C
HAPTER
3
3
I
N
TR
O
D
UC
TI
ON
T
O
FIXEDI
NCO
ME
V
AL
U
A
T
I
ON
SO
L
U
TI
ONS
112 Part II: Solutions
P
+
1
1
()
+
()
3
+
3
1
00
+
)
12
()
+
)
(
+
h
a
l
ve
d
.
e price is
d
etermine
d
in t
h
e
f
o
ll
owing manner
:
P
V
PM
T
PM
T
PM
T
PM
T
=
()
r
+
()
r
+
+
()
r
+
()
r
+
+
)
r
+
(
)
r
+
(
1
234
+
()
+
()
+
)
+
(
PM
P
P
T
PM
TF
V
F
F
56
()
r
1
r
+
()
r
1
1
+
where:
P
V
=
present va
l
ue, or t
h
e price o
f
t
h
e
b
on
d
P
M
T
=
coupon payment per perio
d
F
V
=
f
uture va
l
ue pai
d
at maturity, or t
h
e par va
l
ue o
f
t
h
e
b
on
d
r
= mar
k
et
d
iscount rate, or require
d
rate o
f
return per perio
d
+
+
+
+
+
+
+
+
+
P
=
=
PM
T
PM
T
PM
T
PM
TF
V
r
()
r
()
r
+
+
()
r
+
+
)
r
+
(
)
r
+
(
+
+
3
P
l
h
f
h
b
d
P
T
d
F
f
l
d
h
l
f
h
b
d
k
d
d
f
d
+
+
+
+
+
+
PV
=
PM
T
PM
T
PM
T
PM
TF
V
()
r
()
+
+
()
r
)
r
+
(
)
r
+
+
3
()
+
r
PV
P
T
FV
r
45
45
45
45
45
45
1
00
+
Chapter 3 Introduction to Fixed-Income Valuation 113
25
25
25
25
25
100
+
+
+
+
+
100
+
+
+
+
+
+
+
114 Part II: Solutions
is greater in absolute value when the market discount rate goes down than when it goes
u
p by the same amount (the convexity e ect). If a 100 basis point decrease in the market
d
iscount rate wi
ll
cause t
h
e price o
f
t
h
e
b
on
d
to increase
b
y 5%, t
h
en a 100
b
asis point in
–
crease in t
h
e mar
k
et
d
iscount rate wi
ll
cause t
h
e price o
f
t
h
e
b
on
d
to
d
ec
l
ine
b
y an amount
l
ess t
h
an 5%.
11 . B is correct. Genera
ll
y,
f
or two
b
on
d
s wit
h
t
h
e same time-to-maturity, a
l
ower coupon
b
on
d
wi
ll
experience a greater percentage price c
h
ange t
h
an a
h
ig
h
er coupon
b
on
d
w
h
en
t
h
eir mar
k
et
d
iscount rates c
h
ange
b
y t
h
e same amount. Bon
d
B an
d
Bon
d
C
h
ave t
h
e
f
d
ll
l
w
h
en t
h
eir mar
k
et
d
iscount rates c
h
ange
b
y t
h
e same amount. Genera
ll
y,
f
or t
h
e same
coupon rate, a
l
onger-term
b
on
d
h
as a greater percentage price c
h
ange t
h
an a s
h
orter-term
b
on
d
w
h
en t
h
eir mar
k
et
d
iscount rates c
h
ange
b
y t
h
e same amount. Re
l
ative to Bon
d
C,
B
d
l
d
b
l
h
Chapter 3 Introduction to Fixed-Income Valuation 115
where:
P
V
=
present value, or the price of the bond
P
M
T
=
coupon payment per period
F
V
=
f
uture va
l
ue pai
d
at maturity, or t
h
e par va
l
ue o
f
t
h
e
b
on
d
Z
1
Z
=
spot rate, or t
h
e zero-coupon yie
ld
, or zero rate,
f
or perio
d
1
Z
2
Z
=
spot rate, or t
h
e zero-coupon yie
ld
, or zero rate,
f
or perio
d
2
Z
3
Z
Z
=
spot rate, or t
h
