## rate

The

**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.**

*short rate*Obviously

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

*short rate*A

**having value**

*short rate***- for example 4% or 0.04 - literally means that the lender of**

*r***dollars over a very small time interval**

*N***would earn**

*dt***dollars in annualized terms, but in reality only**

*Nr***over the lending interval**

*Nrdt***.**

*dt*It is possible to calculate the total interest one would learn, if one chose to continuously invest an initial capital

**together with the intermediate interest proceeds over some finite time interval**

*N***using an assumed constant**

*T*

*short rate***.**

*r*We proceed by partitioning the time interval

**into segments, each of length**

*T***.**

*dt*The total capital

**the lender would have after having invested the initila capital**

*N'***over the first time segment with length**

*N***would equal**

*dt***.**

*N' = N + Nrdt = N(1+rdt)*By reinvesting this new capital

**over the next time segment - again with length**

*N'***-, the additional interest earned would be**

*dt***and the total capital at the end of the second interval would be**

*N'rdt***, where we replaced**

*N'' = N' + N'rdt = N'(1+rdt) = N(1+rdt)(1+rdt)***with**

*N'*

*N(1+rdt)*Repeating this process, it follows that at the end of the third time segment the total capital would have reached

**and after**

*N(1+rdt)(1+rdt)(1+rdt)***time segments each of length**

*n***, the capital would equal**

*dt***.**

*N(1+rdt)^n*In the limit where

**and**

*dt -> 0***, such that**

*n -> infinity***, this expression converges to**

*n*dt = T***.**

*N*eᴿᵀ*This formula can be generalised to the case when

**varies over time.**

*r*It follows that knowing the value of the

**over all possible times**

*short rate***, in effect knowing the function**

*t***suffices to recover all simply compounded interest rates over any finite time interval.**

*r(t)*