Sections 6.1, 6.2, 6.3
Chapter 6 is primarily the application of principles from past chapters to
various bonds and securities with some new terminology and formulas to
efficiently handle calculations. Sections 6.1 and 6.2 of the textbook
briefly discuss different types of bonds and stocks.
Example 6.1 illustrates the calculation of a yield rate for a zero coupon
bond.
Example 6.2 illustrates the use of simple discount to calculate the price
for a 13-week Treasury bill (T-bill).
A bond is issued by a borrower to a lender. Terms and symbols are as
follows:
F = the par value or face value of a bond, whose primary purpose is to
define the series of coupon payments to be made by the borrower
C = the redemption value of a bond; often C = F, but not always (Unless
otherwise stated, it is assumed that C = F.)
r = the coupon rate of a bond, with semiannual frequency being most
common
Fr = the amount of the coupon
g = the modified coupon rate of a bond, defined by Fr = Cg ; g = Fr/C is
the coupon rate per unit of redemption value rather than per unit of par
value and is convertible at the same frequency as r; g = r if F = C
n = the number of coupon payment periods from the date of calculation
to the maturity (or redemption) date
NOTE: F, C, r, g, and n are fixed by the terms of the bond.
i = the yield rate (yield to maturity) of a bond, convertible at the same
frequency as the coupon rate, defined to be the interest rate actually
earned by the investor.
P = the price of a bond (which can be defined to be the present value of
future coupons plus the present value of the redemption value)
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. If both bonds are to be
brought to yield 7% convertible semiannually, find the price for each by
(a) finding the present value of future coupons plus the present value of
the redemption value. This is the basic formula.
i = 0.035
Bond X
Bond Y
F = 5000 = C
F = 5000 = C
r = 0.03
r = 0.04
Fr = 150
Fr = 200
n= 4
n= 4
1

P = 150 a –4| 0.035 + 5000 1.0354 =
1

P = 200 a –4| 0.035 + 5000 1.0354 =
$4908.17
$5091.83
Fr = the amount of the coupon
g = the modified coupon rate of a bond, defined by Fr = Cg ; g = Fr/C is
the coupon rate per unit of redemption value rather than per unit of par
value and is convertible at the same frequency as r; g = r if F = C
n = the number of coupon payment periods from the date of calculation
to the maturity (or redemption) date
NOTE: F, C, r, g, and n are fixed by the terms of the bond.
i = the yield rate (yield to maturity) of a bond, convertible at the same
frequency as the coupon rate, defined to be the interest rate actually
earned by the investor.
P = the price of a bond (which can be defined to be the present value of
future coupons plus the present value of the redemption value)
K = Cvn = the present value of the redemption value (with yield rate i)
G = the base amount of a bond, defined by Gi = Fr ; G = Fr/i is the
amount which, if invested at the yield rate i, would produce periodic
interest payments equal to the coupons on the bond
There are four (equivalent) formulas to find the price of a bond:
1

The basic formula is
P = Fr a –n| i + C (1 + i)n =
Fr a –n| + Cvn = Fr a –n| + K
(where interest functions are
understood to be calculated
with yield rate i)
1
=
The premium/discount formula is P = Fr a –
+ C 
n| i
(1 + i)n
Fr a –n| + C(1  i a –n| ) =
C + (Fr  Ci) a –n|
(where interest functions are
understood to be calculated
with yield rate i)
The base amount formula is
1

P = Fr a –n| i + C (1 + i)n =
Gi a –n| + Cvn = G(1  vn) + Cvn =
G + (C  G)vn
(where interest functions are
understood to be calculated
with yield rate i)
1
=
The Makeham formula is
P = Fr a –n| i + C 
(1 + i)n
n
1

v
n
Cv + Cg  =
i
g
n
Cv +  (C  Cvn) =
i
g
K +  (C  K)
i
On page 202 of the textbook the terms nominal yield (note the
ambiguity), current yield, and yield to maturity are defined.
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. If both bonds are to be
brought to yield 7% convertible semiannually, find the price for each by
(a) finding the present value of future coupons plus the present value of
the redemption value. This is the basic formula.
i = 0.035
Bond X
Bond Y
F = 5000 = C
F = 5000 = C
r = 0.03
r = 0.04
Fr = 150
Fr = 200
n= 4
n= 4
1

P = 150 a –4| 0.035 + 5000 1.0354 =
1

P = 200 a –4| 0.035 + 5000 1.0354 =
$4908.17
$5091.83
(b) using the premium/discount formula, the base amount formula, and
the Makeham formula.
i = 0.035
Bond X
Bond Y
F = 5000 = C
F = 5000 = C
r = 0.03
r = 0.04
Fr = 150
Fr = 200
n= 4
n= 4
1

K = 5000 1.0354 = 4357.21
150
G =  = 4285.71
0.035
With the premium/discount
formula, P = C + (Fr  Ci) a –
=
n|
1

K = 5000 1.0354 = 4357.21
200
G =  = 5714.29
0.035
With the premium/discount
formula, P = C + (Fr  Ci) a –
=
n|
5000 + (150  175) a –4| 0.035 =
5000 + (200  175) a –4| 0.035 =
$4908.17
$5091.83
(b) using the premium/discount formula, the base amount formula, and
the Makeham formula.
i = 0.035
Bond X
Bond Y
F = 5000 = C
F = 5000 = C
r = 0.03
r = 0.04
Fr = 150
Fr = 200
n= 4
n= 4
1

K = 5000 1.0354 = 4357.21
150
G =  = 4285.71
0.035
With the base amount
formula, P = G + (C  G)vn =
1

4285.71+(50004285.71) 1.0354 =
1

K = 5000 1.0354 = 4357.21
200
G =  = 5714.29
0.035
With the base amount
formula, P = G + (C  G)vn =
1

5714.29+(50005714.29) 1.0354 =
$4908.17
$5091.83
(b) using the premium/discount formula, the base amount formula, and
the Makeham formula.
i = 0.035
Bond X
Bond Y
F = 5000 = C
F = 5000 = C
r = 0.03
r = 0.04
g = r = 0.03
g = r = 0.04
Fr = 150
Fr = 200
n= 4
n= 4
1

K = 5000 1.0354 = 4357.21
150
G =  = 4285.71
0.035
With the Makeham
g
formula, P = K +  (C  K) =
0.03 i
4357.21+ 
0.035 (50004357.21) =
1

K = 5000 1.0354 = 4357.21
200
G =  = 5714.29
0.035
With the Makeham
g
formula, P = K +  (C  K) =
0.04 i
4357.21+ 
0.035 (50004357.21) =
$4908.17
$5091.83