# Term structure of interest rates and fiscal policy

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## Abstract

As already mentioned in great detail in Chapter 1, the whole mathematical finance literature dealing with the term structure of interest rates has abstracted from studying the influence of fiscal policy on the term structure. Usually, arbitrary stochastic dynamics for a given set of ‘factors’^{1} driving the term structure were introduced by postulating specific forms of diffusion processes representing the evolution of these factors. As a consequence of these ad-hoc approaches, the influence of fiscal policy was either completely neglected or introduced in an ad-hoc way, at best ^{2}. One can, of course, argue that the influence of fiscal policy is already integrated in the factor dynamics used, namely in the way the dynamics are parametrized. This implies that these parameters are seemingly policy invariant. When one argues that the influence of fiscal policy is already embedded in such arbitrary parameters with almost no economic meaning, then estimating these models is subject to criticism similar to the well-known ‘Lucas Critique’ ^{3}. As the respective introductory remarks in Chapter 1 should have made clear, we think that fiscal policy does matter for term structure considerations. We will now discuss the term structure of interest rate and the influence of fiscal policy in the macroeconomic setting laid down in the previous chapter.

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## Notes

- 1.The choice of how many and which factors to use is usually based on an analysis of available market data using a statistical procedure called ‘Principal Component Analysis’. See f.e. Campbell/Lo/MacKinlay (1997), Chapter 6.4, or, for an application, Bühler (1996).Google Scholar
- 2.As examples for such an ad-hoc integration compare Babbs/Webber (1996) or Tice/Webber (1997).Google Scholar
- 3.See Lucas (1976).Google Scholar
- 4.It should be noted that the model in its present form does not lend itself to an analysis what impact anticipated fiscal policy changes may have on the term structure. This would require a different set-up of the household’s decision problem.Google Scholar
- 5.For a classification of second order partial differential equations see Vvedensky (1993).Google Scholar
- 6.See for example Fritz (1982), Evans (1994) or Vvedensky (1993).Google Scholar
- 7.Polyanin/Zaitsev (1996, p. 378) briefly discuss the functional form of such a solution.Google Scholar
- 8.Good references for these methods are Boyce/DiPrima (1997) or Courant/Hilbert (1993).Google Scholar
- 9.For a more detailed discussion of the connections between PDEs and the Feynman-Kac formula see, for example, Duffle (1992, Appendix E), Baxter/Rennie (1996, Chapter 5) or Neftci (1996, Chapter 16).Google Scholar
- 10.See Gaspar/Judd (1997) and Judd (1998) for an extensive discussion of various projection methods, especially their applications in economics.Google Scholar
- 11.The graph of y(r[inD, t) is called „yield curve“, a phrase that we will equivalently use sometimes for yield-to-return and term structure, too.Google Scholar
- 12.The upper value for ξ in the plot is arbitrarily chosen.Google Scholar
- 13.The reason why we chose 30 years as the maximal time to maturity is that on most financial markets one does usually not find private bonds having time to maturity of more than 30 years.Google Scholar
- 14.Note that the ‘hump’ can, at least in principle, be thought to be either a local maximum or a local minimum of the yield curve. In most of the literature, however, local maxima are meant when talking about hump-shaped forms of the term structure.Google Scholar
- 15.However, for a value of ξ = 100, for example, the hump occurs at a time to maturity less than three trading days so that the hump would quite likely no longer be detectable in data and one would observe an inverse term structure although it would theoretically still be hump-shaped.Google Scholar
- 16.Note that for tax rates somewhat smaller than 4% the set of feasible government expenditure ratios z becomes empty since the lower bound on z exceeds the upper bound on z.Google Scholar
- 17.t
_{max}corresponds to the maximal time-to-maturity one can find on bond markets, for example 30 years.Google Scholar - 18.Note that the projection direction choice (A4.1.11a) was made in order to eliminate the exponential term of the residual. Other projection directions not eliminating the exponential term in the residual would result in extremely complex integrals, most of them having no closed-form antiderivative. Additional numerical techniques had to be employed to obtain solutions.Google Scholar
- 19.In order to remain more general, we could also have taken an arbitrary positive constant for the relation between ζ and ξ instead of 1.5. The only necessary restriction is that negativity of the constant has to be ruled out for this could produce an exploding yield curve for any finite time-to-maturity value.Google Scholar
- 20.We exclude the second solution since it is usually negative and smaller than the first one. This is in contradiction to our assumption that is sufficiently high and positive.Google Scholar