Mean Reversion and the Half-Life of Interest Rates
July 19, 2012
In response to a recent article on this site regarding half-life of interest rates, many readers have written to us asking us how to derive the mathematical equation for half-life. Many of our readers want to know that within the context of interest rates, how do we get the formula for half-life?The answer is both straight forward and trivial and those who are familiar with the process of exponential decay – and the math models that explain such decay – will know that the deterministic part of the interest rate process exhibits such decay via the OU stochastic differential equation.
We would like to refer the reader to an article by Marco Antonio Guimaraes Dias*, which is available on the internet and on which the following explanation is based.
Interest rates – short rates – are modeled as a special case of a mean reverting Ornstein Uhlenbeck (OU) process. The most famous of these models is the Vasicek model that describes the short rate as a mean reverting, arithmetic Brownian motion. The governing stochastic differential equation (SDE) for the short rate, , in the Vasicek model is given by:
In the above equation, is a Weiner process. The other terms are: is the speed of mean reversion and , a constant, is the long term average mean towards with the short rate moves. This long term average is the key in estimating the half-life of the rate (by definition, half-life is defined as the time it takes for the interest rate to move half the distance towards its long term average)
Thus, in analyzing the expected value of the rate we can ignore the impact of this Weiner process on the rate. The expected value is given by
Therefore, as far as the deterministic part of the above SDE is concerned we get the following ordinary differential equation (ODE)
If we are analyzing the rate between the time interval and the we can integrate the above equation from and , where, is the expected value of the rate at time, , then we get
Since, is a constant, we can solve the above integral between the given limits.
We know that: .
This gives us,
As noted above, half-life of the interest rate is the time it takes for the interest rate to move half the distance towards its long term average. Therefore, mathematically, half-life of the interest rate is defined as: .
Thus, if is the half-life of the rate then we have
From the above, we get the expression for half-life of the short rate
The above expression for the half-life of interest rate is extremely simply and highly intuitive. Further, it allows us to compute the speed of mean reversion of rates which is not at all an intuitive concept. If we know the half-life of rates, or if we can make a reasonable estimate of it analyzing historical data, then we can easily deduce the speed of mean reversion of rates.
Marco Antonio Guimaraes Dias, Half-Life in Mean Reversion Processes, Pontificia Universidade Catolica Do Rio de Janeiro, 2005
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