e zero-coupon yie
ld
, or zero rate,
f
or perio
d
3
P
=
=
P
P
F
f
l
d
h
l
f
h
b
d
Z
1
Z
h
ld
f
d
Z
2
Z
h
ld
f
d
Z
3
Z
Z
P
=
=
P
P
T
FV
Z
1
Z
Z
2
Z
Z
3
Z
Z
PV
=
=
116 Part II: Solutions
P
V
PM
T
PM
T
PM
TF
V
F
F
=
()
Z
+
()
Z
+
+
()
Z
+
)
Z
+
(
ZZ
1
Z
Z
2
Z
Z
3
w
h
ere:
P
V= present va
l
ue, or t
h
e price o
f
t
h
e
b
on
d
P
MT = coupon payment per perio
d
F
V
=
future value paid at maturity, or the par value of the bond
Z
1
Z
=
s
p
ot rate, or the zero-cou
p
on yield, or zero rate, for
p
eriod 1
Z
2
Z
=
spot rate, or t
h
e zero-coupon yie
ld
, or zero rate,
f
or perio
d
2
Z
3
Z
Z
=
spot rate, or t
h
e zero-coupon yie
ld
, or zero rate,
f
or perio
d
3
P
=
+
+
+
()
()
+
()
6
+
6
+
6
100
+
12
()
+
3
)
(
+
P
=
=
Usin
g
this price, the bond’s yield-to-maturity can be calculated as:
PV
PM
T
PM
T
PM
TF
V
F
F
=
()
r
+
()
r
+
+
()
r
+
)
r
+
(
12
+
()
+
(
3
=
+
+
6
6
6
100
12
3
()
1
+
()
1
1
+
+
()
1
+
)
(
1
+
=
l
price were to
b
e quote
d
b
y
d
ea
l
ers, investors wou
ld
see t
h
e price rise
d
ay a
f
ter
d
ay even
i
f
t
h
e yie
ld
-to-maturity
d
i
d
not c
h
ange.
at is
b
ecause t
h
e amount o
f
accrue
d
interest
increases each day. en after the coupon payment is made the quoted price would drop
dramatically. Using the at price for quotation avoids that misrepresentation. e full
price, at price plus accrued interest, is not usually quoted by bond dealers. Accrued in-
terest is included in, not added to, the full price, and bond dealers do not generally quote
A
s of the beginning of the coupon period on 10 April 2014, there are 2.5 years (5
semiannual periods) to maturity. ese ve semiannual periods occur on 10 October
2014, 10 April 2015, 10 October 2015, 10 April 2016, and 10 October 2016.
PV
PM
T
PM
T
PM
T
PM
T
=
()
r
+
()
r
+
+
()
r
+
()
r
+
+
)
r
+
(
)
r
+
(
1
234
+
()
+
()
+
)
+
(
PM
P
P
TF
V
F
F
()
r
+
5
Chapter 3 Introduction to Fixed-Income Valuation 117
where:
P
V
=
present value
P
M
T
=
coupon payment per period
F
V
=
f
uture va
l
ue pai
d
at maturity, or t
h
e par va
l
ue o
f
t
h
e
b
on
d
r
=
mar
k
et
d
iscount rate, or require
d
rate o
f
return per perio
d
P
=
d
d
=
PV
=
=
=
=
=
P
T
=
=
=
=
118 Part II: Solutions
on a government bond having the same, or close to the same, time-to-maturity. e spread is
the di erence between the yield-to-maturity on the new bond and the benchmark rate. e
R
at
h
er t
h
ey are
k
nown an
d
use
d
to estimate t
h
e require
d
yie
ld
sprea
d
o
f
a new
b
on
d
.
l