1
Finance School of Management
Objective
Explain the principles of bond pricing
Understand the features that affect
bond prices
Chapter 8,Valuation of Known
Cash Flows,Bonds
2
Finance School of Management
Chapter 8 Contents
? Using Present Value Formulas to Value Known Flows
? The Basic Building Blocks,Pure Discount Bonds
? Coupon Bonds,Current Yield,and Yield-To-Maturity
? Reading Bond Listings
? Why Yields for the Same Maturity Differ
? The Behavior of Bond Prices over Time
3
Finance School of Management
Valuation and Fixed-Income Securities
? Essence of valuation process
? Valuation models
? Fixed-income securities and other contracts
promising a stream of known future cash
payments
– Bonds
– Mortgages
– Pension annuities
4
Finance School of Management
Reasons for Valuing Fixed-Income
Securities
? To have an agreed-upon valuation procedure in
setting the terms of the contracts at the outset,
? To revaluate the securities when they are sold
before maturity,
5
Finance School of Management
Using Present Value Formulas to Value
Known Cash Flows
? A fixed-income
security that
promises to pay
$100 each year for
the next three years,
? The appropriate
discount rate is 6%
per year,
? An hour after you
buy the security,the
risk-free interest rate
rises from 6% to 7%
per year,
? ?
i
iP M TPV n???? )1(1
n i PV FV PM T Res ult
3 6%? 0 100 PV= 267.3 0
n i PV FV PM T Res ult
3 7%? 0 100 PV= 262.4 3
6
Finance School of Management
Bond Prices Fall as the Interest
Rates Rise
? Write the PV of the fixed-income security as the
sum terms
n
n
n
n
n
j
j
j
i
p m t
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p m t
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p m t
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1
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1
1
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1
1
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1
1
2
2
1
1
1
7
Finance School of Management
The Difficulty of Valuation of
Known Cash Flows
? We do not know usually which discount rate to use
in the present value formula,
? Is it correct to use the interest rate corresponding to
a three-year maturity in valuing the three-year
annuity in the previous example?
8
Finance School of Management
U S T r e a s u r y Y i l e d C u r v e,J a n 9 7
4, 5 0
5, 0 0
5, 5 0
6, 0 0
6, 5 0
7, 0 0
7, 5 0
0 5 10 15 20 25 30
Y e a r s t o M a t u r i t y
A
n
n
u
a
l
i
z
e
d
Y
i
e
l
d
(
%
)
9
Finance School of Management
The Basic Building Blocks,Pure
Discount Bonds
? The difficulties of finding equivalent fixed-income
securities,or comparables and making adjustments for
differences,
? Any fixed-income security can be decomposed into a
series of known payments at different time points in the
future,
? Pure discount bonds (zero-coupon bonds),Promising a
single payment of cash at the maturity date (in the future),
10
Finance School of Management
Pure Discount Bonds
? The pure discount bond is an example of the
present value of a lump sum equation we
analyzed in Chapter 4,
? Solving this,the yield-to-maturity on a pure
discount bond is given by the relationship,
? ? 11
1
??
?
?
?
?
?
????
nn
P
F
iiPF
11
Finance School of Management
Pure Discount Bonds
? ? 11
1
??
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?
?
?
?
????
nn
P
F
iiPF
?In this equation,
– P is the present value or price of the bond
– F is the face or future value
– n is the investment period
– i is the yield-to-maturity
12
Finance School of Management
Pure Discount Bonds
%60.61
8 8 0
1 0 0 0
1
2
11
???
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???
?
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n
P
F
i
n i PV FV PM T Res ult
2? 880 100 0 0 i= 6.60%
? A two-year pure discount bond with a face
value of $1,000 and a price of $880
13
Finance School of Management
Pricing a Coupon Bond
? A 3-year bond with a face value of $1,000 that makes
annual coupon payments at a coupon rate 10%
? Prices of pure discount bonds
M at ur i t y P r i c e s pe r $1 of
F ac e V al ue
1 ye a r 0,96
2 ye a r s 0,89
3 ye a r s 0,81
P r i c e s of P u r e D i s c ou n t B on d s
14
Finance School of Management
First Solution Method
? ?
