Models for the Term Structure of Interest Rates
Name: Careve E. Williams
Student Number: 169043847
The Eastern Caribbean, where I reside, is plagued by hurricanes originating off the coast of Africa, every year. The paths of these hurricanes are very unpredictable and therefore the weather officials use models, generated by computers, to predict the directions that they travel. These models are different and predict varied paths. Similarly, financiers use different models, also generated by computers, to determine the most appropriate term structure of Interest rates to apply to the financial world.
Models can have various definitions depending on the area of study. However, for the purpose of this paper the definition of “models” as cited by Hickman (2013) is “a set of verifiable mathematical relationships or logical procedures which is used to represent observed, measurable real-world phenomena, to communicate alternative hypotheses about the cause of the phenomena, and to predict future behaviour of the phenomena for the purpose of decision making.” (Jewell 1980). A model uses data gathered related to the purpose of the model to give the user a sense of what may happen in an upcoming event or what a particular situation may look like depending on what parameters are used. Additionally, a model is used to examine the effects of different input model parameters and how sensitive the parameters used are. It can be deterministic or stochastic i.e. non-random or random. It is an approximation using some form of computer program, stimulated or not, and plays an important role in decision making.

The Term Structure of Interest Rates according to Keown, Martin & Petty (2011), refers to “the relationship between a debt security’s rate of return and the length of time until the debt matures.” In other words, it shows the relationship between interest rates or yield to maturity and time to maturity. This is often depicted by a graph known as the Yield Curve with term to maturity shown on the y axis and interest rate on the x axis. The graph in figure 1 below, shows how fluctuation of the yield to maturity behaves in relation to changes in the interest rate over the short and long run.
132842041026010210963175Yield to Maturity
00Yield to Maturity
1543792288216
1983179150857Time to Maturity
0Time to Maturity
133003567491
Figure 1
Over the years numerous models have been used to model the term structure of interest rates. These models should possess the following characteristics: any model of term structure interest rates should be arbitrage free, interest rates should be positive, interest rates should also be mean-reverting, models should fit historical data and should calibrated easily to current data, and they should be flexible, tractable in terms of calculations and realistic. No one model has succeeded in implementing all of these characteristics in the same model at the same time. The characteristics of any given model really depend on the purpose and use of the model. These characteristics will be discussed as it relates to each model that will be referred to in this paper.

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Of the numerous models that have been used to model the Term Structure of Interest Rates, this paper will concentrate on the one factor interest rate models such as Vasicek, Cox-Ingersoll-Ross and Hull-White models. One-factor interest rate models are often expressed in terms of stochastic differential equations for the instantaneous interest rates or short rates. They use a single stochastic differential equation for the short rate and they are also known as short term interest rate models or continuous interest rate models. (Stehlíková 2008).

The first model to be discussed is the Vasicek model. It is one of the earlier models used to model term structure of interest rates. It is used to describe the development of interest rates and the assessment of financial derivatives. According to CT8 (x) (4), it is one of the simplest one-factor models. It illustrates the values of interest rates and is based on the basic risk-free interest rate process:
drt= ??-rtdt+?dW(t)where ?,? and ? are strictly positive constants and ? represents the risk-neutral long-term risk-free rate; ? represents the rate at which r(t) reverts back to this long-term rate and ? represents the diffusion parameter of the short-term interest rates also known as the volatility term. W(t) is the Wiener process i.e. a Markov stochastic process which refers to changes in value over small time periods under the risk-neutral measure Q. It is a special version of the Ornstein-Uhlenbeck process which is a diffusion process. As a result, the diffusion in the model is constant and the model has the characteristic that it is mean reverting. Mean-reversion, which is one of the most important characteristics of a term structure of interest rate model, refers to a tendency of asset prices to return to a trend path (Balvers, Wu & Gilliland 2000). It assumes that the price of a stock or assets will generally in the long run move towards the long run mean. So, when interest rates are high persons tend to stop borrowing monies. Spending and investing decrease and, as a result, the economy slows down. In order for interest rates to be at equilibrium, the rate decreases until supply meet demand. But when interest rates are low people borrow more monies, spend and invest more which causes the economy to improve. As stated before, the economy should be at equilibrium, so the interest rates will go down until supply meets demand. The Vasicek model was the first model to have the mean reverting characteristic, and also has a fixed level of randomness.
Another characteristic that this model holds is that it is relatively simple to implement as the model can be of no use if no one can use or understand it. Additionally, it is time homogenous, this means that rather than being dependant on what present time (t) is, the future dynamics of r(t) is dependent on the current value of r(t). Also, it assumes the characteristic that there are no arbitrage opportunities in the market and it has constant volatility. A major limitation of this model is that, in order to keep the model simple for persons to use and understand, it allows for negative interest rates. This is a characteristic that one generally does not want. Without positive interest rates, banks and other investors would have no desire or motivation to invest as they would be losing monies if they invested at a negative rate of return. Although this does happen, it mostly occurs in a situation where probabilities are used, and the probability of a negative return is low. The model characteristic conditions that the probability of a negative interest rate be non-negative with a positive probability. To account for this failure, an extension of this model was implemented.
This brings us to the second model to be discussed the Cox-Ingersoll-Ross model.
This model is another one factor stochastic model and is an extension of the Vasicek model discussed before. It can be illustrated by using the following formula:
drt= ??-rtdt+?r(t)dW(t).

