数理解析研究所講究録
第 1818 巻 2012 年 122-147

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A Modified Arbitrage-Free Nelson Siegel Model:

An Alternative Affine Term Structure Model of Interest Rates
Masamitsu Ohnishi
Graduate School of Economics & Center for the Study of Finance and Insurance
Osaka University
Dara Sim
Graduate School of Economics
Osaka University

1

Introduction

Term structure modeling has been a topic of research for many decades. This is because it plays
a key role in measuring and managing interest rate risk, as well as in the valuation of fixedincome securities and their derivatives. It is of interest to many economic participants including
corporations, insurance companies, pension funds, hedge funds, investment banks, commercial
banks, and other financial institutions dealing with fixed-income assets and liabilities, swaps
and other exotic interest rate products, credit derivatives, mortgage portfolios with prepayment
options, and so on.
There are at least two approaches to modeling dynamic term structure of interest rates-theforward rate approach and the sport rate approach. One of the most popular dynamic term
structure models (DTSMs) following the forward rate approach is the Heath-Jarrow-Morton
model (also known as the HJM model) by Heath et al. (1992). On the other hand, the DTSMs
following the spot rate approach find their history back to the work of Vasicek (1977). Other wellknown one-factor affine DTSMs include the CIR model of COX, Ingersoll, and Ross (1985), andthe $Hull$ and White $(1990)$ ’s model, an extension of the Vasicek model. Duffie and Kan (1996)
generalize these literatures introducing a general form of continuous-time affine arbitrage-free
DTSMs, while Dai and Singleton (2000) characterize classes of admissible affine DTSMs. Affine
DTSMs are popular among practitioners and academics because of their theoretical tractability.
Other classes of DTSMs are also introduced. The quadratic DTSMs, for instance, are introduced
in order to ensure that the short rate is always positive.1
On the hand, the well-known Nelson-Siegel model (later, $NS$ ) and its extensions are in fact
non-dynamic term structure models. This class of models has been used by central banks forday-to-day yield curve fitting, and by fixed-income portfolio managers to measure interest raterisk as a consequence of non-parallel shifts in the yield curve. Attention has been shifted backto the $NS$ model after Diebold and Li (2006) extended it to incorporate dynamic structure. The
Dynamic Nelson-Siegel (later, DNS) model is empirically tractable and performs empiricallywell. Furthermore, since it inherits the properties of the $NS$ , its factors can be interpreted aslevel, slope, and curvature of the yield curve. However, theoretically it does not satisfy thearbitrage-free condition. Christensen et al. (2011) solve this theoretical flaw by deriving a newclass of models called the Arbitrage-Free Nelson-Siegel (ANFS) model. Since it is in fact belongs
to the $A_{0}(3)$ class of affine DTSMs of Dai and Singleton (2000), it is theoretically rigorous, andsince it is of the $NS$-type, it possesses the same properties as the DNS model.
This paper derives a modified version of the AFNS model in which the level factor in the
AFNS model is replaced by a CIR process. There are three motivations for doing so. Firstof all, for modeling purpose, we sometimes require modeling the long-term yield as a positive
process, and it is natural for doing so. Such a motivation appears when we attempt to modelfor example Lin and Wu(2010)fora survey on term stmcture models.
$\overline{lSee}$123
negative correlation between default intensity and interest rate factors in the reduced-form credit
risk modeling framework, using AFNS as a term structure of interest rates model. Second, we
happen to realize the fact that although changing the dynamic of the first factor would change
its loading in the yield function, such loading can be approximated by one for short time-tomaturity, and is a slightly decreasing function of time-to-maturity, and so the resulted model
can in fact be considered an approximation to AFNS. Finally, whereas AFNS is a subclass oftheclass of affine DTSMs of Dai and Singleton (2000), the modified model (MAFNS) is
in fact a subclass of theclass of affine DTSMs. This has important implication because in
the same manner that AFNS has the edge, in terms of empirical performance and tractability,
over any other models from the sameclass thanks to its $NS$-specified structures, it could
happen that MAFNS has the edge over any other models that belong to theclass in thatregard.
Empirically, we explore the performance and properties of our model using two sets of interest
rates data. One is the $US$ treasury zero-coupon yield data based on Svensson’s method, andthe other one is the Japan Government Bond (JGB) zero-coupon yield data based on Steeley’s
method. We compare our proposed model, in terms of in-sample fitting, and out-of-sample
forecasting performance, with the independent-factor DNS, and AFNS models. We also examine
whether changing the first factor would change the economic interpretation of the latent statevariable factors. For the $US$ treasury data, we find that our model outperforms AFNS in termsof in-sample fit, but that is offset by poorer forecasting performance relative to AFNS. However,
our $mo$ del still maintains some good properties, which includes the fact that one of its factors
can be interpreted as long-term yields, and one as slope, and that the long-term yield factor has
the most persistent dynamic. For JGB data, we do not find significant difference between our
MAFNS and AFNS, which means that both models have similar properties and performances.
Finally, we give an example of using our model in joint credit risk pricing model within the
reduced-form framework.
This paper contributes to the literature by proposing a DTSM that for modeling purpose, canbe used as an alternative to the existing AFNS model. With our model as a DTSM of interest
rates, we are able to model a joint credit risk pricing model that captures some importantstylized facts and fulfills a theoretical requirement. In addition, to our knowledge, this is thefirst study to use the AFNS model on Japan data, and also the first paper to propose using an$NS$ -type model in credit
risk pricing.
The organization of the paper is as following. Section 2 provides an overview of the $NS$
type models. In particular, the $NS$ , DNS, and AFNS, as well as other related models arebriefly reviewed. Section 3 presents our modified AFNS (later, MAFNS) model. Its empiricalperformance and properties in comparison to the DNS and AFNS models, as well as economic
interpretation of its factors, are examined in Section 4. Section 5 illustrates with example how
our proposed model can be used in reduced-form credit risk pricing model to capture some
important stylized facts. Section 6 concludes the paper. Proof of proposition 1 and other details
are illustrated in the appendix.
$\mathbb{A}_{0}(3)$$\mathbb{A}_{1}(3)$$\mathbb{A}_{0}(3)$$\mathbb{A}_{1}(3)$-2Overview of the Nelson-Siegel type models