00.1 0 7 6$
1 0 01 0 0 081.01 0 089.01 0 096.0
?
???????
P
P
15
Finance School of Management
Second Solution Method
91.0 7 5,1$
0 7 2 8.1
1 0 01 0 0 0
0 6 0 0.1
1 0 0
0 4 1 7.1
1 0 0
%28.71
8 1 0
0 0 0,1
%00.61
8 9 0
0 0 0,1
%17.41
9 6 0
0 0 0,1
32
3
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16
Finance School of Management
The YTM of the Coupon Bond
?It would be a mistake to discount all three cash flows
using the same three-year yield of 7.28%,
?The single discount rate that we can use to discount all
three cash flows is the yield-to-maturity (YTM),
?However,can we get it?
n i PV FV PMT R esult
3 7.28 %? 1000 100 PV =$10 71
n i PV FV PMT R esult
3? 1076 1000 100 i=7.10%
17
Finance School of Management
? You would like to create a 2-year synthetic zero-coupon
bond,
? Assume you are aware of the following information,
– 1-year zero-coupon bonds are trading for $0.93 per
dollar of face value,and
– 2-year 7% coupon bonds (annual payments) are selling
at $985.30 (Face value = $1,000),
? Assume you can purchase the 2-year coupon bond and
unbundle the two cash flows and sell them,
– You would receive,93× $70 = $65.10 from the sale of
the first payment,
– To break even,you would need to receive $985.30-
$65.10 = $920.20 from the sale of the 2-year strip,
Coupon Stripping
18
Finance School of Management
Coupon Rate
? Coupon rate is the interest rate applied to the face
value to compute the coupon payment,
– A bond with a face value of $1,000 and a coupon
rate of 10%
– An annuity component of $100 per year and a
“balloon” or,bullet” payment at maturity
19
Finance School of Management
Current Yield and Yield-to-maturity
? Current yield is the annual coupon divided by the
bond’s price,
? Yield-to-maturity is the discount rate that makes
the present value of a bond’s stream of promised
cash payments equal to its price,
20
Finance School of Management
Example 1
? A 20-year-maturity
bond with a face
value of $1,000 and a
coupon rate of 10%
was originally issued
19 years ago,
? At that time,the yield
curve was flat at 10%
per year,
? Now the interest rate
on one-year bonds is
5% per year,
? Its market price will now be
? Its current yield is
? Its yield-to-maturity is
62.1 0 4 7$05.11 1 0 0 ??P
%55.962.1 0 4 7100 ??y i e l dC u r r e n t
%5162.1 0 4 71 1 0 0 ????? m a t u r i t ytoY i e l d
21
Finance School of Management
Example 2
? A bond with a face value of $1,000 and a coupon
rate of 4% will mature in two years,
? Its market price is $950,
? Its current yield is
? Its yield-to-maturity
%21.49 5 040 ??y i e l dC u r r e n t
n i PV FV PMT R esult
2? 950 1000 40 i=6.76%
22
Finance School of Management
Bonds Trading at Par
? Bond Pricing Principle #1,(Par Bonds)
– If a bond’s price equals its face value,then its
yield-to-maturity = current yield = coupon rate,
? Proof,
F
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1
1
1
23
Finance School of Management
Bonds Trading at Premium or Discount
? Bond Pricing Principle #2,(Premium Bonds)
– If a bond has a price higher than its face value,then
its yield-to-maturity < current yield < coupon rate,
? Bond Pricing Principle #3,(Discount Bonds)
– If a bond has a price lower than its face value,then
its yield-to-maturity > current yield > coupon rate,
24
Finance School of Management
Proof,
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1
1
1
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p m t
FiP
ip m t
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p m tni
