Problem Set 9-Solutions PDF

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Econ 136: Financial Economics Problem Set #9 – Solutions Due Date: April 23, 2015 1. In the spreadsheet Markowitz-01.xlsx some of the entries in the long/short and longonly portfolio data sections are missing (the missing data locations are highlighted in yellow). Use Solver to replace this data: (a) The mean excess return, standard deviation, and portfolio weights for a long-short portfolio with an expected excess return of 0.03. (b) The mean excess return, standard deviation, and portfolio weights for the optimum (maximum Sharpe ratio) long-only portfolio. (c) The mean excess return, standard deviation, and portfolio weights for a long-only portfolio with an expected excess return of 0.045. Briefly explain how you could have obtained this answer from the input data for each country without running an optimization. The solutions I obtained are given in the table below. There may be some variation from these results if you started with different initial weights. Such variations are to be expected in a multidimensional optimization problem. Long/Short Long Only E[rport ] = 0.03 Optimum E[rport] = 0.045 E[rport ] 0.0300 0.0575 0.0450 σport 0.1187 0.1402 0.1879 US 0.5771 0.6885 0.0000 UK 1.2130 0.0521 0.0000 France -0.2438 0.0000 0.0000 Germany -0.7711 0.0000 0.0000 Australia 0.0535 0.1336 0.0000 Japan 0.2408 0.1258 1.0000 Canada -0.0694 0.0000 0.0000

The solution to question 1c could have been solved by inspection. Since the Japanese index has the lowest expected return of any of the indices in the portfolio and the portfolio is constrained to be long only, the only possible portfolio solution is that of being completely invested in Japan.

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2. Calculate the expected return of Emerson Electric (EMR) using the mulifactor model given the following estimates for the factor betas β and risk premiums RP = (rfactor − rf ) if the risk-free rate rf is 1.83%. Macroeconomic Factor Measure Level of interest rates T-bond rate Term structure steepness T-bond rate – T-bill rate Inflation rate Consumer Price Index (CPI) Economic growth rate Gross National Product (GNP)

β RP 0.3 1.7 1.4 0.5 1.3 1.9 1.6 3.2

The multifactor model for the expected return on stock j is straightforward generalization of the CAPM and can be written as E(rj ) = rf +

N X

βi × RPi ,

(1)

i=1

where rf is the risk-free rate. In the case of Emerson Electric this becomes E(rEMR ) = rf +

4 X

βi × RPi

(2)

i=1

= 1.83 + 0.3 × 1.7 + 1.4 × 0.5 + 1.3 × 1.9 + 1.6 × 3.2

(3)

= 1.83 + 8.80 = 10.63%.

(4)

3. Given the following equation from the Fama & French study of returns between 1963 and 1990 for monthly return ri = 1.77 − 0.11 ln (MV) + 0.35 ln (BV/MV) where MV is the market value of equity in hundreds of millions of dollars and BV is the book value of equity in hundreds of millions of dollars. (a) What is the expected annual return on Lucent Technologies if its market value equity is $132 billion and its book value of equity is $74.7 billion. This is relatively straightforward, although we need to be careful with the units. This equation is for em monthly returns when the MV is expressed in hundreds of millions of dollars (HM). Taking the latter point first, we note that $132 B = $132 × 109 which, expressed in units of “hundreds of millions” (HM) or 108 is $1320 HM. With this we have that (m)

rLT = 1.77 − 0.11 ln (1320) + 0.35 ln (0.5659) = 0.80 , percent per month. Converting this to an annual rate 12  (a) (m) − 1 = (1.0080)12 − 1 = 0.100 or 10.0%. rLT = 1 + rLT 2

(5)

(6)

(b) Lucent Technologies has a beta of 1.2. If the riskless rate is 1.8% and the risk premium for the market portfolio is 4.5%, what is the expected return? Using CAPM the expected return is E(rLT) = rf + βLT [E(rm ) − rf ] = 1.8% + 1.2 [4.5%] = 7.20% .

