
1 :

Chapter 4 The Valuation of LongTerm Securities © 2001 PrenticeHall, Inc.
Fundamentals of Financial Management, 11/e
Created by: Gregory A. Kuhlemeyer, Ph.D.
Carroll College, Waukesha, WI 

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The Valuation of LongTerm Securities Distinctions Among Valuation Concepts
Bond Valuation
Preferred Stock Valuation
Common Stock Valuation
Rates of Return (or Yields) 

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What is Value? Goingconcern value represents the amount a firm could be sold for as a continuing operating business. 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. 

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What is Value? (2) a firm: total assets minus liabilities and preferred stock as listed on the balance sheet. Book value represents either
(1) an asset: the accounting value of an asset  the asset’s cost minus its accumulated depreciation; 

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What is Value? Intrinsic value represents the price a security “ought to have” based on all factors bearing on valuation. Market value represents the market price at which an asset trades. 

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Bond Valuation Important Terms
Types of Bonds
Valuation of Bonds
Handling Semiannual Compounding 

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Important Bond Terms 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. A bond is a longterm debt instrument issued by a corporation or government. 

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Important Bond Terms The discount rate (capitalization rate) is dependent on the risk of the bond and is composed of the riskfree rate plus a premium for risk. The bond’s coupon rate is the stated rate of interest; the annual interest payment divided by the bond’s face value. 

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Different Types of Bonds A perpetual bond is a bond that never matures. It has an infinite life. (1 + kd)1 (1 + kd)2 (1 + kd)¥ V = + + ... + I I I = S ¥ t=1 (1 + kd)t I or I (PVIFA kd, ¥ ) V = I / kd [Reduced Form] 

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Perpetual Bond Example Bond P has a $1,000 face value and provides an 8% coupon. The appropriate discount rate is 10%. What is the value of the perpetual bond?
I = $1,000 ( 8%) = $80.
kd = 10%.
V = I / kd [Reduced Form]
= $80 / 10% = $800. 

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N: “Trick” by using huge N like 1,000,000!
I/Y: 10% interest rate per period (enter as 10 NOT .10)
PV: Compute (resulting answer is cost to purchase)
PMT: $80 annual interest forever (8% x $1,000 face)
FV: $0 (investor never receives the face value) “Tricking” the Calculator to Solve N I/Y PV PMT FV Inputs Compute 1,000,000 10 80 0 800.0 

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Different Types of Bonds A nonzero couponpaying bond is a couponpaying bond with a finite life. (1 + kd)1 (1 + kd)2 (1 + kd)n V = + + ... + I I + MV I = S n t=1 (1 + kd)t I V = I (PVIFA kd, n) + MV (PVIF kd, n) (1 + kd)n + MV 

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Bond C has a $1,000 face value and provides an 8% annual coupon for 30 years. The appropriate discount rate is 10%. What is the value of the coupon bond? Coupon Bond Example V = $80 (PVIFA10%, 30) + $1,000 (PVIF10%, 30) = $80 (9.427) + $1,000 (.057)
[Table IV] [Table II]
= $754.16 + $57.00 = $811.16. 

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N: 30year annual bond
I/Y: 10% interest rate per period (enter as 10 NOT .10)
PV: Compute (resulting answer is cost to purchase)
PMT: $80 annual interest (8% x $1,000 face value)
FV: $1,000 (investor receives face value in 30 years) N I/Y PV PMT FV Inputs Compute 30 10 80 +$1,000 811.46 Solving the Coupon Bond on the Calculator (Actual, rounding
error in tables) 

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Different Types of Bonds A zerocoupon 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. (1 + kd)n V = MV = MV (PVIFkd, n) 

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V = $1,000 (PVIF10%, 30) = $1,000 (.057) = $57.00 ZeroCoupon Bond Example Bond Z has a $1,000 face value and a 30year life. The appropriate discount rate is 10%. What is the value of the zerocoupon bond? 

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N: 30year zerocoupon bond
I/Y: 10% interest rate per period (enter as 10 NOT .10)
PV: Compute (resulting answer is cost to purchase)
PMT: $0 coupon interest since it pays no coupon
FV: $1,000 (investor receives only face in 30 years) N I/Y PV PMT FV Inputs Compute 30 10 0 +$1,000 57.31 Solving the ZeroCoupon Bond on the Calculator (Actual, rounding
error in tables) 

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Semiannual Compounding (1) Divide kd by 2
(2) Multiply n by 2
(3) Divide I by 2 Most bonds in the U.S. pay interest twice a year (1/2 of the annual coupon).
Adjustments needed: 

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(1 + kd/2 ) 2*n (1 + kd/2 )1 Semiannual Compounding A nonzero coupon bond adjusted for semiannual compounding. V = + + ... + I / 2 I / 2 + MV = S 2*n t=1 (1 + kd /2 )t I / 2 = I/2 (PVIFAkd /2 ,2*n) + MV (PVIFkd /2 , 2*n) (1 + kd /2 ) 2*n + MV I / 2 (1 + kd/2 )2 

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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 Bond C has a $1,000 face value and provides an 8% semiannual coupon for 15 years. The appropriate discount rate is 10% (annual rate). What is the value of the coupon bond? 

