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5-0 CHAPTER 5 How to Value Bonds and Stocks McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-1 Chapter Outline 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 Definition and Example of a Bond How to Value Bonds Bond Concepts The Present Value of Common Stocks Estimates of Parameters in the Dividend-Discount Model Growth Opportunities The Dividend Growth Model and the NPVGO Model (Advanced) Price Earnings Ratio Stock Market Reporting Summary and Conclusions McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-2 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. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-3 5.1 Definition and Example of a Bond A bond is a legally binding agreement between a borrower and a lender: 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) McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-4 5.1 Definition and Example of a Bond Consider a U.S. government bond listed as 6 3/8 of December 2009. 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, 2005 the size and timing of cash flows are: $31.875 $31.875 $31.875 $1,031.875 6 / 30 / 09 12 / 31 / 09 1 / 1 / 05 6 / 30 / 05 McGraw-Hill/Irwin Corporate Finance, 7/e 12 / 31 / 05 © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-5 5.2 How to Value Bonds Identify the size and timing of cash flows. Discount at the correct discount rate. If you know the price of a bond and the size and timing of cash flows, the yield to maturity is the discount rate. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-6 Pure Discount Bonds 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 ) McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-7 Pure Discount Bonds: Example 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 29 0 1 2 29 30 F $1,000 PV $174.11 T 30 (1 r ) (1.06) McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-8 Pure Discount Bonds: Example Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%. N 30 I/Y 6 PV 174.11 PMT FV McGraw-Hill/Irwin Corporate Finance, 7/e 1,000 © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-9 Level-Coupon Bonds 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 McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-10 Level-Coupon Bonds: Example Find the present value (as of January 1, 2004), of a 6-3/8 coupon Tbond with semi-annual payments, and a maturity date of December 2009 if the YTM is 5-percent. On January 1, 2004 the size and timing of cash flows are: $31.875 $31.875 $31.875 $1,031.875 6 / 30 / 09 12 / 31 / 09 1 / 1 / 04 6 / 30 / 04 12 / 31 / 04 $1,000 $31.875 1 PV 1 $1,070.52 12 12 .05 2 (1.025) (1.025) McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-11 Level-Coupon Bonds: Example Find the present value (as of January 1, 2004), of a 6-3/8 coupon T-bond with semi-annual payments, and a maturity date of December 2009 if the YTM is 5-percent. N I/Y PV PMT 12 5 – 1,070.52 31.875 = 1,000×0.06375 2 FV McGraw-Hill/Irwin Corporate Finance, 7/e 1,000 © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-12 5.3 Bond Concepts 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. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-13 YTM and Bond Value When the YTM < coupon, the bond trades at a premium. Bond Value $1400 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. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-14 Bond Value Maturity and Bond Price Volatility 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 McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-15 Bond Value Coupon Rate and Bond Price Volatility 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 Discount Rate Low Coupon Bond McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-16 5.4 The Present Value of Common Stocks Dividends versus Capital Gains Valuation of Different Types of Stocks Zero Growth Constant Growth Differential Growth McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-17 Case 1: Zero Growth Assume that dividends will remain at the same level forever Div1 Div2 Div3 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 1 2 3 (1 r ) (1 r ) (1 r ) Div P0 r P0 McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-18 Case 2: Constant Growth Assume that dividends will grow at a constant rate, g, forever. i.e. Div1 Div0 (1 g ) 2 Div2 Div1 (1 g ) Div0 (1 g ) 3 Div3 Div ( 1 g ) Div ( 1 g ) 0 .2 .. 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 McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-19 Case 3: Differential Growth 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. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-20 Case 3: Differential Growth Assume that dividends will grow at rate g1 for N years and grow at rate g2 thereafter Div1 Div0 (1 g1 ) 2 Div2 Div1 (1 g1 ) Div0 (1 g1 ) .. . N DivN DivN 1 (1 g1 ) Div0 (1 g1 ) DivN 1 DivN (1 g2 ) Div0 (1 g1 )N (1 g2 ) .. . McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-21 Case 3: Differential Growth Dividends will grow at rate g1 for N years and grow at rate g2 thereafter 2 Div0 (1 g1 ) Div0 (1 g1 ) … 0 1 2 Div0 (1 g1 )N DivN (1 g2 ) Div0 (1 g1 )N (1 g2 ) … … N McGraw-Hill/Irwin Corporate Finance, 7/e N+1 © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-22 Case 3: Differential Growth We can value this as the sum of: an N-year annuity growing at rate g1 T C (1 g1 ) PA 1 T r g1 (1 r ) plus the discounted value of a perpetuity growing at rate g2 that starts in year N+1 DivN 1 r g2 PB (1 r )N McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-23 Case 3: Differential Growth To value a Differential Growth Stock, we can use DivN 1 T C (1 g1 ) r g2 P 1 T N r g1 (1 r ) (1 r ) Or we can cash flow it out. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-24 A Differential Growth Example 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? The discount rate is 12%. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-25 With the Formula DivN 1 T C (1 g1 ) r g2 P 1 r g1 (1 r )T (1 r )N $2(1.08)3 (1.04) 3 (1.08) .12 .04 $ 2 ( 1 . 08 ) 1 P 3 3 .12 .08 (1.12) (1.12) ( $32.75 ) P $54 [1 . 8966] (1.12)3 P $5.58 $23.31 P $28.89 McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-26 A Differential Growth Example (continued) $2(1.08) 0 1 $2.16 0 2 $2(1.08) 2 $2.33 $2(1.08)3 $2(1.08)3 (1.04) … 3 $2.62 $2.52 .08 4 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 $28.89 P0 2 3 1.12 (1.12) (1.12) 1 McGraw-Hill/Irwin Corporate Finance, 7/e 2 3 © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-27 A Differential Growth Example (continued) 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 N 3 1.12 I/Y 3.70 = PV – 5.58 PMT FV McGraw-Hill/Irwin Corporate Finance, 7/e $2 = 0 1.08 3 –1 ×100 I/Y 2×1.08 12 PV – 23.31 PMT 0 1.08 FV 2×(1.08)3 ×(1.04) 32.75 = .08 © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-28 5.5 Estimates of Parameters in the Dividend-Discount Model 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? McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-29 Where does g come from? g = Retention ratio × Return on retained earnings McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-30 Where does r come from? 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. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-31 5.6 Growth Opportunities 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 100percent of its earnings as dividends and the net present value of the growth opportunities. EPS NPVGO P r McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-32 5.7 The Dividend Growth Model and the NPVGO Model (Advanced) 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 per-share value of its growth opportunities. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-33 The Dividend Growth Model and the NPVGO Model Consider a firm that has EPS of $5 at the end of the first year, a dividend-payout ratio of 30%, a discount rate of 16-percent, and a return on retained earnings of 20percent. 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 $75 P0 r g .16 .14 McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-34 The NPVGO Model First, we must calculate the value of the firm as a cash cow. Div1 $5 $31.25 P0 r .16 Second, we must calculate the value of the growth opportunities. .20 3 . 50 3.50 .16 $.875 $43.75 P0 r g .16 .14 Finally, McGraw-Hill/Irwin Corporate Finance, 7/e P0 31.25 43.75 $75 © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-35 5.8 Price Earnings Ratio 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 Wall Street Journal uses last 4 quarter’s 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 favor sell at low multiples. Value stocks for example. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-36 Other Price Ratio Analysis 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 McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-37 5.9 Stock Market Reporting 52 WEEKS YLD VOL NET HI LO STOCK SYM DIV % PE 100s HI LO CLOSE CHG 52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75 Gap has been as high as $52.75 in the last year. Gap pays a dividend of 9 cents/share Given the current price, the dividend yield is ½ % Gap has been as low as $19.06 in the last year. McGraw-Hill/Irwin Corporate Finance, 7/e Gap ended trading at $19.25, down $1.75 from yesterday’s close Given the current price, the PE ratio is 15 times earnings 6,517,200 shares traded hands in the last day’s trading © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-38 5.9 Stock Market Reporting 52 WEEKS YLD VOL NET HI LO STOCK SYM DIV % PE 100s HI LO CLOSE CHG 52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75 Gap Incorporated is having a tough year, trading near their 52-week low. Imagine how you would feel if within the past year you had paid $52.75 for a share of Gap and now had a share worth $19.25! That 9-cent dividend wouldn’t go very far in making amends. Yesterday, Gap had another rough day in a rough year. Gap “opened the day down” beginning trading at $20.50, which was down from the previous close of $21.00 = $19.25 + $1.75 Looks like cargo pants aren’t the only things on sale at Gap. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-39 5.10 Summary and Conclusions 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 ) C 2. The value of a perpetuity is PV r McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-40 5.10 Summary and Conclusions (continued) 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 PV r 1 F 1 T T (1 r ) (1 r ) The yield to maturity (YTM) of a bond is that single rate that discounts the payments on the bond to the purchase price. McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-41 5.10 Summary and Conclusions (continued) 5. A stock can be valued by discounting its dividends. There are three cases: Div Zero growth in dividends P0 r Div1 Constant growth in P0 r g dividends DivN 1 Differential growth in dividends r g T C ( 1 g 2 1) P 1 T N r g1 (1 r ) (1 r ) McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 5-42 5.10 Summary and Conclusions (continued) 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 NPVGO P r McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.