STAT 2820
Chapter 7 Interest Rate Forwards
by K.C. Cheung
7.1 Zerocoupon bonds
7.1.1
A zerocoupon bond is a bond that makes only a single payment at its maturity. The
price of a zerocoupon bond
quoted at time
t
, with par value (or maturity value, principal value)
1 and to be purchased at time
t
1
and maturing at
t
2
, is denoted as
P
t
(
t
1
,t
2
). When
t
=
t
1
, we
simply write
P
(
t
1
,t
2
). If
t < t
1
, then
P
t
(
t
1
,t
2
) is the price of the bond to be purchased at time
t
1
you are lock in at time
t
. This is like a forward contract: actual transaction takes place at
a future time point, but the price is ﬁxed now. So when
t < t
1
we may call
P
t
(
t
1
,t
2
) a
forward
bond price
.
7.1.2
We let
r
t
(
t
1
,t
2
) be the annual interest rate (compounded annually) from time
t
1
to
t
2
,
prevailing on date
t
. If
t
=
t
1
, we simply write
r
(
t
1
,t
2
). Remark that buying zerocoupon bond
is the same as lending money, selling zerocoupon bond is the same as borrowing money. So
r
t
(
t
1
,t
2
) is the lending or borrowing rate for the period [
t
1
,t
2
] that one can lock in at time
t
.
7.1.3
In general, we have
P
(0
,n
) =
1
(1 +
r
(0
,n
))
n
7.1.4
For example, for a 1year zerocoupon bond with par value 1, if the bond price today
(
t
= 0) is 0.943396, then
P
(0
,
1) = 0
.
943396
and
r
(0
,
1) =
1
P
(0
,
1)

1 = 0
.
06
7.1.5
For a 2year zerocoupon bond with par value 1, if the bond price today (
t
= 0) is
0.881659, then
P
(0
,
2) = 0
.
881659
and
P
(0
,
2) =
1
(1 +
r
(0
,
2))
2
=
⇒
r
(0
,
2) = 0
.
065
7.1.6
If at time
t
= 0, we put $
A
in a riskfree
n
year deposit, the annual interest rate we earn
is
r
(0
,n
), then after
n
years we receive
A
(1 +
r
(0
,n
))
n
.

{z
}
r
(0
,n
) per year
A
(1 +
r
(0
,n
))
n
A
0
n