Solution Key to Problem Set 4 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 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 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 D1 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 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 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 P0 12 00 PV10 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 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 T C 1 g ii Use growing annuity formula PV Growing Annuity 1 1 r g 1 r 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 C r g 1 g T 114 1 1 r 0 12 0 14 1 0 14 20 114 21 237 2421 02 1 1 0 12 C iii Use the growing perpetuity formula PV Growing Annuity r g 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 1 1 12 20 year 0 by multiplying by PV of the growing perpetuity c 1 r g 1 r 20 1456 8 1 0 12 0 06 1 12 20 0 103667 24280 0 2517 03 2421 02 2517 03 49 3805 This is the Differential Growth Factor As stated 100 iv in 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 billion This 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 billion Pretty 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 10 114 1 0 14 114 0 9 6813 1103 67 1 0 12 0 14 1 0 12 Note dividends in year 11 will be 100 1 14 10 1 06 392 965 392 965 1 PV of growing perpetuity 0 12 0 06 1 12 10 0 321973 6549 42 2108 74 Total PV 1103 67 2108 74 3212 41 Differential Growth Factor 3212 41 32 1241 100 Value or fair price of stock 2 657 32 1241 85 3537 billion Much 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 RoM 0 6 1 045 t 1 v RoM s dividends are given by the expression Dt 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 5 years 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 dividends