Difficulties in Modeling Interest Rates An Honors Thesis (HONORS 499) By Ashley Frey Ball State University Muncie, IN December 2008 lJ~ C/i.-'{ r ~ cJ 7'h L ..:": II Abstract .;l,oo .FePj This paper discusses two types of basic interest rate models: the Vasicek model and Cox- Ingersoll-Ross model. The mathematics behind interest rate modeling is extremely complex, so this paper does not review proofs of the stochastic processes behind these models. Instead, the paper focuses on various techniques to estimate the parameters of the models. It also discusses the difficulties in creating and implementing these models using Microsoft Excel. Acknowledgements I would like to thank Professor Gary Dean for his willingness to help me on this thesis. He provided excellent guidance and insight on this project. Without his help, this paper would not have been possible. Table of Contents I. Introduction II. Vasicek Model III. Cox-Ingersoll-Ross Model IV. Conclusion V. Appendix A: A Proof of Cox-Ingersoll-Ross Parameter Estimation using Maximum Likelihood Estimate VI. Appendix B: Comparison of Historic Rates versus Vasicek and Cox-Ingersoll-Ross Simulated Rates Introduction Any investor knows that financial markets are volatile. For years researchers have studied the financial markets looking for trends in data that may help investors predict the future of these markets. Due to the extremely complex nature and plethora of variables to be considered, investors are still unable to predict the markets with much certainty; however, research hasn't been an entire waste of time. Many models have been created which are able to help investors manage their risk. These models have become especially important in recent years as the derivatives market has exploded. The introduction of these complex financial instruments has commanded the need for better models of prices of their underlying assets. Before issuing a new financial product, a company must understand the cash flows for the product. This often relies on scenario testing of stocks, bonds, interest rates, and other assets. Small changes in interest rates can result in millions of dollars of changes in asset values. For this reason, many models incorporate different scenarios to illustrate cash flows under various potential conditions. This paper examines the implementation of two interest rate models: the Vasicek model, created in 1977, and the Cox~Ingersoll~Ross model from 1985. These models are classified as "short rate models" opposed to "no-arbitrage models." These short rate models fit the parameters of the model to historical data, which may allow for arbitrage. The term structure of interest rates is then determined through the simulation of the model using the estimated parameters. While this paper does not examine the term structure of interest rates, this characteristic distinguishes the Vasicek and Cox~Ingersoll­ Ross short rate models from no arbitrage models and is important to point out. The no arbitrage models, in contrast, use the current term structure of interest rates, as implied by zero coupon bonds of various maturities, to develop a model that does not allow for arbitrage. Both types of models have their advantages and disadvantages which will be examined later in the paper through the simulation of the Vasicek and Cox-Ingersoll-Ross models. These models are also classified as one-factor models. This means that the predicted interest rate is a function of the previous interest rate. Although, one-factor models are elementary among the new multi-factor models, they provide a good introduction to the study of interest rates. The assumptions and details of these models are incorporated into other models; therefore, a study of the both the accomplishments and shortcomings of these models provides a strong background for understanding the dynamics of interest rates. Vasicek Model One of the most basic interest rate models was originally introduced by Vasicek in 1977. The interest rates modeled in this paper are based off of the daily interest rate reported by the 13-week Treasury Bill as stated on Yahoo! Finance. Due to extra risk factors that are not considered in the Vasicek model, only "risk-free" bonds were modeled. (Treasury bills, notes, and bonds issued by the United States government are assumed in most financial studies to be risk-free.) The general equation for this stochastic process is shown below: dr u(r)dt + a(r)dW where: dr change in interest rate u(r) drift rate dt = change in time o-(r) standard deviation of the interest rate 2 dW = random variable distributed normally with mean 0 and variance 0- This can be specialized for the Vasicek model. The equation as used in the modeling of interest rates using the Vasicek model is shown below: where: rk = interest rate in period