Chapter 4 The Valuation of Long-Term Securities Learning Objectives After studying Chapter 4, you should be able to: 1. Distinguish among the various terms used to express value. 2. Value bonds, preferred stocks, and common stocks. 3. Calculate the rates of return (or yields) of different types of long-term securities. 4. List and explain a number of observations regarding the behavior of bond prices. Topics Distinctions Among Valuation Concepts Bond Valuation Preferred Stock Valuation Common Stock Valuation Rates of Return (or Yields) What is Value? Liquidation value represents the amount of money that could be realized if an asset or group of assets is sold separately from its operating organization. Going-concern value represents the amount a firm could be sold for as a continuing operating business. What is Value? Book value represents either (1) an asset: the accounting value of an asset -- the asset's cost minus its accumulated depreciation; (2) a firm: total assets minus liabilities and preferred stock as listed on the balance sheet. What is Value? Market value represents the market price at which an asset trades. Intrinsic value represents the price a security &quot;ought to have&quot; based on all factors bearing on valuation. Bond Valuation Important Terms Types of Bonds Valuation of Bonds Handling Semiannual Compounding Important Bond Terms A bond is a long-term debt instrument issued by a corporation or government. The maturity value (MV) [or face value] of a bond is the stated value. In the case of a U.S. bond, the face value is usually $1,000. Important Bond Terms The bond's coupon rate is the stated rate of interest; the annual interest payment divided by the bond's face value. The discount rate (capitalization rate) is dependent on the risk of the bond and is composed of the risk-free rate plus a premium for risk. Different Types of Bonds A perpetual bond is a bond that never matures. It has an infinite life. I I I V 1 2 [4.1] (1 k d ) (1 k d ) (1 k d ) t 1 I t (1 k d ) [4.2] I kd [4.3] Perpetual Bond Example Bond P has a $1,000 face value and provides an 8% annual coupon. The <a href="/keyword/appropriate-discount-rate/" >appropriate discount rate</a> is 10%. What is the value of the perpetual bond? I kd V = $1,000 ( 8%) = $80. = 10%. = I / kd [Reduced Form] = $80 / 10% = $800. Different Types of Bonds A non-zero coupon-paying bond is a coupon paying bond with a finite life. I I I MV V 1 2 (1 k d ) (1 k d ) (1 k d ) n (1 k d ) n n t 1 I (1 k d ) t MV (1 k d ) n [4.4] [4.5] I ( PVIFAkd ,n ) MV ( PVIFk d ,n ) I ( P / A, k d , n) MV ( P / F , k d , n) Coupon Bond Example Bond C has a $1,000 face value and provides an 8% annual coupon for 30 years. The <a href="/keyword/appropriate-discount-rate/" >appropriate discount rate</a> is 10%. What is the value of the coupon bond? V = $80 (PVIFA10%, 30) + $1,000 (PVIF10%, 30) = $80 (9.427) + $1,000 (.057) = $754.16 + $57.00 = $811.16. Different Types of Bonds A zero coupon bond is a bond that pays no interest but sells at a deep discount from its face value; it provides compensation to investors in the form of price appreciation. MV [4.6] V n (1 k d ) MV ( PVIFkd ,n ) MV ( P / F , k d , n) [4.7] Zero-Coupon Bond Example Bond Z has a $1,000 face value and a 30 year life. The <a href="/keyword/appropriate-discount-rate/" >appropriate discount rate</a> is 10%. What is the value of the zerocoupon bond? V = $1,000 (PVIF10%, 30) = $1,000 (.057) = $57.00 