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N: 15year semiannual coupon bond (15 x 2 = 30)
I/Y: 5% interest rate per semiannual period (10 / 2 = 5)
PV: Compute (resulting answer is cost to purchase)
PMT: $40 semiannual coupon ($80 / 2 = $40)
FV: $1,000 (investor receives face value in 15 years) N I/Y PV PMT FV Inputs Compute 30 5 40 +$1,000 846.28 The Semiannual Coupon Bond on the Calculator (Actual, rounding
error in tables) 

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Semiannual Coupon Bond Example Let us use another worksheet on your calculator to solve this problem. Assume that Bond C was purchased (settlement date) on 12312000 and will be redeemed on 12312015. This is identical to the 15year period we discussed for Bond C.
What is its percent of par? What is the value of the bond? 

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Solving the Bond Problem Press:
2nd Bond
12.3100 ENTER ?
8 ENTER ?
12.3115 ENTER ?
? ? ?
10 ENTER ?
CPT 

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Semiannual Coupon Bond Example What is its percent of par?
What is the value of the bond?
84.628% of par (as quoted in financial papers)
84.628% x $1,000 face value = $846.28 

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Preferred Stock is a type of stock that promises a (usually) fixed dividend, but at the discretion of the board of directors. Preferred Stock Valuation Preferred Stock has preference over common stock in the payment of dividends and claims on assets. 

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Preferred Stock Valuation This reduces to a perpetuity! (1 + kP)1 (1 + kP)2 (1 + kP)¥ V = + + ... + DivP DivP DivP = S ¥ t=1 (1 + kP)t DivP or DivP(PVIFA kP, ¥ ) V = DivP / kP 

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Preferred Stock Example DivP = $100 ( 8% ) = $8.00. kP = 10%. V = DivP / kP = $8.00 / 10% = $80 Stock PS has an 8%, $100 par value issue outstanding. The appropriate discount rate is 10%. What is the value of the preferred stock? 

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Common Stock Valuation Pro rata share of future earnings after all other obligations of the firm (if any remain).
Dividends may be paid out of the pro rata share of earnings. Common stock represents a residual ownership position in the corporation. 

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Common Stock Valuation (1) Future dividends
(2) Future sale of the common stock shares What cash flows will a shareholder receive when owning shares of common stock? 

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Dividend Valuation Model Basic dividend valuation model accounts for the PV of all future dividends. (1 + ke)1 (1 + ke)2 (1 + ke)¥ V = + + ... + Div1 Div¥ Div2 = S ¥ t=1 (1 + ke)t Divt Divt: Cash dividend at time t
ke: Equity investor’s required return 

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Adjusted Dividend Valuation Model The basic dividend valuation model adjusted for the future stock sale. (1 + ke)1 (1 + ke)2 (1 + ke)n V = + + ... + Div1 Divn + Pricen Div2 n: The year in which the firm’s shares are expected to be sold.
Pricen: The expected share price in year n. 

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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 

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Constant Growth Model The constant growth model assumes that dividends will grow forever at the rate g. (1 + ke)1 (1 + ke)2 (1 + ke)¥ V = + + ... + D0(1+g) D0(1+g)¥ = (ke  g) D1 D1: Dividend paid at time 1.
g : The constant growth rate.
ke: Investor’s required return. D0(1+g)2 

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Constant Growth Model Example Stock CG has an expected growth rate of 8%. Each share of stock just received an annual $3.24 dividend per share. The appropriate discount rate is 15%. What is the value of the common stock?
D1 = $3.24 ( 1 + .08 ) = $3.50
VCG = D1 / ( ke  g ) = $3.50 / ( .15  .08 ) = $50 

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Zero Growth Model The zero growth model assumes that dividends will grow forever at the rate g = 0. (1 + ke)1 (1 + ke)2 (1 + ke)¥ VZG = + + ... + D1 D¥ = ke D1 D1: Dividend paid at time 1.
ke: Investor’s required return. D2 

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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 appropriate discount rate 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 

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D0(1+g1)t Dn(1+g2)t Growth Phases Model The growth phases model assumes that dividends for each share will grow at two or more different growth rates. (1 + ke)t (1 + ke)t V =S t=1 n S t=n+1 ¥ + 

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D0(1+g1)t Dn+1 Growth Phases Model Note that the second phase of the growth phases model assumes that dividends will grow at a constant rate g2. We can rewrite the formula as: (1 + ke)t (ke  g2) V =S t=1 n + 1 (1 + ke)n 

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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 appropriate discount rate is 15%. What is the value of the common stock under this scenario? 

