Chapter Six How to Value Bonds and Stocks Prepared by Professor Wei Wang Queen’s University © 2011 McGraw–Hill Ryerson Limited 2-1 6-1 Chapter Outline 6.1 6.2 6.3 6.4 6.5 Definition and Example of a Bond How to Value Bonds Bond Concepts The Present Value of Common Stocks Estimates of Parameters in the DividendDiscount Model 6.6 Growth Opportunities 6.7 The Dividend Growth Model and the NPVGO Model (Advanced) 6.8 Price-Earnings Ratio 6.9 Stock Market Reporting 6.10 Summary and Conclusions © 2011 McGraw–Hill Ryerson Limited 2-2 6-2 Valuation of Bonds and Stock • First Principles: – Value of financial securities = PV of expected future cash flows • To value bonds and stocks we need to: – Estimate future cash flows: • Size (how much) and • Timing (when) – Discount future cash flows at an appropriate rate: • The rate should be appropriate to the risk presented by the security. © 2011 McGraw–Hill Ryerson Limited 2-3 6-3 Definition and Example of a Bond LO6.1 • A bond is a legally binding agreement between a borrower (bond issuer) and a lender (bondholder): – Specifies the principal amount of the loan. – Specifies the size and timing of the cash flows: • In dollar terms (fixed-rate borrowing) • As a formula (adjustable-rate borrowing) © 2011 McGraw–Hill Ryerson Limited 2-4 6-4 Definition and Example of a Bond LO6.1 • Consider a Government of Canada bond listed as 5.000 December 2014. – The Par Value of the bond is $1,000. – Coupon payments are made semi-annually (June 30 and December 31 for this particular bond). – Since the coupon rate is 5.000 the payment is $25.000. – On January 1, 2011 the size and timing of cash flows are: $25 $25 $25 $1,025 6 / 30 / 14 12 / 31 / 14 1 / 1 / 11 6 / 30 / 11 12 / 31 / 11 © 2011 McGraw–Hill Ryerson Limited 2-5 6-5 How to Value Bonds LO6.2 • Bond value is determined by the present value of the coupon payments and par value. • We have to identify the size and timing of cash flows. • Discount at the correct discount rate. – Discount rates are inversely related to present (i.e., bond) values. – If you know the price of a bond and the size and timing of cash flows, the yield to maturity is the discount rate. © 2011 McGraw–Hill Ryerson Limited 2-6 6-6 Pure Discount Bonds LO6.2 • Make no periodic interest payments (coupon rate = 0%) • The entire yield to maturity comes from the difference between the purchase price and the par value. • Cannot sell for more than par value • Sometimes called zeroes, deep discount bonds, or original issue discount bonds (OIDs) • Treasury Bills and principal-only Treasury strips are good examples of zeroes. © 2011 McGraw–Hill Ryerson Limited 2-7 6-7 Pure Discount Bonds LO6.2 Information needed for valuing pure discount bonds: – Time to maturity (T) = Maturity date - today’s date – Face value (F) – Discount rate (r) $0 $0 $0 $F T −1 T 0 1 2 Present value of a pure discount bond at time 0: F PV = T (1 + r ) © 2011 McGraw–Hill Ryerson Limited 2-8 6-8 Pure Discount Bonds: Example LO6.2 Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%. $0 $0 $0 $1,000 $,0$ 1 $00 0 20 9 23 1 0 1 2 30 29 F $1,000 PV = = = $174.11 T 30 (1 + r ) (1.06) © 2011 McGraw–Hill Ryerson Limited 2-9 6-9 Pure Discount Bonds: Example LO6.2 Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%. N I/Y PV 30 6 – 174.11 PMT FV 1,000 © 2011 McGraw–Hill Ryerson Limited 2-10 6-10 Level Coupon Bonds LO6.2 • Make periodic coupon payments in addition to