The simulation of the term structure of interest rates is a common issue for ALM simulations and optimization as well as for fixed income portfolios market VaR. A first possibility is to select few interest rates and risk factors and correlate them. But it would be better to model several rates and simulate a "consistent" term structure of interest rates.

Under an ALM perspective, the issue is to explore what could happen with simple scenarios other than standard shifts of all interest rates or increasing the long-term and short-term spread. ALM simulations of the term structure interest rates provide the faculty of testing, bask-testing and stress-testing what would happen with a large number of simulations of the term structures of interest rates. In ALM, the target variables are the N11 and the economic value. Under a market VaR perspective the horizon is shorter and limited to current conditions, and the target variable is the P & L of the market portfolio. Any methodology relying on factor models, notably the delta-VaR methodology, requires simulating consistent term structure of interest rates.

Interest rate models used for pricing purposes are not so helpful for such simulations. The statistical methodology based on principal component analysis is adequate since a few principal components summarize fairly well the set of highly correlated interest rates^{[1]}.

This chapter explains simulations based on PCA and its applications to ALM policies and market VaR. Section 37.1 is a reminder of the basics of principal component analysis and properties of the factors, or principal components. Section 37.2 details the process of fitting a PCA model to interest rates. A fit of the model to Euro interest rates, using daily observations over a period of around 3.5 years until July 2007 is described. The results from this fit of the PCA model is used for simulations in Section 37.3. As usual in such a case, it is found that a very few factors explain the variance of interest rates. In general, three factors are sufficient and represent parallel shift of the curve, changes in steepness of the curve and a bump of the curve (if any). Section 37.3 uses the fit to conduct simulation of the entire curve by generating random normal standard values of the factors. The last two sections address applications to market VaR (Section 37.4) and ALM (Section 37.5). For market VaR, the simulations of the curve provide the distribution of the daily P & L using given portfolio sensitivities to some selected rates, and converting them into sensitivities to principal components that drive the interest rates. The same process is implemented for ALM simulations of the net interest income using sensitivities with a bank with significant mismatch risk.

[1] The reference paper is that of Frye [30]. Note that the Vasicek model presented in Chapter 13 allows simulations, but the version limited to a single factor does not allow comprehensive simulations on the long-end of the term structure.

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