The present study deals with the analysis and mapping of Swiss franc interest rates. Interest rates depend on time and maturity, defining term structure of the interest rate curves (IRC). In the present study IRC are considered in a two-dimensional feature space - time and maturity. Geostatistical models and machine learning algorithms (multilayer perceptron and Support Vector Machines) were applied to produce interest rate maps. IR maps can be used for the visualisation and patterns perception purposes, to develop and to explore economical hypotheses, to produce dynamic asses-liability simulations and for the financial risk assessments. The feasibility of an application of interest rates mapping approach for the IRC forecasting is considered as well.
Deep Dive into Interest rates mapping.
The present study deals with the analysis and mapping of Swiss franc interest rates. Interest rates depend on time and maturity, defining term structure of the interest rate curves (IRC). In the present study IRC are considered in a two-dimensional feature space - time and maturity. Geostatistical models and machine learning algorithms (multilayer perceptron and Support Vector Machines) were applied to produce interest rate maps. IR maps can be used for the visualisation and patterns perception purposes, to develop and to explore economical hypotheses, to produce dynamic asses-liability simulations and for the financial risk assessments. The feasibility of an application of interest rates mapping approach for the IRC forecasting is considered as well.
The present study deals with an empirical analysis and mapping of Swiss franc (CHF) interest rates (IR). The complete empirical analysis includes comprehensive quantitative analysis and characterisation of daily behaviour of CHF interest rates from October 1998 to December 2005. A mapping part of the paper considers the application of spatial interpolation models for IR mapping in a feature space "maturity-date". In a more general setting interest rates can be considered as functional data (IR curves are formed by different maturities) having specific internal structures (term structure) and parameterised by econometric models.
The main issues related to IR mapping are the following: 1) empirical analysis of IR spatio-temporal patterns, 2) reconstruction and prediction of interest rate curves; 3) incorporation of economical/financial hypotheses into the IR prediction process; 4) development of what-if scenario for financial engineering and risk management. Some of the preliminary ideas elaborated in this study were firstly presented in (Kanevski et. al., 2003).
In general there are two principal approaches to make term-structure predictions (Diebold and Canlin, 2006): no-arbitrage models and 2) equilibrium models. The no-arbitrage models focuses on fitting the term structure at a point in time (one dimensional model depending on maturity) to ensure that no arbitrage possibilities exist. This is important for pricing derivatives. The equilibrium models focuses on modelling the dynamics of the intravenous rate using affine models after which rates at other maturities can be derived under various assumptions about risk premium. Detailed discussion along with corresponding references can be found in (Diebold and Canlin, 2006).
An important and interesting approach complementary to classical empirical analysis of interest rates time series was developed in (Di Matteo et. al., 2005;Di Matteo and Aste, 2002;Cajueiro and Tabak, 2007) where both traditional econophysics studies (power law distributions, etc.) and a coherent hierarchical structure of interest rates were considered in detail. An empirical quantitative analysis of multivariate interest rates time series and their increments (carried out but not presented in this paper) includes study of autocorrelations, cross-correlations, detrending fluctuation analysis, embedding, analysis of distribution of tails, etc. (Kanevski et. al., 2003;Cajueiro and Tabak, 2007;Mantegna and Stanley, 1999;Kantz and Schreiber, 2003).
The most important part of the current study deals with an IR mapping in a two dimensional feature space {maturity(months), time(date/days)} using spatial interpolation/extrapolation models (inverse distance weighting -IDW, geostatistical kriging models), nonlinear artificial neural network models (multilayer perceptron -MLP) and robust approaches based on recent developments in Statistical Learning Theory (Support Vector Regression) (Vapnik, 1999). Embedding of IR data into a two-dimensional space brings us to the application of spatial statistics and its modelling tools. Higher dimensional feature spaces can be considered and applied as well. Simple models, like linear and inverse distance weighting were used mainly for the comparison and visualisation purposes.
Evolution of CHF interest rates data is given in Figure 1 where temporal be- haviour of different maturities is presented. The IRCs are composed of LIBOR interest rates (maturities up to 1 year) and of swap interest rates (maturities from one year to 10 years). Such information is available on the specialised terminals like Reuters, Bloomberg, etc. and is usually provided for some fixed time intervals (daily, weekly, monthly) and for some definite maturities (in this research we use the following maturities: 1 week, 1, 2, 3, 6 and 9 months; 1, 2, 3, 4, 5, 7 and 10 years).
There are some important stylized facts that have to be considered when modelling IR curves (Diebold and Canlin, 2006): the average yield curve is increasing and concave; the yield curve assumes a variety of shapes through time, including upward sloping, downward sloping, humped, and inverted humped; IR dynamics is persistent, and spread dynamics is much less persistent; the short end of curve is more volatile than the long end; long rates are more persistent than short rates. There is a coherence in evolution of interest rate curves (see some typical examples in Figure 1).
In the present research the application of IR mapping is concentrated on: 1) visualisation and perception of complex data: IR data represented as maps are easier for the analysis and interpretation, 2) reconstruction of interest rates at any time and for any maturities by using spatial prediction models and 3) prediction of IR curves. The last problem can be considered from two different points of view: a) predictions without any a priori information (problem of extrapolation using historical IR data) and b) forecasting under some prior hypothe
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