We begin with a bit of history, as what we have done in this book blends the old and the new. In particular, we sketch and contrast what we will call the “traditional” and “modern” approaches to yield curve modeling, to heighten our understanding of where and how DNS/AFNS fits.
Early on, bond markets were few and far between, so the term structure of bond yields remained a dormant issue. But as financial markets developed, and as economic theory and measurement advanced, issues related to the yield curve emerged as central in asset allocation, asset pricing, and risk management.
The earliest work effectively assumes perfect foresight, as in Bohm-Bawerk (1889). Building on that work, early interwar term-structure modeling typically allows for risk but assumes risk neutrality, as in Fisher (1930) and Keynes (1936), culminating in the classic expectations theory of Hicks (1946). Hicks’s most basic theory, sometimes called the “traditional expectations theory,” asserts that current forward rates equal expected future spot rates, in which case current long rates are simple averages of expected future spot rates.
An immediate implication of the traditional expectations theory is that the yield curve will be flat if spot rates are expected to remain unchanged.1 This conflicts with the observed upward-sloping average yield curve and directs attention to the possibility of “term premia” that may separate forward rates from expected future spot rates. The first resulting variant of the traditional expectations theory is the “liquidity preference theory” of Hicks (1946). Hicks posited that lenders prefer short maturities, to avoid the risk associated with holding long-duration bonds, and that borrowers prefer long maturities, to lock in the cost of finance. Lenders then require and receive long-maturity premia from borrowers, producing an upward-sloping average yield curve.
Richer approximations to average yield curves, and time-varying conditional yield curves, soon followed.2 In the “preferred habitat” or “market segmentation” theory of Modigliani and Sutch (1967), different agents prefer to borrow or lend in different regions of the curve. Depending on the distribution of agents among various preferred habitats, the curve can take many shapes, not just upward sloping. Moreover, the “slope factor” generated by preferred habitats can vary over time, as the distribution of agents among various preferred habitats can vary over time, for example, with the level of yields and/or with expected business conditions, as in Meiselman (1962), Kessel (1965), Van Horne (1965), and Nelson (1972).3 Perhaps most notably in that tradition, Nelson (1972) identified expected spot rates using time-series models, subtracted them from current forward rates to get term premia, and then projected the term premia on business conditions indicators.
The “traditional” literature’s concern with dynamic yield curve evolution is very much maintained in the “modern” literature, which in many respects began with Vasicek (1977), and which was significantly extended in classic subsequent contributions by Duffie and Kan (1996), Dai and Singleton (2000), Ang and Piazzesi (2003), among others. The modern literature, however, introduces a key distinguishing characteristic: explicit enforcement of absence of arbitrage.
Interestingly, DNS and AFNS in many respects bridge the traditional and modern literatures. DNS is the penultimate traditional model: explicitly stochastic, dynamic, parsimonious yet flexible, three-factor state-space structure, and so on. AFNS is a “modernized” DNS, explicitly imposing no-arbitrage while maintaining its other appealing features and empirical tractability.
Against this background, several questions emerge, all of which are presently incompletely resolved but very much evident in the evolving literature, related to the role of no-arbitrage constraints, empirical tractability of various approaches to arbitrage-free modeling, and whether and why the DNS/AFNS model is “special.” So, in parallel to Chapter 1, in which we began this book with several questions, we now end with three.
DNS fits well, so if market yields are arbitrage-free, then DNS should also be arbitrage-free, at least up to an accurate approximation. Hence it would appear intuitively that no-arbitrage restrictions, if imposed on DNS, would likely be largely nonbinding. Hence the implicit feeling in Diebold and Li (2006) and subsequent DNS work is that DNS’s lack of imposition of no-arbitrage is not very important, and that attention is better focused on other matters, primarily how to extend DNS in creative ways.
The AFNS theory shows that not only intuition but also rigorous theory suggests that imposition of no-arbitrage restrictions is unlikely to improve DNS forecasts. In particular, as shown in the key Proposition AFNS of Chapter 3, although no-arbitrage strongly restricts risk-neutral dynamics, it puts no restrictions on physical dynamics. Instead, imposition of no-arbitrage delivers only yield-adjustment terms that vary with maturity but are constant over time.
