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HW3sS09 - Solution Key to Problem Set 3 ECN 134 Finance...

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Solution Key to Problem Set 3 ECN 134 Finance Economics Prof. Farshid Mojaver Stock Valuation 1 1. 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 = (D 1 / P 0 ) + g = ($3.10 / $48.00) + .05 = 11.46% 2. Using the constant growth model, we find the price of the stock today is: P 0 = D 1 / (k g ) = $3.60 / (.13 – .045) = $42.35 3. 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 yield Now we know both the dividend yield and capital gains yield. The dividend is simply the stock price times the dividend yield, so: D 1 = .06($70) = $4.20 This 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: D 1 = D 0 (1 + g ) We can solve for the dividend that was just paid: $4.20 = D0 (1 + .06) D 0 = $4.20 / 1.06 = $3.96 4. 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: P 0 = $12.00(PV 10%,8 ) = $64.02 5. 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) = 43.1 Similarly, the stream must be worth 50/(1+0.16) 2 = 37.16 in year one, and 50/(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.
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Dividend in 1 year: 100*(1.14) = 114.0 Dividend in 2 years: 100*(1.14) 2 = 129.96 Dividend in 10 years: 100*(1.14) 10 = 370.72 Dividend in 20 year: 100*(1.14) 20 = 1374.3 Dividend in 21 year: 100*(1.14) 20 (1.06) 1 = 1456.8 Note: These are the actual dividends paid in the corresponding years, not their PV. ii) Use growing annuity formula: + + - - = T r g g r C Annuity Growing PV 1 1 1 1 ) ( 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 . 2421 237 . 21 114 12 . 0 1 14 . 0 1 1 14 . 0 12 . 0 114 1 1 1 20 = × = + + - - = + + - - T r g g r C iii) Use the growing perpetuity formula: g r C Annuity Growing PV - = ) ( 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: 20 12 . 1 1 PV of the growing perpetuity: 03 . 2517 0 . 24280 103667 . 0
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