Sections 6.4, 6.5, 6.6, 6.7, 6.8, 6.10, 6.11
If the purchase price of a bond exceeds its redemption value, the bond is
said to sell at a premium, and P  C (> 0) is called the premium.
If the purchase price of a bond is less than its redemption value, the bond
is said to sell at a discount, and C  P (> 0) is called the discount.
Recall the premium/discount formula to find the price P of a bond:
Note the overuse of
P = C + (Fr  Ci) a –n| i
the word “discount”
For bonds sold at a premium, we have
Premium = (Fr  Ci) a –n| i ; for F = C, Premium = C(g  i) a –n| i and g > i.
For bonds sold at a discount, we have
Discount = (Ci  Fa) a –n| i ; for F = C, Discount = C(i  g) a –n| i and i > g.
Recall that when a loan is being repaid with the amortization method,
each payment is partially a repayment of principal and partially a
payment of interest.
Suppose C = F (g = r) for a bond (i.e., the par value and redemption
values are equal). The bond purchased for price P with yield rate i and
redemption value C can be viewed as a loan/investment of P paid back
with payments/returns equal to the periodic coupons and finally the
redemption value at the end of the life of the bond.
If P = C (g = i), then each coupon is interest on the loan/investment, and
the redemption value is the principal. If P  C (g  i), then each coupon
must be divided into interest earned and principal adjustment. A bond
amortization schedule shows this division. When P > C, the principal
adjustment will be downward, and when C > P, the principal adjustment
will be upward.
In the bond amortization schedule, the value of the bond is continually
adjusted beginning at the price on the purchase date and ending at the
redemption value on the redemption date; these adjusted values are
called the book values of the bond. The book value at any time is not
necessarily equal to the market value which can change as the prevailing
interest rates change.
Let Bt be the book value after t periods. Then B0 = P and Bn = C.
Suppose C = 1; then P = 1 + p where p = (g  i) a –n| i , and each coupon
is g.
At the end of period 1, the interest earned on the balance at the beginning
of the period is iP = i[1 + (g  i) a – ] ,
n| i
and the portion of the coupon that does not count as part of this interest,
called the principal adjustment, is
g  i[1 + (g  i) a –n| i ] = g  i  (g  i)(1  vn) = (g  i)vn .
The book value at the end of the first period equals the book value at the
beginning of the period minus the principal adjustment, which is
n =
B1 = B0  (g  i)vn = [1 + (g  i) a –
]

(g

i)v
1 + (g  i) a –––
.
n| i
n  1| i
Note that if the bond is sold at a premium, then the principal adjustment
is lowering the book value, but if the bond is sold at a discount, then the
principal adjustment is raising the book value.
By successively continuing this reasoning, an amortization table can be
developed. Table 6.1 on page 208 of the textbook displays the format for
such an amortization schedule. Observe each of the following from this
table:
1. The book values displayed on each line match the premium/discount
formula to find the price at the original yield rate.
2. The sum of the principal adjustment column is p = (g  i) a –n| i .
3. The sum of the interest earned column is the difference between the
sum of the coupons and the sum of the principal adjustment column.
4. The principal adjustment column is a geometric progression with
common ratio 1 + i.
The process of obtaining the book values in the last column is called
writing up or writing down book values, depending on whether the book
values are increasing or decreasing.
Bond X and Bond Y are each a two-year bond with a par value of $5000.
Bond X has a coupon rate of 6% payable semiannually, and Bond Y has
a coupon rate of 8% payable semiannually. Both bonds are to be brought
to yield 7% convertible semiannually. It is easy to check (and this was
done already in the handout for Sections 6.1, 6.2, 6.3) that the price for
Bond X is $4908.17, and the price for Bond Y is $5091.83.
(a) Complete the bond amortization schedule for Bond X below.
Interest Principal Adjustment
Book
Period
Amount for Discount
Value
(Half-Year) Coupon Earned
0
4908.17
1
150.00
171.79
21.79
4929.96
2
150.00
172.55
22.55
4952.51
3
150.00
173.34
23.34
4975.85
4
150.00
174.15
24.15
5000.00
(b) Complete the bond amortization schedule for Bond Y below.
Period
(Half-Year)
Coupon
Interest
Earned
Principal Adjustment
Amount for Premium
0
Book
Value
5091.83
1
200.00
178.21
21.79
5070.04
2
200.00
177.45
22.55
5047.49
3
200.00
176.66
23.34
5024.15
4
200.00
175.85
24.15
5000.00
The process of obtaining the book values in the last column is called
writing up or writing down book values, depending on whether the book
values are increasing or decreasing. An approximate but simple method
for writing up or writing down book values is the straight line method
which sets the principal adjustment each period equal to (P  C)/n and
the interest earned each period equal to Fr  (P  C)/n .
For each of Bond X and Bond Y in the earlier exercise, list the book
values that would result from the straight line method.
For Bond X, the principal adjustment each period is
(4908.17  5000)/4 =  22.9575
and the interest earned each period is 150  ( 22.9575) = 172.9575.
The book values are 4908.17, 4931.13, 4954.085, 4977.04, 5000.00.
