# Cox-Ingersoll-Ross Model (CIR) Definition

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

The Cox-Ingersoll-Ross model (CIR) is a mathematical formula used to model interest rate movements. 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 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 and can be utilized, among other things, to calculate prices for bonds and value interest rate derivatives.

### 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 Cox-Ingersoll-Ross Model (CIR)

The CIR 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 toward a long-term normal interest rate level.

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:


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)

## The Cox-Ingersoll-Ross Model (CIR) vs. The Vasicek Interest Rate Model

Like the CIR 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.