Forward rate

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

{\displaystyle r_{1,2}} depends on the rate calculation mode (simpleyearly 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}}

Solving for {\displaystyle r_{1,2}} yields:

Thus {\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) {\displaystyle \Delta _{t}} expressed in years, and rate {\displaystyle r_{t}} for this period being {\displaystyle DF(0,t)={\frac {1}{(1+r_{t}\,\Delta _{t})}}}, the forward rate can be expressed in terms of discount factors: {\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

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

Solving for {\displaystyle r_{1,2}} yields :

{\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{\displaystyle \Delta _{t}} expressed in years, and rate {\displaystyle r_{t}} for this period being {\displaystyle DF(0,t)={\frac {1}{(1+r_{t})^{\Delta _{t}}}}}, 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}

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)}}

Solving for {\displaystyle r_{1,2}} 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)}}
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)}}
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)}
STEP 4→ {\displaystyle r_{1,2}\ast \left(t_{2}-t_{1}\right)={(r}_{2}\ast \ t_{2})-{(r}_{1}\ast \ t_{1})}
STEP 5→ {\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{\displaystyle \Delta _{t}} expressed in years, and rate {\displaystyle r_{t}} for this period being {\displaystyle DF(0,t)=e^{-r_{t}\,\Delta _{t}}}, 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}))}

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

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

Related instruments

  • Forward rate agreement
  • Floating rate note

See also

  • Forward price
  • Spot rate

References

  1. ^ Fabozzi, Vamsi.K (2012), The Handbook of Fixed Income Securities (Seventh ed.), New York: kvrv, p. 148, ISBN 0-07-144099-2.