i
F
ii
p m t
P
25
Finance School of Management
Y i e l d R e l a t i o n s h i p s
0
0, 0 2
0, 0 4
0, 0 6
0, 0 8
0, 1
0, 1 2
0, 1 4
0, 1 6
0, 1 8
0, 2
6 0 0, 0 0 8 0 0, 0 0 1 0 0 0, 0 0 1 2 0 0, 0 0 1 4 0 0, 0 0 1 6 0 0, 0 0 1 8 0 0, 0 0
P r i c e
Y
i
e
l
d
c o u p o n _ y
c u r r e n t _ y
y _ t _ m
26
Finance School of Management
Beware of ―High-Yield‖ US Treasure
Bond Funds
? You have $10,000 to invest for one year,You are
deciding between,
– Putting your money in a one-year government-insured
bank CD offering an interest rate of 5%;
– Investing in the shares of a U.S,Treasure bond fund that
holds one-year bonds with a coupon rate of 8%,
? The bonds are selling at a premium,you must pay
$10,285.71 for $10,000 of face value,
? The fund advertises a yield of 7.78%,
? The fund charges a 1% annual fee for their services,
27
Finance School of Management
Beware of ―High-Yield‖ US Treasure
Bond Funds
%51
71.285,10$
000,10$800$ ?????? m a t u r i t ytoY i e l d
%78.7
71.285,10$
800$ ??y i e l dCu r r e n t
28
Finance School of Management
The Effect of Coupon Rate
? Two different two-year coupon bonds—one with a
coupon rate of 5% and the other with a coupon rate of
10%,
? The current market prices and yields of one- and two-
year pure discount bonds,
M a t u r i t y P r i c e s p e r $ 1 o f
F a c e V a l u e
Y i e l d
( p e r y e a r )
1 y e a r $ 0, 9 6 1 5 3 8 4%
2 y e a r s $ 0, 8 8 9 9 9 6 6%
29
Finance School of Management
The Effect of Coupon Rate
? The market prices of the two coupon bonds should be
– For the 5%-coupon bond,
– For the 10%-coupon bond,
? The yields to maturity on the coupon bonds should be
– For the 5%-coupon bond,the YTM is 5.9500%
– For the 10%-coupon bond,the YTM is 5.9064%
? When the yield curve is not flat,bonds of the same
maturity with different coupon rates have different
yields to maturity,
57.9 8 2$0 5 0,1$8 8 9 9 9 6.50$9 6 1 5 3 8,????
15.0 7 5,1$1 0 0,1$8 8 9 9 9 6.1 0 0$9 6 1 5 3 8,????
30
Finance School of Management
Other Effects on Bond Yields
?Default risk
?Taxes
?Callability
?Convertibility
31
Finance School of Management
The Effect of the Passage of Time
?A 20-year pure discount bond with a face value of $1,000
and a constant yield of 6% should be priced at
n i PV FV PMT R esult
20 6%? 1000 0 PV =$31 1.80
?After one year goes by,its price should be
n i PV FV PMT R esult
19 6%? 1000 0 PV =$33 0.51
?If the yield curve were flat and interest rates did not
change,any default-free discount bond’s price would rise
with the passage of time,and any premium bond’s price
would fall,
32
Finance School of Management
Movement of a Pure discount Bond’s Price over Time
300
400
500
600
700
800
900
1,000
0 2 4 6 8 10 12 14 16 18 20
Y e ar s E l ap s e d s i n c e D ate of I s s u e
P
r
i
c
e
33
Finance School of Management
Interest-Rate Risk
? The concept
? The sensitivity of bond prices to interest rates
? The prices of 30-year 8% coupon par bond would fall
by roughly 10% if the level of interest rates were to rise
from 8% to 9%,
? The prices of 30-year pure discount bond would fall by
roughly 23% if the level of interest rates were to rise
from 8% to 9%,
? Why?
34
Finance School of Management
Sensitivity of Bond Price to Interest Rates
0
0.5
1
1.5
2
2.5
3
3.5
4
4 5 6 7 8 9 10 11 12
I n te r e s t R a te i n % p e r Y e a r
P
r
i
c
e
R
a
ti
o
Zeros
Par Bonds
35
Finance School of Management
?
?
??
T
t
twtD
1
)(
PV
kCF
PV
CFPVtw ttt ???? )1()()(
Dk kdPVd P V ???? 1 )1(
?