Semiannual Compounding Most bonds in the U.S. pay interest twice a year (1/2 of the annual coupon). Adjustments needed: (1) Divide kd by 2 (2) Multiply n by 2 (3) Divide I by 2 Semiannual Compounding A non-zero coupon bond adjusted for semiannual compounding. I 2n MV 2 [4.8] kd t kd 2n t 1 (1 ) (1 ) 2 2 [4.9] ( I )( PVIFAkd ) MV ( PVIFkd ) 2 ,2n ,2n 2 2 I )( P / A, k d , 2n) MV ( P / F , k d , 2n) ( 2 2 2 V Semiannual Coupon Bond Example Bond C has a $1,000 face value and provides an 8% semiannual coupon for 15 years. The <a href="/keyword/appropriate-discount-rate/" >appropriate discount rate</a> is 10% (annual rate). What is the value of the coupon bond? V = $40 (PVIFA5%, 30) + $1,000 (PVIF5%, 30) = $40 (15.373) + $1,000 (.231) [Table IV] [Table II] = $614.92 + $231.00 = $845.92 Semiannual Coupon Bond Example Let us use another worksheet to solve this problem. Assume that Bond C was purchased (settlement date) on 12-31-2004 and will be redeemed on 12-31-2019. This is identical to the 15-year period we discussed for Bond C. What is its percent of par? What is the value of the bond? Semiannual Coupon Bond Example 1. What is its percent of par? 84.628% of par (as quoted in financial papers) 84.628% x $1,000 face value = $846.28 2. What is the value of the bond? Preferred Stock Valuation Preferred Stock is a type of stock that promises a (usually) fixed dividend, but at the discretion of the board of directors. Preferred Stock has preference over common stock in the payment of dividends and claims on assets. Preferred Stock Valuation DP 1 (1 k P ) t 1 V DP DP 2 (1 k P ) (1 k P ) DP ( PVIFAk d , ) [4.10] DP (1 k P ) t DP kP Preferred Stock Example Stock PS has an 8%, $100 par value issue outstanding. The <a href="/keyword/appropriate-discount-rate/" >appropriate discount rate</a> is 10%. What is the value of the preferred stock? DivP kP V = $100 ( 8% ) = $8.00. = 10%. = DivP / kP = $8.00 / 10% = $80 Common Stock Valuation Common stock represents a residual ownership position in the corporation. Pro <a href="/keyword/rata-share/" >rata share</a> of future earnings after all other obligations of the firm (if any remain). Dividends may be paid out of the pro <a href="/keyword/rata-share/" >rata share</a> of earnings. Common Stock Valuation What cash flows will a shareholder receive when owning shares of common stock? (1) Future dividends (2) Future sale of the common stock shares Dividend Valuation Model Basic dividend valuation model accounts for the PV of all future dividends. D1 D2 D V [4.11] 1 2 (1 ke ) (1 ke ) (1 ke ) t 1 Dt t (1 ke ) Dt: Cash Dividend at time t [4.12] ke: Equity investor's required return Adjusted Dividend Valuation Model The basic dividend valuation model adjusted for the future stock sale. V D1 1 (1 ke ) Dn Pn D2 2 n (1 ke ) (1 ke ) n: The year in which the firm's shares are expected to be sold. Pricen: The expected share price in year n. Dividend Growth Pattern Assumptions The dividend valuation model requires the forecast of all future dividends. The following dividend growth rate assumptions simplify the valuation process. Constant Growth No Growth Growth Phases Constant Growth Model The constant growth model assumes that dividends will grow forever at the rate g. 2 D0 (1 g ) D0 (1 g ) D0 (1 g ) V 1 2 [4.13] (1 ke ) (1 ke ) (1 ke ) D1 (ke ke D1 V g) g [4.14] D0: Dividend paid at time 0. g: [4.15] k : e The constant growth rate. Investor's required return. Constant Growth