Does the Formula for Forwards Rates Always Give a Continuously Comounded Value

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 [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]

EQUATION→ e ( r 2 t 2 ) = e ( r 1 t 1 ) e ( r 1 , 2 ( t 2 t 1 ) ) {\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)}}


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}\ast t_{2})}=e^{{(r}_{1}\ast t_{1})+\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\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}\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)}}
STEP 3→ ( r 2 t 2 ) = ( r 1 t 1 ) + ( r 1 , 2 ( t 2 t 1 ) ) {\displaystyle {(r}_{2}\ast \ t_{2})={(r}_{1}\ast \ t_{1})+\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}
STEP 4→ r 1 , 2 ( t 2 t 1 ) = ( r 2 t 2 ) ( r 1 t 1 ) {\displaystyle r_{1,2}\ast \left(t_{2}-t_{1}\right)={(r}_{2}\ast \ t_{2})-{(r}_{1}\ast \ 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}\ast t_{2})-{(r}_{1}\ast 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 = 1 t 2 t 1 ( ln D F ( 0 , t 1 ) ln D F ( 0 , t 2 ) ) {\displaystyle r_{1,2}={\frac {1}{t_{2}-t_{1}}}(\ln DF(0,t_{1})-\ln DF(0,t_{2}))}

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).

[edit]

  • Forward rate agreement
  • Floating rate note

See also [edit]

  • Forward price
  • Spot rate

References [edit]

  1. ^ Fabozzi, Vamsi.K (2012), The Handbook of Fixed Income Securities (Seventh ed.), New York: kvrv, p. 148, ISBN0-07-144099-2 .

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Source: https://en.wikipedia.org/wiki/Forward_rate

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