Zero-Coupon Treasury Rates and Returns using the Volatility Index

TL;DR

This work addresses how stock-market volatility, as measured by the volatility index $V(t)$, can drive the dynamics of Treasury yield curves by modeling the first three principal components (level, slope, curvature) of ten zero-coupon rate series with a multivariate autoregressive stochastic volatility framework. The model uses the observed volatility $V(t)$ in the factor dynamics $\mathbf{X}(t) = \mathbf{a} + \mathbf{B}\mathbf{X}(t-1) + \mathbf{c}V(t) + \xi(t)\mathbf{Z}(t)$ and $\ln V(t) = \alpha + \beta \ln V(t-1) + Z_0(t)$, with continuous-time analogues yielding an Ornstein-Uhlenbeck structure for $\ln V$ and a multivariate Ornstein-Uhlenbeck for $\mathbf{X}$. The paper proves long-term stability, ergodicity, and strong laws of large numbers for both factor processes and total zero-coupon returns, enabling a simple linear representation of bond returns in terms of the PCs and $V(t)$. A key finding is that VIX improves the Gaussianity of slope innovations while having limited impact on the level, highlighting a surprising link between stock-market volatility and the slope of the yield curve, and supporting a CAPM-like interpretation of term premia under a multi-factor yield-curve framework. Overall, the results provide a parsimonious, tractable approach to yield-curve dynamics and bond-return LLNs with potential extensions to broader bond markets and daily data.

Abstract

We study a multivariate autoregressive stochastic volatility model for the first 3 principal components (level, slope, curvature) of 10 series of zero-coupon Treasury bond rates with maturities from 1 to 10 years. We fit this model using monthly data from 1990. Unlike classic models with hidden stochastic volatility, here it is observed as VIX: the volatility index for the S&P 500 stock market index. Surprisingly, this stock index volatility works for Treasury bonds, too. Next, we prove long-term stability and the Law of Large Numbers. We express total returns of zero-coupon bonds using these principal components. We prove the Law of Large Numbers for these returns. All results are done for discrete and continuous time.

Figures (3)

• Figure 1: Principal Components: level (PC1), slope (PC2), curvature (PC3), and their loading factors. In the left, we see the level gradually decreasing before sharply increasing at the end. This corresponds to the overall dynamics of rates since 1990. The slope and the curvature do not exhibit such clear-cut tendency. In the right, the loading factors are $c_{il}$.
• Figure 2: The quantile-quantile plots versus the Gaussian distributions for innovations $Z_1$ of PC1 (level); innovations $Z_2$ of PC2 (slope); and normalized innovations $Z_2/V$ of PC2, taken from \ref{['eq:AR']}. One sees that $Z_1$ is close to Gaussian, but $Z_2$ is not. However, normalizing $Z_2$ (dividing it by $V$) makes it closer to Gaussian.
• Figure 3: Autocorrelation functions for innovations $Z_1$ of PC1 (level); normalized innovations $Z_2/V$ of PC2 (slope); and innovations $Z_3$ of PC3 (curvature), taken from \ref{['eq:AR']}. We see that some values of the ACF are high enough to be outside of the shadowed area. If this value is indeed outside, then applying the white noise test for this lag value, we reject the white noise hypothesis. However, most ACF values in each case are small enough. We consider it reasonable to model $Z_1, Z_2, Z_3$ using IID Gaussian.

Theorems & Definitions (19)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• Theorem 5
• proof
• Theorem 6