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( T o p )
1
F o r w a r d r a t e c a l c u l a t i o n
T o g g l e F o r w a r d r a t e c a l c u l a t i o n s u b s e c t i o n
1 . 1
S i m p l e r a t e
1 . 2
Y e a r l y c o m p o u n d e d r a t e
1 . 3
C o n t i n u o u s l y c o m p o u n d e d r a t e
2
R e l a t e d i n s t r u m e n t s
3
S e e a l s o
4
R e f e r e n c e s
T o g g l e t h e t a b l e o f c o n t e n t s
F o r w a r d r a t e
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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
The forward rate is the future yield on a bond . It is calculated using the yield curve . For example, the yield on a three-month Treasury bill six months from now is a forward rate .[1]
Forward rate calculation [ edit ]
To extract the forward rate, we need the zero-coupon yield curve .
We are trying to find the future interest rate
r
1
,
2
{\displaystyle r_{1,2}}
for time period
(
t
1
,
t
2
)
{\displaystyle (t_{1},t_{2})}
,
t
1
{\displaystyle t_{1}}
and
t
2
{\displaystyle t_{2}}
expressed in years , given the rate
r
1
{\displaystyle r_{1}}
for time period
(
0
,
t
1
)
{\displaystyle (0,t_{1})}
and rate
r
2
{\displaystyle r_{2}}
for time period
(
0
,
t
2
)
{\displaystyle (0,t_{2})}
. To do this, we use the property that the proceeds from investing at rate
r
1
{\displaystyle r_{1}}
for time period
(
0
,
t
1
)
{\displaystyle (0,t_{1})}
and then reinvesting those proceeds at rate
r
1
,
2
{\displaystyle r_{1,2}}
for time period
(
t
1
,
t
2
)
{\displaystyle (t_{1},t_{2})}
is equal to the proceeds from investing at rate
r
2
{\displaystyle r_{2}}
for time period
(
0
,
t
2
)
{\displaystyle (0,t_{2})}
.
r
1
,
2
{\displaystyle 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:
Simple rate [ edit ]
(
1
+
r
1
t
1
)
(
1
+
r
1
,
2
(
t
2
−
t
1
)
)
=
1
+
r
2
t
2
{\displaystyle (1+r_{1}t_{1})(1+r_{1,2}(t_{2}-t_{1}))=1+r_{2}t_{2}}
Solving for
r
1
,
2
{\displaystyle r_{1,2}}
yields:
Thus
r
1
,
2
=
1
t
2
−
t
1
(
1
+
r
2
t
2
1
+
r
1
t
1
−
1
)
{\displaystyle r_{1,2}={\frac {1}{t_{2}-t_{1}}}\left({\frac {1+r_{2}t_{2}}{1+r_{1}t_{1}}}-1\right)}
The discount factor formula for period (0, t)
Δ
t
{\displaystyle \Delta _{t}}
expressed in years, and rate
r
t
{\displaystyle r_{t}}
for this period being
D
F
(
0
,
t
)
=
1
(
1
+
r
t
Δ
t
)
{\displaystyle DF(0,t)={\frac {1}{(1+r_{t}\,\Delta _{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
)
{\displaystyle r_{1,2}={\frac {1}{t_{2}-t_{1}}}\left({\frac {DF(0,t_{1})}{DF(0,t_{2})}}-1\right)}
Yearly compounded rate [ edit ]
(
1
+
r
1
)
t
1
(
1
+
r
1
,
2
)
t
2
−
t
1
=
(
1
+
r
2
)
t
2
{\displaystyle (1+r_{1})^{t_{1}}(1+r_{1,2})^{t_{2}-t_{1}}=(1+r_{2})^{t_{2}}}
Solving for
r
1
,
2
{\displaystyle r_{1,2}}
yields :
r
1
,
2
=
(
(
1
+
r
2
)
t
2
(
1
+
r
1
)
t
