Bonds and Their Valuation
1
What is Bond: a bond is an IOU issued by a
borrower (US government, state and local
governments or corporations). The price (value)
of a bond on the date of issuance is usually equal
to the face value (par value or maturity value,
denoted by M in this chapter) of the bond. When
bond investor buys a bond, she/he expects to
receive interest payment known as coupon
payment each year until the date of maturity of the
bond. On the date of maturity she/he is entitled to
receive the face value of bond. For example,
suppose you buy a bond with $1,000 face value, a
10% coupon rate (denoted by c in this chapter)
and a 15-year maturity (denoted by n in this
chapter). In this case, you receive coupon
payment (interest payment) of $100 each year for
15 years (assuming bond calls for annual
payments). On the date of maturity you would
receive $1,000 and the last coupon payment. The
type of bond that we explained above is called”
fixed coupon payment” bond with annual
2
payments (there are bonds that call for semiannual coupon payments).
Bonds are generally classified as "fixed-income"
securities.
 Bond’s Characteristics
1. Par value (M)
2. Coupon interest rate(c)
3. Coupon pmt = I = M x c
4. Required Rate of Return (rd)
5. Maturity
6. Years to maturity (n)
7. Issue date
8. Default risk
9. Special provisions: call provision ( call
premium = rd(c) - rd (nc) >0)
10. Call price
11. Call Protection
3
 Types of Bonds(in general)
1. Treasury bonds and notes.
2. Corporate bonds
3. Municipal bonds
4. Eurobonds
5. Foreign Bonds

Bond Valuation Model(Fixed Coupon
Rate Bond)
Value of a bond is present value of its
expected cash flows (Is and M):

1
1


n
(
1

r
)
d
VI 

rd




M
+
 (1  rd ) n


Example 1:
The Morrissey Company's bonds mature in 7 years, have a par
value of $1,000, and make an annual coupon payment of $70.
4
The market interest rate for the bonds is 8.5%. What is the
bond's price?

1
1


n
(
1

r
)
d
VB  I 

rd


1

1


(1  0.085) 7
V  $70 
0.085




M
+
 (1  rd ) n




1000
 $923.22
+
7
(
1

0
.
085
)


What is $70?
 $70 is coupon payment (= interest income to
the bond investor) and is calculated as
follows:
1,000 x 0.07= $70
Example 2
Suppose we have a bond with the following
characteristics M =$1,000, c = 10%, rd=8% and n
= 3 years. The fundamental value of this bond is
calculated as:
5

1
1 
(1  rd ) n

VB  I

rd




+ M
 (1  rd ) n



1
1 
3
(
1

0
.
08
)
V  $100 

0.08




 + 1000
 257.71  793.83  1051.54
 (1  0.08) 3


Example 3:
Jack Weber calls his broker to inquire about purchasing
a bond of Just-in-Time Technologies. His broker quotes
a price of $1,180. Jack is concerned that the bond might
be overpriced based on the facts involved. The $1,000
face value bond pays 14% interest (coupon rate)
annually, and it has 25 years to maturity. The present
interest rate on similar bonds is 12 percent. What is the
fundamental value this bond? Is this bond overpriced?
Please explain.
M = $1,000; I = $140(=> coupon rate of 14%); n= 25
years;
rd = 12%
1


1

 (1  0.12) 25 
1000

VB  140
 1098.04  58.82  $1,156.86
25
0.12

 (1  0.12)


Current market price =$1,180.00
6
 Yield to Maturity (YTM)
YTM is the discount rate that forces sum of
present values of the future expected cash flows
of a bond to be price of the bond, if bond is
bought today and is held until the maturity. YTM
can be viewed as internal rate of return of
investment on this bond. It is the promised
yield(ex-ante) if bond is purchased to-day and
held to maturity.
Or Yield to Maturity (YTM) = ex-ante yield; is the interest
rate (discount rate/ex-ante yield) that equates the PV of
all cash inflows from a bond to the price of bond, if it is
held until the maturity.
Example 1:
Ezzell Enterprises’ non-callable bonds currently sell for $1,165.
They have a 15-year maturity, an annual coupon of $95, and a
par value of $1,000. What is their yield to maturity?