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Growth Phases Model Example 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. ? 0 1 2 3 4 5 6 D1 D2 D3 D4 D5 D6 Growth of 16% for 3 years Growth of 8% to infinity! 

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Growth Phases Model Example Note that we can value Phase #2 using the Constant Growth Model ? 0 1 2 3 D1 D2 D3 D4 D5 D6 0 1 2 3 4 5 6 Growth Phase
#1 plus the infinitely long Phase #2 

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Growth Phases Model Example Note that we can now replace all dividends from Year 4 to infinity with the value at time t=3, V3! Simpler!! ? V3 = D4 D5 D6 0 1 2 3 4 5 6 D4
kg We can use this model because
dividends grow at a constant 8%
rate beginning at the end of Year 3. 

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Growth Phases Model Example Now we only need to find the first four dividends to calculate the necessary cash flows. 0 1 2 3 D1 D2 D3 V3 0 1 2 3 New Time
Line D4
kg Where V3 = 

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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 

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Growth Phases Model Example Now we need to find the present value of the cash flows. 0 1 2 3 3.76 4.36 5.06 78 0 1 2 3 Actual
Values 5.46
.15.08 Where $78 = 

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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 

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D0(1+.16)t D4 Growth Phases Model Example Finally, we calculate the intrinsic value by summing all the cash flow present values. (1 + .15)t (.15.08) V = S t=1 3 + 1 (1+.15)n V = $3.27 + $3.30 + $3.33 + $51.32 V = $61.22 

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Solving the Intrinsic Value Problem using CF Registry Steps in the Process (Page 1)
Step 1: Press CF key
Step 2: Press 2nd CLR Work keys
Step 3: For CF0 Press 0 Enter ? keys
Step 4: For C01 Press 3.76 Enter ? keys
Step 5: For F01 Press 1 Enter ? keys
Step 6: For C02 Press 4.36 Enter ? keys
Step 7: For F02 Press 1 Enter ? keys 

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Solving the Intrinsic Value Problem using CF Registry RESULT: Value = $61.18! (Actual, rounding error in tables) Steps in the Process (Page 2)
Step 8: For C03 Press 83.06 Enter ? keys
Step 9: For F03 Press 1 Enter ? keys
Step 10: Press ? ? keys
Step 11: Press NPV
Step 12: Press 15 Enter ? keys
Step 13: Press CPT


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Calculating Rates of Return (or Yields) 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. Steps to calculate the rate of return (or yield). 

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Determining Bond YTM Determine the YieldtoMaturity (YTM) for the couponpaying bond with a finite life. P0 = S n t=1 (1 + kd )t I = I (PVIFA kd , n) + MV (PVIF kd , n) (1 + kd )n + MV kd = YTM 

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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? 

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YTM Solution (Try 9%) $1,250 = $100(PVIFA9%,15) + $1,000(PVIF9%, 15)
$1,250 = $100(8.061) + $1,000(.275)
$1,250 = $806.10 + $275.00
= $1,081.10 [Rate is too high!] 

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YTM Solution (Try 7%) $1,250 = $100(PVIFA7%,15) + $1,000(PVIF7%, 15)
$1,250 = $100(9.108) + $1,000(.362)
$1,250 = $910.80 + $362.00
= $1,272.80 [Rate is too low!] 

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.07 $1,273
.02 IRR $1,250 $192
.09 $1,081
X $23 .02 $192 YTM Solution (Interpolate) $23 X = 

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.07 $1,273
.02 IRR $1,250 $192
.09 $1,081
X $23 .02 $192 YTM Solution (Interpolate) $23 X = 

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.07 $1273
.02 YTM $1250 $192
.09 $1081
($23)(0.02) $192 YTM Solution (Interpolate) $23 X X = X = .0024 YTM = .07 + .0024 = .0724 or 7.24% 

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N: 15year annual bond
I/Y: Compute  Solving for the annual YTM
PV: Cost to purchase is $1,250
PMT: $100 annual interest (10% x $1,000 face value)
FV: $1,000 (investor receives face value in 15 years) N I/Y PV PMT FV Inputs Compute 15 1,250 100 +$1,000 7.22% (actual YTM) YTM Solution on the Calculator 

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Determining Semiannual Coupon Bond YTM P0 = S 2n t=1 (1 + kd /2 )t I / 2 = (I/2)(PVIFAkd /2, 2n) + MV(PVIFkd /2 , 2n) + MV [ 1 + (kd / 2) ]2 1 = YTM Determine the YieldtoMaturity (YTM) for the semiannual couponpaying bond with a finite life. (1 + kd /2 )2n 

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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? 