the maturity value • The payments are equal each period. Therefore, the bond is just a combination of an annuity and a terminal (maturity) value. • Coupon payments are typically semiannual. © 2011 McGraw–Hill Ryerson Limited 2-11 6-11 Level-Coupon Bonds LO6.2 Information needed to value level-coupon bonds: – Coupon payment dates and time to maturity (T) – Coupon payment (C) per period and Face value (F) – Discount rate $C $C $C $C + $ F T −1 T 0 1 2 Value of a Level-coupon bond = PV of coupon payment annuity + PV of face value C 1 F PV = 1 − + T r (1 + r ) (1 + r )T © 2011 McGraw–Hill Ryerson Limited 2-12 6-12 Level-Coupon Bonds: Example LO6.2 • Find the present value (as of January 1, 2011), of a 6-3/8 coupon Government of Canada bond with semi-annual payments, and a maturity date of December 31, 2016 if the YTM is 5-percent. – The Par Value of the bond is $1,000. – Coupon payments are made semi-annually (June 30 and December 31 for this particular bond). – Since the coupon rate is 6 3/8%, the payment is $31.875. – On January 1, 2011 the size and timing of cash flows are: $31.875 $31.875 $31.875 $1,031.875 6 / 30 / 16 12 / 31 / 16 1 / 1 / 11 6 / 30 / 11 12 / 31 / 11 © 2011 McGraw–Hill Ryerson Limited 2-13 6-13 Level Coupon Bond: Example LO6.2 • On January 1, 2011, the required annual yield is 5%. • The present value of the bond is: $1,000 $31.875 1 PV = 1− + = $1,070.52 12 12 .05 2 (1.025) (1.025) © 2011 McGraw–Hill Ryerson Limited 2-14 6-14 Level-Coupon Bonds: Example LO6.2 Find the present value (as of January 1, 2011), of a 6-3/8 coupon Government of Canada bond with semi-annual payments, and a maturity date of December 31, 2016 if the YTM is 5-percent. N I/Y PV PMT 12 5 – 1,070.52 31.875 = 1,000×0.06375 2 FV 1,000 © 2011 McGraw–Hill Ryerson Limited 2-15 6-15 Bond Pricing with a Spreadsheet LO6.2 • There are specific formulas for finding bond prices and yields on a spreadsheet. – PRICE(Settlement,Maturity,Rate,Yld,Redemption, Frequency,Basis) – YIELD(Settlement,Maturity,Rate,Pr,Redemption, Frequency,Basis) – Settlement and maturity need to be actual dates – The redemption and Pr need to be given as % of par value • Click on the first Excel icon for pricing the bond in the 6-3/8 December 2016 bond. • Click on the second Excel icon for another example. © 2011 McGraw–Hill Ryerson Limited 2-16 6-16 Bond Market Reporting LO6.2 CANADA Canada Coupon 6.375 Mat. Date Dec 31/16 The Government of Canada issued this bond The bond pays an annual coupon rate of 6.375% Bid $ 107.05 Yld% 5.00 The bond’s quoted annual yield to maturity is 5% The bond matures on December 31, 2016 The bond is selling at 107.05% of the face value of $1,000 © 2011 McGraw–Hill Ryerson Limited 2-17 6-17 Consols LO6.2 • Not all bonds have a final maturity. • British consols pay a set amount (i.e., coupon) every period forever. • These are examples of a perpetuity. C PV = R © 2011 McGraw–Hill Ryerson Limited 2-18 6-18 Bond Concepts LO6.3a 1. Bond prices and market interest rates move in opposite directions. 2. When coupon rate = YTM, price = par value. When coupon rate > YTM, price > par value (premium bond) When coupon rate < YTM, price < par value (discount bond) 3. A bond with longer maturity has higher relative (%) price change than one with shorter maturity when interest rate (YTM) changes. All other features are identical. 