For Bond Y, the principal adjustment each period is
(5091.83  5000)/4 = 22.9575
and the interest earned each period is 150  22.9575 = 127.0425.
The book values are 5091.83, 5068.87, 5045.915, 5022.96, 5000.00.
Investor/Purchaser/Lender
“loans” amount P at time 0
Issuer/Borrower receives
amount P at time 0
Investor/Purchaser/Lender
receives periodic modified
coupon payments on the “loan”
from Issuer/Borrower
P > C (g > i) implies a larger n
is more favorable to the investor
P < C (g < i) implies a larger n
is more favorable to the issuer
Investor/Purchaser/Lender
receives amount C at time n
Issuer/Borrower pays
amount C at time n
When a bond is sold at a premium, it can be said that the investor
(purchaser or lender) experiences a loss equal to the premium at the
redemption date; when a bond is sold at a discount, it can be said that the
investor (purchaser or lender) experiences a profit equal to the discount
at the redemption date. The profit or loss is reflected in the yield rate i.
In the case where the bond is sold at a premium, suppose the investor is
able to have the premium (loss) replaced by periodic deposits of the
difference Cg  iP into a sinking fund earning interest rate j convertible
at the same frequency as the yield rate i.
Example 6.4 in the textbook gives the formula needed to appropriately
adjust the price of the bond, and also provides a specific example.
Suppose we want to determine a price/book value for a bond between
coupon payment dates t and t + 1. Let t + k be the time for which the
price/book value is to be determined, i.e., 0 < k < 1. We define the
following:
m
Bt + k
= market value of the bond, which is based on the present value of
future coupons plus the present value of the redemption value minus a
portion of the coupon payment at time t + 1
f
Bt + k
= flat price of the bond, which is the amount actually paid for the
bond
time over which market value is based
t
t+k
t+1
time over which portion of coupon payment at t + 1 is based to
be subtracted from the market value.
n
m
Bt + k
= market value of the bond, which is based on the present value of
future coupons plus the present value of the redemption value minus a
portion of the coupon payment at time t + 1
f
Bt + k
= flat price of the bond, which is the amount actually paid for the
bond
Frk = accrued coupon, which is the portion of coupon payment at t + 1 to
be subtracted from the market value, since the purchaser will receive the
entire coupon payment at time t + 1
time over which market value is based
t
t+k
t+1
time on which portion of coupon payment at t + 1 to be subtracted
from the market value is based
n
f
Bt + k
m
Bt + k
m
Bt + k
f
Bt + k
We must have that
=
+ Frk , or
=
three methods for computing these values:
Flat Price Accrued Coupon
Btf + k
Frk
k1
(1
+
i)
Theoretical
Bt(1 + i)k
Fr 
i
Method
 Frk . There are
Market Price
Btm+ k
Bt(1 + i)k
(1 + i)k  1
 Fr 
i
Practical
Method
Bt(1 + ki)
kFr
Bt(1 + ki)  kFr
Semi-theoretical
Method
Bt(1 + i)k
kFr
Bt(1 + i)k  kFr
Another issue which can be considered is the amount of premium or
discount between coupon dates. This is easily calculated from Btm+ k  C
m
if g > i or from C  Bt + k if i > g.
From the amortization schedule of Bond X constructed earlier, use all
three methods to find the flat price, accrued coupon, and market price for
the bond four months after purchase.
First, we calculate the following:
Bt(1 + i)k = B0(1 + 0.035)2/3 = (4908.17)(1.035)2/3 = $5021.86
Bt(1 + ki) = B0(1 + (2/3)0.035) = (4908.17)(1 + (2/3)0.035) = $5022.52
2/3  1
(1.035)
k
+ i)  1
= $99.43
Fr (1
 = 150 
0.035
i
kFr = (2/3)(150) = $100.00
First, we calculate the following:
Bt(1 + i)k = B0(1 + 0.035)2/3 = (4908.17)(1.035)2/3 = $5021.86
Bt(1 + ki) = B0(1 + (2/3)0.035) = $5022.52
2/3  1
(1.035)
k
+ i)  1
= $99.43
Fr (1
 = 150 
0.035
i
kFr = (2/3)(150) = $100.00
Flat Price
B0f + 2/3
Accrued Coupon
Fr2/3
Market Price
B0m+ 2/3
Theoretical
Method
$5021.86
$99.43
$4922.43
Practical
Method
$5022.52
$100.00
$4922.52
Semi-theoretical
Method
$5021.86
$100.00
$4921.86
From the amortization schedule of Bond Y constructed earlier, use all
three methods to find the flat price, accrued coupon, and market price for
the bond 13 months after purchase.
