recession, normal times and expansion are 0.1, 0.8 and 0.1, respectively. Stock A pays no dividend.
Assume the CAPM is true. Other information about the market includes: σ (m) = 10% σ (B) = 12 % E(Rb) = 9 %
Correlation Coefficient (A, m) = 0.8 Correlation Coefficient (B, m) = 0.2 Correlation Coefficient (A, B) = 0.6 a. If you are a typical risk averse investor, which stock would you prefer? Why? b. What are the expected return and standard deviation of a portfolio consisting of 70 % of stock A and 30 % of stock B? c. What is the beta of the portfolio in part (b)?
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Solution:
(a)
Price of Security A today =
$50
Expected Price of Security A next year = (0.1)($40) + (0.8)($55) + (0.1)($60) = $54 $54 − $50 $50
∴ Ε (~ rA ) =
= 8%
σ A2 = 0.1 × [− 0.20 − 0.0 8]2 + 0.8 × [0.10 − 0.0 8]2 + 0.1 × [0.20 − 0.0 8]2 =
0.0096
σA
=
σ A2
βA
=
= 9 .8 %
ρ A, m × σ m × σ A σ
2 m
=
0.8 × 0.1 × × 0.098 = 0 .1 × 0 .1
0.784
The beta of stock B is:
B
B,M B M
B,M M B 2M 0.20.12 0.10
0. 24
The return on stock B is higher than the return on stock A. The risk of stock B, as measured by its beta, is lower than the risk of A. Thus, a typical risk-averse investor will prefer stock B. b.
E (Rp ) = 70% x E (Ra ) + 30% x E (Rb ) = 70% x 8% + 30% x 9% = 8.3%
~ ~
σ P = x12σ 12 + (1 − x1 ) 2 σ 22 + 2 x1 (1 − x1 ) cov( R1 , R2 ) = [(0.7)2(0.098)2 + (0.3)2(0.12)2 + (2) (0.7)(0.3)(0.6*0.098*0.12) ]1/2 = (0.0089655)1/2 = 9.47% c.
β P =0.7*0.784+0.3*0.24=0.62
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Weighted Average Cost of Capital (WACC) Problem 5. (Ross, Westerfield & Jaffe) The equity beta of Adobe Online Company is 1.29. Adobe online has a debt-to-equity ratio of 1. The expected return on the market is 13%. The risk free rate is 7%. The cost of debt capital is 7%. The corporate tax rate is 35%.
(a) What is Adobe Online’s cost of equity? (b) What is Adobe Online’s weighted average cost of capital? Solution:
(a) Cost of Equity: 7 + 1.29 (13 – 7) = 14.74% (b) B/(S+B) = S/(S+B) = 0.5 WACC = 0.5(7)(0.65) +0.5(14.74) = 9.645%
(*) Problem 6. (Ross, Westerfield & Jaffe) Calculate the WACC of the Luxury Porcelain Company. The book value of Luxury’s outstanding debt is $60 million. Currently debt is trading at 120 percent of book value and is priced to yield 12 percent. The 5 million outstanding shares of Luxury stocks are selling for $20 per share. The required return on Luxury stock is 18 percent. The tax rate is 25 percent.
Solution:
B = $60million * 1.2 = $72million S = $20*5million = $100 million B/(S+B) = 72 / 172 = 0.4186 S/(S+B) = 100/172 = 0.5814 WACC = 0.4186(12%)(0.75) + 0.5814(18%) = 14.23%
(*) Problem 7. (Ross, Westerfield & Jaffe) Calgary Industries, Inc., is considering a new project that costs $25 million. The project will generate after-tax (year-end) cash flows of $7 million for 5 years. The firm has a debt-to-equity ratio of 0.75. The cost of equity is 15% and the cost of debt is 9%. The corporate tax rate is 35%. It appears that the project has the same risk as that of the overall firm. Should Calgary take on the project?
Solution:
B/S = 0.75 B/(S+B) = 3/7 S/(S+B) = 4/7
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WACC = (4/7)15% + (3/7)(9%)(1-0.35) = 11.08% 5
NPV = −$25million + ∑ t =1
$7million
(1 + 0.1108)t
= $819,299.04 Therefore, the project should be undertaken.
(*) Problem 8. (Ross, Westerfield & Jaffe) National Electricity Company (NEC) is considering a $20 million modernization expansion project in the power station division. Tom Edison, the CFO, has evaluated the project; he determined that the project’s after-tax cash flow will be $8 million, in perpetuity. In addition, Mr. Edison has devised two possibilities for raising the necessary $20 million: • •
Issue 10-year, 10% debt; Issue common stock.
NEC’s cost of debt is 10% and its cost of equity is 20%. The firm’s target debt-equity ratio is 200%. The expansion project has the same risk as the existing business, and it will support the same amount of debt. NEC is in the 34% tax bracket. Mr. Edison has advised the company to undertake the expansion. He suggests they use debt to finance 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 of the expense of the debt? Why or why not? Solution:
(a) NEC’s debt equity ratio is 2. That means for every dollar of equity the firm has, it has two dollars of debt. The debt to value ratio of the firm is B/(S+B) which is equal to 2/(2+1) = 2/3. The equity to value ratio is 1/3. Thus, the WACC is: WACC = (2/3)(0.10)(1-0.34) + (1/3)(0.20) = 0.1107 Thus, NPV = -$20million + ($8million/0.1107) = $52,267,389.34 Therefore, NEC should accept the project.
