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 Research in Motion’s growth opportunities moderate, investors may justifiably believe that Research in Motion 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 a P0 = r = + g r = + 04 = 14 = 14% 6-1 Copyright © 2006 McGraw-Hill Ryerson Limited 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 Two such behavioral biases are alluded to in the text a.) Attitudes toward risk: Psychologists have observed the tendency of investors to be particularly averse to the possibility of incurring losses Consequently, when investors sustain a loss they tend to exercise excessive conservatism in their subsequent investment decisions to avoid worsening their deficit Conversely, when investors have secured a gain they are more eager to take on risky bets because they take comfort in their cushion of profit This behavioral bias has been implicated as a cause of stock market “bubbles”: a phenomenon where market prices far exceed intrinsic firm values b.) Beliefs about probabilities: This bias is twofold Firstly, investors tend to look at previous periods and assume that future market fluctuations will follow suit This is incorrect since stock prices follow a random walk and ought to be independent of previous market movements To further compound their error, investors tend to be myopic and place greater emphasis on recent market developments; as such, they overlook valuable information from the distant past (i.e market reactions to long-term economic cycles) Secondly, investors tend to fall prey to overconfidence When investors profit in the market they wrongly attribute their success to skill rather than mere luck Such excessive optimism may cause prices to greatly deviate from intrinsic share values 6-2 Copyright © 2006 McGraw-Hill Ryerson Limited 11 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 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) 12 Dividend growth rate, g = return on equity × plowback ratio: g = 15 40 = 06 r = + g = + 06 = 16 = 16% 13 a P0 = = = $21 b P0 = = $30 The lower discount rate makes the present value of future dividends higher, raising the value of the stock c P0 = (2 x (1 – 05) / (.15 – (-.05)) = $9.50 The price is the present value of the future dividends before they are eroded by the negative growth rate 14 r = + g g = r - = 14 – = 04 = 4% 15 a r = + g = + 03 = 0926 = 9.26% 6-3 Copyright © 2006 McGraw-Hill Ryerson Limited b If r = 10, then 10 = 1.64(1.03)/27 + g c g = Return on equity plowback ratio 5% = Return on equity Return on equity = = = 12.5% So g = 0374 = 3.74% 16 P0 = DIV1/(r g) = [$.50 x 1.06 /(.12 - 06)] / [(1 + 12) ^ 4] = $5.61 17 a P0 = DIV1/(r g) = 3/[.15 – (.10)] = 3/.25 = $12 b P1 = DIV2/(r g) = 3(1 10)/.25 = $10.80 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 18 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 6-4 Copyright © 2006 McGraw-Hill Ryerson Limited 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 19 Stock exchange information Example information: Click on the pin in the Atlantic Ocean to get the Bermuda Stock Exchange: 2007 Listed companies: 53 Types of securities traded: equity and debt The website has limited current stock price information, unlike the Toronto Stock Exchange Click on the daily trade report and see how variable is trading volumes 20 Hollywood stock exchange Expected result: A fun experience trading securities Students won’t have trouble figuring out how to set up an account 21 P0 = DIV1/(r g) where g = plowback ratio x ROE $50 = $2.50 / 15 – (.6 x ROE) ROE = [($2.5/$50) – 15] / 60 = 16.667 % 22 a P0 = + + = 18.10 b DIV1/P0 = $1/18.10 = 0552 = 5.52% a b Payout ratio g = ROE plowback 23 Stock A $1/$2 = 50 15% = 7.5% 6-5 Copyright © 2006 McGraw-Hill Ryerson Limited Stock B $1/$1.50 = 67 10% 333 = 3.33% 24 c 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 25 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 26 a 27 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% 6-6 Copyright © 2006 McGraw-Hill Ryerson Limited 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 28 29 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 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 30 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 PV = 2/1.10 = 1.818 PV = 2.40/1.102 = 1.983 PV = 2.88/1.103 = 2.164 6-7 Copyright © 2006 McGraw-Hill Ryerson Limited 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 31 a Book value = $100 million 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 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 32 P0 = [1.819/1.089] + [1.95/1.0892] + [2.08/1.0893] + [2.23/1.0894] + [2.38/1.0895] + [2.49/ (.089 - 0445)]/1.0895 = $ 44.61 33 P0 = + 34 a DIV1 = $2 1.20 = $2.40 b DIV1 = $2.40 + = $16.59 DIV2 = $2.88 DIV3 = $3.456 6-8 Copyright © 2006 McGraw-Hill Ryerson Limited P3 = = $32.675 P0 = + + = $28.02 c P1 = + = $29.825 d Capital gain = P1 P0 = $29.825 $28.02 = 1.805 r = = 15 = 15% 35 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 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 6-9 Copyright © 2006 McGraw-Hill Ryerson Limited 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 36 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 37 a BCE Series T is listed as BCE.PR.T-T Click on the actual share name and go to the page describing the preferred share The dividend will be listed as $1.13 due to rounding Suppose the traded price was $15 Then the expected rate of return, or yield to maturity is: r = = $1.1255 =.075 or 7.5% $15 