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Fitting Yield Curves

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Fixed-Income Portfolio Analytics

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Abstract

The yield curve is unquestionably the central concept in the fixed-income world. It represents the relationship between investment yields and tenor for a given issuer at a given point in time. A wide range of important information is embedded in this relationship ranging from fundamental issues such as the time value of money, expected monetary policy actions, and inflationary expectations to more complicated, but equally important ideas such as risk premia, assessments of creditworthiness, and relative liquidity.

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Notes

  1. 1.

    The approach used is a multivariate adaptive regression spline. See Friedman [17], Bolder and Rubin [9], or Lewis and Stevens [21] for more detail.

  2. 2.

    See Lancaster and Salkauskas [20] is a good introduction to the general realm of curve-fitting.

  3. 3.

    During some periods the curve is flat, or even inverted, with short rates equal to or greater than long-term yields. During most periods, however, long-term yields dominate the short rates. The magnitude of the difference between short and long yields—what is often termed the slope or steepness of the yield curve—varies substantially over the 10 years displayed in Fig. 5.3.

  4. 4.

    There are many very good references on fitting yield curves. A good start would include Anderson et al. [3], Dierckyx [14], and deBoor [13].

  5. 5.

    Sadly, a broad range of such instruments does not exist. The closest thing to a pure-discount bond in the real world is a treasury bill. Unfortunately, most sovereign issuers do not issue such instruments beyond the 1 year tenor. In some markets, interest and principal payments from coupon bonds can be separately sold, thus creating long-dated pure-discount bonds—see Bolder and Boisvert [5] for more details on this practice. These are rarely used for fitting yield curves, however, due to their relative poor liquidity and lack of transparent pricing.

  6. 6.

    One can think of the bond yield, therefore, as a kind of complicated weighted average of the individual zero-coupon rates.

  7. 7.

    Going in the other direction, however, is no problem. That is, given the set of zero-coupon rates or pure-discount bond prices, we can easily compute the corresponding unique set of bond yields.

  8. 8.

    One occasionally suspects that a child could efficiently fit the UST curve with a pencil.

  9. 9.

    It may be more realistic to think of them as the hard and easy approaches.

  10. 10.

    Bond yields are the typical focus in casual conversation among financial practitioners. In the classic curve-fitting literature, however, one typically does works with bond prices rather than yields.

  11. 11.

    Given the price, \(V (t,y_{\tau _{k}})\), one alternatively solves for this yield.

  12. 12.

    The optimization details obviously matter, but gaining a conceptual understanding of the approach is much more important.

  13. 13.

    In other words, our parameter set is extremely simple: θ = { a}. 

  14. 14.

    We use continuously compounded rates for mathematical simplicity, although you have approximately the same result using Eq. (5.2).

  15. 15.

    There are myriad details involved, for non-trivial choices of g θ , in efficiently and accurately solving this optimization problem. There are many good references for help on this point. See Bolder and Stréliski [10], Bolder and Gusba [7], Cairns [11], Cairns [12], or Bliss [4] to get started.

  16. 16.

    The interested reader is referred to Hurn et al. [18] or Bolder and Liu [8, Appendix C] for a derivation and appropriate references.

  17. 17.

    Its stylized view of the yield curve, along with a generally reasonable fit to bond prices, was one of the reasons the Nelson–Siegel model has found favour with central banks. Central banks are generally more interested in understanding general macroeconomic trends than the detailed over- or under-pricing of specific securities.

  18. 18.

    Numerous good references, for the interested reader, include McCulloch [23], Fisher et al. [16], Anderson and Sleath [2], Shumaker [28], Eilers and Marx [15], Wegman and Wright [32], Ahlberg and Nilsen [1], Nürnberger [26], and Bolder and Gusba [7].

  19. 19.

    Without getting into the gory details, this essentially amounts to a certain degree of independence between these functions making them more efficient for our application. For more detail on the exponential spline model in general, and orthogonal functions in particular, the reader is referred to Li et al. [22] and Bolder and Gusba [6].

  20. 20.

    This immediately reduces the data burden thus leading to greater simplicity. Many of these computations can easily be implemented in a simple spreadsheet.

  21. 21.

    See Ralston and Rabinowitz [27] for a formal description of the general notions of interpolation.

  22. 22.

    The reason is that some errors are positive and others are negative and one can have a small sum of errors overall, but sizeable individual errors. This can be resolved by minimizing the sum of the absolute value of the errors. This is also challenging. The derivative of the absolute-value function is unfortunately discontinuous at zero and, as such, makes the use of standard optimization methods a bit tricky.

  23. 23.

    It is thus classified as a non-parametric approach. Most of the effort involved in using kernel regression is finding a reasonable value for h. See Nadaraya [24] and Watson [31] for more details.

  24. 24.

    Often, the weighting is defined as a decreasing function of the distance from the point, Ï„. This has the logical effect of increasing the importance of nearby yield observations and decreasing the importance of yields that are further away.

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Authors and Affiliations

  1. Bank for International Settlements, Basel, Switzerland

    David Jamieson Bolder

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Bolder, D.J. (2015). Fitting Yield Curves. In: Fixed-Income Portfolio Analytics. Springer, Cham. https://doi.org/10.1007/978-3-319-12667-8_5

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