Instantaneous Forward Rates

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.


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13 thoughts on “Instantaneous Forward Rates

  1. I am assuming it is f(1Y,1Y+t) where t is the tenor on the x-axis… i.e. I am assuming you have plotted the 1 year forward curve, as mentioned in the text.. but I am worried I\’ve misunderstood.

  2. In your blue Forward curve, what exactly are you plotting there? For your forward rates.. f(n,m) .. what are the values of n and m you are using?

  3. I just looked at the spreadsheet, and now I see I did completely misunderstand 🙂

    The rates being plotted here are the forward rates on a tenor-to-tenor basis.

    So, at the 20Y point on the x-axis, the value plotted is f(15Y,20Y).. because 15Y is the nearest neighbouring tenor for which there is a spot rate.

    I realise this is still very unclear in words, so some examples might help clarify:

    At x=0, you have no data, so assume f(0,0) = spot(0)

    For all the others, you can use the formula for f given in the article, matching x position to [n,m] values in this way:

    At x=7d, plot f(0,7d)
    At x=1M, plot f(7d,1M)
    ..
    At x=9M, plot f(6M,9M)
    At x=1Y, plot f(9M,1Y)
    ..
    At x=15Y, plot f(12Y,15Y)
    At x=20Y, plot f(15Y,20Y)
    ..

    It\’s not quite clear to me how you would use this chart to gain an real insight, but it is helpful to know how it was constructed!

  4. Hi Phil. Thank you for the feedback. The chart is not meant to give any insight of economic significance, but rather to display how the spot rate values differ from forwards and instantaneous forwards, i.e. prove to readers that these rate types are not to be mixed. It is very important to distinguish these 3 methods when it comes to pricing or financial modelling by known approaches that refer to one of the rate types. We hope that this article has been helpful.

  5. Yes, it is great. I would recommend that anyone wanting more clarification download the spreadsheet – it is well put together and fairly easy to understand.
    Many thanks!

  6. Does that mean (in Phil\’s notation) that at x=9M, you\’re plotting the 3M forward rate from time 6M and at time 15Y (i.e. x=15Y) you\’re plotting the 3Y forward rate from time 12Y?

    Thanks

  7. Also just to confirm, if you were to plot \’consistent forward rates\’, i.e. only 3 month forward rates, you would need to interpolate between the spot rates so that you have a spot rate for every tenor (i.e. every 3 months). Is this correct? are the calculated forward rates then annualised forward rates?

    Many thanks

  8. Very good article, and very good spreadsheet!

    But excuse me for the maybe trivial question, but in Bloomberg how to find/download the Spot Curve (the initial input of your spreadsheet)?

    Thanks!

  9. You can find USD spot curve by using ICVS 23 commnad -> 21) Stripped curve under the column Spot curve. Also, in this tab you can see the source used, i.e. for short maturities there is Cash deposits, Eurodollars for up to 3 years and swaps are used for long maturities.

  10. Thanks a lot Alexander! But so do you also mean (sorry again for the trivial question), that there are several Spot Curve? I mean, there is the USD Spot Curve (ICVS 23 in BL), the EONIA curve (ICVS 133), and so on? I was thinking the \”spot curve\” was intended (in the literature) only for the Term Structure of Zero Coupon Bonds..

    So, for example, if I want to use the Spot Curve to extract the Instantaneous Forward Rate, by using your spreadsheet, to calibrate the theta function of the Hull-White model (as in Brigo-Mercurio book, 2nd ed. page 73), to simulate future EONIA rates, I may refer to the Eonia Swap Curve (ICVS 133 in Bloomberg) as the current Spot Curve, right?

    Thanks!

  11. Marco,
    you are right. For Eur you can take EONIA spot curve, ICVS 133, for USD take ICVS 23. These curves are built using bootstraping.
    what I mean is: in order to build spot curve you need to extract yields from different instruments, such as Deposits, Eurodollars, swaps etc.
    and of cause, there is \”common used\” spot curves, like ICVS23, but you can build your own if you take, for example, bonds instead of swaps))). Actiually, the market takes those instruments that are very liquid to bootstrap the yield.