Future yield on a bond
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[]
To extract the forward rate, we need the zero-coupon yield curve.
We are trying to find the future interest rate
for time period
,
and
expressed in years, given the rate
for time period
and rate
for time period
. To do this, we use the property that the proceeds from investing at rate
for time period
and then reinvesting those proceeds at rate
for time period
is equal to the proceeds from investing at rate
for time period
.
depends on the rate calculation mode (simple, yearly compounded or continuously compounded), which yields three different results.
Mathematically it reads as follows:
Simple rate[]
![{\displaystyle (1+r_{1}t_{1})(1+r_{1,2}(t_{2}-t_{1}))=1+r_{2}t_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46e3b6434209cf7b550d107e52a526482549c1af)
Solving for
yields:
Thus
The discount factor formula for period (0, t)
expressed in years, and rate
for this period being
,
the forward rate can be expressed in terms of discount factors:
Yearly compounded rate[]
![{\displaystyle (1+r_{1})^{t_{1}}(1+r_{1,2})^{t_{2}-t_{1}}=(1+r_{2})^{t_{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2c1e58bf3474e4ae3045201db104ca7befa2b9)
Solving for
yields :
![{\displaystyle r_{1,2}=\left({\frac {(1+r_{2})^{t_{2}}}{(1+r_{1})^{t_{1}}}}\right)^{1/(t_{2}-t_{1})}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffbf2dee53ea9f52692039ec246a9796471dbc2)
The discount factor formula for period (0,t)
expressed in years, and rate
for this period being
, the forward rate can be expressed in terms of discount factors:
![{\displaystyle r_{1,2}=\left({\frac {DF(0,t_{1})}{DF(0,t_{2})}}\right)^{1/(t_{2}-t_{1})}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb6424b0ed3da3f70049bd7a8bb19c26e3173d4)
Continuously compounded rate[]
- EQUATION→
![{\displaystyle e^{{(r}_{2}\ast t_{2})}=e^{{(r}_{1}\ast t_{1})}\ast \ e^{\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c97a2f9dce0b92dcc4c4f8294006379c817af62)
Solving for
yields:
- STEP 1→
![{\displaystyle e^{{(r}_{2}\ast t_{2})}=e^{{(r}_{1}\ast t_{1})+\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7043a9395347d4e274e502372c14fce632c848d)
- STEP 2→
![{\displaystyle \ln {\left(e^{{(r}_{2}\ast t_{2})}\right)}=\ln {\left(e^{{(r}_{1}\ast t_{1})+\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca57ebf78c18ca192b445653de24fc40dba6a28)
- STEP 3→
![{\displaystyle {(r}_{2}\ast \ t_{2})={(r}_{1}\ast \ t_{1})+\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faeee6f8d66bfcb8839ce073b45f7cd0132665e8)
- STEP 4→
![{\displaystyle r_{1,2}\ast \left(t_{2}-t_{1}\right)={(r}_{2}\ast \ t_{2})-{(r}_{1}\ast \ t_{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76fad8059513e595f3370ffe3d5961b696c58a91)
- STEP 5→
![{\displaystyle r_{1,2}={\frac {{(r}_{2}\ast t_{2})-{(r}_{1}\ast t_{1})}{t_{2}-t_{1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/530548d9162cad0384da70fffc1b66c08b5269f5)
The discount factor formula for period (0,t)
expressed in years, and rate
for this period being
,
the forward rate can be expressed in terms of discount factors:
![{\displaystyle r_{1,2}={\frac {1}{t_{2}-t_{1}}}(\ln DF(0,t_{1})-\ln DF(0,t_{2}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9932f18c134fe18b36d4c61c12b636b5409bc4)
is the forward rate between time
and time
,
is the zero-coupon yield for the time period
, (k = 1,2).
Related instruments[]
See also[]
References[]
- ^ Fabozzi, Vamsi.K (2012), The Handbook of Fixed Income Securities (Seventh ed.), New York: kvrv, p. 148, ISBN 0-07-144099-2.