(7)

4. Given a market index trading at $1052 where the average dividend yield of the stocks in the index is 5.85% and where earnings and dividends are expected to grow at 3.15% per year forever. If the market risk premium is 4.63% and the risk-free rate is 1.46% what is the value of the index according to the Gordon growth model? Since the market index is the market it has a β of 1.0. With this the value according to the Gordon growth model is: D0 (1 + g) . (8) V0 = (E(rIndex ) − g) We are given the growth rate, but need to calculate the dividend amount and the cost of capital. The problem gives us the dividend as a dividend yield. To convert this into a dividend amount: D0 = (dividend yield) × (value of the index) = 0.0585 × $1052 = $61.54 .

(9)

For the cost of equity we use the CAPM: E (rIndex ) = rf + β [E (rm ) − rf ] = 1.46% + 1.0 × 4.63% = 6.09% , | {z }

(10)

market risk premium

and the value of the index follows as V0 =

D0 (1 + g) $61.54 (1.0315) = $2159.13 . = (0.0609 − 0.0315) (E(rIndex ) − g)

(11)

5. BP PLC (NYSE: BP) has a current stock price of $35 and current dividend of $1.35. The dividend is expected to grow at 3.45% annually. BP’s beta is 0.89. The risk-free rate is 1.5%, and the market risk premium is 4.5%. (a) What is next year’s projected dividend? The projected dividend D1 is D1 = D0 (1 + g) = $1.35 × 1.0345 = $1.3966 or $1.40 .

(12)

(b) What is BP’s cost of equity capital based on the CAPM? The cost of equity capital rBP is rBP = rf + β [E(rm ) − rf ] = 1.5% + 0.89 [4.5%] = 1.5% + 4.01% = 5.51% . (13)

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(c) Using the Gordon growth model, what is the value of BP? $1.40 $1.40 D1 = = = $67.96 . (14) rBP − g 0.0551 − 0.0345 0.0206 (d) Assuming the Gordon growth model is valid, what dividend growth rate would result in a model value of BP equal to it’s market price? V0 =

Beginning with the Gordon growth model D0 (1 + g) V0 = (rBP − g) V0 (rBP − g) = D0 (1 + g) V0 rBP − V0 g = D0 + D0 g

(15) (16) (17)

V0 rBP − D0 = g (D0 + V0 ) $35.0 × 0.0551 − $1.35 V0 rBP − D0 = g= (D0 + V0 ) $1.35 + $35.0 = 0.0159 or 1.59% .

(18) (19) (20)

6. For three utility stocks, the table below provides the expected dividend for the next year, the current market price, the expected dividend growth rate, and the beta. The risk-free rate is currently 2.11%, and the market risk premium is 3.2%. Stock American Electric (NYSE: AEP) Exelon Corp. (NYSE: EXC) Dominion Resources (NYSE: D)

D1 1.60 1.79 2.48

V0 45.17 66.12 60.15

g(%) 4.0 5.5 5.7

β 0.50 0.80 0.65

(a) Calculate the expected rate of return using the Gordon growth model. In the Gordon growth model the expected rate of return is ri = D1 /V0 + g : rAEP = 1.60/45.17 + 4.0% = 3.54% + 4.0% = 7.54%

(21)

rEXC = 1.79/66.12 + 5.5% = 2.71% + 6.5% = 8.21% rD = 2.48/60.15 + 5.7% = 4.12% + 4.7% = 9.82%

(22) (23)

(b) Calculate the required rate of return using the CAPM. In the CAPM the expected rate of return is ri = rf + βi [E(rm ) − rf ]: rAEP = 2.11% + 0.50 × 3.2% = 2.11% + 1.60% = 3.71% rEXC = 2.11% + 0.80 × 3.2% = 2.11% + 2.56% = 4.67%

(24) (25)

rD = 2.11% + 0.65 × 3.2% = 2.11% + 2.08% = 4.19%

(26)

