Term structure shapes and their consistent dynamics in the Svensson family

Abstract

We examine the shapes attainable by the forward- and yield-curve in the widely-used Svensson family, including the Nelson-Siegel and Bliss subfamilies. We provide a complete classification of all attainable shapes and partition the parameter space of each family according to these shapes. Building upon these results, we then examine the consistent dynamic evolution of the Svensson family under absence of arbitrage. Our analysis shows that consistent dynamics further restrict the set of attainable shapes, and we demonstrate that certain complex shapes can no longer appear after a deterministic time horizon. Moreover a single shape (either inverse of normal curves) must dominate in the long-run.

Figures (9)

• Figure 1: Two differently shaped yield curves for AAA-rated bonds in the Euro area. Plots generated through the ECB's website https://www.ecb.europa.eu for (A) Sep 16, 2022 and (B) Sep 27, 2023.
• Figure 2: Shapes of the forward curve (A) and the yield curve (B) for different regions of the parameter space $\Theta'$ of the Svensson family in the scale-regular regime $r > 1$. Red and green curves indicate the augmented envelope $\hat{\eta}$ (see Def. \ref{['defn:augmented']}). Parameters used are $\tau_1 = 1, \tau_2 = 1/2$ and region labels (see Table \ref{['tab:shape']}) correspond to the case $\beta_3 > 0$ . For $\beta_3 < 0$ the plot stays the same, but region labels must be changed, see Remark \ref{['rem:polarity']}
• Figure 3: Values of $r = \tau_1/\tau_2$ visiting different regimes (see Def. \ref{['defn:augmented']}) in the Svensson family for Euro Area AAA-rated bond data over time. Dots are colored black if $\beta_3 > 0$ and blue if $\beta_3 < 0$.
• Figure 4: Shapes of the forward curve (red) and yield curve (green) for Euro Area AAA-rated bond data over time. Abbreviations are explained in Table \ref{['tab:shape']}
• Figure 5: This diagram shows the shapes that are attained with strictly positive probability for the forward curve in the consistent Svensson model as time $t$ progresses, illustrating Thm. \ref{['thm:main_consistency']}, Cor. \ref{['cor:trapped']} and Thm. \ref{['thm:ergodic']}. The branch $\beta_2 > 0$ or $\beta_2 < 0$ is selected when the model is calibrated to the initial term structure. Note that each of the times $T_\dagger^{\rm f}, T_*^{\rm f}$ may be zero, in which case the corresponding part of the diagram collapses.

Theorems & Definitions (54)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7
• Theorem 3.1: See KR21
• Theorem 3.2
• Lemma 3.3