What Is the Cox-Ingersoll-Ross Model (CIR)?

The Cox-Ingersoll-Ross model (CIR) is a mathematical formula used to model interest rate movements. It can also be used to calculate prices for bonds. The CIR model is an example of a "one-factor model" because it describes interest movements as driven by a sole source of market risk. It is used as a method to forecast interest rates and is based on a stochastic differential equation.

The Cox-Ingersoll-Ross (CIR) model was developed in 1985 by John C. Cox, Jonathan E. Ingersoll, and Stephen A. Ross as an offshoot of the Vasicek Interest Rate model.

Key Takeaways

  • The CIR is used to forecast interest rates and in bond pricing models.
  • The CIR is a one-factor equilibrium model that uses a square-root diffusion process to ensure that the calculated interest rates are always non-negative.
  • The CIR model was developed in 1985 by John C. Cox, Jonathan E. Ingersoll, and Stephen A. Ross as an offshoot of the Vasicek Interest Rate model.

Understanding the CIR Model

The Cox-Ingersoll-Ross model determines interest rate movements as a product of current volatility, the mean rate, and spreads. Then, it introduces a market risk element. The square root element does not allow for negative rates and the model assumes mean reversion towards a long-term normal interest rate level. The Cox-Ingersoll-Ross model is often used in the valuation of interest rate derivatives.

An interest rate model is, essentially, a probabilistic description of how interest rates can change over time. Analysts using expectation theory take the information acquired from short-term interest rate models in order to more accurately forecast long-term rates. Investors use this information on the change in short- and long-term interest rates to protect themselves from risk and market volatility.

CIR Model Formula

The equation for the CIR model is expressed as follows:

d r t = a ( b r t ) d t + σ r t d W t where: r t = Instantaneous interest rate at time  t a = Rate of mean reversion b = Mean of the interest rate W t = Wiener process (random variable modeling the market risk factor) σ = Standard deviation of the interest rate (measure of volatility) \begin{aligned}&dr_{t}=a(b-r_{t})\,dt+\sigma {\sqrt {r_{t}}}\,dW_{t} \\&\textbf{where:} \\&rt = \text{Instantaneous interest rate at time } t \\&a = \text{Rate of mean reversion} \\&b = \text{Mean of the interest rate} \\&W_t = \text{Wiener process (random variable} \\&\text{modeling the market risk factor)} \\&\sigma = \text{Standard deviation of the interest rate} \\&\text{(measure of volatility)} \\\end{aligned} drt=a(brt)dt+σrtdWtwhere:rt=Instantaneous interest rate at time ta=Rate of mean reversionb=Mean of the interest rateWt=Wiener process (random variablemodeling the market risk factor)σ=Standard deviation of the interest rate(measure of volatility)


rt = the instantaneous interest rate at time t

a = rate of mean reversion

b = mean of the interest rate

Wt = Wiener process (random variable modeling the market risk factor)

 sigma = the standard deviation of the interest rate (a meas

The Difference Between the CIR and the Vasicek Interest Rate Model

Like the Cox-Ingersoll-Ross model, the Vasicek model is also a one-factor modeling method. However, the Vasicek model allows for negative interest rates as it does not include a square root component.

It was long thought that the inability of the model to produce negative rates was a big advantage of the Cox-Ingersoll-Ross model over the Vasicek model, but in recent years as many European central banks have introduced negative rates this stance has been rethought.

Limitations of Using the CIR Model

While interest rate models like the CIR model are an important tool for financial companies trying to manage risk and price complicated financial products, actually implementing these models can be quite difficult. The CIR model, in particular, is very sensitive to the parameters chosen by the analyst. Thus, during a period of low volatility, the CIR can be an incredibly useful and accurate model. However, if the model is used to predict interest rates during a timeframe in which volatility extends beyond the parameters chosen by the researcher, the CIR is limited in its scope and reliability.