# Yield Curve Modeling and Forecasting

## The Dynamic Nelson-Siegel Approach

## Francis X. Diebold, Glenn D. Rudebusch

# Yield Curve Modeling and Forecasting

## The Dynamic Nelson-Siegel Approach

## Francis X. Diebold, Glenn D. Rudebusch

## About This Book

Understanding the dynamic evolution of the yield curve is critical to many financial tasks, including pricing financial assets and their derivatives, managing financial risk, allocating portfolios, structuring fiscal debt, conducting monetary policy, and valuing capital goods. Unfortunately, most yield curve models tend to be theoretically rigorous but empirically disappointing, or empirically successful but theoretically lacking. In this book, Francis Diebold and Glenn Rudebusch propose two extensions of the classic yield curve model of Nelson and Siegel that are both theoretically rigorous and empirically successful. The first extension is the dynamic Nelson-Siegel model (DNS), while the second takes this dynamic version and makes it arbitrage-free (AFNS). Diebold and Rudebusch show how these two models are just slightly different implementations of a single unified approach to dynamic yield curve modeling and forecasting. They emphasize both descriptive and efficient-markets aspects, they pay special attention to the links between the yield curve and macroeconomic fundamentals, and they show why DNS and AFNS are likely to remain of lasting appeal even as alternative arbitrage-free models are developed.

Based on the Econometric and Tinbergen Institutes Lectures, Yield Curve Modeling and Forecasting contains essential tools with enhanced utility for academics, central banks, governments, and industry.

## Information

## Facts, Factors, and Questions

### 1.1 Three Interest Rate Curves

*P*(

*τ*) denote the price of a

*τ*-period discount bond, that is, the present value of $1 receivable

*τ*periods ahead. If

*y*(

*τ*) is its continuously compounded yield to maturity, then by definition

*P*(

*τ*),

*y*(

*τ*), and

*f*(

*τ*) implies knowledge of the other two, the three are effectively interchangeable. Hence with no loss of generality one can choose to work with

*P*(

*τ*),

*y*(

*τ*), or

*f*(

*τ*). In this book, following much of both academic and industry practice, we work with the yield curve,

*y*(

*τ*). But again, the choice is inconsequential in theory.

### 1.2 Zero-Coupon Yields

^{1}The fitted discount curve, however, diverges at long maturities due to the polynomial structure, and the corresponding yield curve inherits that unfortunate property. Hence such curves can provide poor fits to yields that flatten out with maturity, as emphasized by Shea (1984).

### 1.3 Yield Curve Facts

^{2}In this section we address the obvious descriptive question: How do yields tend to behave across different maturities and over time?

*curves*.

^{3}In Figure 1.1 we show the resulting three-dimensional surface for the United States, with yields shown as a function of maturity, over time. The figure reveals a key yield curve fact: yield curves move a lot, shifting among different shapes: increasing at increasing or decreasing rates, decreasing at increasing or decreasing rates, flat, U-shaped, and so on.

*spreads*relative to the 10-year bond. Yield spread dynamics contrast rather sharply with those of yield levels; in particular, spreads are noticeably less volatile and less persistent. As with yields, the 1-month spread autocorrelations are very large, but they decay more quickly, so that the 12-month spread autocorrelations are noticeably smaller than those for yields. Indeed many strategies for active bond trading (sometimes successful and sometimes not!) are based on spread reversion.

*Notes*: We present descriptive statistics for end-of-month yields at various maturities. We show sample mean, sample standard deviation, and first- and twelfth-order sample autocorrelations. Data are from the Board of Governors of the Federal Reserve System. The sample period is January 1985 through December 2008.

### 1.4 Yield Curve Factors

*restricted*vector autoregression, displaying

*factor structure.*Factor structure is said to be operative in situations where one sees a high-dimensional object (e.g., a large set of bond yields), but where that high-dimensional object is driven by an underlying lower-dimensional set of objects, or “factors.” Thus beneath a high-dimensional seemingly complicated set of observations lies a much simpler reality.

^{4}Similarly, Aruoba and Diebold (2010) discuss empirical factor structure in macroeconomic fundamentals, and Diebold and Rudebusch (1996) discuss theoretical factor structure in macroeconomic models.

*Notes*: We present descriptive statistics for end-of-month yield spreads (relative to the 10-year bond) at various maturities. We show sample mean, sample standard deviation, and first- and twelfth-order sample autocorrelations. Data are from the Board of Governors of the Federal Reserve System, based on Gürkaynak et al. (2007). The sample period is January 1985 through December 2008.

^{5}Most early studies involving mostly long rates implicitly adopt a single-factor world view (e.g., Macaulay (1938)), where the factor is the level (e.g., a long rate). Similarly, early arbitrage-free models like Vasicek (1977) involve only a single factor. But single-factor structure severely limits the scope for interesting term structure dynamics, which rings hollow in terms of both introspection and observation.

*do*tend to move noticeably together, but at the same time, it’s clear that more than just a common level factor is operative. In the real world, term structure data—and, correspondingly, modern empirical term structure models—involve

*multiple*factors. This classic recognition traces to Litterman and Scheinkman (1991), Willner (1996), and Bliss ...

## Table of contents

*Yield Curve Modeling and Forecasting*([edition unavailable]). Princeton University Press. Retrieved from https://www.perlego.com/book/735622/yield-curve-modeling-and-forecasting-the-dynamic-nelsonsiegel-approach-pdf (Original work published 2013)

*Yield Curve Modeling and Forecasting*. [Edition unavailable]. Princeton University Press. https://www.perlego.com/book/735622/yield-curve-modeling-and-forecasting-the-dynamic-nelsonsiegel-approach-pdf.

*Yield Curve Modeling and Forecasting*. [edition unavailable]. Princeton University Press. Available at: https://www.perlego.com/book/735622/yield-curve-modeling-and-forecasting-the-dynamic-nelsonsiegel-approach-pdf (Accessed: 14 October 2022).

*Yield Curve Modeling and Forecasting*. [edition unavailable]. Princeton University Press, 2013. Web. 14 Oct. 2022.