where ?,? and ? are strictly positive constants and W(t) is the Wiener process under the risk-neutral measure Q as in the above model. Even though this model was not a modification but an addition of the Vasicek model, we can see here that the only difference between the Vasicek model and Cox-Ingersoll-Ross model in terms of the formula used is the volatility term used. In the first model it was ? and now it is ?r(t). The change of the volatility term accounts for the none negative interest rates that the first model had as it decreases the volatility when interest is low of the short rate and as a result it is always positive and increases when r increases. It describes the valuation of interest rates. This model is also time homogenous and stochastic. Both the Vasicek and Cox-Ingersoll-Ross models are known as equilibrium models since the interest rate does not move too far from the mean and always returns there. None of these two models will perfectly fit the term structure of interest rates since too many different equations will have to be solved and there are too many unknowns or random variables involved. The Cox-Ingersoll-Ross model has the other characterises of the Vasicek model including that it is arbitrage free and has mean reverting properties. The mean reversion property is one of the most important characteristics of both models. When compared to the Vasicek model’s normal distribution, the Cox-Ingersoll-Ross model distribution is more complicated. The equation for the Cox-Ingersoll-Ross model is not tractable in terms of calculations and thus it may be more difficult to implement than the Vasicek model. Even if one was able to solve it, the results may be inconsistent and may cause arbitrage opportunities to arise. This would contradict the characteristics of the model of being arbitrage free and ease of calculations.
According to CT8 (x) (4), since both models are time homogenous, the bond prices at time t are dependent on the risk-free rate at time t and at the maturity rate. Even though the formula is relatively simple it lacks flexibility when it comes to pricing bonds and options. These models rely on the specification of the stochastic process for the underlying risk-free interest rate. Thus, it leads to an explicit equation for the price of the zero-coupon bond. A zero-coupon bond or discount bond is a bond that “provides only a bond’s face value, and it will be sold at a discount to the face value in order to provide a return and compensate for the risks related to holding it.” (Pfau 2017). The martingale and state-price deflator approaches are used to calculate the pricing of zero-coupon bonds and interest rate derivatives for the Vasicek, Cox-Ingersoll-Ross and Hull-White models of the risk-free rate of interest. For the purposes of this paper the martingale approach will be used.

In the Vasicek model, the pricing of zero-coupon bonds can be formulated as follows
PVt, T= eaT-t-bT-tr(t)and is considered to be simple and controllable in bond and option pricing. The price of the zero-coupon bond varies with term for a range of values of . The formula for the Vasicek model is identical to the one for the Cox-Ingersoll-Ross model. The difference comes with how a (T) and b (T) are defined. According to Ordura, Lin and Larochelle (2015), a key advance of these models is that the bond prices at any maturity will have an analytical form (i.e. there is an explicit formula to define zero coupon bond prices at any time t) from which the yield curve can be derived. These models are instantaneous spot rate or spot rate based, that is, the rate that any entity can borrow money for an infinitely small period of time. The structures are also simplistic and could produce unintended term structures.
The final model to be discussed is the Hull-White Model which is also a one factor model. It is an extension of both the Vasicek and the Cox-Ingersoll-Ross models and is formulated as follows:
drt= ??(t)-rtdt+?dW(t)where ? and ? are positive constants whereas ?(t) is deterministic functions of t. All of the parameters are time-inhomogeneous which mean they are time dependent. However, if there is a quick change with time in the actual underlying risk-free rate, routine recalculation may be mandatory. This may cause the model to become time homogeneous. It is like the other models exhibiting mean-reverting behaviour. This model is arbitrage free which means that the models are constant and that one should not be able to use one model to profit off of another. As a result, it is known as a no arbitrage model and adjusts the market price of the principal assets to the price that is provided by the model. Unlike the previous two models, this model is able to fit the term structure of interest rates by using time changing parameters. Thus, this model is flexible so that if any changes or unexpected circumstances were to arise, the model would be able to adapt. There is also ease of calibration to current market data. It has a normal distribution which may cause it to have negative interest rates at times but the chance of it having a negative probability is less than some other models not discussed in this discourse. The White-Hull model also does not produce realistic dynamics as it cannot realistically produce features similar to those used in the past with reasonable probability.

For the Hull-White model, the price of bonds formula is given as:
PHWt,T= eat,T-bt,Tr(t)where PHWt,T is the price at time t of a zero-coupon bond maturing at time T, a(t,T) and b(t,T) are functions of t and T, and r is the short rate at time t. The function a(t,T) is determined from the initial values of discount bonds P(0,T). Bonds and certain derivative contracts are calculated with ease.

