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 calculationEdit

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 rateEdit

 

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 rateEdit

 

Solving for   yields :

 

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:

 

Continuously compounded rateEdit

EQUATION→  


Solving for   yields:


STEP 1→  


STEP 2→  


STEP 3→  


STEP 4→  


STEP 5→  

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:

 

  is the forward rate between time   and time  ,

  is the zero-coupon yield for the time period  , (k = 1,2).

Related instrumentsEdit

See alsoEdit

ReferencesEdit

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