?
???
T
t
t
t kCFPV
1
)1(
)1(* kDD ??
Duration
Finance School of Management
Objective
Explain the principles of bond pricing
Understand the features that affect
bond prices
Chapter 8,Valuation of Known
Cash Flows,Bonds
2
Finance School of Management
Chapter 8 Contents
? Using Present Value Formulas to Value Known Flows
? The Basic Building Blocks,Pure Discount Bonds
? Coupon Bonds,Current Yield,and Yield-To-Maturity
? Reading Bond Listings
? Why Yields for the Same Maturity Differ
? The Behavior of Bond Prices over Time
3
Finance School of Management
Valuation and Fixed-Income Securities
? Essence of valuation process
? Valuation models
? Fixed-income securities and other contracts
promising a stream of known future cash
payments
– Bonds
– Mortgages
– Pension annuities
4
Finance School of Management
Reasons for Valuing Fixed-Income
Securities
? To have an agreed-upon valuation procedure in
setting the terms of the contracts at the outset,
? To revaluate the securities when they are sold
before maturity,
5
Finance School of Management
Using Present Value Formulas to Value
Known Cash Flows
? A fixed-income
security that
promises to pay
$100 each year for
the next three years,
? The appropriate
discount rate is 6%
per year,
? An hour after you
buy the security,the
risk-free interest rate
rises from 6% to 7%
per year,
? ?
i
iP M TPV n???? )1(1
n i PV FV PM T Res ult
3 6%? 0 100 PV= 267.3 0
n i PV FV PM T Res ult
3 7%? 0 100 PV= 262.4 3
6
Finance School of Management
Bond Prices Fall as the Interest
Rates Rise
? Write the PV of the fixed-income security as the
sum terms
n
n
n
n
n
j
j
j
i
p m t
i
p m t
i
p m t
i
p m t
i
p m tPV
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1
1
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1
1
*...
1
1
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1
1
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1
1
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1
1
2
2
1
1
1
7
Finance School of Management
The Difficulty of Valuation of
Known Cash Flows
? We do not know usually which discount rate to use
in the present value formula,
? Is it correct to use the interest rate corresponding to
a three-year maturity in valuing the three-year
annuity in the previous example?
8
Finance School of Management
U S T r e a s u r y Y i l e d C u r v e,J a n 9 7
4, 5 0
5, 0 0
5, 5 0
6, 0 0
6, 5 0
7, 0 0
7, 5 0
0 5 10 15 20 25 30
Y e a r s t o M a t u r i t y
A
n
n
u
a
l
i
z
e
d
Y
i
e
l
d
(
%
)
9
Finance School of Management
The Basic Building Blocks,Pure
Discount Bonds
? The difficulties of finding equivalent fixed-income
securities,or comparables and making adjustments for
differences,
? Any fixed-income security can be decomposed into a
series of known payments at different time points in the
future,
? Pure discount bonds (zero-coupon bonds),Promising a
single payment of cash at the maturity date (in the future),
10
Finance School of Management
Pure Discount Bonds
? The pure discount bond is an example of the
present value of a lump sum equation we
analyzed in Chapter 4,
? Solving this,the yield-to-maturity on a pure
discount bond is given by the relationship,
? ? 11
1
??
?
?
?
?
?
????
nn
P
F
iiPF
11
Finance School of Management
Pure Discount Bonds
? ? 11
1
??
?
?
?
?
?
????
nn
P
F
iiPF
?In this equation,
– P is the present value or price of the bond
– F is the face or future value
– n is the investment period
– i is the yield-to-maturity
12
Finance School of Management
Pure Discount Bonds
%60.61
8 8 0
1 0 0 0
1
2
11
???
?
?
?
?
?
???
?
?
?
?
?
?
n
P
F
i
n i PV FV PM T Res ult
2? 880 100 0 0 i= 6.60%
? A two-year pure discount bond with a face
value of $1,000 and a price of $880
13
Finance School of Management
Pricing a Coupon Bond
? A 3-year bond with a face value of $1,000 that makes
annual coupon payments at a coupon rate 10%
? Prices of pure discount bonds
M at ur i t y P r i c e s pe r $1 of
F ac e V al ue
1 ye a r 0,96
2 ye a r s 0,89
3 ye a r s 0,81
P r i c e s of P u r e D i s c ou n t B on d s
14
Finance School of Management
First Solution Method
? ?
