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**Solutions** to **Chapter** Valuing Stocks No The dividend discount model allows for the fact that firms may not currently pay dividends As the market matures, and Rogers Wireless Communication’s growth opportunities moderate, investors may justifiably believe that Rogers Wireless will enjoy high future earnings and will pay dividends then The stock price today can still reflect the present value **of** the expected per share stream **of** dividends Dividend yield = Expected dividend/Price = DIV1/P0 So: P0 = DIV1/dividend yield P0 = $2.4/.08 = $30 a The typical preferred stock pays a level perpetuity **of** dividends The expected dividend next year is the same as this year’s dividend, $7 Thus the dividend growth rate is zero and the price today is: P0 = D1/r = 7/.12 = $58.33 b The expected dividend in two years is this year’s dividend, $7 P1= D2/r = 7/.12 = $58.33 c Dividend yield = $7/$58.33 = 12 =12% Expected capital gains = Expected rate **of** return = 12% r = DIV1/P0 + g = 8% + 5% = 13% The value **of** a common stock equals the present value **of** dividends received out to the investment horizon, plus the present value **of** the forecast stock price at the horizon But the stock price at the horizon date depends on expectations **of** dividends from that date forward So even if an investor plans to hold a stock for only a year for two, the price ultimately received from another investor depends on dividends to be paid after the date **of** purchase Therefore, the stock’s present value is the same for investors with different time horizons 6-1 Copyright © 2006 McGraw-Hill Ryerson Limited a P0 = r = + g r = + 04 = 14 = 14% b P0 = 2.50/(.165 04) = $20 The dividend yield is defined as the annual dividend (or the annualized current dividend) divided by the current price The current annual dividend is ($2 4) = $8 and the dividend yield is: DIV1/P0 = 048 $8/ P0 = 048 P0 = $8/.048 = $166.7 To work with the quarterly dividend, divide the dividend yield by and repeat the above steps: Quarterly DIV/P0 = 048/4 = 012 $2/ P0 = 012 P0 = $2/.012 = $166.7 Weak, semi-strong, strong, fundamental, technical True The search for information and insightful analysis is what makes investor assessments **of** stock values as reliable as possible Since the rewards accrue to the investors who uncover relevant information before it is reflected in stock prices, competition among these investors means that there is always an active search on for mispriced stocks 10 a DIV1 = $1 1.04 = $1.04 DIV2 = $1 1.042 = $1.0816 DIV3 = $1 1.043 = $1.1249 b P0 = DIV1/(r g) = = $13 c P3 = DIV4/(r g) = = $14.6237 6-2 Copyright © 2006 McGraw-Hill Ryerson Limited d Your payments are: DIV Sales Price Total cash flow PV **of** cash flow Year 1.04 Year 1.0816 1.04 1.0816 Year 1.1249 14.6237 15.7486 9286 8622 11.2095 Sum **of** PV = $13.00, the same as your answer to (b) 11 Dividend growth rate, g = return on equity × plowback ratio: g = 15 40 = **06** r = + g = + **06** = 16 = 16% 12 a P0 = = = $21 P0 = = $30 The lower discount rate makes the present value **of** future dividends higher, raising the value **of** the stock 13 r = + g g = r - = 14 – = 04 = 4% 14 a r = + g = + 03 = 0926 = 9.26% b If r = 10, then 10 = 1.64(1.03)/27 + g c g = Return on equity plowback ratio 5% = Return on equity Return on equity = = = 12.5% So g = 0374 = 3.74% 15 P0 = DIV1/(r g) = $2/(.12 – 06) = $33.33 16 a P0 = DIV1/(r g) = 3/[.15 – (.10)] = 3/.25 = $12 b P1 = DIV2/(r g) = 3(1 10)/.25 = $10.80 6-3 Copyright © 2006 McGraw-Hill Ryerson Limited c return = = = 150 = 15.0% d “Bad” companies may be declining, but if the stock price already reflects this fact, the investor still can earn a fair rate **of** return, as we saw in part c 17 a (i) reinvest 0% **of** earnings: g = and DIV1 = $5: P0 = = = $33.33 (ii) reinvest 40%: g = 15% 40 = 6% and DIV1 = $5 (1 – 40) = $3: P0 = $3/(.15 – 06) = $33.33 (iii) reinvest 60%: g = 15% 60 = 9% and DIV1 = $5 (1 – 60) = $2: P0 = 2/(.15 – 09) = $33.33 b (i) reinvest 0%: P0 = 5/(.15 – 0) = $33.33 PVGO = $0 (ii) reinvest 40%: P0 = = $42.86 PVGO = $42.86 – $33.33 = $9.53 (iii) reinvest 60%: P0 = = $66.67 PVGO = $66.67 – $33.33 = $33.34 c In part (a), the return on reinvested earnings is equal to the discount rate Therefore, the NPV **of** the firm’s new projects is zero, and PVGO is zero in all cases, regardless **of** the reinvestment rate While higher reinvestment results in higher growth rates, it does not result in a higher value **of** growth opportunities This example illustrates that there is a difference between growth and growth opportunities In part (b), the return on reinvested earnings is greater than the discount rate Therefore, the NPV **of** the firm’s new projects is positive, and PVGO is positive PVGO is higher when the reinvestment rate is higher in this case, since the firm is taking greater advantage **of** its opportunities to invest in positive NPV projects 6-4 Copyright © 2006 McGraw-Hill Ryerson Limited 18 Stock exchange information Tips: When accessing stock exchange information at www.fibv.com, you don’t actually click on “Member” but rather place your cursor on the word You click on the region and stock exchange **of** interest Also, remind the students to click on the symbol **of** the stock exchange to connect to the Web site **of** that exchange Expected results: Students learn about variety **of** stock exchanges around the world and the types **of** information provided 19 Hollywood stock exchange Expected result: A fun experience trading securities 20 a P0 = + + = 18.10 b DIV1/P0 = $1/18.10 = 0552 = 5.52% 21 a b c 22 Stock A Stock B Payout ratio $1/$2 = 50 $1/$1.50 = 67 g = ROE plowback 15% = 7.5% 10% 333 = 3.33% Price = DIV1/(r g) = $14.33 = $8.85 Note: We interpret “recent” to mean in the past The current stock price depends on future dividends – so the next dividend must be + g times higher a ROE plowback ratio = 20% = 6% b E = $2, plowback ratio = 3, r = 12, g = **06** P0 = = $23.33 c No-growth value = E/r = $2/.12 = $16.67 PVGO = P0 no-growth value = $23.33 $16.67 = $6.66 d P/E = 23.33/2 = 11.665 e If all earnings were paid as dividends, price would equal the no-growth value, $16.67, and P/E would be 16.67/2 = 8.335 23 f High P/E ratios reflect expectations **of** high PVGO a =$60 b No-growth value = E/r = $6.20/.12 = $51.67 PVGO = P0 no-growth value = $60 51.67 = $8.33 6-5 Copyright © 2006 McGraw-Hill Ryerson Limited 24 a 25 Earnings = DIV1 = $4 Growth rate g = P0 = = $33.33 P/E = 33.33/4 = 8.33 b If r = 10, P0 = = 40, and P/E increases to 40/4 = 10 A decrease in the required rate **of** return, holding dividends constant, raises the stock price and the P/E ratio a Plowback ratio = implies DIV1 = $3 and g = Therefore, P0 = = $30 and the P/E ratio is 30/3 = 10 b Plowback ratio = 40 implies DIV1 = $3(1 – 40) = $1.80, and g = 10% 40 = 4% Therefore P0 = $1.80/(.10 – 04) = $30 and the P/E ratio is 30/3 = 10 c Plowback ratio = 80 implies DIV1 = $3(1 – 80) = $.60, and g = 10% 80 = 8% Therefore P0 = $.60/(.10 – 08) = $30 and the P/E ratio is 30/3 = 10 Regardless **of** the plowback ratio, the stock price = $30 because all projects offer return on equity just equal to the opportunity cost **of** capital 26 27 a P0 = DIV1/(r g) = $5/(.10 – 06) = $125 b If Trendline followed a zero-plowback strategy, it could pay a perpetual dividend **of** $8 Its value would be $8/.10 = $80, and therefore, the value **of** assets in place is $80 The remainder **of** its value must be due to growth opportunities, so PVGO = $125 – $80 = $45 a g = 20% 30 = 6% DIV1 = $2(1 – 30) = $1.40 P0 = DIV1/(r g) = $1.40/(.12 06) = $23.33 P/E = 23.33/2 = 11.665 6-6 