Instantaneous Forward Rates

Note: Instantaneous forward rate calculations can be downloaded here.

The main focus of this article is to clarify the difference between spot, forward and instantaneous forward rates, define the meaning of the latter and outline its application.

The main feature of interest rates as a class is that they do not represent any specific financial instrument such as a contract or security, but are rather used as a reference category that determines prices of interest rate linked securities and their derivatives.
The simplest type of rates are the spot rates, which determine the time value of money, i.e. they indicate the return an investment can yield over a respective time period starting from today (t = 0).
The population of such interest rates constitutes a term structure which describes the dependency between tenor and yield, which is also known as the spot rate yield curve.
We can sometimes ask ourselves, how much would an investment made in a years’ time be worth a certain period after it has been made.
This is a situation when we encounter the forward rate.
Forward rates indicate the yield generated in between 2 future dates and are fairly easy to determine provided the spot rate yield curve is known:

Rearranging (1):

where f(n,m) is the forward rate between time n and m, s_n and s_m are the respective spot rates and the periods satisfy inequality m > n.
The chart below illustrates a sample spot yield curve and the derived forward rates.

So far we have discussed interest rates on a discrete time scale.
It is also possible to analyze rates using continuous time, which is a common thing for the quant world.
This can be useful especially for financial modeling and simulation purposes, for example if there is a requirement to derive a yield for a discretely small timeframe such that ∆t→0, which can practically mean hours, minutes, seconds or less. In this case we incorporate the concept of instantaneous forward rates, which are sometimes called short rates. The instantaneous forward rate curve is displayed on the given chart and is based on the equation:

which means:

if (ds/dt)*t exists, the instantaneous rate R should be given by:

The derivative part of the equation contributes to the “continuousness” of interest rates, but at the same time it captures the slope of different areas of the spot curve, which immediately affects the twist of the instantaneous forward rate curve.
For example if we look at the far end of the spot curve, we will notice that the term structure tends to be nearly flat with slightly negative convexity at 3%. The calculated derivative will return negative values and brings the instantaneous forward rate below the spot level.
Instantaneous forward rates can be used in quantitative approaches to yield curve modelling such as Heath, Jarrow & Morton (HJM) model.