第五章 债券和股票的定价
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PV (1 Fr)T($ 11 .0 ,0)6 3000$17.141
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
High Coupon Bond Discount Rate
Low Coupon Bond
5.4 The Present Value of Common Stocks
• Dividends versus Capital Gains • Valuation of Different Types of Stocks
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
$31.875 $31.875 $31.875
1/1/02 6/30/02 12/31/02 6/30/09
$1,031.875
12/31/09
Level-Coupon Bonds: Example
Find the present value (as of January 1, 2002), 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.
– Time to maturity (T) = Maturity date - today’s date – Face value (F) – Discount rate (r)
$0
$0
$0
$F
0
1
2
T 1
T
Present value of a pure discount bond at time 0:
– 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, 2002 the size and timing of cash flows are:
PV
F (1 r)T
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
1
$0
$0
2
29
$1,000
$0$$1,00 1023209
4. A lower coupon bond has a higher relative price change than a higher coupon bond when YTM changes. All other features are identical.
YTM and Bond Value
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:
$31.875 $31.875 $31.875
1/1/02 6/30/02 12/31/02 6/30/09
$1,031.875
12/31/09
5.2 How to Value Bonds
• Identify the size and timing of cash flows. • Discount at the correct discount rate.
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)
$C
$C
$C
$C$F
0
1
2
T 1
T
Value of a Level-coupon bond
= PV of coupon payment annuity + PV of face value
PV C r1(11r)T(1Fr)T
Level-Coupon Bonds: Example
Find the present value (as of January 1, 2002), 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. – On January 1, 2002 the size and timing of cash flows are:
$1400
When the YTM < coupon, the bond
1300
trades at a premium.
Bond Value
1200
1100
When the YTM = coupon, the
bond trades at par.
1000
800 0
0.01 0.02 0.03 0.ຫໍສະໝຸດ Baidu4 0.05 0.06 0.07 0.08 0.09 0.1
6 3/8
Discount Rate
When the YTM > coupon, the bond trades at a discount.
Maturity and Bond Price Volatility
Bond Value
Consider two otherwise identical bonds.
– If you know the price of a bond and the size and timing of cash flows, the yield to maturity is the discount rate.
Pure Discount Bonds
Information needed for valuing pure discount bonds:
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:
P0
Div 1 rg
Case 3: Differential Growth
Discount Model 5.6 Growth Opportunities 5.7 The Dividend Growth Model and the NPVGO
Model (Advanced) 5.8 Price Earnings Ratio 5.9 Stock Market Reporting 5.10 Summary and Conclusions
• In dollar terms (fixed-rate borrowing) • As a formula (adjustable-rate borrowing)
5.1 Definition and Example of a Bond
• Consider a U.S. government bond listed as 6 3/8 of December 2009.
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:
Case 2: Constant Growth
Assume that dividends will grow at a constant rate, g, forever. i.e. D1i vD0i(1 vg)
D 2 iD v 1 (1 i g v ) D 0 (1 i g v )2
D 3 iD v 2 (1 i g ...v ) D 0 (1 i g v )3
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.
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
– 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.
– Zero Growth – Constant Growth – Differential Growth
Case 1: Zero Growth
• Assume that dividends will remain at the same level forever
D 1 iD v2 iD v3 i v
• Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity:
P0
Div1 (1r)1
Div2 (1r)2
Div3 (1r)3
P0
Div r
Chapter Outline
5.1 Definition and Example of a Bond 5.2 How to Value Bonds 5.3 Bond Concepts 5.4 The Present Value of Common Stocks 5.5 Estimates of Parameters in the Dividend-
• PV= $31.875 [ 1- 1/ (1.025)16 ] + $1,000 = $1,089.75
.05/2
(1.025)16
5.3 Bond Concepts
1. Bond prices and market interest rates move in opposite directions.
• Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter.
PV (1 Fr)T($ 11 .0 ,0)6 3000$17.141
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
High Coupon Bond Discount Rate
Low Coupon Bond
5.4 The Present Value of Common Stocks
• Dividends versus Capital Gains • Valuation of Different Types of Stocks
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
$31.875 $31.875 $31.875
1/1/02 6/30/02 12/31/02 6/30/09
$1,031.875
12/31/09
Level-Coupon Bonds: Example
Find the present value (as of January 1, 2002), 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.
– Time to maturity (T) = Maturity date - today’s date – Face value (F) – Discount rate (r)
$0
$0
$0
$F
0
1
2
T 1
T
Present value of a pure discount bond at time 0:
– 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, 2002 the size and timing of cash flows are:
PV
F (1 r)T
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
1
$0
$0
2
29
$1,000
$0$$1,00 1023209
4. A lower coupon bond has a higher relative price change than a higher coupon bond when YTM changes. All other features are identical.
YTM and Bond Value
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:
$31.875 $31.875 $31.875
1/1/02 6/30/02 12/31/02 6/30/09
$1,031.875
12/31/09
5.2 How to Value Bonds
• Identify the size and timing of cash flows. • Discount at the correct discount rate.
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)
$C
$C
$C
$C$F
0
1
2
T 1
T
Value of a Level-coupon bond
= PV of coupon payment annuity + PV of face value
PV C r1(11r)T(1Fr)T
Level-Coupon Bonds: Example
Find the present value (as of January 1, 2002), 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. – On January 1, 2002 the size and timing of cash flows are:
$1400
When the YTM < coupon, the bond
1300
trades at a premium.
Bond Value
1200
1100
When the YTM = coupon, the
bond trades at par.
1000
800 0
0.01 0.02 0.03 0.ຫໍສະໝຸດ Baidu4 0.05 0.06 0.07 0.08 0.09 0.1
6 3/8
Discount Rate
When the YTM > coupon, the bond trades at a discount.
Maturity and Bond Price Volatility
Bond Value
Consider two otherwise identical bonds.
– If you know the price of a bond and the size and timing of cash flows, the yield to maturity is the discount rate.
Pure Discount Bonds
Information needed for valuing pure discount bonds:
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:
P0
Div 1 rg
Case 3: Differential Growth
Discount Model 5.6 Growth Opportunities 5.7 The Dividend Growth Model and the NPVGO
Model (Advanced) 5.8 Price Earnings Ratio 5.9 Stock Market Reporting 5.10 Summary and Conclusions
• In dollar terms (fixed-rate borrowing) • As a formula (adjustable-rate borrowing)
5.1 Definition and Example of a Bond
• Consider a U.S. government bond listed as 6 3/8 of December 2009.
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:
Case 2: Constant Growth
Assume that dividends will grow at a constant rate, g, forever. i.e. D1i vD0i(1 vg)
D 2 iD v 1 (1 i g v ) D 0 (1 i g v )2
D 3 iD v 2 (1 i g ...v ) D 0 (1 i g v )3
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.
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
– 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.
– Zero Growth – Constant Growth – Differential Growth
Case 1: Zero Growth
• Assume that dividends will remain at the same level forever
D 1 iD v2 iD v3 i v
• Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity:
P0
Div1 (1r)1
Div2 (1r)2
Div3 (1r)3
P0
Div r
Chapter Outline
5.1 Definition and Example of a Bond 5.2 How to Value Bonds 5.3 Bond Concepts 5.4 The Present Value of Common Stocks 5.5 Estimates of Parameters in the Dividend-
• PV= $31.875 [ 1- 1/ (1.025)16 ] + $1,000 = $1,089.75
.05/2
(1.025)16
5.3 Bond Concepts
1. Bond prices and market interest rates move in opposite directions.
• Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter.