## forward interest rate

A

**is always in relation to some specific underlying interest rate**

*forward interest rate***.**

*r*Given the contract

**that starts at**

*C***and defines the underlying interest rate**

*T***, the**

*r(T)***at time**

*forward interest rate***is the number**

*t < T***with the following meaning:**

*f(t,T)*It makes economic sense for any two parties at time

**to enter into the contract**

*t***with the respective interest rate within the contract stipulated to equal exactly**

*C***.**

*f(t,T)*Note that at the later time

**it may make no sense to enter into the contract**

*T > t***any more using the interest rate value of**

*C***.**

*f(t,T)*As an example, let

**be a certain ibor rate prevailing at time**

*r(T)***.**

*T*In simplifing terms, the associated contract

**is a lending/borrowing contract between two banks over a term that starts at**

*C***and ends a certain time interval later, for example at**

*T***.**

*T + 0.5*Then the

*forward interest rate***is the number agreed between two parties at time**

*f(t,T)***, such that they will enter into the contract**

*t***at the latter time**

*C***using the rate**

*T***even if that rate is not fair any more.**

*f(t,T)*It follows that

**as**

*f(t,T) -> r(T)***.**

*t -> t*Similar to the interest rate function

**, we may also speak of a function**

*r(t)***that - keeping**

*f(t,T)***fixed - maps each time**

*T***to the respective**

*t***value**

*forward interest rate***.**

*f(t,T)*Assuming

**designates the time now, the value**

*t = 0***is not a simple number but rather a random variable, since it is not possible to know with certainty the**

*f(t,T), t > 0 and t < T***that is going to prevail at the future tinme**

*forward interest rate***.**

*t*It follows, the function

**represents a mapping from**

*f(t,T)***to some random variable, and therefore is a stochastic process.**

*t*