Chapter 5
BONDS
• Price of a Bond
• Book Value
• Bond Amortization Schedule
• Other Topics
5.1 Price of a Bond
• Bonds
• certificates issued by a corporation or government
• are sold to investors
• in return, the borrower (i.e. corporation or government) agrees:
– to pay interest at a specified rate (the coupon rate) until a specified
date (the maturity date)
– and, at that time, to pay a fixed sum (the redemption value)
• Usually:
– the coupon rate is a nominal rate convertible semiannually and is
applied to the face (or par) value stated on the front of the bond
– the face and redemption values are equal (not always)
• Thus we have:
– regular interest payments
– lump-sum payment at the end
Example of a bond
• Face amount = 500
• Redemption value = 500
• Redeemable in 10 years with semiannual coupons
at rate 11%, compounded semiannually
• Then in return investor receives:
• 20 half-yearly payments of (.055)(500) = 27.50 interest
• a lump-sum payment of 500 at the end of the 10 years
Notations
• F = the face value (or the par value)
• r = the coupon rate per interest period (we assume
that the quoted rate will be a nominal rate 2r
convertible semiannually)
• Note: the amount of each interest payment (coupon)
is Fr
• C = the redemption value (often C = F, i.e. bond
“redeemable at par”)
• i = the yield rate per interest period
• n = the number of interest periods until the
redemption date (maturity date)
• P = the purchase price of a bond to obtain yield rate i
redemption value C
coupon
(interest)
Fr
Fr
Fr
…..
0
1
2
Note: time is
measured in
half-years
n
purchase price P
Equation to find yield rate i
P  ( Fr )an|i  C (1  i)
Note: often C = F
n
Examples (p. 94 – p. 96)
• A bond of 500, redeemable at par after 5
years, pays interest at 13% per year
convertible semiannually. Find the price to
yield an investor
– 8% effective per half-year
– 16% effective per year
• Remarks
– P < C since the yield rate is higher than the
coupon rate, i > r
– therefore the investor is buying the bond at a
discount
– otherwise (if i < r) we would have P > C and
then the investor would have to buy the bond
at a premium of P - C
• A corporation decides to issue 15-years
bonds, redeemable at par, with face
amount of 1000 each. If interest payments
are to be made at a rate of 10% convertible
semiannually, and if George is happy with a
yield of 8% convertible semiannually, what
should he pay for one of these bonds?
• A 100 par-value 15-year bond with coupon
rate 9% convertible semiannually is selling
for 94. Find the yield rate.
5.2 Book Value
• The book value of a bond at a time t is an analog of an
outstanding balance of a loan
• The book value Bt is the present value of all future
payments
• At time t (the tth coupon has just been paid) we have:
Bt  ( Fr )ant|  Cv
n t
• Remarks
– Usually C = F
– In the last formula, an-t and v are computed using the
yield rate i
– P = B0 < Bt < Bt+1< Bn = C or P = B0 > Bt > Bt+1 > Bn = C
Examples (p. 96 - 97)
1. Find the book value immediately after
the payment of 14th coupon of a 10-year
1,000 par-value bond with semiannual
coupons, if r =.05 and the yield rate is
12% convertible semiannually.
2. Let Bt and Bt+1 be the book values just
after the tth and (t+1)th coupons are paid.
Show that Bt+1 = Bt (1+i) – Fr
3. Find the book value in 1) exactly 2
months after the 14th coupon is paid.
How do we find the book value
between coupon payment dates?
Assume simple interest at rate i per period
between adjacent coupon payments
Example
Find the book value exactly 2 months after the 14th
coupon is paid of a 10-year 1,000 par-value bond
with semiannual coupons, if r =.05 and the yield rate
is 12% convertible semiannually.
Alternative approach
•
•
•
•
Since Bt+1 = Bt (1+i) – Fr we can view Bt (1+i) as the book value just
before next (i.e. (t+1)th) coupon is paid
Book value calculated using simple interest between coupon dates
is called the flat price of a bond
Using linear interpolation between Bt+1 and Bt we obtain the market
price (or the amortized value) of the bond
Clearly market price ≤ flat price at any given moment
(1+i) Bt
Fr
Bt
Bt+1
t
t+1
Example (p. 98)
Find the market price exactly 2 months
after the 14th coupon is paid of a
10-year 1,000 par-value bond with
semiannual coupons, if r =.05 and the
yield rate is 12% convertible
semiannually.
5.3 Bond Amortization Schedule
• Goal: trace changes of the book value
• Bond amortization schedule – table, containing the following
columns:
Time Coupon Interest
Principal Book Value
– time
adjustment
– coupon
0
1037.17
– interest
1
40
31.12
8.88
1028.29
– principal adjustment
2
40
30.85
9.15
1019.14
– book value
3
40
30.57
9.43
1009.71
Example
4
40
30.29
9.71
1000.00
1000 par value two-year bond which pays interest at 8%
convertible semiannually; yield rate is 6% convertible
semiannually
Algorithm
•
•
•
•
Book value at time t is Bt
Amount of coupon at time t+1 is Fr
The amount of interest contained in this coupon is iBt
Fr – iBt represents the change in the book value
between these dates
Time Coupon Interest
Principal Book Value
adjustment
0
1037.17
1
40
31.12
8.88
1028.29
2
40
30.85
9.15
1019.14
3
40
30.57
9.43
1009.71
4
40
30.29
9.71
1000.00
Example (p. 99)
• Consider 1000 par-value 10-year bond
with semiannual coupons, r = .05 and the
yield rate i = 0.06 effective semiannually.
Find the amount of interest and change in
book value contained in the 15th coupon
of the bond.
Example (p. 99)
• Construct a bond amortization schedule
for a 1000 par-value two-year bond which
pays interest at 8% convertible
semiannually, and has a yield rate of 6%
convertible semiannually
Time
0
1
2
3
4
Coupon
Interest
Principal Adjustment
40
31.12
8.88
Book Value
1037.17
1028.29
5.4 Other Topics
• Different frequency of coupon payments
• Increasing or decreasing coupon
payments
• Different yield rates
• Callable bonds
Examples (p. 101 – p. 102)
• (Different frequency) Find the price of a 1000 par-value
10-year bond which has quarterly 2% coupons and is
bought to yield 9% per year convertible semiannually
• (Increasing coupon payments) Find the price of a 1000
par-value 10-year bond which has semiannual coupons of
10 the first half-year, 20 the second half-year,…, 200 the
last half-year, bought to yield 9% effective per year
• (Different yield rates) Find the price of a 1000 par-value
10-year bond with coupons at 11% convertible
semiannually, and for which the yield rate is 5% per halfyear for the first 5 years and 6% per half-year for the last 5
years
Callable bonds
• A borrower (i.e. corporation, government etc.) has the right
to redeem the bond at any of several time points
• The earliest possible date is the call date and the latest is
the usual maturity date
• Once the bond is redeemed, no more coupons will be paid
possible
redemption
Purchase Date
Call Date
Maturity Date
Examples (p. 103 – p. 105)
• Consider a 1000 par-value 10-year bond with semiannual
5% coupons. Assume this bond can be redeemed at par at
any of the last 4 coupon dates. Find the price which will
guarantee an investor a yield rate of
– 6% per half-year
– 4% per half-year
• Consider a 1000 par-value 10-year bond with semiannual
5% coupons. This bond can be redeemed for 1100 at the
time of the 18th coupon, for 1050 at the time of the 19th
coupon, or for 1000 at the time of 20th coupon. What price
should an investor pay to be guaranteed a yield rate of
– 6% per half-year
– 4% per half-year