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UCD ECN 134 - HW4s-S10

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Solution Key to Problem Set 4ECN 134Finance Economics Prof. Farshid Mojaver Stock Valuation 11. We need to find the required return of the stock. Using the constant growth model, we can solve the equation for k. Doing so, we find:k = (D1 / P0) + g = ($3.10 / $48.00) + .05 = 11.46%2. Using the constant growth model, we find the price of the stock today is:P0 = D1 / (k – g) = $3.60 / (.13 – .045) = $42.353. We know the stock has a required return of 12 percent, and the dividend and capital gains yield are equal, so:Dividend yield = 1/2(.12) = .06 = Capital gains yieldNow we know both the dividend yield and capital gains yield. The dividend is simply the stock price times the dividend yield, so:D1 = .06($70) = $4.20This is the dividend next year. The question asks for the dividend this year. Using the relationship between the dividend this year and the dividend next year:D1 = D0 (1 + g)We can solve for the dividend that was just paid:$4.20 = D0 (1 + .06)D0 = $4.20 / 1.06 = $3.964. The price of any financial instrument is the PV of the future cash flows. The future dividends of this stock are an annuity for eight years, so the price of the stock is the PVA, which will be:P0 = $12.00(PV10%,8) = $64.025. i) Suppose we were in year three, then use the perpetuity formula:8/0.16=50. This is the value of the stream in year three.ii) Then the same stream must be additionally discounted by 1/(1+r) in year two (discount one): 50/(1+0.16) = 43.1Similarly, the stream must be worth50/(1+0.16)2 = 37.16 in year one, and50/(1+0.16)3 = 32.04 in year zero.In year four, the ex-dividend price will be 50 again.6. i) The dividends grow by 14% for the next 20 years, and then by 6% every year after that, forever:In 1996 the dividend was $100. Note: We do not count this in our PV calculations, we only use this as a reference point from which we make our calculations.Dividend in 1 year: 100*(1.14) = 114.0Dividend in 2 years: 100*(1.14)2 = 129.96Dividend in 10 years: 100*(1.14)10 = 370.72Dividend in 20 year: 100*(1.14)20 = 1374.3Dividend in 21 year: 100*(1.14)20(1.06)1 = 1456.8Note: These are the actual dividends paid in the corresponding years, not their PV.ii) Use growing annuity formula: TrggrCAnnuityGrowingPV1111)(Note: This give us the PV of the growing annuity the year before the payments start. In this case, the dividends start in year 1, so the formula will give us the value in year 0, which is what we want.02.2421237.2111412.0114.01114.012.011411120TrggrC iii) Use the growing perpetuity formula: grCAnnuityGrowingPV)(Note: The formula gives the PV for the period before the first payment of the growing perpetuity. In our problem, the growing perpetuity starts in year 21, so the formula will give us the value of the growing perpetuity in year 20. Thus, to get the PV in year 0, we must further discount the value from the formula, which is given in year 20 dollars, to year 0 by multiplying by: 2012.11PV of the growing perpetuity:03.25170.24280103667.012.1106.012.08.1456112020 rgrciv) 3805.4910003.251702.2421. This is the “Differential Growth Factor”. As statedin the problem, you can use this number to multiply by 1996 dividends to get the PV of the stock.v) Given the answer to the last part of this question, we multiply the 1996 dividends by the “Differential Growth Factor” to get the total PV of Coca-Cola stock.1.25*49.3805 = $ 61.7256 billionThis is less than half of the market value!vi) In this part, we use “gross dividends” in our stock valuation procedure. Here the fair price turns out to be: 2.657*49.3805 = $ 131.204 billionPretty close approximation!vii) Now we recomputed the PV of the dividend stream, then recompute the “Differential Growth Factor”, and finally recompute the value of the stock using “gross dividends”:PV of growing annuity: 67.11036813.90.11412.0114.01114.012.011410Note: dividends in year 11 will be: 100*(1.14)10*(1.06) = 392.965PV of growing perpetuity: 74.210842.6549321973.012.1106.012.0965.39210 Total PV: 1103.67 + 2108.74 = 3212.41Differential Growth Factor: 1241.3210041.3212Value or “fair price” of stock: 2.657 * 32.1241 = $85.3537 billionMuch less than market value!viii) 2 out of ounces of total fluid intake is a very large number. Indeed, Coca-Cola has already 48% markets share of the world soft-drink market. How many more soft-drinks will people ever drink? While the above calculations indicate that Coca-Cola will need to sell a lot of more soft-drinks in the future to justify its current share price, not that much room for growth seems to be left. Could it be that Wall Street bets on rapid global warming??7. i) Bovine’s fair stock price is 2.40/0.06 = $40 .ii) Since RoM invests at its opportunity cost of capital, its fair stock price must also be $40.iii) RoM’s dividends next year will be 0.25 × 2.40 = $0.60.iv) RoM’s dividends will grow at the rate g = ROE.b = 0.06 × 0.75 = 0.045.v) RoM’s dividends are given by the expression)1(045.16.0tRoMtD.To surpass Bovine Cash’s dividends, they will need to quadruple over time; that is, dividends will need to double twice. At g = 4.5%, it takes approximately 0.7/0.045 ≈ 15.5years to double, and thus 31 years to quadruple. Hence RoM will surpass Bovine Cash’s dividends in about 32 years from now.Remark: the exact solution is found by solving the equation 0.6 × 1.045(t−1) = 2.4.vi) The PV of all future dividends is 0.6/(0.06−0.045) = $40. This must come out to the same value as ii), since the fair stock price is always equal to the PV of all future


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