Model Example Stock LVN's dividend per share at t=1 is expected to be $4, that it is expected to grow at a 6% rate forever, and that the <a href="/keyword/appropriate-discount-rate/" >appropriate discount rate</a> is 14%. What is the value of the common stock? VLVN = D1 / ( ke - g ) = $4 / ( .14 - .06 ) = $50 Constant Growth Model Conversion to an Earnings Multiplier Approach: Let b = constant retained earning proportion (1 b) V (ke D1 E1 D1 g) (1 b) E1 (ke g ) V E1 (1 b) (ke g ) [4.16] [4.17] Earning Multiplier [4.18] Constant Growth Model Example Stock LVN has a retention rate of 40% and earning per share for period 1 are expected to be $6.67 and growing at 6%. The <a href="/keyword/appropriate-discount-rate/" >appropriate discount rate</a> is 14%. What is the value of the common stock? V (1 b) E1 (ke g ) (1 0.4)$6.67 (0.14 0.06 ) $50 Zero Growth Model The zero growth model assumes that dividends will grow forever at the rate g = 0. V D0 1 (1 ke ) D1 (ke ) D0 D0 2 (1 ke ) (1 ke ) [4.19] D1: ke: Dividend paid at time 1. Investor's required return. Zero Growth Model Example Stock ZG has an expected growth rate of 0%. Each share of stock just received an annual $3.24 dividend per share. The <a href="/keyword/appropriate-discount-rate/" >appropriate discount rate</a> is 15%. What is the value of the common stock? D1 = $3.24 ( 1 + 0 ) = $3.24 VZG = D1 / ( ke - 0 ) = $3.24 / ( .15 - 0 ) = $21.60 Growth Phases Model The growth phases model assumes that dividends for each share will grow at two or more different growth rates. n V t 1 n t 1 D0 (1 g1 ) t (1 ke ) D0 (1 g1 ) t (1 ke ) t t n t Dn (1 g 2 ) t (1 ke ) 1 ke t n [4.20] 1 n (1 ke ) Dn 1 [4.21] g2 Growth Phases Model Example Stock GP has an expected growth rate of 16% for the first 3 years and 8% thereafter. Each share of stock just received an annual $3.24 dividend per share. The <a href="/keyword/appropriate-discount-rate/" >appropriate discount rate</a> is 15%. What is the value of the common stock under this scenario? Growth Phases Model Example 0 1 D1 2 D2 3 D3 4 D4 5 D5 6 D6 Growth of 16% for 3 years Growth of 8% to infinity! Stock GP has two phases of growth. The first, 16%, starts at time t=0 for 3 years and is followed by 8% thereafter starting at time t=3. We should view the time line as two separate time lines in the valuation. Growth Phases Model Example 0 1 D1 2 D2 3 D3 Growth Phase #1 plus the infinitely long Phase #2 0 1 2 3 4 D4 5 D5 6 D6 Note that we can value Phase #2 using the Constant Growth Model Growth Phases Model Example V3 0 D4 k g 1 2 We can use this model because dividends grow at a constant 8% rate beginning at the end of Year 3. 3 4 D4 5 D5 6 D6 Note that we can now replace all dividends from year 4 to infinity with the value at time t=3, V3! Simpler!! Growth Phases Model Example 0 1 D1 2 D2 3 D3 New Time Line 0 1 2 3 Where V3 V3 D4 k g Now we only need to find the first four dividends to calculate the necessary cash flows. Growth Phases Model Example Determine the annual dividends. D0 = $3.24 (this has been paid already) D1 = D0(1+g1)1 = $3.24(1.16)1 =$3.76 D2 = D0(1+g1)2 = $3.24(1.16)2 =$4.36 D3 = D0(1+g1)3 = $3.24(1.16)3 =$5.06 D4 = D3(1+g2)1 = $5.06(1.08)1 =$5.46 Growth Phases Model Example 0 1 2 3 Actual Values 3.76 4.36 5.06 0 1 2 3 78 5.46 Where $78 = .15-.08 Now we need to find the present value of the cash flows. Growth Phases Model Example We determine the PV of cash flows. PV(D1) = D1(PVIF15%, 1) = $3.76 (.870) = $3.27 PV(D2) = D2(PVIF15%, 2) = $4.36 (.756) = $3.30 PV(D3) = D3(PVIF15%, 3) = $5.06 (.658) = $3.33 P3 = $5.46 / (.15 - .08) = $78 [CG Model] PV(P3) = P3(PVIF15%, 3) = $78 (.658) = $51.32 Growth Phases Model Example Finally, we calculate the intrinsic value by summing all of cash flow present values. V = $3.27 + $3.30 + $3.33 + $51.32 V = $61.22 3 V t 1 D0 (1 .16) t (1 .15) t 1 3 (1 .15) D4 .15 .08 Calculating Rates of Return (or Yields) Steps to calculate the rate of return (or Yield). 