### Abstract

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. NELSON–SIEGEL TERM STRUCTURE MODELS
- 3. EXTENSIONS OF THE NELSON–SIEGEL MODEL
- 4. FIVE SPECIFIC NELSON–SIEGEL MODELS
- 5. ESTIMATION OF THE MODELS
- 6. CONCLUSION
- ACKNOWLEDGMENTS
- REFERENCES
- Appendix

**Summary ** The Svensson generalization of the popular Nelson–Siegel term structure model is widely used by practitioners and central banks. Unfortunately, like the original Nelson–Siegel specification, this generalization, in its dynamic form, does not enforce arbitrage-free consistency over time. Indeed, we show that the factor loadings of the Svensson generalization cannot be obtained in a standard finance arbitrage-free affine term structure representation. Therefore, we introduce a closely related generalized Nelson–Siegel model on which the no-arbitrage condition can be imposed. We estimate this new AFGNS model and demonstrate its tractability and good in-sample fit.

### 1. INTRODUCTION

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. NELSON–SIEGEL TERM STRUCTURE MODELS
- 3. EXTENSIONS OF THE NELSON–SIEGEL MODEL
- 4. FIVE SPECIFIC NELSON–SIEGEL MODELS
- 5. ESTIMATION OF THE MODELS
- 6. CONCLUSION
- ACKNOWLEDGMENTS
- REFERENCES
- Appendix

To investigate yield-curve dynamics, researchers have produced a vast literature with a wide variety of models. Many of these models assume that at observed bond prices there are no remaining unexploited opportunities for riskless arbitrage. This theoretical assumption is consistent with the observation that bonds of various maturities all trade simultaneously in deep and liquid markets. Rational traders in such markets should enforce a consistency in the yields of various bonds across different maturities—the yield curve at any point in time—and the expected path of those yields over time—the dynamic evolution of the yield curve. Indeed, the assumption that there are no remaining arbitrage opportunities is central to the enormous finance literature devoted to the empirical analysis of bond pricing. Unfortunately, as noted by Duffee (2002), the associated arbitrage-free (AF) models can demonstrate disappointing empirical performance, especially with regard to out-of-sample forecasting. In addition, the estimation of these models is problematic, in large part because of the existence of numerous model likelihood maxima that have essentially identical fit to the data but very different implications for economic behavior (Kim and Orphanides, 2005).^{1}

In contrast to the popular finance AF models, many other researchers have employed representations that are empirically appealing but not well grounded in theory. Most notably, the Nelson and Siegel (1987) curve provides a remarkably good fit to the cross section of yields in many countries and has become a widely used specification among financial market practitioners and central banks. Moreover, Diebold and Li (2006) develop a dynamic model based on this curve and show that it corresponds exactly to a modern factor model, with yields that are affine in three latent factors, which have a standard interpretation of level, slope and curvature. Such a dynamic Nelson–Siegel (DNS) model is easy to estimate and forecasts the yield curve quite well. Despite its good empirical performance, however, the DNS model does not impose the presumably desirable theoretical restriction of absence of arbitrage (e.g. Filipović, 1999, and Diebold et al., 2005).

In Christensen et al. (2007), henceforth CDR, we show how to reconcile the Nelson–Siegel model with the absence of arbitrage by deriving an affine AF model that maintains the Nelson–Siegel factor loading structure for the yield curve. This arbitrage-free Nelson–Siegel (AFNS) model combines the best of both yield-curve modeling traditions. Although it maintains the theoretical restrictions of the affine AF modeling tradition, the Nelson–Siegel structure helps identify the latent yield-curve factors, so the AFNS model can be easily and robustly estimated. Furthermore, our results show that the AFNS model exhibits superior empirical forecasting performance.