Since all the models discussed were one factor model which means that there is only one source of randomness accounted for in each model; this can sometimes cause some limitations to arise which we will now discuss.

According to Ahlgrim (2001), one factor term structures models may place too much rigidity on the yield curve movements. Specifically, the short-term rate exclusively determines all other points on the yield curve, that is, long term yields are a one-to one mapping of the short-term rate. All the models discussed incorporate one stochastic factor and all bond prices are subsequently related to the level of the short-term rate. As a result, bond yields of all maturities are perfectly correlated, severely constraining implied yield curve movements. This can be fixed by using two factor models which will not be discussed here.

According to CT8 (x) (5), research on the historical interest-rate data indicates that changes in the prices of bonds with different terms are not perfectly interconnected as would be expected if one-factor models were accurate. Occasionally one can see that these are negatively correlated since short dated bonds fall in price while long-dated bonds rise in price. Three factors, rather than one, are required to capture most of the randomness in bonds of different durations according to research.

When one looks at the long run of historical data, it can be seen that there has been continued periods of both high and low interest rates, each with periods of both high and low volatility. This behaviour would also be difficult to capture in a one-factor model. Therefore, derivative contracts which are more complex than the standard European options may need more multifaceted models to deal effectively with them.
The term structure of interest rate models plays an important role in financial analysis as it stimulates the movement of interest over time. These models can be applied to insurance/actuarial science since in life insurance or endowments customers are offered a guaranteed interest rate. Since customers are becoming more aware of the financial market and the rates that are offered, insurance companies have to deal with the ultimate effects of the volatility of life insurers as relates to the changing market. Insurance liabilities are affected by interest rates because many products contain a number of fixed interest rate options. The cost of providing these guarantees is related to the movements in the interest rate over the policies lifetime. Insurers can set aside an appropriate surplus to protect against adverse circumstances. They may also use the projected distribution of surplus when defining capital requirements. If the interest rate given is too high, this can lead to the company becoming bankrupt, since they may have to dip into their own funds to make up for the difference. Also, the fair value of fixed liabilities, with no option characteristics, must accurately reflect the current observed term-structure of interest rates. According to Ordura, Lin and Larochelle (2015), policyholder’s behaviour is commonly tied to the projected interest rates, which causes the relevance of real-world interest rate models. Actuaries have to understand the complexity and specifics of the interest rate model used, given the intended purpose as parameters may have to be calibrated to obtain suitable historical data.
In conclusion, when modelling term structure of interest rates many different models can be used. The three discussed in this paper were one factor models: Vasicek, Cox-Ingersoll-Ross and Hull-White model. The choice depends on the purpose of the model. The purpose will determine what characteristics that model has, as all models do not have all the characteristics at any given time. As a result, most models have some limitations to them. Models can be used in calculating bond prices as well as interest-rate derivates. These models can be used in the actuarial and insurance industries to predict interest rates for polices with fixed rates. Additionally, they can serve to prevent insurance companies from having insolvency problems if the interest rate given to policy holders is higher than what the company got on investments.

References
Ahlgrim, K. C. (2001) “The Effects of Multifactor Term Structure Models on the Valuation of Insurance,” Ph.D. Retrieved from https://www.casact.org/education/oncourses/module4/ahlgrim.pdfBalvers, R., Wu, Y., Gilliland, E. (2000)
Mean Reversion across National Stock Markets and Parametric Contrarian Investment Strategies. The Journal of Finance Vol. LV. No. 2. Retrieved from
http://andromeda.rutgers.edu/~yangruwu/mr_jf.pdf
CT8 (x) (4) – Actuarial CT. Particular Models for the Term Structure of Interest Rates. The Vasicek Model. Retrieved from https://wiki.lamp.le.ac.uk/actuarial/index.php?title=CT8_(x)_(4);printable=yesCT8 (x) (4) – Actuarial CT. Particular Models for the Term Structure of Interest Rates. The Cox-Ingersoll—Ross Model. Retrieved from
https://wiki.lamp.le.ac.uk/actuarial/index.php?title=CT8_(x)_(4);printable=yesCT8 (x) (4) – Actuarial CT. Particular Models for the Term Structure of Interest Rates. The Hull – White model. Retrieved from
https://wiki.lamp.le.ac.uk/actuarial/index.php?title=CT8_(x)_(4);printable=yesCT8 (x) (5) – Actuarial CT. Limitations of One-factor models. Retrieved from notes given 8/16/2018 https://wiki.lamp.le.ac.uk/actuarial/index.php?title=CT8_(x)_(5);printable=yesHickman, J. C. (2013). Introduction to Actuarial Modeling. Retrieved from
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Keown, A. J., Martin, J. D., William Petty, J. (2011) Foundations of Finance (7th ed). USA: Pearson Education, Inc.

Orduna, F., Lin, M. and Larochelle, J. (2015). The Modeling Platform Real World Interest Rates Models and Current Practices. Retrieved from https://www.soa.org/Library/Newsletters/The-Modeling-Platform/2015/july/modeling-platform-2015-vol-1-iss-1.pdf 