Deriscope ## The Excel Derivatives Periscope

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forward_interest_rate

A *forward interest rate* is always in relation to some specific underlying interest rate *r*.

Given the contract *C* that starts at *T* and defines the underlying interest rate *r(T)*, the *forward interest rate* at time *t < T* is the number *f(t,T)* with the following meaning:

It makes economic sense for any two parties at time *t* to enter into the contract *C* with the respective interest rate within the contract stipulated to equal exactly *f(t,T)*.

Note that at the later time *T > t* it may make no sense to enter into the contract *C* any more using the interest rate value of *f(t,T)*.

As an example, let *r(T)* be a certain ibor rate prevailing at time *T*.

In simplifing terms, the associated contract *C* is a lending/borrowing contract between two banks over a term that starts at *T* and ends a certain time interval later, for example at *T + 0.5*.

Then the *forward interest rate* *f(t,T)* is the number agreed between two parties at time *t*, such that they will enter into the contract *C* at the latter time *T* using the rate *f(t,T)* even if that rate is not fair any more.

It follows that *f(t,T) -> r(T)* as *t -> t*.

Similar to the interest rate function *r(t)*, we may also speak of a function *f(t,T)* that - keeping *T* fixed - maps each time *t* to the respective *forward interest rate* value *f(t,T)*.

Assuming *t = 0* designates the time now, the value *f(t,T), t > 0 and t < T* is not a simple number but rather a random variable, since it is not possible to know with certainty the *forward interest rate* that is going to prevail at the future tinme *t*.

It follows, the function *f(t,T)* represents a mapping from *t* to some random variable, and therefore is a stochastic process.