Forward rate

Forward rate calculation

To extract the forward rate, we need the zero-coupon yield curve.

We are trying to find the future interest rate r 1 , 2 for time period ( t 1 , t 2 ) , t 1 and t 2 expressed in years, given the rate r 1 for time period ( 0 , t 1 ) and rate r 2 for time period ( 0 , t 2 ) . To do this, we use the property that the proceeds from investing at rate r 1 for time period ( 0 , t 1 ) and then reinvesting those proceeds at rate r 1 , 2 for time period ( t 1 , t 2 ) is equal to the proceeds from investing at rate r 2 for time period ( 0 , t 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:

Simple rate

( 1 + r 1 t 1 ) ( 1 + r 1 , 2 ( t 2 − t 1 ) ) = 1 + r 2 t 2

Solving for r 1 , 2 yields:

Thus r 1 , 2 = 1 t 2 − t 1 ( 1 + r 2 t 2 1 + r 1 t 1 − 1 )

The discount factor formula for period (0, t) Δ t expressed in years, and rate r t for this period being D F ( 0 , t ) = 1 ( 1 + r t Δ 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 )

Yearly compounded rate

( 1 + r 1 ) t 1 ( 1 + r 1 , 2 ) t 2 − t 1 = ( 1 + r 2 ) t 2

Solving for r 1 , 2 yields : r 1 , 2 = ( ( 1 + r 2 ) t 2 ( 1 + r 1 ) t 1 ) 1 t 2 − t 1 − 1

The discount factor formula for period (0, t) Δ t expressed in years, and rate r t for this period being D F ( 0 , t ) = 1 ( 1 + r t ) Δ 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

Continuously compounded rate

e r 1 t 1 e r 1 , 2 ( t 2 − t 1 ) = e r 2 t 2

Solving for r 1 , 2 yields : r 1 , 2 = r 2 t 2 − r 1 t 1 t 2 − t 1

The discount factor formula for period (0, t) Δ t expressed in years, and rate r t for this period being D F ( 0 , t ) = e − r t Δ 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 ) )

r 1 , 2 is the forward rate between time t 1 and time t 2 ,

r k is the zero-coupon yield for the time period ( 0 , t k ) , (k=1, 2).

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