4. A lower coupon bond has a higher relative price change than a higher coupon bond when YTM changes. All other features are identical. © 2011 McGraw–Hill Ryerson Limited 2-19 6-19 YTM and Bond Value LO6.3a Bond Value $1400 When the YTM < coupon, the bond trades at a premium. 1300 1200 When the YTM = coupon, the bond trades at par. 1100 1000 800 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 6 3/8 0.08 0.09 0.1 Discount Rate When the YTM > coupon, the bond trades at a discount. © 2011 McGraw–Hill Ryerson Limited 2-20 6-20 Bond Example Revisited LO6.3a • Using our previous example, now assume that the required yield is 11%. • How does this change the bond’s price? $31.875 $31.875 $31.875 $1,031.875 6 / 30 / 16 12 / 31 / 16 1 / 1 / 11 6 / 30 / 11 12 / 31 / 11 $1,000 $31.875 1 PV = 1− + = $800.70 12 12 .11 2 (1.055) (1.055) • The bond trades at a discount. © 2011 McGraw–Hill Ryerson Limited 2-21 6-21 Bond Value Maturity and Bond Price Volatility LO6.3a Consider two otherwise identical bonds. The long-maturity bond will have much more volatility with respect to changes in the discount rate Par Short Maturity Bond C Discount Rate Long Maturity Bond © 2011 McGraw–Hill Ryerson Limited 2-22 6-22 Bond Value Coupon Rate and Bond Price Volatility LO6.3a Consider two otherwise identical bonds. The low-coupon bond will have much more volatility with respect to changes in the discount rate High Coupon Bond Low Coupon Bond Discount Rate © 2011 McGraw–Hill Ryerson Limited 2-23 6-23 Holding Period Return LO6.3b • Suppose that on January 1, 2011, you purchased 6.375 coupon Government of Canada bond with semi-annual payments, and a maturity date of December 31, 2018. • At that time the YTM was 5-percent, and you paid $1,089.75 (the PV of the bond). • Six months later (July 1, 2011), You sold the bond when the YTM was 4-percent. The size and timing of cash flows (as of July 1, 2011 ) were: $31.875 $31.875 $31.875 $1,031.875 6 / 30 / 18 12 / 31 / 18 7 / 1 / 11 12 / 31 / 11 6 / 30 / 11 © 2011 McGraw–Hill Ryerson Limited 2-24 6-24 Holding Period Return (cont.) LO6.3b • Given that the YTM at that time was 4-percent, you sold the bond for: $31.875 1 $1,000 PV = 1− + = $1,152.59 15 15 .04 2 (1.02) (1.02) • Your holding period return was: $31.875 + $1,152.59 − $1,089.75 = 8.69% $1,089.75 • This annualizes to an effective rate of: 2 (1.0869) − 1 = 18.14% © 2011 McGraw–Hill Ryerson Limited 2-25 6-25 Computing Yield to Maturity LO6.3b • Yield to maturity is the rate implied by the current bond price. • Finding the YTM requires trial and error if you do not have a financial calculator and is similar to the process for finding R with an annuity. • If you have a financial calculator, enter N, PV, PMT, and FV, remembering the sign convention (PMT and FV need to have the same sign, PV the opposite sign). © 2011 McGraw–Hill Ryerson Limited 2-26 6-26 YTM with Annual Coupons LO6.3b • Find YTM for a bond with a 10% annual coupon rate, 15 years to maturity, and a par value of $1,000. The current price is $928.09. N 15 I/Y 11.00 PV – 928.09 PMT 100 FV 1,000 © 2011 McGraw–Hill Ryerson Limited 2-27 6-27 YTM with Semiannual Coupons LO6.3b • Find TM for a bond with a 10% coupon rate and semiannual coupons has a face value of $1,000, 20 years to maturity, and is selling for $1,197.93. – What is the semiannual coupon payment? – How many periods are there? © 2011 McGraw–Hill Ryerson Limited 2-28 6-28 YTM with Semiannual Coupons (cont.) LO6.3b • Find TM for a bond with a 10% coupon