In this paper, we consider some important generalizations of the Nelson–Siegel yield curve that are also widely used in central banks and industry (e.g. De Pooter, 2007).^{2} Foremost among these is the Svensson (1995) extension to the Nelson–Siegel curve, which is used at the Federal Reserve Board (see Gürkaynak et al., 2007, 2008), the European Central Bank (see Coroneo et al., 2008) and many other central banks (see Söderlind and Svensson, 1997, and Bank for International Settlements, 2005). The Svensson extension adds a second curvature term, which allows for a better fit at long maturities. Following Diebold and Li (2006), we first introduce a dynamic version of this model, which corresponds to a modern four-factor term structure model. Unfortunately, we show that it is not possible to obtain an arbitrage-free ‘approximation’ to this model in the sense of obtaining analytically identical factor loadings for the four factors. Intuitively, such an approximation requires that each curvature factor must be paired with a slope factor that has the same mean-reversion rate. This pairing is simply not possible for the Svensson extension, which has one slope factor and two curvature factors. Therefore, to obtain an arbitrage-free generalization of the Nelson–Siegel curve, we add a second slope factor to pair with the second curvature factor. The simple dynamic version of this model is a generalized version of the DNS model. We also show that the result in CDR can be extended to obtain an arbitrage-free approximation to that five-factor model, which we refer to as the arbitrage-free generalized Nelson–Siegel (AFGNS) model.

Finally, we show that this new AFGNS model of the yield curve not only displays theoretical consistency but also retains the important properties of empirical tractability and fit. We estimate the independent-factor versions of the four-factor and five-factor non-AF models and the independent-factor version of the five-factor arbitrage-free AFGNS model. We compare the results to those obtained by CDR for the DNS and AFNS models and find good in-sample fit for the AFGNS model.

The remainder of the paper is structured as follows. Section 2 briefly describes the DNS model and its arbitrage-free equivalent as derived in CDR. Section 3 contains the description of the AFGNS model. Section 4 describes the five specific models that we analyze, while Section 5 describes the data, estimation method and estimation results. Section 6 concludes the paper, and an Appendix contains some additional technical details.

### 3. EXTENSIONS OF THE NELSON–SIEGEL MODEL

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. NELSON–SIEGEL TERM STRUCTURE MODELS
- 3. EXTENSIONS OF THE NELSON–SIEGEL MODEL
- 4. FIVE SPECIFIC NELSON–SIEGEL MODELS
- 5. ESTIMATION OF THE MODELS
- 6. CONCLUSION
- ACKNOWLEDGMENTS
- REFERENCES
- Appendix

The main in-sample problem with the regular Nelson–Siegel yield curve is that, for reasonable choices of λ (which are empirically in the range from 0.5 to 1 for U.S. Treasury yield data), the factor loading for the slope and the curvature factor decay rapidly to zero as a function of maturity. Thus, only the level factor is available to fit yields with maturities of ten years or longer. In empirical estimation, this limitation shows up as a lack of fit of the long-term yields, as described in CDR.

To address this problem in fitting the cross section of yields, Svensson (1995) introduced an extended version of the Nelson–Siegel yield curve with an additional curvature factor,

Just as Diebold and Li (2006) replaced the three β coefficients with dynamic factors in the regular Nelson–Siegel model, we can replace the four β coefficients in the Svensson model with dynamic processes (*L*_{t}, *S*_{t}, *C*^{1}_{t}, *C*^{2}_{t}) interpreted as a level, a slope and two curvature factors, respectively. Thus, the dynamic factor model representation of the Svensson yield curve, which we label the DNSS model, is given by

along with the processes describing factor dynamics. The factor loadings of the four state variables in the yield function of the DNSS model are illustrated in Figure 1(a) with λ_{1} and λ_{2} set equal to our estimates described in Section 5. The left-hand figure shows the factor loadings of the four state variables in the yield function of the DNSS model with λ_{1} and λ_{2} equal to 0.8379 and 0.09653, respectively.

The critique raised by Filipović (1999) against the dynamic version of the Nelson–Siegel model also applies to the dynamic version of the Svensson model introduced in this paper. Thus, this model is not consistent with the concept of absence of arbitrage. Ideally, we would like to repeat the work in CDR and derive an arbitrage-free approximation to the DNSS model. However, from the mechanics of Proposition 2.1 for the arbitrage-free approximation of the regular Nelson–Siegel model, it is clear that we can only obtain the Nelson–Siegel factor loading structure for the slope and curvature factors under two specific conditions. First, each pair of slope and curvature factors must have identical own mean-reversion rates. Second, the impact of deviations in the curvature factor from its mean on the slope factor must be scaled with a factor equal to that own mean-reversion rate (λ). Thus, it is impossible in an arbitrage-free model to generate the factor loading structure of two curvature factors with only one slope factor. Consequently, it is impossible to create an arbitrage-free version of the Svensson extension to the Nelson–Siegel model that has factor loadings analytically identical to the ones in the DNSS model.