The Nelson-Siegel model has been extensively used by central banks and policy makers to estimate zero-coupon yield curve. The fixed-income portfolio manager uses these models to calculate
the interest rate risk in order to immunize their portfolios. According to BIS (2005), the countries
whose central bank uses $NS$ -type models for zero-coupon yield curve fitting, include Belgium,
Finland,, Germany, Italy, Norway, Spain and Switzerland. Their popularities can probably be attributed to at least 3 important characteristics. First, they are easy to estimate.
$\mathbb{R}ance$

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Second, the factors in these models have intuitive economic interpretation, namely level, slope,and curvature. Finally, the most important thing is that these models exhibit good empiricalperformance.
In this section, the $NS$ -type models are briefly reviewed. In the next section, we present a
modified version of he AFNS model which is the model proposed in this paper.

The $NS$ model
In the $NS$ model of Nelson and Siegel (1987), the zero-coupon yield with time-to-maturity
is given by the functional form,$\tau\geq 0$$y( \tau)=\alpha_{1}+\frac{1-e^{-\lambda\tau}}{\lambda\tau}\alpha_{2}+[\frac{1-e^{-\lambda\tau}}{\lambda\tau}-e^{-\lambda\tau}]\alpha_{3},$which is correspondent to the forward rate of the form
$f(\tau)=\alpha_{1}+e^{-\lambda\tau}\alpha_{2}+\lambda\tau e^{-\lambda\tau}\alpha_{3}.$is interpreted as the level
The instantaneous short rate is then $r=f(O)=\alpha_{1}+\alpha_{2}$ , where
easilyThiscanseen by letting the timebe
parameter, which represents the long-term yield.
to-maturity go to infinity at which the value of the long-term yield is equal to . On thecan be interpreted as negative slope of the zero-coupon yield curve, since it is theother hand,. Positive (resp. negative)
difference between the short rate and the long term rate
resp. $>0)$ , (i.e., the long term interest rate is higher (resp. lower) than the shortslope
term rate) corresponds to the normal (resp. inverted) yield curve which indicates investors’
expectation of future economic growth (resp. downturn). Meanwhile, when the slope is zero(the long term rate is equal to the short term rate), the yield curve can be flat or humped, in
is interpreted
either case there is high uncertainty about future economic condition. Also,
as the curvature of the term structure over the medium term. The term structure is concave. Finally, is the shape parameter which determines the, and convex when
when
speed of decay of the second and third loadings. It also determines at which maturity the factor
loading on the curvature parameter reaches its maximum value. The higher the value of , thefaster the speed of convergence towards zero of the second and third loadings, and thus leaves
to describe the long term rate.
only the first parameter$\alpha_{1}$$\alpha_{1}$$\tau$$\alpha_{2}$$(\alpha_{2}=r-\alpha_{1})$$(\alpha_{2}<0$$\alpha_{3}$$\alpha_{3}<0$$\alpha_{3}>0$$\lambda$$\lambda$$\alpha_{1}$The DNS model
The $NS$ model is static because level, slope, and curvature parameters are not time-varying.
Diebold and Li (2006) extend this model by introducing time-varying factors instead of the. The zero-coupon yield at time with time-to-maturity is then of theparametersfunctional form:$t$$\alpha_{1},$$\alpha_{2},$$\tau$$\alpha_{3}$$y_{t}( \tau)=L_{t}+\frac{1-e^{-\lambda\tau}}{\lambda\tau}s_{t}+[\frac{1-e^{-\lambda\tau}}{\lambda\tau}-e^{-\lambda\tau}]C_{t}$wherecan be interpreted as the level, the slope, and the curvature factors and can beassumed to have a Vector Auto-Regressive (VAR) dynamic. is a constant parameter. Its value
is estimated to range between 0.5 and 1 when the time-to-maturity is measured in year for $US$data.
$L_{t},$ $S_{t},$ $C_{t}$$\lambda$