1. Determine the expected cash flows. 2. Replace the intrinsic value (V) with the market price (P0). 3. Solve for the market required rate of return that equates the discounted cash flows to the market price. Determining Bond YTM Determine the Yield-to-Maturity (YTM) for the annual coupon paying bond with a finite life. n P0 t 1 I t (1 k d ) MV n (1 k d ) [4.22] I ( PVIFAkd ,n ) MV ( PVIFkd ,n ) I ( P / A, k d , n) MV ( P / F , k d , n) kd = YTM Determining the YTM Julie Miller want to determine the YTM for an issue of outstanding bonds at Basket Wonders (BW). BW has an issue of 10% annual coupon bonds with 15 years left to maturity. The bonds have a current market value of $1,250. What is the YTM? YTM Solution (Try 9%) $1,250 = $1,250 = $1,250 = = $100(PVIFA9%,15) + $1,000(PVIF9%, 15) $100(8.061) + $1,000(.275) $806.10 + $275.00 $1,081.10 [Rate is too high!] YTM Solution (Try 7%) $1,250 = $1,250 = $1,250 = = $100(PVIFA7%,15) + $1,000(PVIF7%, 15) $100(9.108) + $1,000(.362) $910.80 + $362.00 $1,272.80 [Rate is too low!] YTM Solution (Interpolate) .07 $1,273 IRR $1,250 .09 $1,081 $23 $192 .02 X $23 $192 X .02 = YTM Solution (Interpolate) .07 $1273 YTM $1250 .09 $1081 .02 X $23 $192 X= ($23)(0.02) $192 X = .0024 YTM = .07 + .0024 = .0724 or 7.24% Determining Semiannual Coupon Bond YTM Determine the Yield-to-Maturity (YTM) for the semiannual coupon paying bond with a finite life. I 2n MV 2 [4.23] P0 kd t kd 2n t 1 (1 ) (1 ) 2 2 ( I )( PVIFAkd ) MV ( PVIFkd ) 2 ,2n ,2n 2 2 I )( P / A, k d , 2n) MV ( P / F , k d , 2n) ( 2 2 2 [ 1 + (kd / 2) ] 2 -1 = YTM Determining the Semiannual Coupon Bond YTM Julie Miller want to determine the YTM for another issue of outstanding bonds. The firm has an issue of 8% semiannual coupon bonds with 20 years left to maturity. The bonds have a current market value of $950. What is the YTM? Determining Semiannual Coupon Bond YTM Determine the Yield-to-Maturity (YTM) for the semiannual coupon paying bond with a finite life. [ 1 + (kd / 2)2 ] -1 = YTM [ 1 + (.042626)2 ] -1 = .0871 or 8.71% Note: make sure you utilize the calculator answer in its DECIMAL form. Determining Semiannual Coupon Bond YTM This technique will calculate kd. You must then substitute it into the following formula. [ 1 + (kd / 2)2 ] -1 = YTM [ 1 + (.0852514/2)2 ] -1 = .0871 or 8.71% (same result!) Bond Price - Yield Relationship Discount Bond -- The market required rate of return exceeds the coupon rate (Par &gt; P0 ). Premium Bond -- The coupon rate exceeds the market required rate of return (P0 &gt; Par). Par Bond -- The coupon rate equals the market required rate of return (P0 = Par). Bond Price - Yield Relationship 1600 BOND PRICE ($) 1400 1200 1000 Par 600 0 0 2 4 6 8 10 12 Coupon Rate 14 16 18 5 Year 15 Year MARKET REQUIRED RATE OF RETURN (%) Bond Price-Yield Relationship When interest rates rise, then the market required rates of return rise and bond prices will fall. Assume that the required rate of return on a 15 