However, this discussion suggests that we can create a generalized AF Nelson–Siegel model by including a fifth factor in the form of a second slope factor. The yield function of this model takes the form

This dynamic generalized Nelson–Siegel model, which we denote as the DGNS model, is a five-factor model with one level factor, two slope factors and two curvature factors. (Note that we impose the restriction that λ_{1} > λ_{2}, which is non-binding due to symmetry.^{6}) The factor loadings of the five state variables in the yield function of the DGNS model are illustrated in Figure 1(b) with λ_{1} and λ_{2} set equal to our estimates in Section 5. The right-hand figure shows the factor loadings of the five state variables in the yield function of the DGNS model with λ_{1} and λ_{2} equal to 1.190 and 0.1021, respectively. These λ_{i} values equal the estimated values obtained below, and they require maturity to be measured in years.

A straightforward extension of Proposition 2.1 delivers the arbitrage-free approximation of this model, which we denote as the AFGNS model.

**Proposition 3.1. ***Assume that the instantaneous risk-free rate is defined by*

*In addition, assume that the state variables* *X*_{t}= (*X*^{1}_{t}, *X*^{2}_{t}, *X*^{3}_{t}, *X*^{4}_{t}, *X*^{5}_{t}) *are described by the following system of SDEs under the risk-neutral Q-measure:*

*where* λ_{1} > λ_{2} > 0.

*Then, zero-coupon bond prices are given by*

*where* *B*^{1}(*t*, *T*), *B*^{2}(*t*, *T*), *B*^{3}(*t*, *T*), *B*^{4}(*t*, *T*), *B*^{5}(*t*, *T*) *and* *C*(*t*, *T*) *are the unique solutions to the following system of ODEs:*

- ((3.1))

*and*

- ((3.2))

*with boundary conditions* *B*^{1}(*T*, *T*) = *B*^{2}(*T*, *T*) = *B*^{3}(*T*, *T*) = *B*^{4}(*T*, *T*) = *B*^{5}(*T*, *T*) = *C*(*T*, *T*) = 0. *The unique solution for this system of ODEs is:*

*and*

*Finally, zero-coupon bond yields are given by*

The proof is a straightforward extension of CDR.

Similar to the AFNS class of models, the yield-adjustment term will have the following form:^{7}

Following arguments similar to the ones provided for the AFNS class of models in the previous section, the maximally flexible specification of the volatility matrix that can be identified in estimation is given by a triangular matrix

### 4. FIVE SPECIFIC NELSON–SIEGEL MODELS

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. NELSON–SIEGEL TERM STRUCTURE MODELS
- 3. EXTENSIONS OF THE NELSON–SIEGEL MODEL
- 4. FIVE SPECIFIC NELSON–SIEGEL MODELS
- 5. ESTIMATION OF THE MODELS
- 6. CONCLUSION
- ACKNOWLEDGMENTS
- REFERENCES
- Appendix

In general, all the models considered in this paper are silent about the *P*-dynamics, and an infinite number of possible specifications could be used to match the data. However, for continuity with the existing literature, our econometric analysis focuses on independent-factor versions of the five different models we have described. These models include the DNS and AFNS models from CDR and the generalized DNSS, DGNS and AFGNS models introduced in Section 3.

In the *independent-factor DNS model*, all three state variables are assumed to be independent first-order autoregressions, as in Diebold and Li (2006). Using their notation, the state equation is given by

where the error terms η_{t}(*L*), η_{t}(*S*) and η_{t}(*C*) have a conditional covariance matrix given by

In this model, the measurement equation takes the form

where the measurement errors ɛ_{t}(τ_{i}) are assumed to be independently and identically distributed (i.i.d.) white noise.

The corresponding AFNS model is formulated in continuous time and the relationship between the real-world dynamics under the *P*-measure and the risk-neutral dynamics under the *Q*-measure is given by the measure change

where Γ_{t} represents the risk premium specification. To preserve affine dynamics under the *P*-measure, we limit our focus to essentially affine risk premium specifications (see Duffee, 2002). Thus, Γ_{t} will take the form

With this specification, the SDE for the state variables under the *P*-measure,

- ((4.1))

remains affine. Due to the flexible specification of Γ_{t}, we are free to choose any mean vector θ^{P} and mean-reversion matrix *K*^{P} under the *P*-measure and still preserve the required *Q*-dynamic structure described in Proposition 2.1. Therefore, we focus on the *independent-factor AFNS model*, which corresponds to the specific DNS model from earlier in this section and assumes all three factors are independent under the *P*-measure

In this case, the measurement equation takes the form

where, again, the measurement errors ɛ_{t}(τ_{i}) are assumed to be i.i.d. white noise.