1
1 
(1  YTM )15
1,165  95 

YTM


N
PV
PMT
FV
I/YR


 + 1,000
 (1  YTM )15


15
$1,165
$95
$1,000
7.62%=YTM
7
Example 2:
Suppose a bond is now selling for $877.07(VB=
$877.07) with coupon rate of 8% (c = 8%), maturity
value of $1,000(M =$1,000), and years to maturity
of 10 years (n = 10 i.e. years that is bond has 10
years to maturity), what is the YTM of this bond?
Solution:
1


1

 (1  YTM )10 
1000
877.07  80

10
YTM

 (1  YTM )


,
Using financial calculator (HP, 10B), we key the
following:
PV= -877.07; FV=1000;PMT = 80; N=10; we solve
for I/YR which is YTM = 10%
Using financial calculator => YTM = 10%
8
 Change in Bond’ value over time.
Value of Bond(VB)
----------------------------------------------------------------------------------------------------------rd(c=10%)
today(n=3)
n=2
n=1
n=0(Just before maturity)
8%
$1,051.54
$1,035.76
$1,018.52
$1,000.00
10%
1,000.00
1,000.00
1,000.00
1,000.00
12%
951.96
966.21
982.14
1,000.00
How bond values in the above table are calculated?
Here are 2 examples: a) Consider the first bond in the
above table ($1,051.54). This bond has the following
characteristics:
M =$1,000, c = 10%, rd=8% and n = 3 years and its value
is calculated as:

1
1 
(1  rd ) n
VB  I 

rd




M
+
 (1  rd ) n



1
1 
(1  0.08) 3
VB  $100 

0.08




1000
+
 257.71  793.83  1051.54
 (1  0.08) 3


b) Consider the last number in the 2nd column of the
above table:

1
1 
(1  rd ) n

VB  I

rd




M
+

(1  rd ) n


9

1
1 
(1  0.12) 3

VB  $100

0.12




 + 1000  240.18  711.78  951.96
 (1  0.12)3


Value (VB)
rd=8%<c=10%
rd =10%=c=10%
rd=12%>c=10%
Years to Maturity
 Discount bonds: rd > c => V < M
 Premium bonds: rd < c => V > M
 Par bonds: rd = c => V = M
 Total Rate of Return on Bond Held from t to t+1 :
Total yield = Current Yield (CY) + Capital Gains Yield (CGY)
10
RET  Total rate of return  Total yield 
note :
I
VB ( t )
VB ( t )

VB (t 1)  VB ( t )
VB ( t )
 iC  Current yield
VB ( t 1)  VB ( t )
VB ( t )
I
 g  Capital gains yield
If stays the same(constant):
t0
t1
M=$1,000
M=$1,000
I =$150
I =$150
rd = 15%
rd = 15%
n=15
n=14
VB0 =$1,000
VB1=$1,000
t2
M=$1,000
I =$150
rd = 15%
n=13
VB2 =$1,000
RET  Total rate of return  Total yield 

150 1,000  1,000

 15.00%  (0.0%)  15%
1,000
1,000
If interest rate falls:
t0
t1
F=$1,000
F=$1,000
I =$150
I =$150
rd = 15%
rd = 10%
n=15
n=14
VB0 =$1,000
VB1 =$1,368.33
t2
F=$1,000
I =$150
rd = 10%
n=13
VB2 =$1,355.16
RET  Total rate of return  Total yield 

150
1,355.16  1,368.33

 10.96%  (0.96%)  10%
1,368.33
1,368.33
11
If interest rate rises:
t0
t1
F=$1,000
F=$1,000
I =$150
I =$150
rd = 15%
rd = 20%
n=15
n=14
VB0 =$1,000
VB1 =$769.48
t2
F=$1,000
I =$150
rd = 20%
n=13
VB2 =$773.37
RET  Total rate of return 
150
773.37  769.48


