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

1
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
2
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)
3
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 F r r Fr
Fr n n P P
5000 C
5000 C
0.03
0.04
150
200
4
4
1 ??? 1.0354
1 ??? 1.0354
150 5000
200 5000
a 4 0.035
a 4 0.035
4908.17
5091.83
4
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
5
There are four (equivalent) formulas to find the
price of a bond
1 ??? (1 i)n
P Fr C
The basic formula is
a n i
Fr Cvn
Fr K
a n
a n
(where interest functions are understood to be
calculated with yield rate i)
1 ??? (1 i)n
The premium/discount formula is
P Fr C
a n i
Fr C(1 ? i )
a n
a n
C (Fr ? Ci)
a n
(where interest functions are understood to be
calculated with yield rate i)
6
1 ??? (1 i)n
The base amount formula is
P Fr C
a n i
Gi Cvn G(1 ? vn) Cvn
a n
G (C ? G)vn
(where interest functions are understood to be
calculated with yield rate i)
1 ??? (1 i)n
The Makeham formula is
P Fr C
a n i
Cvn Cg
1 ? vn ??? i
g ? i
Cvn (C ? Cvn)
g ? i
K (C ? K)
On page 202 of the textbook the terms nominal
yield (note the ambiguity), current yield, and
yield to maturity are defined.
7
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 F r r Fr
Fr n n P P
5000 C
5000 C
0.03
0.04
150
200
4
4
1 ??? 1.0354
1 ??? 1.0354
150 5000
200 5000
a 4 0.035
a 4 0.035
4908.17
5091.83
8
(b) using the premium/discount formula, the base
amount formula, and the Makeham formula.
i 0.035
Bond X Bond Y F F r r Fr
Fr n n K K G G
5000 C
5000 C
0.03
0.04
150
200
4
4
4
1 ??? 1.0354
1 ??? 1.0354
4357.21
4357.21
5000
5000
150 ??? 0.035
200 ??? 0.035
4285.71
5714.29
With the premium/discount formula, P
With the premium/discount formula, P
C (Fr ? Ci)
C (Fr ? Ci)
a n
a n
5000 (150 ? 175)
5000 (200 ? 175)
a 4 0.035
a 4 0.035
4908.17
5091.83
9
(b) using the premium/discount formula, the base
amount formula, and the Makeham formula.
i 0.035
Bond X Bond Y F F r r Fr
Fr n n K K G G
5000 C
5000 C
0.03
0.04
150
200
4
4
1 ??? 1.0354
1 ??? 1.0354
4357.21
4357.21
5000
5000
150 ??? 0.035
200 ??? 0.035
4285.71
5714.29
With the base amount formula, P
With the base amount formula, P
G (C ? G)vn
G (C ? G)vn
1 ??? 1.0354
1 ??? 1.0354
4285.71(5000?4285.71)
5714.29(5000?5714.29)
4908.17
5091.83
10
(b) using the premium/discount formula, the base
amount formula, and the Makeham formula.
i 0.035
Bond X Bond Y F F r r Fr
Fr n n K K G G
5000 C
5000 C
0.03
0.04
g r 0.03
g r 0.04
150
200
4
4
1 ??? 1.0354
1 ??? 1.0354
4357.21
4357.21
5000
5000
150 ??? 0.035
200 ??? 0.035
4285.71
5714.29
With the Makeham formula, P
With the Makeham formula, P
g ? i
g ? i
K (C ? K)
K (C ? K)
0.03 ??? 0.035
0.04 ??? 0.035
4357.21 (5000?4357.21)
4357.21 (5000?4357.21)
4908.17
5091.83