# Forward rate

Not to be confused with forward price or forward exchange 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 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 rate

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 rate

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 rate

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:

is the forward rate between time and time ,

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

## References

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

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