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7. EverGrow corporation has a beta of 1.10 just paid a dividend of $1.45. The risk-free rate is 1.45% and the expected market return is 6.55%. Your growth analysts have concluded that the growth rate earnings per share and dividends will be 6.5% per year for the next three years and 3.2% per year thereafter. Calculate the value of EverGrow using the 2-stage dividend discount model. The 2-stage dividend discount model for this problem can be written 3 X Dt V3 . (27) V0 = t + (1 + r)3 t=1 (1 + r) Since we are not given the cost of equity capital r, but are given the needed data for a CAPM calculation, we proceed using the CAPM: r = rf + β [E(rm ) − rf ] = 1.45% + 1.1 (6.55% − 1.45%) = 7.06% . (28) The estimated future dividends are D1 = $1.45 × 1.065 = $1.54

(29)

D2 = $1.54 × 1.065 = $1.65 D3 = $1.65 × 1.065 = $1.75 D4 = $1.75 × 1.032 = $1.80

(30) (31) (32)

The terminal stock price comes from the forward Gordon growth model: $1.81 D4 = V3 = = $46.89 . r − g 0.0706 − 0.032 The present values of the first three dividends and the terminal value are $1.64 $1.75 $46.83 $1.54 + + + V0 = 2 3 1.0706 1.0706 1.0706 1.07063 = $1.44 + $1.44 + $1.43 + $38.16 = $42.47 . Alternatively, we can use the equation from Slide 17 of Lecture 21: "    T   #  1 + gh T 1 + gs 1 + gh 1 + gh , V0 = D0 1 − + D0 r h − gh r s − gs 1 + rh 1 + rh {z } | {z } |

(33)

(34) (35) (36)

(37)

stable growth state

high growth phase

with D0 = $1.45, gh = 0.065, gs = 0.032 and rh = rs = 0.0706 as used above and T = 3 since the the high-growth phase is for 3 years. Substituting we obtain "    3    # 1.065 3 1.065 1.032 1.065 V0 = $1.45 1 − + $1.45 , 0.0706 − 0.065 1.0706 0.0706 − 0.032 1.0706 (38) = $4.30 + $38.16 = $42.46

(39)

The difference between the values is due to rounding. If you do either in a spreadsheet you will get $42.47. 5

8. Referring to the financial statement for MasterToy shown in Table 1 below: (a) Identify the three components of the DuPont formula. ROE = (Profit Margin ) (Asset Turnover) (Financial Leverage), or ROE = (Net Income / Sales) (Sales / Assets) (Assets / Equity) (b) Calculate the ROE for 1998 using the three components of the DuPont formula. ROE = (475 / 4,750) (4,750 / 2,950) (2,950 / 2,100) = 22.62% . (c) Calculate the sustainable growth rate for 1998. g = b × ROE = (1 − dividend payout ratio) × ROE (40)   475 0.55 × = 15.67% . (41) = 1− 1.79 2, 100 Table 1: MasterToy Inc. Actual 1998 and Estimated 1999 Financial Statements for FY Ending December 31 ($ millions except per-share data) 1998

1999

Income Statement Revenue (sales)

$4,750

$5,140

Cost of goods sold Selling, general, and administrative Depreciation Goodwill amortization Operating income

$2,400 1,400 180 10 $760

$2,540 1,550 210 10 $830

Interest expense Income before taxes Income taxes Net income

20 $740 265 $475

25 $805 295 $510

Earnings per share Average shares outstanding (millions)

$1.79 265

$1.96 260

Balance Sheet Cash Accounts receivable Inventories Net property, plant, and equipment Inangibles Total Assets

$400 680 570 800 500 $2,950

$400 700 600 870 530 $3,100

Current liabilities Long-term debt Total Liabilities Stockholder’s equity Total liabilities and equity

$550 300 $850 2,100 $2,950

$600 300 $900 2,200 $3,100

$7.92 $0.55

$8.46 $0.60

Book value per share Annual dividend per share

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