# Vasicek Interest Rate Model: Predicting Future Interest Rates

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## 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](https://www.investopedia.com/terms/i/interestrate.asp). It is a single-factor short-rate model that is based on market risk. The Vasicek 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 model predicts interest rate movements using market risk, time, and long-term equilibrium interest rate values.
*   It is a stochastic model, showing how interest rates revert to a mean value over time.
*   The model is widely used to value interest rate futures and complex bonds.
*   The Vasicek model can account for the occurrence of negative interest rates.
*   Alternative interest rate models include the Cox-Ingersoll-Ross and Hull-White models.

## Understanding the Mechanics of the Vasicek Model

Predicting interest rate changes can be challenging. 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](https://www.investopedia.com/terms/e/economy.asp). 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](https://www.investopedia.com/terms/s/stochastic-modeling.asp), 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](https://www.investopedia.com/terms/m/marketrisk.asp), time, and equilibrium value. The rate usually moves back toward the average 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](https://www.investopedia.com/terms/v/volatility.asp), the long-run mean interest rate value, and a given market risk factor.

The Vasicek model uses this equation to value the instant interest rate:

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

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 _d_Wt = 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](https://www.investopedia.com/terms/e/equilibrium.asp).  
  

> **Important**
>
> 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.

## Important Insights on the Vasicek Model

As noted earlier, the Vasicek model is a 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](https://www.investopedia.com/terms/m/marketrisk.asp) is what affects interest rate changes.

This model also accounts for [negative interest rates](https://www.investopedia.com/terms/n/negative-interest-rate.asp). 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)](https://www.investopedia.com/terms/b/bankofjapan.asp), which pushed its interest rate into negative territory in 2016.  

## Comparing the Vasicek Model with Alternative Interest Rate Models

The Vasicek Model isn't the only single-factor model available. The following are some of the other common models:

*   **Merton's Model:** This model helps determine the level of a company's [credit risk](https://www.investopedia.com/terms/c/creditrisk.asp). Analysts and investors can use the [Merton Model](https://www.investopedia.com/terms/m/mertonmodel.asp) 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](https://www.investopedia.com/terms/c/cox-ingersoll-ross-model.asp) does so through current volatility, the mean rate, and [spreads](https://www.investopedia.com/terms/s/spread.asp).
*   **Hull-While Model:** The [Hull-While Model](https://www.investopedia.com/terms/h/hullwhite-model.asp) assumes that volatility will be low when short-term interest rates are near the zero-mark. This is used to price interest rate [derivatives](https://www.investopedia.com/terms/d/derivative.asp).  
    

## The Bottom Line

The Vasicek Interest Rate Model is a single-factor short-rate model that predicts future interest rate changes based on market risk, time, and equilibrium value. It's useful in the valuation of interest rate futures and in the pricing of complex bonds. The Vasicek Model can account for negative interest rates, which are significant in economic uncertainty.

It's important that investors understand different models to make informed financial decisions and manage investments effectively. Others include the Cox-Ingersoll-Ross Model and the Hull-White Model, each with their unique features and applications.

> **Article Sources**
>
> 1.  Vasicek, Oldrich A. "[An Equilibrium Characterization of the Term Structure](https://www.ressources-actuarielles.net/EXT/ISFA/1226.nsf/0/e11e4bc52747d1ddc125772400459afe/$FILE/Vasicek77.pdf)." _Journal of Financial Economics_, vol. 5, 1977, pp. 177-188.
>     
> 2.  Andrew Lesniewski. "[Interest Rate and Credit Models; 10. Term Structure Models: Short Rate Models](https://mfe.baruch.cuny.edu/wp-content/uploads/2019/12/IRC_Lecture10_2019.pdf)," Page 11. Baruch College, New York, Spring 2019.
>     
> 3.  World Economic Forum. "[Negative Interest Rates: Absolutely Everything You Need to Know](https://www.weforum.org/agenda/2016/11/negative-interest-rates-absolutely-everything-you-need-to-know/)."
>     
> 4.  Wang, Yu. "[Structural Credit Risk Modeling: Merton and Beyond](https://www.soa.org/globalassets/assets/Library/newsletters/risk-management-newsletter/2009/june/jrm-2009-iss16-wang.pdf)." _Society of Actuaries: Risk Management_, no. 16, June 2009, pp. 30-33.
>     
> 5.  Cox, John C., and Jonathan E. Ingersoll, Jr., Stephen A. Ross. "[A Theory of the Term Structure of Interest Rates](https://pages.stern.nyu.edu/~dbackus/BCZ/discrete_time/CIR_Econometrica_85.pdf)." _Econometrica_, vol. 53, no. 2, March 1985, pp. 385-407.
>     
> 6.  Hull, John, and Alan White. "[The General Hull-White Model and Super Calibration](https://w4.stern.nyu.edu/finance/docs/WP/2000/pdf/wpa00024.pdf)." _Joseph L. Rotman School of Management, University of Toronto_, August 2000, pp. 1-20.