Forward rate (original) (raw)

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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

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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}} $r_{1,2}$ for time period ( t 1 , t 2 ) {\displaystyle (t_{1},t_{2})} $(t_{1},t_{2})$ , t 1 {\displaystyle t_{1}} $t_{1}$ and t 2 {\displaystyle t_{2}} $t_{2}$ expressed in years, given the rate r 1 {\displaystyle r_{1}} $r_{1}$ for time period ( 0 , t 1 ) {\displaystyle (0,t_{1})} $(0,t_{1})$ and rate r 2 {\displaystyle r_{2}} $r_{2}$ for time period ( 0 , t 2 ) {\displaystyle (0,t_{2})} $(0,t_{2})$ . To do this, we use the property that the proceeds from investing at rate r 1 {\displaystyle r_{1}} $r_{1}$ for time period ( 0 , t 1 ) {\displaystyle (0,t_{1})} $(0,t_{1})$ and then reinvesting those proceeds at rate r 1 , 2 {\displaystyle r_{1,2}} $r_{1,2}$ for time period ( t 1 , t 2 ) {\displaystyle (t_{1},t_{2})} $(t_{1},t_{2})$ is equal to the proceeds from investing at rate r 2 {\displaystyle r_{2}} $r_{2}$ for time period ( 0 , t 2 ) {\displaystyle (0,t_{2})} $(0,t_{2})$ .

r 1 , 2 {\displaystyle r_{1,2}} $r_{1,2}$ 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 {\displaystyle (1+r_{1}t_{1})(1+r_{1,2}(t_{2}-t_{1}))=1+r_{2}t_{2}} $(1+r_{1}t_{1})(1+r_{1,2}(t_{2}-t_{1}))=1+r_{2}t_{2}$

Solving for r 1 , 2 {\displaystyle r_{1,2}} $r_{1,2}$ 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)} $r_{1,2}={\frac {1}{t_{2}-t_{1}}}\left({\frac {1+r_{2}t_{2}}{1+r_{1}t_{1}}}-1\right)$

The discount factor formula for period (0, t) Δ t {\displaystyle \Delta _{t}} $\Delta _{t}$ expressed in years, and rate r t {\displaystyle r_{t}} $r_{t}$ 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})}}} $DF(0,t)={\frac {1}{(1+r_{t}\,\Delta _{t})}}$ , 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)} $r_{1,2}={\frac {1}{t_{2}-t_{1}}}\left({\frac {DF(0,t_{1})}{DF(0,t_{2})}}-1\right)$

Yearly compounded rate

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( 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}}} $(1+r_{1})^{t_{1}}(1+r_{1,2})^{t_{2}-t_{1}}=(1+r_{2})^{t_{2}}$

Solving for r 1 , 2 {\displaystyle r_{1,2}} $r_{1,2}$ 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} $r_{1,2}=\left({\frac {(1+r_{2})^{t_{2}}}{(1+r_{1})^{t_{1}}}}\right)^{1/(t_{2}-t_{1})}-1$

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} $r_{1,2}=\left({\frac {DF(0,t_{1})}{DF(0,t_{2})}}\right)^{1/(t_{2}-t_{1})}-1$

Continuously compounded rate

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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)}} $e^{r_{2}\cdot t_{2}}=e^{r_{1}\cdot t_{1}}\cdot \ e^{r_{1,2}\cdot \left(t_{2}-t_{1}\right)}$

Solving for r 1 , 2 {\displaystyle r_{1,2}} $r_{1,2}$ 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)}} $e^{r_{2}\cdot t_{2}}=e^{r_{1}\cdot t_{1}+r_{1,2}\cdot \left(t_{2}-t_{1}\right)}$

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)} $\ln \left(e^{r_{2}\cdot t_{2}}\right)=\ln \left(e^{r_{1}\cdot t_{1}+r_{1,2}\cdot \left(t_{2}-t_{1}\right)}\right)$

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)} $r_{2}\cdot t_{2}=r_{1}\cdot t_{1}+r_{1,2}\cdot \left(t_{2}-t_{1}\right)$

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}} $r_{1,2}\cdot \left(t_{2}-t_{1}\right)=r_{2}\cdot t_{2}-r_{1}\cdot t_{1}$

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}}}} $r_{1,2}={\frac {r_{2}\cdot t_{2}-r_{1}\cdot t_{1}}{t_{2}-t_{1}}}$

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}}}} $r_{1,2}={\frac {\ln \left(DF\left(0,t_{1}\right)\right)-\ln \left(DF\left(0,t_{2}\right)\right)}{t_{2}-t_{1}}}={\frac {-\ln \left({\frac {DF\left(0,t_{2}\right)}{DF\left(0,t_{1}\right)}}\right)}{t_{2}-t_{1}}}$

r 1 , 2 {\displaystyle r_{1,2}} $r_{1,2}$ is the forward rate between time t 1 {\displaystyle t_{1}} $t_{1}$ and time t 2 {\displaystyle t_{2}} $t_{2}$ ,

r k {\displaystyle r_{k}} $r_{k}$ is the zero-coupon yield for the time period ( 0 , t k ) {\displaystyle (0,t_{k})} $(0,t_{k})$ , (k = 1,2).

^ Fabozzi, Vamsi.K (2012), The Handbook of Fixed Income Securities (Seventh ed.), New York: kvrv, p. 148, ISBN 978-0-07-144099-8.