rate and semiannual coupons has a face value of $1,000, 20 years to maturity, and is selling for $1,197.93. N 40 I/Y 4.00 PV – 1,197.93 PMT FV => YTM = 4%*2 = 8% 50 1,000 © 2011 McGraw–Hill Ryerson Limited 2-29 6-29 The Present Value of Common Stocks LO6.4a • The value of any asset is the present value of its expected future cash flows. • Stock ownership produces cash flows from: – Dividends – Capital Gains • Dividends versus Capital Gains • Valuation of Different Types of Stocks – Zero Growth – Constant Growth – Differential Growth © 2011 McGraw–Hill Ryerson Limited 2-30 6-30 Case 1: Zero Growth LO6.4b • Assume that dividends will remain at the same level forever Div1 = Div 2 = Div 3 = • Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity: Div1 Div 2 Div 3 P0 = + + + 1 2 3 (1 + r ) (1 + r ) (1 + r ) Div P0 = r © 2011 McGraw–Hill Ryerson Limited 2-31 6-31 A Zero Growth Example LO6.4b ABC Corp. is expected to pay $0.75 dividend per annum, starting a year from now, in perpetuity. If stocks of similar risk earn 12% annual return, what is the price of a share of ABC stock? The stock price is given by the present value of the perpetual stream of dividends: $0.75 $0.75 $0.75 $0.75 … 0 1 2 3 4 P0 = Div1 / r = 0.75/0.12 = $6.25 © 2011 McGraw–Hill Ryerson Limited 2-32 6-32 Case 2: Constant Growth LO6.4C Assume that dividends will grow at a constant rate, g, forever. i.e. Div1 = Div 0 (1 + g ) Div 2 = Div1 (1 + g ) = Div 0 (1 + g ) 2 Div 3 = Div 2 (1 + g ) = Div 0 (1 + g )3 .. . Since future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity: Div1 P0 = r−g © 2011 McGraw–Hill Ryerson Limited 2-33 6-33 A Constant Growth Example LO6.4C XYZ Corp. has a common stock that paid its annual dividend this morning. It is expected to pay a $3.60 dividend one year from now, and following dividends are expected to grow at a rate of 4% per year forever. If stocks of similar risk earn 16% effective annual return, what is the price of a share of XYZ stock? © 2011 McGraw–Hill Ryerson Limited 2-34 6-34 A Constant Growth Example (cont.) LO6.4C The stock price is given by the the present value of the perpetual stream of growing dividends: $3.60 $3.60´1.04 $3.60´1.042 $3.60´1.043 … 0 1 2 3 4 P0 = Div1 / (r-g) = 3.60/(0.16-0.04) = $30.00 © 2011 McGraw–Hill Ryerson Limited 2-35 6-35 Case 3: Differential Growth LO6.4d • Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter. • To value a Differential Growth Stock, we need to: – Estimate future dividends in the foreseeable future. – Estimate the future stock price when the stock becomes a Constant Growth Stock (case 2). – Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate. © 2011 McGraw–Hill Ryerson Limited 2-36 6-36 Case 3: Differential Growth LO6.4d • Assume that dividends will grow at rate g1 for N years and grow at rate g2 thereafter Div1 = Div 0 (1 + g1 ) Div 2 = Div1 (1 + g1 ) = Div 0 (1 + g1 ) 2 .. . Div N = Div N −1 (1 + g1 ) = Div 0 (1 + g1 ) N Div N +1 = Div N (1 + g 2 ) = Div 0 (1 + g1 ) N (1 + g 2 ) .. . © 2011 McGraw–Hill Ryerson Limited 2-37 6-37 Case 3: Differential Growth LO6.4d • Dividends will grow at rate g1 for N years and grow at rate g2 thereafter Div 0 (1 + g1 ) Div 0 (1 + g1 ) 2 … 0 1 2 Div 0 (1 + g1 ) N Div N (1 + g 2 ) = Div 0 (1 + g1 ) N (1 + g 2 ) … … N N+1 © 2011 McGraw–Hill Ryerson Limited 2-38 6-38 Case 3: Differential Growth LO6.4d We can value this as the sum of: an N-year annuity growing at rate