Important subsequent work effectively extends the “predictive irrelevance” flavor of Proposition AFNS to the general context of maximally flexible canonical models. In particular, Duffee (2011a) and Joslin et al. (2011b) advance arguments for predictive irrelevance in general yields-only environments. Joslin et al. (2011a), moreover, extend those arguments to environments that include macroeconomic fundamentals.
Hence, and perhaps ironically, several literatures have in a sense come full circle. From a financial economic perspective, one might feel uncomfortable about arbitrage possibilities in various “traditional” models, so Vasicek (1977), Duffie and Kan (1996), and Dai and Singleton (2000) made them arbitrage-free. Similarly, one might feel uncomfortable simply fitting unrestricted vector autoregressions, so Ang and Piazzesi (2003) made them arbitrage-free. Finally, one might feel uncomfortable fitting unrestricted DNS, so Christensen et al. (2011a) made it arbitrage-free. Presumably for certain central tasks (e.g., pricing) one should feel uncomfortable with models that don’t impose no-arbitrage, but the emerging theoretical recognition is that, for at least one central task (forecasting), imposition of no-arbitrage appears theoretically unlikely to help.4
We hasten to add, however, that as discussed in section 3.7, empirical work has sometimes found predictive gains from imposing no-arbitrage in DNS environments. Whether such gains are largely real (e.g., coming from the AFNS yield-adjustment term) or largely just good luck (e.g., sample selection across various yields at various times in various countries) remains to be seen, and we look forward to additional research. Perhaps the current best-practice recommendation should be always to work with AFNS, as it is almost as simple as DNS to estimate, and imposition of no-arbitrage is clearly important for many, if not all, tasks.
The short answer is “until recently, but interestingly, not anymore.” In our view, the claim that, until very recently, AFNS was the only empirically tractable modern model is not exaggeration. In particular, the maximally flexible canonical A0(N) models have notoriously recalcitrant likelihood surfaces, notwithstanding the rarely published acknowledgments of such. In recent insightful work, Hamilton and Wu (2010b) summarize the situation well:
Buried in the footnotes of this literature and in the practical experience of those who have used these models are tremendous numerical challenges in estimating the necessary parameters from the data due to highly non-linear and badly-behaved likelihood surfaces.
They proceed to clarify the precise reason for the empirical nightmares associated with the canonical models, showing in particular that three leading canonical models are unidentified.5
In addition to A0(N) identification problems, there appear to be A0(N) data-mining problems. In particular, Duffee (2010) shows that most standard affine arbitrage-free models with profligate parameterization (i.e., maximally flexible models) imply Sharpe ratios absurdly higher than can actually be obtained in practice. It would appear, then, that for several reasons the maximally flexible A0(N) models are ill-suited for empirical work, at least as traditionally empirically implemented.
Fortunately, however, the recent work of Joslin et al. (2011b) remedies the situation and opens important new doors. They develop a well-behaved (among other things, identified) family of Gaussian term structure models, for which trustworthy estimation is very simple, just as with AFNS. Moreover, it turns out that AFNS is a nested special case of their canonical form, corresponding to three extra constraints relative to the maximally flexible model, including a repeated eigenvalue in the risk-neutral dynamic matrix KQ.6
The obvious question, then, is whether AFNS, although clearly an important member and early discovery in what was later expanded to the Joslin et al. (2011b) class, continues to merit elevated status; that is, whether it is special in some way. There are many issues and nuances, and it is too soon to offer a definitive answer. Indeed there may never be a definitive answer, but in what follows we offer several insights that bear on the question. All told, we believe that AFNS is special, based on at least three classes of considerations.