year, 10% annual coupon paying bond rises from 10% to 12%. What happens to the bond price? Bond Price - Yield Relationship 1600 BOND PRICE ($) 1400 1200 1000 Par 600 0 0 2 4 6 8 10 12 Coupon Rate 14 16 5 Year 15 Year 18 MARKET REQUIRED RATE OF RETURN (%) Bond Price-Yield Relationship (Rising Rates) The required rate of return on a 15 year, 10% annual coupon paying bond has risen from 10% to 12%. Therefore, the bond price has fallen from $1,000 to $864. Bond Price-Yield Relationship When interest rates fall, then the market required rates of return fall and bond prices will rise. Assume that the required rate of return on a 15 year, 10% annual coupon paying bond falls from 10% to 8%. What happens to the bond price? Bond Price - Yield Relationship 1600 BOND PRICE ($) 1400 1200 1000 Par 600 0 0 2 4 6 8 10 12 Coupon Rate 14 16 18 5 Year 15 Year MARKET REQUIRED RATE OF RETURN (%) Bond Price-Yield Relationship (Declining Rates) The required rate of return on a 15 year, 10% coupon paying bond has fallen from 10% to 8%. Therefore, the bond price has risen from $1000 to $1171. The Role of Bond Maturity The longer the bond maturity, the greater the change in bond price for a given change in the market required rate of return. Assume that the required rate of return on both the 5 and 15 year, 10% annual coupon paying bonds fall from 10% to 8%. What happens to the changes in bond prices? Bond Price - Yield Relationship 1600 BOND PRICE ($) 1400 1200 1000 Par 600 0 0 2 4 6 8 10 12 Coupon Rate 14 16 18 5 Year 15 Year MARKET REQUIRED RATE OF RETURN (%) The Role of Bond Maturity The required rate of return on both the 5 and 15 year, 10% annual coupon paying bonds has fallen from 10% to 8%. The 5 year bond price has risen from $1,000 to $1,080 for the 5 year bond (+8.0%). The 15 year bond price has risen from $1,000 to $1,171 (+17.1%). Twice as fast! The Role of the Coupon Rate For a given change in the market required rate of return, the price of a bond will change by proportionally more, the lower the coupon rate. Example of the Role of the Coupon Rate Assume that the market required rate of return on two equally risky 15 year bonds is 10%. The annual coupon rate for Bond H is 10% and Bond L is 8%. What is the rate of change in each of the bond prices if market required rates fall to 8%? Example of the Role of the Coupon Rate The price on Bond H and L prior to the change in the market required rate of return is $1,000 and $848 respectively. The price for Bond H will rise from $1,000 to $1,171 (+17.1%). The price for Bond L will rise from $848 to $1,000 (+17.9%). Faster Increase! Determining the Yield on Preferred Stock Determine the yield for preferred stock with an infinite life. DP [4.24] P0 kP Solving for kP such that kP DP P0 [4.25] Preferred Stock Yield Example Assume that the annual dividend on each share of preferred stock is $10. Each share of preferred stock is currently trading at $100. What is the yield on preferred stock? kP = $10 / $100. kP = 10%. Determining the Yield on Common Stock Assume the constant growth model is appropriate. Determine the yield on the common stock. P0 D1 ke g [4.26] Solving for ke such that ke D1 P0 g [4.27] Common Stock Yield Example Assume that the expected dividend (D1) on each share of common stock is $3. Each share of common stock is currently trading at $30 and has an expected growth rate of 5%. What is the yield on common stock? ke = ( $3 / $30 ) + 5% ke = 10% + 5% = 15%