Copyright © 2006 McGraw-Hill Ryerson Limited b If the plowback ratio is reduced to 20, g = 20% 20 = 4% DIV1 = $2(1 – 20) = $1.60 P0 = DIV1/(r g) = $1.60/(.12 – 04) = 20 P/E = 20/2 = 10 P/E falls because the firm’s value **of** growth opportunities is now lower: It takes less advantage **of** its attractive investment opportunities 28 29 c If the plowback ratio = 0, g = 0, and DIV1 = $2, P0 = $2/.12 = 16.67 and E/P = 2/16.67 = 12 a DIV1 = 2.00 DIV2 = 2(1.20) = 2.40 DIV3 = 2(1.20)2 = 2.88 b This could not continue indefinitely If it did, the stock would be worth an infinite amount Another way to think about the feasible perpetual growth rate is to compare the company’s growth rate with the growth rate **of** the economy The economy grows about 3% a year To grow faster than the economy as a whole is feasible when the company is small However, to continue to grow at 20%, the company must take over other companies and eventually become the entire economy But in the long run, it still can only grow as quickly as the entire economy So it is impossible to grow at 20% in perpetuity Think about Microsoft – it has had phenomenal growth partly by acquiring other companies and partly by growing its own businesses However, even if it were to own all **of** the companies in the world, eventually its growth rate would fall to the growth rate **of** the world economy We are assuming that Bill Gates is not able to successfully market his software to still to be discovered alien worlds!! Finally, note too that the constant dividend growth model fails when the assumed perpetual growth rate is greater than the discount rate a Book value = $100 million PV = 2/1.10 = 1.818 PV = 2.40/1.102 = 1.983 PV = 2.88/1.103 = 2.164 First year earnings = $100 million 24 = $24 million Dividends = Earnings (1 – plowback ratio) = $12 million g = return on equity plowback ratio = 24 50 = 12 Market value = = $400 million 6-7 Copyright © 2006 McGraw-Hill Ryerson Limited Market-to-book ratio = $400/$100 = b Now g falls to 10 50 = 05, first year earnings decline to $10 million (=$100 million × 1), and dividends decline to $5 million (=$10 million × 5) Market value = = $50 million Market-to-book ratio = ½ This makes sense, because the firm now earns less than the required rate **of** return on its investments Its project is worth less than it costs 30 P0 = + + = $16.59 31 a DIV1 = $2 1.20 = $2.40 b DIV1 = $2.40 DIV2 = $2.88 DIV3 = $3.456 P3 = = $32.675 P0 = + + = $28.02 c P1 = + = $29.825 d Capital gain = P1 P0 = $29.825 $28.02 = 1.805 r = = 15 = 15% 32 a Note: If students carry at least decimal places, the results will be clearer Also, it is easier to solve the prices in reverse order DIV1 = $.5 DIV2 = $.5 DIV4 = $.5 × 1.04 = $.52 DIV3 = $.5 DIV5 = $.5 × 1.042 = $.5408 P4 = = = $7.7257 P3 = = = $7.4286 P2 = = = $7.1429 P1 = = = $6.8855 P0 = = = $6.6536 b Year 6-8 Copyright © 2006 McGraw-Hill Ryerson Limited Dividend yield = = = 07515 Capital gains yield = = = 03485 Dividend yield + capital gains yield = 07515 + 03485 = 11 Year Dividend yield = = = 07262 Capital gains yield = = = 03738 Dividend yield + capital gains yield = 07262 + 03738 = 11 Year Dividend yield = = = 0700 Capital gains yield = = = 0400 Dividend yield + capital gains yield = 07 + 04 = 11 Year Dividend yield = = = 0700 Capital gains yield = = = 0400 Dividend yield + capital gains yield = 07 + 04 = 11 Yes, each year the sum **of** the dividend yield and the capital gains yield equal 11 percent, the required rate **of** return Once the company hits constant growth rate **of** percent, both the dividend yield and the capital gains yield also become constant 33 DIV1 = dividend payout × earnings1 = × $3 = $1.2 DIV2 = dividend payout × earnings2 = × $3 × 1.1 = $1.32 DIV3 = dividend payout × earnings3 = × $3 × 1.12 = $1.452 DIV4 = dividend payout × earnings4 = × $3 × 1.13 = $1.5972 