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(b) Mr. Edison’s conclusion is incorrect. Even though the issuing costs of debt are far lower than those of equity, the firm must try and maintain its optimal capital structure. Thus, anytime all debt financing is used, the firm will have to issue more equity in the future to bring the capital structure back to the optimal.
Revision Problems Problem 9. (Ross, Westerfield & Jaffe) You are saving for the college education of 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 $21,000 per year per child. The annual interest rate is 15%, and it will remain 15% throughout the next 25 years. How much money must you place in an account each year to fund your children’s educations? You will begin payments one year from today. You will discontinue payments when your oldest child enters college. Assume that the duration of the college degree is 4 years for each child and that the fees are due and are paid at the beginning of the college year.
Solution :
You have to pay $21,000 each year for 4 years, for each of your two children. The first 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 fees are due and paid at the beginning of the college year. Given an annual interest rate of 15% which is expected to remain constant through the next 25 years, you are required to calculate the yearly payments (the first starting one year from now) that you need to invest so as to be able to pay for 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’ from now) The first thing you need to do is to calculate the present value of your total financial obligation, as follows : PV (first child’s fees) =
$21,000 * PVAnnuityFactor (4 yrs,15%)
PV (second child’s fees) =
14
(1.15)
= $8,473.36
$21,000 * PVAnnuityFactor (4 yrs,15%) 16
(1.15)
= $6,407.08
PV of total obligation = $8,473.36 + $6,407.08 = $14,880.44
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The next step is to find the annuity payments starting one year from today. Let the unknown annuity payment be $X, then $14,880.44 = $X * PV Annuity Factor (15 years, 15%) = $X * 5.8474 $14,880.44 Re-arranging, $X = = $2,544.80 5.8474 (*) Problem 10. (Ross, Westerfield & Jaffe) After extensive medical and marketing research Pill, Inc. believes it can penetrate the pain reliever market. It can follow one of two strategies. The first is to manufacture 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 of 4 $ per package in real terms. The broader remedy would probably sell 10 million packages a year. This is twice the sales rate for the headache-only medication. Cash costs of production in the first year are expected to be 1.50$ per package in real terms for the headache-only brand. Production costs are expected to be 1.70$ in real terms for the more general pill. All prices and costs are expected to rise at the general inflation rate of 5%.
Either strategy will require further investment in plant. The headache-only pill could be produced using equipment that would cost 10.2 $ million, last three years and have no resale value. The machinery required to produce the cure-all would cost 12$ million and last three years. At this time the firm would be able to sell it for $1 million in real terms. The production machinery would need to be replaced every three years at constant real costs. Suppose that for both projects the firm will use straight-line depreciation. The firm faces a corporate tax rate of 34%. The firm believes the appropriate real discount rate is 13%. Capital gains are taxed at the ordinary corporate tax rate of 34%. Which pain reliever should the firm produce?
Solution: We will solve the question using nominal quantities. The nominal discount rate is (1 + inflation rate) x (1 + real rate) = 18.65% Headache only After tax operating income (using nominal prices and costs) is
qty price revenue costs/unit total cost op. income tax NOPLAT
1 5000000 4.2 21000000 1.575 7875000 13125000 4462500 8662500
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2 5000000 4.41 22050000 1.65375 8268750 13781250 4685625 9095625
3 5000000 4.6305 23152500 1.736438 8682188 14470313 4919906 9550406
7
The total PV comes out to be: PV = 8662500/1.1865 + 9095625/(1.1865)^2 + 9550406/(1.1865)^3 = $19,479,508.93 Depreciation tax shield: In nominal terms, it is expected to be (10.2 million/3)*(0.34) = $1,156,000 Its total PV is $2,487,521.65 The NPV of the project, therefore, is NPV = - $10,200,000 + $19,479,508.93 + $2,487,521.65 = $11,767,030 Headache & Arthritis only After tax operating income (using nominal prices and costs) is
qty price revenue costs/unit total cost op. income tax NOPLAT
1 10000000 4.2 42000000 1.785 17850000 24150000 8211000 15939000
2 10000000 4.41 44100000 1.87425 18742500 25357500 8621550 16735950
3 10000000 4.6305 46305000 1.967963 19679625 26625375 9052628 17572748
Total PV comes out to be $35,842,296.44 Depreciation tax shield: In nominal terms, it is expected to be (12 million/3)*(0.34) = $1,360,000 Its total PV is $2,926,495.488 Revenue from the machine sale: The machine would sell for $1,000,000 (1.05)3 = $1,157,625. After tax value = $1,157,625(0.66) = $764,032.5 PV = $764,032.5/(1.1865^3) = $457,413.1071 NPV of this option is therefore: -$12,000,000 + $35,842,296.44 + $2,926,495.488 + $457,413.1071 = $27,226,205.03 The firm should choose to manufacture the Headache & Arthritis drug.
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