1
)
1
/
(
t
2
−
t
1
)
−
1
{\displaystyle r_{1,2}=\left({\frac {(1+r_{2})^{t_{2}}}{(1+r_{1})^{t_{1}}}}\right)^{1/(t_{2}-t_{1})}-1}
The discount factor formula for period (0,t )
Δ
t
{\displaystyle \Delta _{t}}
expressed in years, and rate
r
t
{\displaystyle r_{t}}
for this period being
D
F
(
0
,
t
)
=
1
(
1
+
r
t
)
Δ
t
{\displaystyle DF(0,t)={\frac {1}{(1+r_{t})^{\Delta _{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
{\displaystyle r_{1,2}=\left({\frac {DF(0,t_{1})}{DF(0,t_{2})}}\right)^{1/(t_{2}-t_{1})}-1}
Continuously compounded rate [ edit ]
e
r
2
⋅
t
2
=
e
r
1
⋅
t
1
⋅
e
r
1
,
2
⋅
(
t
2
−
t
1
)
{\displaystyle e^{r_{2}\cdot t_{2}}=e^{r_{1}\cdot t_{1}}\cdot \ e^{r_{1,2}\cdot \left(t_{2}-t_{1}\right)}}
Solving for
r
1
,
2
{\displaystyle r_{1,2}}
yields:
STEP 1→
e
r
2
⋅
t
2
=
e
r
1
⋅
t
1
+
r
1
,
2
⋅
(
t
2
−
t
1
)
{\displaystyle e^{r_{2}\cdot t_{2}}=e^{r_{1}\cdot t_{1}+r_{1,2}\cdot \left(t_{2}-t_{1}\right)}}
STEP 2→
ln
(
e
r
2
⋅
t
2
)
=
ln
(
e
r
1
⋅
t
1
+
r
1
,
2
⋅
(
t
2
−
t
1
)
)
{\displaystyle \ln \left(e^{r_{2}\cdot t_{2}}\right)=\ln \left(e^{r_{1}\cdot t_{1}+r_{1,2}\cdot \left(t_{2}-t_{1}\right)}\right)}
STEP 3→
r
2
⋅
t
2
=
r
1
⋅
t
1
+
r
1
,
2
⋅
(
t
2
−
t
1
)
{\displaystyle r_{2}\cdot t_{2}=r_{1}\cdot t_{1}+r_{1,2}\cdot \left(t_{2}-t_{1}\right)}
STEP 4→
r
1
,
2
⋅
(
t
2
−
t
1
)
=
r
2
⋅
t
2
−
r
1
⋅
t
1
{\displaystyle r_{1,2}\cdot \left(t_{2}-t_{1}\right)=r_{2}\cdot t_{2}-r_{1}\cdot t_{1}}
STEP 5→
r
1
,
2
=
r
2
⋅
t
2
−
r
1
⋅
t
1
t
2
−
t
1
{\displaystyle r_{1,2}={\frac {r_{2}\cdot t_{2}-r_{1}\cdot t_{1}}{t_{2}-t_{1}}}}
The discount factor formula for period (0,t )
Δ
t
{\displaystyle \Delta _{t}}
expressed in years, and rate
r
t
{\displaystyle r_{t}}
for this period being
D
F
(
0
,
t
)
=
e
−
r
t
Δ
t
{\displaystyle DF(0,t)=e^{-r_{t}\,\Delta _{t}}}
,
the forward rate can be expressed in terms of discount factors:
r
1
,
2
=
ln
(
D
F
(
0
,
t
1
)
)
−
ln
(
D
F
(
0
,
t
2
)
)
t
2
−
t
1
=
−
ln
(
D
F
(
0
,
t
2
)
D
F
(
0
,
t
1
)
)
t
2
−
t
1
{\displaystyle r_{1,2}={\frac {\ln \left(DF\left(0,t_{1}\right)\right)-\ln \left(DF\left(0,t_{2}\right)\right)}{t_{2}-t_{1}}}={\frac {-\ln \left({\frac {DF\left(0,t_{2}\right)}{DF\left(0,t_{1}\right)}}\right)}{t_{2}-t_{1}}}}
r
1
,
2
{\displaystyle r_{1,2}}
is the forward rate between time
t
1
{\displaystyle t_{1}}
and time
t
2
{\displaystyle t_{2}}
,
r
k
{\displaystyle r_{k}}
is the zero-coupon yield for the time period
(
0
,
t
k
)
{\displaystyle (0,t_{k})}
, (k = 1,2).
Related instruments [ edit ]
See also [ edit ]
References [ edit ]
^ Fabozzi, Vamsi.K (2012), The Handbook of Fixed Income Securities (Seventh ed.), New York: kvrv, p. 148, ISBN 978-0-07-144099-8 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Forward_rate&oldid=1128168902 "
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