 19.49%  (0.51%)  20%
769.48
769.48
 Reading Quotations:
I.
Treasury Bonds and Notes
II.
Corporate Bonds
 Yield to Call(YTC)
 Yield to Call (YTC):Callability is a provision:
the bondholder would not have the option of
holding callable bond if bond called by issuer.
 Rationale: XYZ corporation issues a set
of annual bonds with M=$1,000;coupon rate =12%; n=15
year ;N=5 years.
After 5 years interest on similar bonds falls to 6%;
corporation then issues now bonds with the rate of 6%
and saves interest cost:
$120 – $60 = $60 per bond per year
12
 Calculation of YTC:
1

1

N


1  YTC 
VB  I 
YTC




CP

3
 (1  YTC )

Where N is number of years until bond becomes
callable; and CP is call price
Example 1: Consider the following annual callable bond:
Original Maturity = 10 years, issued 1 years ago,
coupon rate = 8%, M = $1000, Price 110.961% of the
face value and call price = $1,080 with call protection
period of 5 year. To calculate YTC, we have:

1
1 
1  YTC 4
$1109.61  $80

YTC




  $1080
 (1  YTC ) 4


I use my financial calculator and I key the
followings:
PV= -1,109.61; FV=1080;PMT = 80; N=4; I solve
for I/YR then YTC = 6.61%
Example 2 : Consider the following callable bond:
13
Original Maturity = 20 years, issued 2 years ago, c =
10.5%, M = $1000, Price 115.174% of the face value
(i.e.V = $1,151.74 calculated as 1.15174 x
1000=$1,151.74), call price = $1,100, with call protection
period of 5 years. To calculate YTC, we solve the
following equation:

1
1


3

1  YTC 

$1,151.74  $105

YTC




  $1100
 (1  YTC ) 3


Using financial calculator (HP, 10B), we key the
following:
PV= -1,151.74; FV=1100;PMT = 105; N=3; we solve
for I/YR which is YTC = 7.73%
Example 3:
Sadik Inc.'s bonds currently sell for $1,280 and have a par value of $1,000.
They pay a $135 annual coupon and have a 15-year maturity, but they can
be called in 5 years at $1,050. What is their yield to call (YTC)?
1

1


1  YTC 5
V  135
YTC





1050

(1  YTC ) 5



14
N
PV
PMT
FV
I/YR = YTC
5
$1,280
$135
$1,050
7.45%
 Note that, generally, if a bond sells at a
premium (i.e. its coupon rate is higher than
current YTM of similar bonds) a call is likely.
It follows you should expect to earn:
- YTC on premium bonds
- YTM on par and discount bonds
 Value of Bond with Semi-annual Coupon
Payments:
1

1


r
(1  d ) 2 n

I
2
VB  
rd
2

2




M

 (1  rd ) 2 n

2

Example 1
Assume that you are considering the purchase of a 15-year bond
with an annual coupon rate of 9.5%. The bond has face value of
$1,000 and makes semiannual interest payments. If you require
an 11.0% nominal yield to maturity on this investment, what is
the maximum price you should be willing to pay for the bond?
15
Par value
$1,000
Coupon rate
9.5%
Periods/year
2
Yrs to maturity
15
Annual rate
11.0%
===================
N = periods
Periodic rate
PMT/period
FV
PV
30
5.50%
$47.50
$1,000
$891.00
Example 2:
M= $1000, c=10%, n=20 years, rd =YTM=12%
1 

1

 (1.06) 40 
1000

V  $50
 $849.52
40
.
06
(
1
.
06
)




 YTM(ex-ante yield) of semi-annual bond is
solution the following equation:
1

2n
1 
YTM



1 

I 
2 

VB  
YTM
2

2












M
YTM 

1 

2 

2n
16
Consider a 3-year, 5%, $1000 per value semi-annual
bond at a 97.291% price, what is YTM (Ex-ante yield) of
this bond?
1


1

6 

YTM  


1 

50 
1000
2  

972.91 

6

YTM
2 
YTM



 1 

2
2 

 


