مقایسه روش های برآورد منحنی بازده با استفاده از داده UK

I compare different methods of estimating the term structure of interest rates on a daily UK treasury bill and gilt data that spans the period from January 1995 to January 1999. In-sample and out-of-sample statistics reveal the superior pricing ability of certain methods characterised by an exponential functional form. In addition to these standard goodness of fit statistics, model performance is judged in terms of two trading strategies based on model residuals. Both strategies reveal that parsimonious representations of the term structure perform better than their spline counterparts characterised by a linear functional form. This is valid even when abnormal returns are adjusted for market movements. Linear splines overfit the data and are likely to give misleading results.

An important “tool” in the development and testing of financial theory is the term structure of interest or forward rates. This relationship between rates and term to maturity has proved to be critical to policy makers and to market practitioners. In particular, forward rates may serve as indicators of monetary policy and as inputs to a pricing model. Indeed, the examination of the term structure theories empowered by theory of contingent claims has led to the derivation of term structure models characterised by the absence of arbitrage. There are two approaches1 to term structure modelling. The first approach is the “term structure consistent” approach pioneered by Ho and Lee (1986) and Heath et al. (1992). Recently, Fisher and Gilles (1998) have constructed term structure models in the spirit of Heath et al. (1992) that are consistent with the unbiased expectation hypothesis. This approach works only if an estimate of the initial forward curve is inputted to the Heath et al. (1992)2 framework. Under such conditions, the pricing and hedging of all types of interest rate claims is possible and consistent with an initial estimate of the forward or spot yield curve. The second approach identifies a state variable with every yield of maturity τ of zero coupon bonds. This approach attributed to Duffie and Kan (1996) constructs a multifactor model of the term structure in which the yields estimated using a term structure estimation method are the state variables. Although, the idea 3 of using yields as the state variables captures almost all of the dynamics of the term structure, Steeley (1997) conducts a principal component analysis on the Bank of England's term structure estimates to find that at least two factors 4 are enough to capture almost all of the dynamics of the UK term structure. In a different study, Elton et al. (1990) employ McCulloch's (1990) term structure estimates to identify suitable proxies for the unobserved state variables that drive the US term structure of interest rates. Furthermore, Elton and Green (1998) adopt a term structure estimation method, which they augment with tax and liquidity variables 5 to account for the deviations between the actual and the model bond prices. Despite the widespread use of the term structure of interest rates, a fundamental issue that other researchers seem to ignore is which term structure estimation method should be used in the first place to imply these estimates. Chambers et al. (1984) point out that the use of linear least squares may result in a singular matrix arising from columns that are perfectly collinear. This problem is more acute with longer maturities. An alternative problem arises when the estimation method employs cubic splines to fit the term structure of forward rates. Occasionally, such an approach may drift off to negative values. Another crucial issue in the estimation of the term structure lies in the use of an over-parameterised term structure estimation method, which raises concerns about overfitting rather than genuinely fitting the bond dataset. Here, I am interested in a term structure estimation method that not only fits the most salient features of the historical bond dataset but also recognises the most stable relationships between the parameters that are useful for out of sample pricing. I also seek to recognise those over-parameterised term structure estimation methods. These methods fit accidental or random features of the bond data that will not recur and are of no use in out of sample pricing. I investigate two mainstream approaches of estimating the term structure of interest rates, a parsimonious representation defined by an exponential decay term and a spline representation classified into a parametric and non-parametric splines. The former approach is developed by Nelson and Siegel (1987) and extended by Svensson (1994). The parametric cubic spline is introduced to finance by McCulloch, 1971 and McCulloch, 1975 and the non-parametric splines are developed by Adams and van Deventer (1994), Fisher et al. (1995), Tanggaard (1995) and Waggoner (1997). Here, I concentrate on seven popular term structure estimation techniques.6 The Nelson and Siegel and Svensson functions, the McCulloch cubic spline, three non-parametric splines and a non-parametric spline characterised by a maturity dependent penalty. I refer to the three non-parametric splines as linear, exponential and integrated B-splines. I estimate each method on daily UK government bond data spanning a period of four years, from January 1995 to January 1999. Hence, each day I observe seven different estimates of the term structure of interest rates. Second, I choose the “best” function in terms of its ability to replicate market prices of bonds, by performing in-sample and out-of-sample tests. Specifically, I compare estimated term structures by the differences between their in-sample and out-of-sample mean absolute error statistics. Earlier work by Buono et al. (1992) examined the ability of three term structure estimation techniques, namely the ordinary least squares method, the exponential polynomial and a recursive technique, to accurately estimate the forward curve. Unlike this paper, which uses real market data, Buono et al. based their comparison on Monte Carlo simulations. This work resembles the work of Bliss (1997), who compared four alternate term structure estimation techniques on the basis of their out of sample performance and on the randomness of their fitted price errors. This work departures from the work of Buono et al. (1992) and Bliss (1997) by applying the trading and filter rule tests of Sercu and Wu (1997). I construct a trading strategy by buying (selling) undervalued (overvalued) bonds according to a given term structure estimation technique and compute the abnormal returns (ARs) as defined by three different benchmarks. These alternative definitions of ARs filter out the general market movements and seek out the mere noise in returns. In this way, I identify potential misspecification or mis-estimation of a given term structure estimation technique and isolate “genuine” mispricing. I seek to identify the best model in terms of its ability to capture information in bond residuals and generate ARs.7 This study is the first to combine and apply this approach to the UK market. Overall, the performance of both parsimonious functions is favoured over their spline counterparts. Both the in-sample fit and the out-of-sample pricing ability of these techniques to replicate the existing term structure is considerably better than splines defined by linear functional form and marginally better than splines defined by an exponential functional form. In the trading rule test, I render a function superior to another competing alternative when this function produces ARs across all three benchmarks. I would then expect to see that these ARs do not persist. On the other hand, I characterise a function as misspecified or mis-estimated only when I observe ARs in one or two rather than in all three benchmarks. With this in mind, I identify the parsimonious specifications and the non-parametric spline with maturity dependent penalty as superior approaches to the competing alternatives of McCulloch's cubic spline, the linear, exponential and integrated B-splines. However, the performance of exponential and integrated B-splines is marginally close to the performance of the parsimonious methods. Finally, to confirm the results of the trading rule test, we conduct a filter rule test to reveal that the weighting scheme utilised by the trading rule test is not optimal. The following section describes the alternative methods of the term structure and deals with estimation method and analysis of in-sample and out-of-sample results. Section 3 presents the model specification tests and results and Section 4 concludes.

I compare different methods of estimating the term structure of interest rates on daily UK Treasury bill and gilt data. I examine the Nelson and Siegel and Svensson functions, McCulloch's cubic spline, the linear, exponential and integrated exponential B-splines and the VRP method, a total of seven methods. The major conclusions stemming from in-sample and out-of-sample analysis of residuals suggest that the parsimonious specifications and VRP method perform better than the linear spline counterparts. In terms of the out-of-sample performance, the non-linear B-splines and the VRP function produced a lower mean absolute pricing error than the Nelson and Siegel specification, but these functions are second best to Svensson. This suggests that the specification of the functional form of the model is important for pricing. To verify this conclusion, I conduct two further tests to investigate genuine pricing errors. The application of the contrarian trading rule indicates that the best term structure estimation techniques are the parsimonious ones. Of the spline specifications, the integrated B-spline and the VRP function produce equally good results. However, the implementation of the filter rule shows for all functions with the exception of the linear and exponential B-splines that increasing our stake in a bond on the basis of the initial mispricing understates profits.