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 Example: From the CIBC website, details on CM.PR.A are found: Ticker Symbol CUSIP Number Quarterly Dividend Initial Dividend Amount Initial Payment Date CM.PR.A 136069648 $0.33125 $0.33125 Apr 28/01 6-10 Copyright © 2006 McGraw-Hill Ryerson Limited At Globeinvestor.com on the preferred stock list, click on CM.PR.A and see details on the preferred stock The annual dividend is listed as as $1.33 The actual annual dividend is $1.325, times the quarterly dividend of $.33125 If the traded stock price is $25.82, the expected annual rate of return on preferred share is $1.325/$25.82 = 0.0513 or 5.13% 38 Expected Results: MB– T is Mega Brands Inc 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 Example: Mega Brands manufactures and markets over 60 different toys, many of which are molded plastic Mega Blocks is a competitor to LEGO They also have paper products such as puzzles and crafts The investor page provides access to financial reports, such as quarterly and annual reports, press releases, audio broadcasts of analyst conference call and a link to their stock listing at the Toronto Stock Exchange 39 Before-tax rate of return: = = = 078 = 7.8% After-tax rate of return: = = = 0596 = 5.96% 40 a b 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 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 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 6-11 Copyright © 2006 McGraw-Hill Ryerson Limited 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 41 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 42 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 43 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 44 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 45 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 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% 6-12 Copyright © 2006 McGraw-Hill Ryerson Limited 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 46 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 c DIV1/P0 = 1.20/34.74 = 0345 = 3.45% d Next year the price will be: + + = $37.01 f return = = = 10 The expected return equals the discount rate (as it should if the stock is fairly priced) 6-13 Copyright © 2006 McGraw-Hill Ryerson Limited 47 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 = = 48 = 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 1/3 - = 0573 = 5.73% 6-14 Copyright © 2006 McGraw-Hill Ryerson Limited 49 a 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% 50 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 growing annuity formula is × – T Years – r = 12%, g = 10%, T = 4, 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 6-15 Copyright © 2006 McGraw-Hill Ryerson Limited 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 6-16 Copyright © 2006 McGraw-Hill Ryerson Limited 51 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 52 a Price on May 1, 2007 = 2.50/(.1 - 06) = 62.50 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% 53 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 54 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 6-17 Copyright © 2006 McGraw-Hill Ryerson Limited because it multiplies the number of shares, but because it reveals management’s confidence about the future 55 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! 56 Standard & Poor's HEINZ: Current Dividend: $1.555 Year Growth Rate: 5.5% Current Stock Price: $39.57 $39.57 = ($1.555 x 1.055) / r - 055 r = [($1.555 x 1.055) / 39.57] + 055 = 9.63% Expected results: An opportunity to apply the dividend growth model to actual companies 57 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 6-18 Copyright © 2006 McGraw-Hill Ryerson Limited = 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 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 2012 equals book value five years ago, at the beginning of 2008 (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, 2012, were $7.7 million and they are projected to be $8 million next year, in 2013, if no changes are made To verify that this growth in dividends is feasible, calculate the sustainable growth rate Starting in 2013, 2/3 of earnings will be paid out as dividends, and 1/3 will be reinvested Therefore, the sustainable growth rate as of 2013 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 2012 under the assumption that the past growth is continued is therefore Value2012 = = = $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 6-19 Copyright © 2006 McGraw-Hill Ryerson Limited If the firm reinvests all earnings in 2013, 2014, 2015 and 2016, dividends paid at the end of 2017 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 the plowback ratio, 15% × 1/3 The value of the firm in 2016 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%: Value2016 = = = $233.3 million The value of the firm as of today, the end of 20012, is the present value of this amount Remember, there will be no dividend flows in the years leading up to 2017 Value20012 = = $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 ... ratio: g = 15 40 = 06 r = + g = + 06 = 16 = 16% 13 a P0 = = = $21 b P0 = = $30 The lower discount rate makes the present value of future dividends higher, raising the value of the stock c P0 =... 1.581228 6-15 Copyright © 2 006 McGraw-Hill Ryerson Limited Present value of Years – 14 dividends at the end of Year [ () ] = × [1 – () ]= 12.0523 = × 1– 10 10 Present value of Years – 14 dividends... 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