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# Vasicek Interest Rate Model Definition, Formula, Other Models

## What Is the Vasicek Interest Rate Model?

The term Vasicek Interest Rate Model refers to a mathematical method of modeling the movement and evolution of interest rates. It is a single-factor short-rate model that is based on market risk. The Vasicek interest model is commonly used in economics to determine where interest rates will move in the future. Put simply, it estimates where interest rates will move in a given period of time and can be used to help analysts and investors figure out how the economy and investments will fare in the future.

### Key Takeaways

• The Vasicek Interest Rate Model is a single-factor short-rate model that predicts where interest rates will end up at the end of a given period of time.
• It outlines an interest rate's evolution as a factor composed of market risk, time, and equilibrium value.
• The model is often used in the valuation of interest rate futures and in solving for the price of various hard-to-value bonds.
• The Vasicek Model values the instantaneous interest rate using a specific formula.
• This model also accounts for negative interest rates.

## How the Vasicek Interest Rate Model Works

Predicting how interest rates evolve can be difficult. Investors and analysts have many tools available to help them figure out how they'll change over time in order to make well-informed decisions about how their investments and the economy. The Vasicek Interest Rate Model is among the models that can be used to help estimate where interest rates will go.

As noted above, the Vasicek Interest Rate model, which is commonly referred to as the Vasicek model, is a mathematical model used in financial economics to estimate potential pathways for future interest rate changes. As such, it's considered a stochastic model, which is a form of modeling that helps make investment decisions.

It outlines the movement of an interest rate as a factor composed of market risk, time, and equilibrium value. The rate tends to revert toward the mean of these factors over time. The model shows where interest rates will end up at the end of a given period of time by considering current market volatility, the long-run mean interest rate value, and a given market risk factor.

The Vasicek interest rate model values the instantaneous interest rate using the following equation:

\begin{aligned} &dr_t = a ( b - r^t ) dt + \sigma dW_t \\ &\textbf{where:} \\ &W = \text{Random market risk (represented by}\\ &\text{a Wiener process)} \\ &t = \text{Time period} \\ &a(b-r^t) = \text{Expected change in the interest rate} \\ &\text{at time } t \text{ (the drift factor)} \\ &a = \text{Speed of the reversion to the mean} \\ &b = \text{Long-term level of the mean} \\ &\sigma = \text{Volatility at time } t \\ \end{aligned}

The model specifies that the instantaneous interest rate follows the stochastic differential equation, where d refers to the derivative of the variable following it. In the absence of market shocks (i.e., when dWt = 0) the interest rate remains constant (rt = b). When rt < b, the drift factor becomes positive, which indicates that the interest rate will increase toward equilibrium.

The Vasicek model is often used in the valuation of interest rate futures and may also be used in solving for the price of various hard-to-value bonds.

## Special Considerations

As mentioned earlier, the Vasicek model is a one- or single-factor short rate model. A single-factor model is one that only recognizes one factor that affects market returns by accounting for interest rates. In this case, market risk is what affects interest rate changes.

This model also accounts for negative interest rates. Rates that dip below zero can help central bank authorities during times of economic uncertainty. Although negative rates aren't commonplace, they have been proven to help central banks manage their economies. For instance, Denmark's central banks lowered interest rates below zero in 2012. European banks followed two years later followed by the Bank of Japan (BOJ), which pushed its interest rate into negative territory in 2016.

## Vasicek Interest Rate Model vs. Other Models

The Vasicek Interest Rate Model isn't the only one-factor model that exists. The following are some of the other common models:

• Merton's Model: This model helps determine the level of a company's credit risk. Analysts and investors can use the Merton Model to find out how positioned the company is to fulfill its financial obligations.
• Cox-Ingersoll-Ross Model: This one-factor model also looks at how interest rates are expected to move in the future. The Cox-Ingersoll-Ross Model does so through current volatility, the mean rate, and spreads.
• Hull-While Model: The Hull-While Model assumes that volatility will be low when short-term interest rates are near the zero-mark. This is used to price interest rate derivatives.
Article Sources
Investopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our editorial policy.
1. World Economic Forum. "Negative interest rates: absolutely everything you need to know." Accessed Dec. 28, 2021.

2. CFI. "Short Rate Model." Accessed Dec. 28, 2021.

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