AFNS’s structure conveys several important and useful characteristics, which are presently difficult or impossible to achieve in competing frameworks. First, as regards specializations, its parametric simplicity makes it easy to impose restrictions. Second, as regards extensions, it is similarly easy to increase the number of AFNS latent factors, for example, with the five-factor AFGNS model of section 4.2.2.7 Third, as regards varied uses, its flexible continuous basis functions facilitate relative pricing, curve interpolation between observed yields, and risk measurement for arbitrary bond portfolios.
Nelson and Siegel (1987) originally motivated their functional form via solutions to second-order differential equations:
A class of functions that readily generates the typical yield curve shapes is that associated with solutions to differential … equations … if the instantaneous forward rate … is given by the solution to a second-order differential equation with real and unequal roots, we would have … a family of forward rate curves that take on monotonic, humped, or S shapes … , and the implied yield curve displays the same range of shapes. (pp. 474–475)
Interestingly, moreover, Nelson and Siegel introduce a restriction that leads them to an explicit justification in terms of function approximation:
A more parsimonious model that can generate the same range of shapes is given by the solution equation for the case of equal roots.… This model may also be derived as an approximation to the solution in the unequal roots case by expanding in a power series in the difference between the roots.… [It] may also be viewed as a constant plus a Laguerre function … [which is a] polynomial times an exponential decay term and is a mathematical class of approximating functions. (p. 475)
We note that the equal-root assumption is not unrelated to the repeated eigenvalue in the AFNS risk-neutral dynamic matrix KQ.
Recent important work takes the function-approximation justification significantly further. In particular, Krippner (2011b) shows that the Nelson-Siegel form provides a Taylor-series approximation to arbitrary A0(N) term-structure dynamics. To be more precise, Krippner shows that the level, slope, and curvature components common to all Nelson-Siegel models arise explicitly from low-ordered Taylor-series expansions around central measures of the eigenvalues for the generic Gaussian affine term-structure model. Hence any yield curve from the A0(N) class can be approximated parsimoniously by an AFNS model.
As mentioned earlier, AFNS is a nested special case of the Joslin et al. (2011b) canonical form, corresponding to three restrictions. Hence likelihood-ratio and related tests of the AFNS null hypothesis are immediately applicable, and Joslin et al. (2011b) perform them. Interestingly, when they normalize their model such that their yield factors are the first three principal components—a natural and obvious benchmark in their view and ours—they find no evidence against AFNS.8 We are not surprised by the nonrejection. Indeed, for the many reasons that we have emphasized throughout this book, we expect AFNS to fit well. Moreover, even if somehow rejected, we find it unlikely that AFNS would be far in any meaningful economic sense from an as-yet-unknown “preferred” three-factor model.
1This would occur, for example, if the spot rate is a martingale and expectations are formed rationally, in which case the optimal spot rate forecast at all horizons is “no change.”
2In addition to theory, measurement also advanced rapidly during this time. High-quality yield curve data, for example, were constructed by Durand (1958) and Malkiel (1966), among others.
3The early literature’s emphasis on linking term premia to yield levels evidently traces to Keynes (1936), pp. 201–202. Yield levels, however, are of course just one aspect of business conditions, and the key issue is relating term premia to broad business conditions.
4We say “appears theoretically unlikely to help” as opposed to “cannot help” because the AFNS yield-adjustment term, despite its time constancy, could help forecasts. It might, for example, serve to provide a bias correction.
5See also Hamilton and Wu (2011).
6In related work, Siegel (2009) considers arbitrage-free models for bond prices rather than yields. Several benefits accrue relative to Duffie-Kan. The model admits a general closed-form solution, and the drift is not affected by the diffusion, which simplifies the incorporation of stochastic volatility. Roughly, then, it seems that Siegel’s family is to discount curves what the Joslin et al. (2011b) family is to yield curves. Moreover, a specific member of Siegel’s family parallels closely the AFNS yield model in structure, fit, and tractability.
7Krippner (2011b) tabulates and discusses a range of existing and potential extensions to the AFNS model, and he characterizes the relationship between each extension and the generic Gaussian affine arbitrage-free model, A0(N).
8Indeed the p-value of their likelihood-ratio test statistic is greater than 0.5.