00.1 0 7 6$
1 0 01 0 0 081.01 0 089.01 0 096.0
?
???????
P
P
15
Finance School of Management
Second Solution Method
91.0 7 5,1$
0 7 2 8.1
1 0 01 0 0 0
0 6 0 0.1
1 0 0
0 4 1 7.1
1 0 0
%28.71
8 1 0
0 0 0,1
%00.61
8 9 0
0 0 0,1
%17.41
9 6 0
0 0 0,1
32
3
1
3,0
2
1
2,0
1
1
1,0
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?
P
P
i
i
i
16
Finance School of Management
The YTM of the Coupon Bond
?It would be a mistake to discount all three cash flows
using the same three-year yield of 7.28%,
?The single discount rate that we can use to discount all
three cash flows is the yield-to-maturity (YTM),
?However,can we get it?
n i PV FV PMT R esult
3 7.28 %? 1000 100 PV =$10 71
n i PV FV PMT R esult
3? 1076 1000 100 i=7.10%
17
Finance School of Management
? You would like to create a 2-year synthetic zero-coupon
bond,
? Assume you are aware of the following information,
– 1-year zero-coupon bonds are trading for $0.93 per
dollar of face value,and
– 2-year 7% coupon bonds (annual payments) are selling
at $985.30 (Face value = $1,000),
? Assume you can purchase the 2-year coupon bond and
unbundle the two cash flows and sell them,
– You would receive,93× $70 = $65.10 from the sale of
the first payment,
– To break even,you would need to receive $985.30-
$65.10 = $920.20 from the sale of the 2-year strip,
Coupon Stripping
18
Finance School of Management
Coupon Rate
? Coupon rate is the interest rate applied to the face
value to compute the coupon payment,
– A bond with a face value of $1,000 and a coupon
rate of 10%
– An annuity component of $100 per year and a
“balloon” or,bullet” payment at maturity
19
Finance School of Management
Current Yield and Yield-to-maturity
? Current yield is the annual coupon divided by the
bond’s price,
? Yield-to-maturity is the discount rate that makes
the present value of a bond’s stream of promised
cash payments equal to its price,
20
Finance School of Management
Example 1
? A 20-year-maturity
bond with a face
value of $1,000 and a
coupon rate of 10%
was originally issued
19 years ago,
? At that time,the yield
curve was flat at 10%
per year,
? Now the interest rate
on one-year bonds is
5% per year,
? Its market price will now be
? Its current yield is
? Its yield-to-maturity is
62.1 0 4 7$05.11 1 0 0 ??P
%55.962.1 0 4 7100 ??y i e l dC u r r e n t
%5162.1 0 4 71 1 0 0 ????? m a t u r i t ytoY i e l d
21
Finance School of Management
Example 2
? A bond with a face value of $1,000 and a coupon
rate of 4% will mature in two years,
? Its market price is $950,
? Its current yield is
? Its yield-to-maturity
%21.49 5 040 ??y i e l dC u r r e n t
n i PV FV PMT R esult
2? 950 1000 40 i=6.76%
22
Finance School of Management
Bonds Trading at Par
? Bond Pricing Principle #1,(Par Bonds)
– If a bond’s price equals its face value,then its
yield-to-maturity = current yield = coupon rate,
? Proof,
F
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pm t
P
ii
pm t
i
P
FP
i
F
ii
pm t
P
nn
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1
1
1
1
1
1
&
1
1
1
1
1
23
Finance School of Management
Bonds Trading at Premium or Discount
? Bond Pricing Principle #2,(Premium Bonds)
– If a bond has a price higher than its face value,then
its yield-to-maturity < current yield < coupon rate,
? Bond Pricing Principle #3,(Discount Bonds)
– If a bond has a price lower than its face value,then
its yield-to-maturity > current yield > coupon rate,
24
Finance School of Management
Proof,
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25
