# What distribution to assume for interest rates?

I am writing a paper with a case study in financial maths. I need to model an interest rate $(I_n)_{n\geq 0}$ as a sequence of non-negative i.i.d. random variables. Which distribution would you advise me to use? Currently I am considering the exponential distribution, but I am not sure that it is the right choice, though it is quite easy to work with.

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You could try a discretisation of CIR process, which should give you a non central chi-square distribution if I remember well. –  TheBridge Nov 8 '11 at 13:04
@TheBridge: thanks, but I wonder if there any benchmark distributions for iid interest rates –  Ilya Nov 8 '11 at 14:17
What is the best distribution to describe a financial time series is up for serious debate.....Every one will say to use something different –  pyCthon Aug 30 '12 at 19:40

## 5 Answers

Exponential distribution, although it's a good distribution for modeling non-negative numbers, doesn't make sense here since it's mode is 0.

From a pure statistical point of view, without any knowledge of interest rate, I'd recommend log-normal as in modeling stock prices and inverse-gamma or gamma distribution which are used to model variance or other scale parameters which is a non-negative distribution with mode greater than zero.

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Interest rates in general are far from independent and identically distributed. A high interest rate observation is quite likely to be followed by another high observation, and the volatility is likely to be higher as well. Interest rates are also mean reverting, as in most real-world situations (at least for developed markets) interest rates rarely rise too high or dip too low.

Since you are looking for the simplest possible solution for a case study, I would recommend you start with a lognormal distribution, which implicitly assumes interest rates follow a geometric brownian motion. The problem with this distribution is that it assumes the interest rate can get arbitrarily high. The next simplest solution would be a Cox-Ingersoll-Ross process, which has a noncentral chi-squared distribution of innovations. The following matlab function includes a simple simulation of a CIR process. The underlying distribution the function uses is noncentral chi-square, and the algorithm itself is quite clear even if you don't use or know matlab.

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I used the matlab function (translated to R), and it gives quite reasonable numbers. –  Owe Jessen Aug 29 '12 at 6:55
"Interest rates in general are far from independent and identically distributed" -- a statistician's least favorite phrase, but so very true in this case. –  jlowin Aug 29 '12 at 12:33

You could try using the Gaussian Affine Term Structure Models (GATSM), with the right boundary conditions to stop rates being negative (in the style of their Black implementation). See, for example, Monika Piazzesi, the "Affine Term Structure Models" if you want to enter/modify the basis or the work of Krippner, for example "Measuring the stance of monetary policy in zero lower bound environments".

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Do you have any references? –  Bob Jansen Aug 28 '12 at 18:09
Hello @Bob Jansen : You could try Monika Piazzesi, the "Affine Term Structure Models" chapter, if you want to enter/modify the basis. Or you can see the work of Krippner, for example "Measuring the stance of monetary policy in zero lower bound environments", if you want to cut to some final ideas. –  user7056 Aug 30 '12 at 11:56
Thanks, I've added some links to your answer. –  Bob Jansen Aug 30 '12 at 16:14

General rate trading wisdom shows that if anything, normal distribution fits developed markets better. For example, most swaption traders talk about implied volatilities in basis points (per day or annualized).

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If you manage to get some data fitting your subject, one solution could be to try an empirical distribution.

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