k a rate of mean reversion b mean 0- standard deviation 2 Z = random variable distributed normally with mean 0 and variance 0- dt = change in time For ro, the interest rate used is the historic interest rate for the beginning day of the time period modeled as found on Yahoo! Finance. For periods I through n, the interest rate used is the interest rate from period k-l as generated by the equation. The change in time can be assumed to be 1 throughout the Vasicek models used in this paper. The random variable Z is particularly important in the development of the model. It accounts for Vasicek's idea that interest rates follow a stochastic process known as the Ornstein- Uhlenbeck process. While the details of this process are beyond the scope of this paper, some of the assumptions of this process are important underlying assumptions of this model. The first assumption states that changes in interest rates are continuous and therefore interest rates will pass through every intermediate value before reaching a new value. Obviously, in practice this is not true; interest rates can and do jump around. In the modeling observed in this paper, this assumption does not seem to strongly hinder the model. The second assumption states that the random term, Z in the model, is distributed normally with mean zero and variance 0'2 . This assumption of a normal distribution allows calculations to be performed much easier. Finally, the last assumption states that interest rates follow a process known as mean reversion. Mean reversion requires that if an interest rate becomes too high in relation to its historic mean, it will fall, and conversely if an interest rate becomes too low in relation to its historic mean, it will rise. This reversion is shown in the term b- rk , and the rate of reversion is measured by the parameter a. The larger a is, the faster the interest rate reverts back to its mean. It must be reiterated that these are not all the assumptions of the Omstein-Uhlenbeck process, but they are the assumptions that are easily described and most visible in the Vasicek model. F our measures for the estimation of parameters a, b, and 0' were taken, First, the standard deviation of the historic interest rates for the time period being modeled was used as a measure of 0' and similarly, the mean of the historic interest rates was used as a measure of b. The rate at which the interest reverted back to the mean, or a, was found by taking the average of the growth of the absolute value of b-rk • This model produced highly volatile results as illustrated in the graph of simulated interest rates versus historic daily interest rates for the one year time period of the I3-week Treasury Bill shown below: Figure 1: Daily Historic and Simulated Interest Rates for 13-Week Treasury Bill (Version One) Table I: Results from Simulated Interest Rates for 13-Week Treasury Bill (Version One) Statistical Results for Parameters for rk historic interest rate Statistical Results for rk a 0.3589 mean 1.7252 mean 1.7651 b 1.7252 standard standard C1 0.7716 0.7716 0.9646 deviation deviation It's important to note that the graph of the simulated rate is just one example of many possible outcomes. Because the Excel model uses a random number generator as part of its input, there are an infinite number of possibilities. Though the actual interest rate values produced will vary with each simulation, the mean reversion trend will not. The graph above clearly shows that the model over/under predicts the interest rate each time and then over/under corrects itself. This is due to a high estimation for the standard deviation term (J' . To fix this problem, a linear regression was performed on the historic data. If the Vasicek equation is rewritten in the form: YI = mx, +h+u where: y, rk+1 - r k Xi r k m a h ab u (J'Z a simple linear regression can be performed using only the historic interest rate from the observed time period. Parameters a and b can be solved for using the coefficient and intercept from the regression results, and the standard error of the y-value can be used as an estimate for (J' . Using these parameters in the model for the I3-week Treasury Bill from the same time period as previously simulated, gives the graph below: Figure 2: Daily Historic and Simulated Interest Rates for 13-Week Treasury Bill (Version Two) 4 3.5 3 25 ~ 2 I 1.5 0.5 0 -0.5 #' (\ #> ,# ,""ff' q)"<Y ,," Date Table 2: Results from Simulated Interest Rates for 13-Week Treasury Bill (Version Two) Statistical Results for Parameters for rk historic interest rate Statistical Results for rk a 0.0187 mean 1.7252 mean 2.1271 b 1.0787 standard standard C1 0.0187 0.7716 0.6159 deviation deviation This graph appears to be a much more accurate representation of a possible random outcome of the interest rate.