g1 Div1 (1 + g1 ) N PA = 1 − N r − g1 (1 + r ) plus the discounted value of a perpetuity growing at rate g2 that starts in year N+1 Div N +1 r − g2 PB = N (1 + r ) © 2011 McGraw–Hill Ryerson Limited 2-39 6-39 Case 3: Differential Growth LO6.4d To value a Differential Growth Stock, we can use Div N +1 N Div1 (1 + g1 ) r − g 2 P= + 1 − N N r − g1 (1 + r ) (1 + r ) Or we can cash flow it out. © 2011 McGraw–Hill Ryerson Limited 2-40 6-40 A Differential Growth Example LO6.4d A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity. If stocks of similar risk earn 12% effective annual return, what is the stock worth? © 2011 McGraw–Hill Ryerson Limited 2-41 6-41 With the Formula Div N +1 T C (1 + g1 ) r − g 2 P= + 1 − T N r − g1 (1 + r ) (1 + r ) LO6.4d $2(1.08) 3 (1.04) 3 .12 − .04 $2 × (1.08) (1.08) P= + 1 − 3 3 .12 − .08 (1.12) (1.12) ( $32.75) P = $54 × [1 − .8966] + 3 (1.12) P = $5.58 + $23.31 P = $28.89 © 2011 McGraw–Hill Ryerson Limited 2-42 6-42 A Differential Growth Example (cont) LO6.4d $2(1.08) $2(1.08) 2 3 3 $2(1.08) $2(1.08) (1.04) … 0 1 $2.16 0 1 2 3 4 $2.33 $2.62 $2.52 + .08 2 3 The constant growth phase beginning in year 4 can be valued as a growing perpetuity at time 3. $2.62 P3 = = $32.75 .08 $2.16 $2.33 $2.52 + $32.75 P0 = + + = $28.89 2 3 1.12 (1.12) (1.12) © 2011 McGraw–Hill Ryerson Limited 2-43 6-43 A Differential Growth Example (cont) LO6.4d A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity. What is the stock worth? $28.89 = 5.58 + 23.31 First find the PV of the supernormal dividend stream then find the PV of the steady-state dividend stream. N I/Y 3.70 = PV – 5.58 PMT FV $2 = 0 3 N 3 1.12 1.08 –1 ×100 2×1.08 1.08 I/Y 12 PV – 23.31 PMT 0 FV 2×(1.08)3 ×(1.04) 32.75 = .08 © 2011 McGraw–Hill Ryerson Limited 2-44 6-44 Estimates of Parameters in the DividendDiscount Model LO6.5 • The value of a firm depends upon its growth rate, g, and its discount rate, r. – Where does g come from? – Where does r come from? © 2011 McGraw–Hill Ryerson Limited 2-45 6-45 Where does g come from? LO6.5 Formula for Firm’s Growth Rate (g): • The firm will experience earnings growth if its net investment (total investment-depreciation) is positive. • To grow, the firm must retain some of its earnings. • This leads to: Earnings Earnings = + next Year this Year Retained Return on earnings ´ retained this Year earnings © 2011 McGraw–Hill Ryerson Limited 2-46 6-46 Where does g come from? (con.) LO6.5 • Dividing both sides by this year’s earnings, we get: Earnings next Year = 1+ Earnings this Year 1+g Retained earnings Return on this Year ´ retained Earnings this Year earnings Retention ratio • This leads to the formula for the firm’s growth rate: g = Retention ratio × Return on retained earnings • The return on retained earnings can be estimated using the firm’s historical return on equity (ROE) © 2011 McGraw–Hill Ryerson Limited 2-47 6-47 Where does g come from? An Example Ontario Book Publishers (OBP) just reported earnings of $1.6 million, and it plans to retain 28-percent of its earnings. If OBP’s historical ROE was 12-percent, what is the expected growth rate for OBP’s earnings? With the above formula: g = 0.28 × 0.12 = 0.0336 = 3.36% Or: (0.28×$1.6 million)×0.12 Change in earnings = = 0.0336 $1.6 million Total earnings © 2011 McGraw–Hill Ryerson Limited 2-48 6-48 Where does r come from? LO6.5 • From the constant growth cas, we can write: Div1 r= +g P0 • The discount