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short_rateThe 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.