To extract the forward rate, we need the zero-coupon yield curve.
We are trying to find the future interest rate
r
1
,
2
for time period
(
t
1
,
t
2
)
,
t
1
and
t
2
expressed in years, given the rate
r
1
for time period
(
0
,
t
1
)
and rate
r
2
for time period
(
0
,
t
2
)
. To do this, we use the property that the proceeds from investing at rate
r
1
for time period
(
0
,
t
1
)
and then reinvesting those proceeds at rate
r
1
,
2
for time period
(
t
1
,
t
2
)
is equal to the proceeds from investing at rate
r
2
for time period
(
0
,
t
2
)
.
r
1
,
2
depends on the rate calculation mode (simple, yearly compounded or continuously compounded), which yields three different results.
Mathematically it reads as follows:
(
1
+
r
1
t
1
)
(
1
+
r
1
,
2
(
t
2
−
t
1
)
)
=
1
+
r
2
t
2
Solving for
r
1
,
2
yields:
Thus
r
1
,
2
=
1
t
2
−
t
1
(
1
+
r
2
t
2
1
+
r
1
t
1
−
1
)
The discount factor formula for period (0, t)
Δ
t
expressed in years, and rate
r
t
for this period being
D
F
(
0
,
t
)
=
1
(
1
+
r
t
Δ
t
)
, the forward rate can be expressed in terms of discount factors:
r
1
,
2
=
1
t
2
−
t
1
(
D
F
(
0
,
t
1
)
D
F
(
0
,
t
2
)
−
1
)
(
1
+
r
1
)
t
1
(
1
+
r
1
,
2
)
t
2
−
t
1
=
(
1
+
r
2
)
t
2
Solving for
r
1
,
2
yields :
r
1
,
2
=
(
(
1
+
r
2
)
t
2
(
1
+
r
1
)
t
1
)
1
t
2
−
t
1
−
1
The discount factor formula for period (0, t)
Δ
t
expressed in years, and rate
r
t
for this period being
D
F
(
0
,
t
)
=
1
(
1
+
r
t
)
Δ
t
, the forward rate can be expressed in terms of discount factors:
r
1
,
2
=
(
D
F
(
0
,
t
1
)
D
F
(
0
,
t
2
)
)
1
t
2
−
t
1
−
1
e
r
1
t
1
e
r
1
,
2
(
t
2
−
t
1
)
=
e
r
2
t
2
Solving for
r
1
,
2
yields :
r
1
,
2
=
r
2
t
2
−
r
1
t
1
t
2
−
t
1
The discount factor formula for period (0, t)
Δ
t
expressed in years, and rate
r
t
for this period being
D
F
(
0
,
t
)
=
e
−
r
t
Δ
t
, the forward rate can be expressed in terms of discount factors:
r
1
,
2
=
1
t
2
−
t
1
(
ln
D
F
(
0
,
t
1
)
−
ln
D
F
(
0
,
t
2
)
)
r
1
,
2
is the forward rate between time
t
1
and time
t
2
,
r
k
is the zero-coupon yield for the time period
(
0
,
t
k
)
, (k=1, 2).
Forward rate agreement
Floating rate note