We now turn to the three generalized Nelson–Siegel models. In the *independent-factor DNSS model*, all four state variables are assumed to be independent first-order autoregressions, as in Diebold and Li (2006). Using their notation, the state equation is given by

where the error terms η_{t}(*L*), η_{t}(*S*), η_{t}(*C*^{1}) and η_{t}(*C*^{2}) have a conditional covariance matrix given by

In the DNSS model, the measurement equation takes the form

where the measurement errors ɛ_{t}(τ_{i}) are assumed to be i.i.d. white noise.

In the *independent-factor DGNS model*, all five state variables are assumed to be independent first-order autoregressions, and the state equation is given by

where the error terms η_{t}(*L*), η_{t}(*S*^{1}), η_{t}(*S*^{2}), η_{t}(*C*^{1}) and η_{t}(*C*^{2}) have a conditional covariance matrix given by

In the DGNS model, the measurement equation takes the form

where the measurement errors ɛ_{t}(τ_{i}) are assumed to be i.i.d. white noise.

Finally, as for the AFNS model, the AFGNS model is formulated in continuous time and the relationship between the real-world dynamics under the *P*-measure and the risk-neutral dynamics under the *Q*-measure is given by the measure change

where Γ_{t} represents the risk premium specification. Again, to preserve affine dynamics under the *P*-measure, we limit our focus to essentially affine risk premium specifications (see Duffee, 2002). Thus, Γ_{t} takes the form

With this specification, the SDE for the state variables under the *P*-measure,

- ((4.2))

remains affine. Due to the flexible specification of Γ_{t}, we are free to choose any mean vector θ^{P} and mean-reversion matrix *K*^{P} under the *P*-measure and still preserve the required structure for the *Q*-dynamics described in Proposition 3.1. Therefore, we focus on the AFGNS model that corresponds to the specific DGNS model we have described earlier. In this *independent-factor AFGNS model*, all five factors are assumed to be independent under the *P*-measure

For the AFGNS model, the measurement equation takes the form

where, again, the measurement errors ɛ_{t}(τ_{i}) are assumed to be i.i.d. white noise.

### 6. CONCLUSION

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. NELSON–SIEGEL TERM STRUCTURE MODELS
- 3. EXTENSIONS OF THE NELSON–SIEGEL MODEL
- 4. FIVE SPECIFIC NELSON–SIEGEL MODELS
- 5. ESTIMATION OF THE MODELS
- 6. CONCLUSION
- ACKNOWLEDGMENTS
- REFERENCES
- Appendix

The Nelson and Siegel (1987) curve and the associated dynamic DNS model of Diebold and Li (2006) both have trouble fitting long-maturity yields (in large part because of convexity effects). In this paper, we solve that problem while *simultaneously* imposing an absence of arbitrage. We argue that although the popular Svensson (1995) extension of the Nelson–Siegel curve may improve long-maturity fit, there does not exist an arbitrage-free yield-curve model that matches its factor loadings. However, we show that there is a natural five-factor generalization, which adds a second slope factor to join the additional curvature factor in the Svesson extension, that *does* achieve freedom from arbitrage. Finally, we show that the estimation of this new AFGNS model is tractable and provides good fit to the yield curve. The empirical tractability is especially important because, as noted in the introduction, it would be very difficult to estimate the maximally flexible five-factor affine arbitrage-free term structure model.

Going forward, the AFGNS model may be a useful addition to the tool kit of central banks and practitioners who now use the non-AF Svensson extension of the Nelson–Seigel yield curve. Furthermore, we envision much future research that employs the underlying arbitrage-free Nelson–Seigel structure. In particular, given its tractable estimation, the basic AFNS model can be easily extended to incorporate other elements, such as stochastic volatility, inflation-indexed bond yields, or interbank lending rates (Christensen et al., 2008a, b, c). These extensions would be difficult to include in an estimated maximally flexible affine model but may help illuminate various important issues.