Fixed Income Securities - Essay Example

Introduction The term structure of fixed income securities indicates the relationship between different bonds having different maturities. The term structure is usually depicted through a yield curve which indicates as the relationship between the interest rates on the fixed income security against their terms. The longer the term of the interest rates the higher the interest rates will be as in comparison to short term interest rates which are mostly lower than the long term interest rates. Yield curve analysis is considered as one of the most valuable bond valuation techniques and is heavily used for measuring the future expectations of the movement of the interest rates. The market participants usually consider the risk free investments such as government treasuries as well as other securities as bench mark to measure and compare the term structure of the different fixed income securities. Depending upon the nature of the relationship, different shapes of the term structure can be obtained in order to analyze how the interest rates are behaving against their maturities and how the market may respond to the future changes. Though above description may seem easy to follow however, determining the term structure exactly or with relative accuracy is really difficult. Different models have been developed to understand the term structure in better way and as such efforts by Vasicek and Cox- Ingersoll-Ross seem to be two of the most important theoretical advances ever made in this direction. This paper is divided into two parts, in first part, a detailed description of the Term Structure models of Vasicek and Cox- Ingersoll-Ross will be discussed along with their benefits whereas in second part of this essay, an analysis will be provided as to whether both the models can empirically be supported or not and which model is more suitable for empirical testing. Vasicke’s Model Vasicke’s model is one of the earliest term structure models and is probably the first serious effort towards studying the term structure models from an academic point of view. Vasicke’s model is based on two basic assumptions while dealing with the term structure of the interest rates. First, it assumes that the whole term structure depends upon the given interest rates for a very short period of time. This interest rate is considered as the spot rate and as such the spot rates are considered as mean reversing in nature.(Beninniga & Czaczkes,2000). One of the most important characteristics of the Vasicke’s model is the fact that it is a one factor model and is a yield based model. This model is also based on the assumption that the short interest rates are normally distributed.(Choudhry,2005). What is also however, critical to note that this model is really practical and is in use by a large number of practitioners who prefer to use this mode in place of other more sophisticated models because of its practical tractability? What is also important to note that this model does not provide arbitrage free environment in relation with the actual prices of the bonds within the market? One of the major disadvantages of this model, however is the assumption of the negative interest rates as under this model, it is possible to experience negative interest rates which are technically impossible in the actual bond market. However, this weakness of the model is overcome through the practical tractability offered by this model as it has been argued that this model is relatively easy to use and provide quick results to analyze the term structure of interest rates to value different fixed income securities. It is also important to note that under this model, the volatility of the interest rates is constant for short term interest rates however; this weakness was overcome by the CIR which assumes that the volatility increase as the short term interest rates increase. This, therefore also means that the volatility assumptions under this model are relatively weak. Further, the original model of the Vasicke’s model is the fact that it failed to include the different time dependent variables which were later included and modified by Hull & White. The non-inclusion of time dependent variables is also because the model assumed time intervals at the relatively very short period. Therefore, time dependent variables are not properly integrated into the model.(Wilmott, Howison, Dewynne,1995). Cox-Ingersoll-Ross Model Cox-Ingersoll-Ross Model is one of the efforts to correct different weaknesses of the Vasicke’s model specially the fact that the Vasicke’s model was predicting negative interest rates and as such this was not considered as a plausible possibility under real market conditions. Under the assumptions of mathematical finance, Cox-Ingersoll-Ross or CIR as it is called maps the behavior of the interest rates as this is also considered as the one factor model however; later variations also include the multi-factor model assumptions also. CIR is also considered as a diffusion process and such utilized for the purpose of studying the behavior of the interest rates with integration of the possibility of having positive interest rates. What is also however, critical to note that this model provides complete integration of interest rate risk premiums charged by different investors according to the perception of the market as well as complete characterization of the different expectations of the market participants? In this regard, this model resembles with that of the Vasicke as both the models are not only one factor models but also take many assumptions which are similar in nature except the fact that the CIR does not predict negative interest rates. What is also important for this model is the fact that it provides a link between the intertemporal asset pricing theory as well as the term structure of the interest rates. Thus this model attempts to link the two and then by integrating different risk premia and by negating the negative interest rates, this model predicts the behavior of the interest rates against their maturities. Subsequent variations of this model, however, also allowed the integration of the two factor assumptions and this model seems to be robust enough to account for the multiple factors. Some research studies also confirmed that the CIR model works best in three factors model environment and as such results predicted by this model seems to be empirically more plausible as well as can be tested with a relative degree of accuracy as well as practical adoptability.(Chen & Scott, 2002). Empirical Validity Both the models are considered as practically adaptable as well as empirically provable as over the period of time, both the models seem to have proven enough about their practical implementation. As discussed above that most of the practitioners often prefer to use Vasicek’s model and as such are getting enormous results which are proving important as for as the investors are concerned. However, from the academic literature point of view, Vasicek’s model carries one of the most critical weaknesses which are that of the negative interest rates. This model has the capability to give negative interest rates therefore considering this, the asset values may inflate to the point of infinity. As such the results produced by this model therefore may not be entirely plausible or even sometime misleading too. Further, Vasicek’s model is one factor model with a very little effort being made to create variations of the model to include multi factor possibilities. This weakness, as discussed above, was overcome by the CIR. CIR, however, also seems to have certain weaknesses when it comes to the regime changes as the assumption of short interest rates as well as short time intervals tend to make it more difficult for the regime change under the CIR. However, this weakness was later overcome by other variations made to the model specially by Zhu and Bansal. Zhu and Bansal proposed certain other variations to the model in a bid to make it more plausible for empirical testing.(Bansal, Zhu, 2002). From the available literature on the topic, it is therefore important to note that no one method is empirically more valid though as CIR is a later effort and has different variations therefore it is considered as empirically more valid model as in comparison to Vasicek which seems to be losing its ground mostly based on the assumption of negative interest rates. Empirically results under this model may therefore, skew to a larger degree because this model assumes negative interest rates and as such asset values may not be correctly projected. Conclusion Vasicek as well as CIR models are initial efforts to determine the term structure of interest rates however both the models have not been entirely successful in accurately predicting the behavior of the interest rates under the normal circumstances. Both the models seem to have some empirical validity however, CIR is considered as more empirically valid model because later variations of this model tend to have overcome most of its weaknesses and as such have been successful in integrating more mathematical variation to the model so that a larger and more comprehensive behavior prediction can be made. Reference list   1. Choudhry,Moorad (2005). Fixed-income securities and derivatives handbook: analysis and valuation. New York: Bloomberg Press. 2. Paul Wilmott, Sam Howison, Jeff Dewynne (1995). he mathematics of financial derivatives: a student introduction. Cambrige: Cambridge Universiy Press. 3. REN-RAW CHEN & LOUIS SCOTT (2002). Multi-Factor Cox-Ingersoll-Ross Models of the Term Structure: Estimates and Tests From a Kalman Filter Model. http://www.bnet.fordham.edu [on line]. 0, [Accessed 13 April 2009], p.1-36. Available from World Wide Web: . 4. Simon Benninga, Benjamin Czaczkes (2000). Financial modeling. MIT: MIT Press.[Assessed 13 April 2009]. Available from World Wide Web: . Read More