First, we calculate the following:
Bt(1 + i)k = B2(1 + 0.035)1/6 = (5047.49)(1.035)1/6 = $5076.51
Bt(1 + ki) = B2(1 + (1/6)0.035) = (5047.49)(1 + (1/6)0.035) = $5076.93
1/6  1
(1.035)
k
+ i)  1 = 200  = $32.86
Fr (1

0.035
i
kFr = (1/6)(200) = $33.33
First, we calculate the following:
Bt(1 + i)k = B2(1 + 0.035)1/6 = (5047.49)(1.035)1/6 = $5076.51
Bt(1 + ki) = B2(1 + (1/6)0.035) = (5047.49)(1 + (1/6)0.035) = $5076.93
1/6  1
(1.035)
k
+ i)  1
= $32.86
Fr (1
 = 200 
0.035
i
kFr = (1/6)(200) = $33.33
Flat Price
B2f + 1/6
Accrued Coupon
Fr1/6
Market Price
B2m+ 1/6
Theoretical
Method
$5076.51
$32.86
$5043.65
Practical
Method
$5076.93
$33.33
$5043.60
Semi-theoretical
Method
$5076.51
$33.33
$5043.18
In practice, instead of using number of months, the exact number of days
most likely would be used with the either the actual/actual method or the
actual/360 method.
Look at Example 6.6. The calculations in #1 and #2 are straightforward.
Note how the calculations in #4 can be done with a TI calculator as
follows:
Press the APPS button, and selecting the Finance option.
Select the TMV Solver option.
Set N = 20, PV = 90, PMT = 4, FV = 100
Press the ALPHA button followed by the SOLVE button.
These results should match the results in the example.
A callable bond is a bond for which the issuer (borrower) has an option
to redeem prior to the normal maturity date. To calculate the price,
assume that the issuer (borrower) will exercise the most advantageous
option, which is least advantageous to the lender (investor):
if the yield rate is less than the modified coupon rate (i.e., the bond sells
at a premium), assume that the redemption date will be the earliest
possible date, since the issuer (borrower) will prefer the lender (investor)
experience the “loss of principle” as soon as possible;
if the yield rate is greater than the modified coupon rate (i.e., the bond
sells at a discount), assume that the redemption date will be the latest
possible date, since the issuer (borrower) will prefer the lender (investor)
experience the “gain of principle” as late as possible.
A putable (put) bond is a bond for which the owner (lender) has an
option to redeem prior to the normal maturity date.
To calculate a price, reverse the rules for callable bonds.
For a given value if i, a –n| i is an increasing function of n. Consequently,
a callable bond will typically sell at a lower price (a higher yield rate)
than an otherwise identical non-callable bond, and a putable bond will
typically sell at a higher price (a lower yield rate) than an otherwise
identical non-putable bond.
Look at Examples 6.8 and 6.9.
Serial bonds are a series of bonds with staggered redemption dates
instead of a common maturity date. If the bonds are redeemable at m
different dates, then the price, redemption value, and present value of the
redemption value all corresponding to time t can be denoted respectively
by Pt , Ct , and Kt for t = 1, 2, …, m. The valuation of the serial bonds is
most efficiently done using Makeham’s formula on each bond in the
series as follows:
P1 + P2 + … + Pm =
g
g
g
K1 +  (C1  K1) + K2 +  (C2  K2) + … + Km +  (Cm  Km) =
i
i
i
g
/
K +  (C /  K /) where
i
C / = C1 + C2 + … + Cm and K / = K1 + K2 + … + Km
Look at Example 6.10.
Preferred stock and perpetual bonds are each types of fixed income
securities without fixed redemption dates. The price must be equal to the
present value of future dividends or coupons forever, i.e. the
dividends/coupon form a perpetuity. The price is
Fr
P=  .
i
Common stock is not a fixed income security, which implies dividends
are not known in advance, and these dividends can fluctuate widely. The
dividend discount model is based on obtaining a theoretical price by
using projected dividends to calculate the present value of projected
dividends. For instance, if a corporation is planning to pay a dividend of
D at time 1, and it is projected that future periodic dividends will be paid
indefinitely and will change geometrically with common ratio 1 + k
where 1 < k < i (i.e., dividends increase at rate k if k > 0 and decrease at
rate | k | if k < 0), then the theoretical price of the stock (from Chapter 4)
is
D
2
3
2
4
3
.
P = vD + v (1 + k)D + v (1 + k) D + v (1 + k) D + … = 
ik
If a corporation is planning to pay a dividend of D at time 1, and it is
projected that future periodic dividends will be paid up to and including
at time n and will change geometrically with common ratio 1 + k where
1 < k < i, then the theoretical price of the stock (from Chapter 4) is
P = vD + v2(1 + k)D + v3(1 + k)2D + v4(1 + k)3D + … + vn(1 + k)n1D =
n
1+k
1  
1+i
D  .
ik
Dividends on stocks in the United States are usually paid quarterly.
Look at Example 6.14.
Look at Example 6.15, and note the ambiguity in the description of the
payments. The solution is based on the following payments:
…
payments
2
0
time
2(1.052)
2(1.05)
1
2
2(1.053)
3
2(1.054)
4
5
2(1.055)
2(1.055)(1.025)
6
7
payments
2(1.055)(1.0253)
2(1.055)(1.0255)
2(1.055)(1.0255)
2(1.055)(1.0255)
2(1.055)(1.0255)
2(1.055)(1.0252)
2(1.055)(1.0254)
time
8
9
10
11
12
13
14
…
15
Look at Example 6.16.
Section 6.11 briefly describes the different approaches to valuation of
securities on which universal agreement does not exist.