DIV5 = dividend payout × earnings5 = × $3 × 1.14 = $1.75692 P0 = + + + + × = $13.95 34 a BCE preferred shares Expected results: Students use real data to calculate the current expected (or 6-9 Copyright © 2006 McGraw-Hill Ryerson Limited required) rate **of** return b CIBC preferred shares Expected results: Students will quickly learn that the www.globeinvestor.com does not a good job on preferred shares The dividend payouts are rounded to digits and hence not match well the actual dividends listed on the CIBC Web site 35 Expected Results: ATY-T is ATI Technologies and SRF-T is Sun-Rype Foods Emphasize to the students that they should describe the businesses **of** these companies in their own words Encourage students to think about how each company presents its products Students will look at the investor pages to see the information they provide 36 Tips: The Web site has changed since the textbook went to press Instruct students to click on “Access Public Filings” to get to the insider information General information on the insiders is found at “View Insider Information” To see current trading by the insiders, click on “View Summary Reports” Pick “insider transaction detail” and click on “next” Then identify the insider (you will see “insider family” as an option) and go to the bottom **of** the page and hit “search” On the next screen select “view” The summary report shows the insider trading **of** each security **of** various companies (“issuers”) For example, in November 2005, the first issuer listed for Frank Stronach was Decoma International Inc Mr Stronach is director or senior officer holding at least 10% **of** the Class A subordinate voting shares The report shows transactions **of** Mr Stronach Expected results: Students will see the information available and wonder whether they can trade on this – stock purchases and sales provide information **of** value to outside investors? 37 Before-tax rate **of** return: = = = 078 = 7.8% After-tax rate **of** return: = = = 0596 = 5.96% 38 a An individual can crazy things, but still not affect the efficiency **of** markets An irrational person can give assets away for free or offer to pay twice the market value However, when the person’s supply **of** assets or money runs out, the price 6-10 Copyright © 2006 McGraw-Hill Ryerson Limited will adjust back to its prior level (assuming there is no new, relevant information released by his/her action) If you are lucky enough to trade with such a person you will receive a positive gain at that investor’s expense You had better not count on this happening very often though Fortunately, an efficient market protects irrational investors in cases less extreme than the above Even if they trade in the market in an “irrational” manner, they can be assured **of** getting a fair price since the price reflects all information b Yes, and how many people have dropped a bundle? Or more to the point, how many people have made a bundle only to lose it later? People can be lucky and some people can be very lucky; efficient markets not preclude this possibility Furthermore, how much risk did they take? You expect to earn a higher return if you take on more market (beta) risk c Investor psychology is a slippery concept, more often than not used to explain price movements which the individual invoking it cannot personally explain Even if it exists, is there any way to make money from it? If investor psychology drives up the price one day, will it so the next day also? Or will the price drop to a “true” level? Almost no one can tell you beforehand what “investor psychology” will Theories based on it have no content 39 There are several thousand mutual funds in Canada and the United States With so many professional managers, it is no surprise that some managers will demonstrate brilliant performance over various periods **of** time As an analogy, consider a contest in which 10,000 people flip a coin 20 times It would not surprise you if someone managed to flip heads 18 out **of** 20 times But it would be surprising if he could repeat that performance Similarly, while many investors have shown excellent performance over relatively short time horizons, and have received favourable publicity for their work, far fewer have demonstrated consistency over long periods 40 If the firm is stable and well run, its price will reflect this information, and the stock may not be a bargain There is a difference between a “good company” and a “good stock.” The best buys in the stock market are not necessarily the best firms; instead, you want firms that are better than anyone else realizes When the market catches up to your assessment and prices adjust, you will profit 41 Remember the first lesson **of** market efficiency: The market has no memory Just because long-term interest rates are high relative to past levels doesn’t mean they won’t go higher still Unless you have special information indicating that long-term rates are too high, issuing long-term bonds should be a zero-NPV transaction So should issuing short-term debt or common stock 42 The stock price will fall The original price would reflect an anticipation **of** a 25% increase in earnings The actual increase is a disappointment compared to original expectations 6-11 Copyright © 2006 McGraw-Hill Ryerson Limited 43 50 = g = 15 – = 10 g = 10 = return on equity plowback ratio = return on equity 60 return on equity = 10/.60 = 1667 = 16.67% 44 a P0 = + DIV1 = $1 DIV2 = $2 P2 = = = $30 P0 = + = $26.40 b Next year, P1 = = = $28.57 c return = = = 12 = + = 12 45 DIV1 = $1 DIV2 = $2 DIV3 = $3 g = **06** Therefore P3 = 3(1.06)/(.14 – 06) = $39.75 P0 = + + = $31.27 46 a b DIV1 = $4, and g = 4% Expected return = DIV1/P0 + g = 4/100 + 4% = 8% Since DIV1 = Earnings (1 – plowback ratio), Earnings = DIV1/(1 – plowback ratio) = 4/(1 – 4) = $6.667 6-12 Copyright © 2006 McGraw-Hill Ryerson Limited If the discount rate is 8% (the expected return on the stock), then the no-growth value **of** the stock is 6.667/.08 = $83.34 Therefore PVGO =$100 – $83.34 = $16.66 c For the first years, g = 10% = 8% Thereafter, g = 10% = 4% Year Earnings plowback DIV g 6.67 80 1.33 08 7.20 80 1.44 08 7.78 80 1.56 08 8.40 80 1.68 08 9.07 80 1.81 08 9.80 40 5.88 04 After year 6, the plowback ratio falls to and the growth rate falls to percent [We assume g = 8% in year (i.e., from t = to t = 6), since the plowback ratio in year is still high at b = 80 Notice the big jump in the dividend when the plowback ratio falls.] By year 6, the firm enters a steadygrowth phase, and the constant-growth dividend discount model can be used to value the stock The stock price in year will be P6 = = = $152.88 P0 = + + + + + = $106.22 47 a DIV1 = 1.00 1.20 = $1.20 DIV2 = 1.00 (1.20)2 = $1.44 DIV3 = 1.00 (1.20)3 = $1.728 DIV4 = 1.00 (1.20)4 = $2.0736 b P4 = DIV5/(r g) = DIV4(1 + g)/(r g) = 2.0736(1.05)/(.10 – 05) = $43.55 c P0 = + + + = $34.74 d DIV1/P0 = 1.20/34.74 = 0345 = 3.45% e Next year the price will be: + + = $37.01 f return = = = 10 6-13 Copyright © 2006 McGraw-Hill Ryerson Limited The expected return equals the discount rate (as it should if the stock is fairly priced) 48 Before-tax rate **of** return = = = = 10% After-tax dividend: Grossed up dividend = 1.25 × = 2.50 Gross federal tax = 22 × 2.50 = 55 Federal tax credit = 1333 × 2.50 = 3333 Net federal tax = 55 - 3333 = 2167 Gross provincial tax = 119 × 2.50 = 2975 Provincial tax credit = 066 × 2.50 = 165 Net provincial tax = 2975 - 165 = 1325 Total dividend tax = 2167 + 1325 = 3492 After-tax dividend = – 3492 = 1.6508 Capital gains tax = × (.22 + 119) × (53 – 50) = 5085 After-tax capital gains = 53 – 50 – 5085 = 2.4915 After-tax rate **of** return = = 49 = 0828 = 8.28% Assume taxes not change We make the easiest reinvestment assumption: dividends are spent as they are received and not earn any interest Thus, the future value **of** dividends received is × = The selling price is $55 The before-tax rate **of** return is = () 1/3 -1= () 1/3 - = 0685 = 6.85% Annual after-tax dividends = – 3492 = 1.6508 (from question 48) Future value **of** dividends received = 1.6508 × = 4.9524 Capital gain tax = × (.22 + 119) × (55 – 50) = 8475 The after-tax rate **of** return is = () 1/3 -1 6-14 Copyright © 2006 McGraw-Hill Ryerson Limited = 50 a () 1/3 - = 0573 = 5.73% Both **of** these instruments are perpetuities Recall the price **of** a perpetuity is P0 = Rearranging the equation to find the required rate **of** return The consol’s annual cash flow is its coupon payment and the preferred share’s annual cash flow is its dividend payment Consol rate **of** return = bond coupon/P0 = 04 × 1000/800 = 05 = 5% Preferred share rate **of** return = preferred dividend/P0 = 6/120 = 05 = 5% b Convert the annual cash flows to their after-tax amounts: After-tax bond coupon = (1 - 35) × 04 × 1000 = 65 × 40 = 26 After-tax preferred dividend = (1 - 29) × = 4.26 Consol after-tax rate **of** return = after-tax coupon/P0 = 26/800 = 0325 = 3.25% Preferred after-tax rate **of** return = after-tax dividend/P0 = 4.26/120 = 0355 = 3.55% 51 c With a 35% **corporate** tax rate, the after-tax rate **of** return on the consol is the same as we calculated in (b), 3.25% However, the **corporate** tax rate on dividends received from other Canadian corporations is zero Thus the rate **of** return on the preferred shares is the before-tax rate from (a), 5% d Corporations with spare cash to invest will prefer to purchase dividend-paying securities **of** Canadian corporations than **corporate** bonds to take advantage **of** preferential dividend tax treatment The first dividend comes in years from today and thereafter grows at a constant annual rate **of** 6% Use the constant dividend growth model to calculate the price two years from today, P2, and then discount that price to today to today’s price, P0: P2 = = = 12.5 P0 =× P2 = × 12.5 = 10.33 52 [ () ] The growing annuity formula is × – Years – r = 12%, g = 10%, T = 4, 6-15 Copyright © 2006 McGraw-Hill Ryerson Limited T C1 = DIV1 = (1 + g) × DIV0 = 1.1 × = 1.1 Present value **of** Year – dividends [ () ] = × [1 – () ]= 3.82 = × 1– T Years – 14 r = 12%, g = 8%, T = 10 To get the Year dividend (which is the first cash flow in the second interval **of** constant growth), figure out the Year dividend first Since dividends are expected to grow 10% a year for years and then grow at 8%: DIV4 = 1.14 × DIV0 = 1.14 × = 1.4641 DIV5 = (1 + g) × DIV4 = 1.08 × 1.4641 = 1.581228 Present value **of** Years – 14 dividends at the end **of** Year [ () ] = × [1 – () ]= 12.0523 = × 1– 10 10 Present value **of** Years – 14 dividends today [ () ] = × 12.0523 = 7.66 =× × 1– 10 Year 15 and on r = 12%, g = 5% DIV14 = (1.08)10 × DIV4 = (1.08)10 × 1.4641 = 3.1609 DIV15 = (1.05) × DIV14 = 1.05 × 3.1609 = 3.3190 Present value **of** Year 15 and on dividends at the end **of** Year 14 = = = 47.41 Present value **of** Year 15 and on dividends today = = × 47.41 = 9.70 Price today = present value **of** all dividends to be received P0 = 3.82 + 7.66 + 9.70 = 21.18 53 Event High quality gold Medium quality gold No gold PV **of** dividends × annuity factor(9%, 20 years) = 73.03 × annuity factor(9%, 20 years) = 18.26 P0 = × 73.03 + × 18.26 = $38.34 54 a Price on May 1, 2007 = 2.50/(.1 - 06) = 62.50 6-16 Copyright © 2006 McGraw-Hill Ryerson Limited b c d Price on May 1, 2007 = 1.248/(.1 - 04) = 20.8 Price on May 1, 2006 = [1.20 + × 62.50 + × 20.8] = $31.37 Rate **of** return if R&D is successful = - = 1.031 = 103.1% Rate **of** return if R&D is unsuccessful = - = -0.299 = -29.9% Expected