rate can be broken into two parts. – The dividend yield – The growth rate (in dividends) • In practice, there is a great deal of estimation error involved in estimating r. © 2011 McGraw–Hill Ryerson Limited 2-49 6-49 Where does r come from? An Example Manitoba Shipping Co. (MSC) is expected to pay a dividend next year of $8.06 per share. Future Dividends for MSC are expected to grow at a rate of 2% per year indefinitely. If an investor is currently willing to pay $62.00 per one MSC share, what is her required return for this investment? With the above formula: r = (8.06/62) + 0.02 = 0.15 = 15% © 2011 McGraw–Hill Ryerson Limited 2-50 6-50 Growth Opportunities LO6.6 • Growth opportunities are opportunities to invest in positive NPV projects. • The value of a firm can be conceptualized as the sum of the value of a firm that pays out 100-percent of its earnings as dividends and the net present value of the growth opportunities. EPS P= + NPVGO r © 2011 McGraw–Hill Ryerson Limited 2-51 6-51 Growth Opportunities LO6.6 • Two conditions must be met in order to increase value: – Earnings must be retained so that projects can be funded. – The projects must have positive NPV. • A firm’s value increases when it invests in growth opportunities with positive NPVGOs. © 2011 McGraw–Hill Ryerson Limited 2-52 6-52 The No-Dividend Firm LO6.6 • Firms that pay no dividends sell at positive prices (e.g. Amazon.com and RIM) • These firms have many growth opportunities and investors believe these firms will pay dividends at some point. • The model for constant growth of dividends does not apply. © 2011 McGraw–Hill Ryerson Limited 2-53 6-53 The Dividend Growth Model and the NPVGO Model (Advanced) LO6.7 • We have two ways to value a stock: – The dividend discount model. – The price of a share of stock can be calculated as the sum of its price as a cash cow plus the pershare value of its growth opportunities. © 2011 McGraw–Hill Ryerson Limited 2-54 6-54 The Dividend Growth Model and the NPVGO Model LO6.7 Consider a firm that has EPS of $5 at the end of the first year, a dividend-payout ratio of 30-percent, a discount rate of 16-percent, and a return on retained earnings of 20-percent. • The dividend at year one will be $5 × .30 = $1.50 per share. • The retention ratio is .70 ( = 1 -.30) implying a growth rate in dividends of 14% = .70 × 20% From the dividend growth model, the price of a share is: Div1 $1.50 P0 = = = $75 r − g .16 − .14 © 2011 McGraw–Hill Ryerson Limited 2-55 6-55 The NPVGO Model LO6.7 First, we must calculate the value of the firm as a cash cow. EPS $5 = = $31.25 r .16 Second, we must calculate the value of the growth opportunities. 3.50 × .20 − 3.50 + .16 $.875 NPVGO = = = $43.75 r−g .16 − .14 Finally, P0 = 31.25 + 43.75 = $75 © 2011 McGraw–Hill Ryerson Limited 2-56 6-56 Price Earnings Ratio LO6.8 • Many analysts frequently relate earnings per share to price. • The price earnings ratio is a.k.a the multiple – Calculated as current stock price divided by annual EPS – The National Post uses last 4 quarters’ earnings Price per share P/E ratio = EPS • Firms whose shares are “in fashion” sell at high multiples. Growth stocks for example. • Firms whose shares are out of favour sell at low multiples. Value stocks for example. © 2011 McGraw–Hill Ryerson Limited 2-57 6-57 Other Price Ratio Analysis LO6.8 • Many analysts frequently relate earnings per share to variables other than