Finance School of Management
Y i e l d R e l a t i o n s h i p s
0
0, 0 2
0, 0 4
0, 0 6
0, 0 8
0, 1
0, 1 2
0, 1 4
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P r i c e
Y
i
e
l
d
c o u p o n _ y
c u r r e n t _ y
y _ t _ m
26
Finance School of Management
Beware of ―High-Yield‖ US Treasure
Bond Funds
? You have $10,000 to invest for one year,You are
deciding between,
– Putting your money in a one-year government-insured
bank CD offering an interest rate of 5%;
– Investing in the shares of a U.S,Treasure bond fund that
holds one-year bonds with a coupon rate of 8%,
? The bonds are selling at a premium,you must pay
$10,285.71 for $10,000 of face value,
? The fund advertises a yield of 7.78%,
? The fund charges a 1% annual fee for their services,
27
Finance School of Management
Beware of ―High-Yield‖ US Treasure
Bond Funds
%51
71.285,10$
000,10$800$ ?????? m a t u r i t ytoY i e l d
%78.7
71.285,10$
800$ ??y i e l dCu r r e n t
28
Finance School of Management
The Effect of Coupon Rate
? Two different two-year coupon bonds—one with a
coupon rate of 5% and the other with a coupon rate of
10%,
? The current market prices and yields of one- and two-
year pure discount bonds,
M a t u r i t y P r i c e s p e r $ 1 o f
F a c e V a l u e
Y i e l d
( p e r y e a r )
1 y e a r $ 0, 9 6 1 5 3 8 4%
2 y e a r s $ 0, 8 8 9 9 9 6 6%
29
Finance School of Management
The Effect of Coupon Rate
? The market prices of the two coupon bonds should be
– For the 5%-coupon bond,
– For the 10%-coupon bond,
? The yields to maturity on the coupon bonds should be
– For the 5%-coupon bond,the YTM is 5.9500%
– For the 10%-coupon bond,the YTM is 5.9064%
? When the yield curve is not flat,bonds of the same
maturity with different coupon rates have different
yields to maturity,
57.9 8 2$0 5 0,1$8 8 9 9 9 6.50$9 6 1 5 3 8,????
15.0 7 5,1$1 0 0,1$8 8 9 9 9 6.1 0 0$9 6 1 5 3 8,????
30
Finance School of Management
Other Effects on Bond Yields
?Default risk
?Taxes
?Callability
?Convertibility
31
Finance School of Management
The Effect of the Passage of Time
?A 20-year pure discount bond with a face value of $1,000
and a constant yield of 6% should be priced at
n i PV FV PMT R esult
20 6%? 1000 0 PV =$31 1.80
?After one year goes by,its price should be
n i PV FV PMT R esult
19 6%? 1000 0 PV =$33 0.51
?If the yield curve were flat and interest rates did not
change,any default-free discount bond’s price would rise
with the passage of time,and any premium bond’s price
would fall,
32
Finance School of Management
Movement of a Pure discount Bond’s Price over Time
300
400
500
600
700
800
900
1,000
0 2 4 6 8 10 12 14 16 18 20
Y e ar s E l ap s e d s i n c e D ate of I s s u e
P
r
i
c
e
33
Finance School of Management
Interest-Rate Risk
? The concept
? The sensitivity of bond prices to interest rates
? The prices of 30-year 8% coupon par bond would fall
by roughly 10% if the level of interest rates were to rise
from 8% to 9%,
? The prices of 30-year pure discount bond would fall by
roughly 23% if the level of interest rates were to rise
from 8% to 9%,
? Why?
34
Finance School of Management
Sensitivity of Bond Price to Interest Rates
0
0.5
1
1.5
2
2.5
3
3.5
4
4 5 6 7 8 9 10 11 12
I n te r e s t R a te i n % p e r Y e a r
P
r
i
c
e
R
a
ti
o
Zeros
Par Bonds
35
Finance School of Management
?
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Dk kdPVd P V ???? 1 )1(
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Duration