rate **of** return = × 1.031 + × (-.299) = 10 = 10% 55 The difference in rates on long- versus short-term bonds does not necessarily reflect a profit opportunity that would be a contradiction **of** the efficient market hypothesis The **text** points out that the slope **of** the yield curve might be a reflection **of** investors’ expectations **of** future short-term interest rates For example, a downward sloping yield curve, in which long-term rates are below short-term rates, might indicate that investors anticipate that future short-term rates will be lower than today’s values Today’s longterm rates, which reflect beliefs about the future, are therefore lower than current shortterm rates 56 a No The split is purely a “paper transaction,” in which more shares are printed and distributed to shareholders b No Operating profits continue as before the split c Earnings per share will fall by half With unchanged earnings and double the number **of** shares, all per share values will fall by 50 percent d The firm’s stock price will fall by half With unchanged total value, and double the number **of** shares, price per share will fall by 50 per cent e The shareholder’s wealth should be unaffected Each owns double the number **of** shares, but the value **of** each share is only half **of** the original value Note: These answers are actually only part **of** the story Most stock split announcements are regarded as good news by investors, and stock prices generally increase This is not because investors suffer from financial illusions, but because managers commit to splits only when they are confident that they can maintain or increase earnings Thus the split is good news, not because it multiplies the number **of** shares, but because it reveals management’s confidence about the future 57 Internet: Online dividend discount model INTERNET UPDATE: The main page **of** this website has changed To access the model, you must now enter a stock ticker symbol If you don't know what it is, you can look it 6-17 Copyright © 2006 McGraw-Hill Ryerson Limited up Expected results: Students will see how **finance** practitioners implement the dividend discount model 58 Standard & Poor's Expected results: An opportunity to apply the ideas **of** the **chapter** to the real world Students might not be able to see sensible patterns The real world is not as tidy as one would like! 59 Standard & Poor's Expected results: An opportunity to apply the dividend growth model to actual companies 60 Bonds Bonds are perpetual, never mature The only cash flow is interest: Annual cash flow to bonds: Interest = $10 million/year Required rate **of** return on bonds = 5% Use the perpetuity formula to value the bonds: Total bond market value = annual cash flow to bonds/required rate **of** return = $10 million/.05 = $200 million Current bond price = total bond market value/number **of** bonds = $200 million/150,000 = $1,333.33 per bond Shares Shares are perpetual and cash flow never grows The only cash flow are dividends Dividends = dividend payout ratio × Net income Net income = (EBIT - Interest) - **Corporate** taxes = (70 million - 10 million) - taxes **Corporate** taxes = tax rate × (EBIT - Interest) = × (70 million - 10 million) = 18 m Dividends = Net income = (70 million - 10 million) - 18 million = 42 million Required rate **of** return on equity = 11% Use the perpetuity to value the shares: Total equity value = annual cash flow to shares/required rate **of** return = 42 million/ 11 = $381.82 million Current share price = total equity market value/number **of** shares = $381.82 million/15 million =$25.45 per share **Solution** to Minicase for **Chapter** 6-18 Copyright © 2006 McGraw-Hill Ryerson Limited The goal is to value the company under both investment plans and to choose the better investment plan Valuation based on past growth scenario The firm has been growing at 5% per year Notice that book value in 2009 equals book value five years ago, at the beginning **of** 2005 (1.05)5 This is calculated as (80/62.7)1/5 – = 0499 = 5% Dividends are proportional to book value and also have grown at 5% annually over the past four years, (7.7/6.3)1/4 – = 05 Dividends paid in the most recent year, 2009, were $7.7 million and they are projected to be $8 million next year, in 2010, if no changes are made To verify that this growth in dividends is feasible, calculate the sustainable growth rate Starting in 2010, 2/3 **of** earnings will be paid out as dividends, and 1/3 will be reinvested Therefore, the sustainable growth rate as **of** 2010 is return on equity plowback ratio = 15% 1/3 = 5% Finally, the relevant discount rate for the company’s cash flows is the 11% that investors believe they can earn on similar-risk investments, not the 15% return on book equity It is great that the company earns more on its investments than investors must earn on similar-risk investments The share price will reflect this better performance However, the relevant cost **of** capital is the opportunity cost, the return on investments **of** similar risk to Prairie Home Stores The value **of** the firm at the end **of** 2009 under the assumption that the past growth is continued is therefore Value2009 = = = $133.33 million and the value per share is $133.33 million/400,000 = $333.33 Therefore, it is clear that Mr Breezeway was correct in advising his relative not to sell for book value **of** $200 per share Valuation based on rapid growth scenario If the firm reinvests all earnings in 2010, 2011, 2012 and 2013, dividends paid at the end **of** 2014 will reach $14 million, far greater than in the constant growth scenario Forgoing dividends allows the company to achieve this rapid growth At the end **of** the rapid growth period, Prairie Homes will re-establish its dividend policy **of** paying two-thirds **of** its earnings as dividends At this point, the dividend growth rate will fall to sustainable growth rate **of** 5%, which is the return on equity times 6-19 Copyright © 2006 McGraw-Hill Ryerson Limited the plowback ratio, 15% × 1/3 The value **of** the firm in 2013 will be the present value **of** the dividend in one year, $14 million, divided by the required rate **of** return, 11%, minus the sustainable growth rate **of** 5%: Value2013 = = = $233.3 million The value **of** the firm as **of** today, the end **of** 2009, is the present value **of** this amount Remember, there will be no dividend flows in the years leading up to 2014 Value2009 = = $153.70 million or a per share value **of** $153.70 million/400,000 = $384.25 Thus, it appears that the rapid growth plan is in fact preferable If the firm follows this plan, it will be able to go public — that is, sell its shares to the public — at a higher price One potential obstacle will be the family members who are not willing to give up their dividends However, they should be persuaded that with an active secondary market for their shares, they will be able to sell off part **of** their holdings for cash 6-20 Copyright © 2006 McGraw-Hill Ryerson Limited ... a profit opportunity that would be a contradiction of the efficient market hypothesis The text points out that the slope of the yield curve might be a reflection of investors’ expectations of. .. the required rate of return, 11%, minus the sustainable growth rate of 5%: Value2013 = = = $233.3 million The value of the firm as of today, the end of 2009, is the present value of this amount... exchange of interest Also, remind the students to click on the symbol of the stock exchange to connect to the Web site of that exchange Expected results: Students learn about variety of stock

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