price, e.g.: – Price/Cash Flow Ratio • cash flow = net income + depreciation = cash flow from operations or operating cash flow – Price/Sales • current stock price divided by annual sales per share – Price/Book (a.k.a. Market to Book Ratio) • price divided by book value of equity, which is measured as assets - liabilities © 2011 McGraw–Hill Ryerson Limited 2-58 6-58 Stock Market Reporting 52W high 42.10 LO6.9 52W Yield Vol low Stock Ticker Div % P/E 00s High Low 32.25 BCE Inc BCE 1.20 3.5 11.8 19210 34.59 33.80 BCE pays a BCE has dividend of 1.2 dollars/share been as Given the high as $42.10 in current price, the dividend the last yield is 3½ % year. BCE has Given the been as low current price, the as $32.25 in P/E ratio is 11.8 the last year. times earnings Close 34.50 Net chg -0.47 BCE ended trading at $34.50, down $0.47 from yesterday’s close 1,921,000 shares traded hands in the last day’s trading © 2011 McGraw–Hill Ryerson Limited 2-59 6-59 Stock Market Reporting 52W high 42.10 52W Yield Vol low Stock Ticker Div % P/E 00s High Low 32.25 BCE Inc BCE 1.20 3.5 11.8 19210 34.59 33.80 Close 34.50 Net chg -0.47 BCE Incorporated is having a tough year, trading near their 52week low. Imagine how you would feel if within the past year you had paid $42.10 for a share of BCE and now had a share worth $34.50! That $1.20 dividend wouldn’t go very far in making amends. Yesterday, BCE had another rough day in a rough year. BCE “opened the day down” beginning trading at $34.59, which was down from the previous close of $34.97 = $34.50 + $0.47 © 2011 McGraw–Hill Ryerson Limited 2-60 6-60 Summary and Conclusions LO6.10 In this chapter, we used the time value of money formulae from previous chapters to value bonds and stocks. 1. The value of a zero-coupon bond is F PV = T (1 + r ) 2. The value of a perpetuity is C PV = r © 2011 McGraw–Hill Ryerson Limited 2-61 6-61 Summary and Conclusions (cont.) LO6.10 3. The value of a coupon bond is the sum of the PV of the annuity of coupon payments plus the PV of the par value at maturity. C 1 F PV = 1 − + T r (1 + r ) (1 + r )T 4. The yield to maturity (YTM) of a bond is that single rate that discounts the payments on the bond to the purchase price. © 2011 McGraw–Hill Ryerson Limited 2-62 6-62 Summary and Conclusions (cont.) LO6.10 5. A stock can be valued by discounting its dividends. There are three cases: Div 1) Zero growth in dividends P0 = r Div1 2) Constant growth in dividends P0 = r−g Div N +1 N Div1 (1 + g1 ) r − g 2 P= + 1 − N N r − g1 (1 + r ) (1 + r ) 3) Differential growth in dividends © 2011 McGraw–Hill Ryerson Limited 2-63 6-63 Summary and Conclusions (cont.) LO6.10 6. The growth rate can be estimated as: g = Retention ratio × Return on retained earnings 7. An alternative method of valuing a stock was presented. The NPVGO values a stock as the sum of its “cash cow” value plus the present value of growth opportunities. EPS P= + NPVGO r © 2011 McGraw–Hill Ryerson Limited 2-64 6-64 Quick Quiz • How do you find the value of a bond, and why do bond prices change? • What is a bond indenture, and what are some of the important features? • What determines the price of a share of stock? • What determines g and r in the Dividend Growth Model ? • Decompose a stock’s price into constant growth and NPVGO values. • Discuss the importance of the P/E ratio. © 2011 McGraw–Hill Ryerson Limited

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