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N: 20year semiannual bond (20 x 2 = 40)
I/Y: Compute  Solving for the semiannual yield now
PV: Cost to purchase is $950 today
PMT: $40 annual interest (8% x $1,000 face value / 2)
FV: $1,000 (investor receives face value in 15 years) N I/Y PV PMT FV Inputs Compute 40 950 40 +$1,000 4.2626% = (kd / 2) YTM Solution on the Calculator 

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Determining Semiannual Coupon Bond YTM [ 1 + (kd / 2) ]2 1 = YTM Determine the YieldtoMaturity (YTM) for the semiannual couponpaying bond with a finite life. [ 1 + (.042626) ]2 1 = .0871
or 8.71% 

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Solving the Bond Problem Press:
2nd Bond
12.3100 ENTER ?
8 ENTER ?
12.3120 ENTER ?
? ? ? ?
95 ENTER ?
CPT 

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Determining Semiannual Coupon Bond YTM [ 1 + (kd / 2) ]2 1 = YTM This technique will calculate kd. You must then substitute it into the following formula. [ 1 + (.0852514/2) ]2 1 = .0871
or 8.71% (same result!) 

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Bond PriceYield Relationship Discount Bond  The market required rate of return exceeds the coupon rate (Par > P0 ).
Premium Bond  The coupon rate exceeds the market required rate of return (P0 > Par).
Par Bond  The coupon rate equals the market required rate of return (P0 = Par). 

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Bond PriceYield Relationship Coupon Rate
MARKET REQUIRED RATE OF RETURN (%) BOND PRICE ($) 1000
Par 1600 1400 1200 600 0 0 2 4 6 8 10 12 14 16 18 5 Year 15 Year 

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Bond PriceYield Relationship Assume that the required rate of return on a 15year, 10% couponpaying bond rises from 10% to 12%. What happens to the bond price? When interest rates rise, then the market required rates of return rise and bond prices will fall. 

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Bond PriceYield Relationship Coupon Rate
MARKET REQUIRED RATE OF RETURN (%) BOND PRICE ($) 1000
Par 1600 1400 1200 600 0 0 2 4 6 8 10 12 14 16 18 15 Year 5 Year 

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Bond PriceYield Relationship (Rising Rates) Therefore, the bond price has fallen from $1,000 to $864.
The required rate of return on a 15year, 10% couponpaying bond has risen from 10% to 12%. 

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Bond PriceYield Relationship Assume that the required rate of return on a 15year, 10% couponpaying bond falls from 10% to 8%. What happens to the bond price? When interest rates fall, then the market required rates of return fall and bond prices will rise. 

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Bond PriceYield Relationship Coupon Rate
MARKET REQUIRED RATE OF RETURN (%) BOND PRICE ($) 1000
Par 1600 1400 1200 600 0 0 2 4 6 8 10 12 14 16 18 15 Year 5 Year 

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Bond PriceYield Relationship (Declining Rates) Therefore, the bond price has risen from $1,000 to $1,171.
The required rate of return on a 15year, 10% couponpaying bond has fallen from 10% to 8%. 

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The Role of Bond Maturity Assume that the required rate of return on both the 5 and 15year, 10% couponpaying bonds fall from 10% to 8%. What happens to the changes in bond prices? The longer the bond maturity, the greater the change in bond price for a given change in the market required rate of return. 

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Bond PriceYield Relationship Coupon Rate
MARKET REQUIRED RATE OF RETURN (%) BOND PRICE ($) 1000
Par 1600 1400 1200 600 0 0 2 4 6 8 10 12 14 16 18 15 Year 5 Year 

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The Role of Bond Maturity The 5year bond price has risen from $1,000 to $1,080 for the 5year bond (+8.0%).
The 15year bond price has risen from $1,000 to $1,171 (+17.1%). Twice as fast! The required rate of return on both the 5 and 15year, 10% couponpaying bonds has fallen from 10% to 8%. 

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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.


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Example of the Role of the Coupon Rate Assume that the market required rate of return on two equally risky 15year bonds is 10%. The 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%? 

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Example of the Role of the Coupon Rate 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%). It rises faster! The price on Bonds H and L prior to the change in the market required rate of return is $1,000 and $848, respectively. 

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Determining the Yield on Preferred Stock Determine the yield for preferred stock with an infinite life.
P0 = DivP / kP
Solving for kP such that
kP = DivP / P0 

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Preferred Stock Yield Example kP = $10 / $100.
kP = 10%. 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? 

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Determining the Yield on Common Stock Assume the constant growth model is appropriate. Determine the yield on the common stock.
P0 = D1 / ( ke  g )
Solving for ke such that
ke = ( D1 / P0 ) + g 

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Common Stock Yield Example ke = ( $3 / $30 ) + 5%
ke = 15% 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? 
