Deriscope ## The Excel Derivatives Periscope

##### Coverage

short__rate

The *short rate* is a particular type of interest rate that represents the percentage gain in annualized terms one can achieve by lending in a risk-free manner - i.e. with perfect creditworthiness - over an infinitesimally small time interval.

Obviously *short rate* is an idealized concept, not existing in the actual markets.

A *short rate* having value *r* - for example 4% or 0.04 - literally means that the lender of *N* dollars over a very small time interval *dt* would earn *Nr* dollars in annualized terms, but in reality only *Nrdt* over the lending interval *dt*.

It is possible to calculate the total interest one would learn, if one chose to continuously invest an initial capital *N* together with the intermediate interest proceeds over some finite time interval *T* using an assumed constant *short rate* *r*.

We proceed by partitioning the time interval *T* into segments, each of length *dt*.

The total capital *N'* the lender would have after having invested the initila capital *N* over the first time segment with length *dt* would equal *N' = N + Nrdt = N(1+rdt)*.

By reinvesting this new capital *N'* over the next time segment - again with length *dt* -, the additional interest earned would be *N'rdt* and the total capital at the end of the second interval would be *N'' = N' + N'rdt = N'(1+rdt) = N(1+rdt)(1+rdt)*, where we replaced *N'* with *N(1+rdt)*

Repeating this process, it follows that at the end of the third time segment the total capital would have reached *N(1+rdt)(1+rdt)(1+rdt)* and after *n* time segments each of length *dt*, the capital would equal *N(1+rdt)^n*.

In the limit where *dt -> 0* and *n -> infinity*, such that *n*dt = T*, this expression converges to *N*exp(rT)*.

This formula can be generalised to the case when *r* varies over time.

It follows that knowing the value of the *short rate* over all possible times *t*, in effect knowing the function *r(t)* suffices to recover all simply compounded interest rates over any finite time interval.