Using MY financial calculator (HP10-B):
PV= -972.91; FV=1000; PMT = 25; N=3x2; we solve
for I/YR which is YTM/2 = 3.00% => YTM=6%
How about YTC in the case of semi-annual
payment? easy:
1

2N
1 
YTC



1 

I
2 

V 
YTC
2

2





CP


2N

YTC


 1 

2 
 

17
Example 1: Hood Corporation recently issued 20-year
bonds. The bonds have a coupon rate of 8 percent and pay
interest semiannually. Also, the bonds are callable in 6
years at a call price equal to 115 percent of par value. The
par value of the bonds is $1,000. If the yield to maturity is 7
percent, what is the yield to call?
Solution: First, calculate the price of the bond as
follows:
1


1

220 

0.07 



1 

80 
1000
2



Pb 

 $1,106.78
220

0.07
2 
0
.
07


 1 

2
2 

 


Now, we can calculate the YTC as follows,
recognizing that the bond can be called in 6 years
at a call price of 115% x1,000 = 1,150:
1

1

26

YTC


 1 

80  
2 
$1,106.78 
YTC
2 

2





 1.15  1000
26

YTC


 1 

2 
 

Using my calculator: N = 6 x 2 = 12, PV = -1,106.78,
PMT = 40, FV = 1,150, and solve for I/YR = ? =
3.8758% x 2 = 7.75%.
18
Example2: Keenan Industries has a bond outstanding with 15
years to maturity, an 8.75% coupon paid semiannually, and a
$1,000 par value. The bond has a 6.50% nominal yield to
maturity, but it can be called in 6 years at a price of $1,050.
What is the bond’s nominal yield to call?
First, use the given data to find the bond's current price.
Then use that price to find the YTC.
Coupon rate
8.75%
YTM
6.50%
Maturity
15
Par value
$1,000
Periods/year
2
Determine the bond's price
PMT/period
$43.75
N
30
I/YR
3.25%
FV
$1,000.00
PV = Price
$1,213.55
Yrs to call
Call price
6
$1,050.00
Determine the bond's YTC
N
12
PV
$1,213.55
PMT
$43.75
FV
$1,050.00
I/YR
2.64%
Nom. YTC
5.27%
Zero-coupon Bonds (Zeros): Zero coupon bonds
are type of bonds that do not offer any coupon
payments. This means that investor (bondholder)
receives only the face value (maturity value) of the
zero-bond on the date of maturity. The value of a
zero is calculated using the following formula:
M
Vzero 
(1  rd ) n
Example: suppose you are looking at a zero with
the face value of $5,000 and 10 years to maturity.
19
What is the value of this zero if your required rate
of return ( rd) is 8%?
Vzero 
$5,000
 $2,315.97
10
(1  0.08)
Perpetual Bonds: Perpetual bonds are type of
bonds that never mature. They call for coupon
payments (interest payments) forever. The value
of a perpetual bond is calculated using the
following formula:
V perp 
I
rd
Example: suppose you are looking at a perpetual
bond that calls for $50 interest payment per year.
What is the maximum price that you are willing to
pay for this bond if your required rate of return is
7%?
V perp 
$50
 $714.29
0.07
20
 Interest Rate Risk of Bond
The effect of Δs in i on the realized rate of return of a
bond causing its return to deviate from it YTM is
referred as “interest rate risk”.
1.
Price risk: the effect of Δs in interest rate on
price of a bond is referred as “price risk”. That is
the risk of interest rate rising causing market
price of bond falling if bond is sold prior to it
maturity thus resulting in a lower realized rate of
return (ex-post rate of return).
2.
Reinvestment risk: the effect of Δs in interest rate
on reinvestment income of a bond is referred as
“reinvestment risk”. That is the risk of interest
rate falling and having to reinvest the coupon
payments at a lower rate than YTM resulting in
lower realized rate of return (ex-post rate of
return).
 Short Term Bonds Versus Long Term Bonds
Short Term Bonds: Reinvestment risk
Long Term Bonds: Price Risk
= Reading and understanding T-notes and Tbonds quotations from WSJ
21