The AFNS model125
Both $NS$ and DNS show great empirical performance; nevertheless, as shown by Bj\"ork andChristensen (1999), there is no affine arbitrage-free model consistent with these models. In
other word, they do not satisfy the theoretical arbitrage-free condition.2
Chistensen et al. (2011) extend the result of Diebold and Li (2006) by deriving an arbitragefree $NS$ model (AFNS), where the yield equation takes the form of the DNS $mo$ del with anadditional term they refer to as an arbitrage-free adjustment term. In AFNS, the zero-coupon
yield curve equation now becomes$y_{t}( \tau)=X_{1,t}+\frac{1-e^{-\lambda\tau}}{\lambda\tau}X_{2,t}+[\frac{1-e^{-\lambda\tau}}{\lambda\tau}-e^{-\lambda\tau}]X_{3,t}-\frac{A(\tau)}{\tau},$

where $-A(\tau)/\tau$ is the deterministic arbitrage-free adjustment term. Note that AFNS differsfrom DNS only by this term.
They show empirically that this model outperforms the DNS model for out-of-sample forecast. In addition, though the family of AFNS models is just a subfamily of themodels asclassified by Dai and Singleton (2000), by imposing parameter restrictions to obtain the structureof the Nelson-Siegel, it is not only empirically tractable, but also provides better out-of-sample
performance.
$\mathbb{A}_{0}(3)$Other $NS$ -type models
The Svensson (1995) yield curve model is an extension of the $NS$ model, and thus can be considered as a family of $NS$ -type. Whereas the original $NS$ model has only one curvature, the Svensson
has two curvatures and so it fits the long-term yield better than the $NS$ model. Christensen etal. (2009), in their attempt to derive another arbitrage-free version of Svensson (1995) models,
derive a generalized AFNS (AFGNS) model in which two additional factors, one for slope andthe other one for curvature, are included. They show empirically that their model provides
better in sample fit than any other corresponding dynamic models that do not rule out thearbitrage opportunity.

3
3.1

$A$

modified version of the AFNS model

The modified AFNS model (MAFNS)In this section, we present a modified model of the AFNS model in which its Gaussian level factor
is replaced by a CIR process. Following Christensen et al. (2011), we start from the standard
arbitrage-free affine term structure of Duffie and Kan (1996). To represent an affine diffusion
process, define a filtered probability space, where the filtration
satisfies
the usual condition. The short rate is assumed to be an affine function as follow:$(\Omega, \mathcal{F}, (\mathcal{F}_{t})_{t\geq 0}, \mathbb{Q})$$r_{t}=\rho_{0}(t)+\rho_{1}(t)’X_{t}$$(\mathcal{F}_{t})_{t\geq 0}$,(3.1.1)

where
:
, and
:
are bounded continuous functions, and $T\geq 0$ isa finite time horizon. The state variable X is an -dimensional Markov process defined on a
, satisfying the following Stochastic Differential Equations (SDEs):
subset$\rho_{0}$$[0, T]arrow \mathbb{R}$$\rho_{1}$$[0, T]arrow \mathbb{R}^{n}$$n$$M\subset \mathbb{R}^{n}$$dX_{t}=K^{\mathbb{Q}}(t)(\theta^{\mathbb{Q}}(t)-X_{t})dt+\Sigma(t)\sqrt{S(t,X_{t})}dW_{t}^{\mathbb{Q}}$,(3.1.2)2However, Coroneo, Nyholm, and Vidova-Koleva (2008) show with 95% confident interval that the $NS$ model
is compatible with the arbitrage-freeness.

126:
are bounded continuous::
is diagonal with the ith element as an affine function
functions, and::are bounded, continuous:of the formof
. From now on, the squareis an -dimensional standard Brownian Motion infunction.
root of any diagonal matrices will denote diagonal matrices with square root on every diagonal
where$K^{\mathbb{Q}}$$[0, T]arrow \mathbb{R}^{n\cross n},$$S$$\theta^{\mathbb{Q}}$$W_{t}^{\mathbb{Q}}$$[0,T]arrow \mathbb{R}^{n\cross n}$$\Sigma$$[0, T]\cross Marrow \mathbb{R}^{nxn}$$\delta_{0}^{i}(t)+\delta_{1}^{i}(t)’X_{t},$$X_{t}$$[0,T]arrow \mathbb{R}^{n},$$\delta_{0}^{i}$$[0, T]arrow \mathbb{R},$$[0, T]arrow \mathbb{R}^{n}$$\delta_{1}^{i}$$\mathbb{R}^{n}$$n$elements.
To obtain the Nelson-Siegel structure in the sense that the factor loadings are identical to
the $NS$ model, Christensen et al. (2011) gives specification of a 3-factor model as follow:$r_{t}=X_{1,t}+X_{2,t}$(3.1.3),.

$d(\begin{array}{l}X_{1,t}X_{2,t}X_{3,t}\end{array})=(\begin{array}{lll}0 0 00 \lambda -\lambda 0 0 \lambda\end{array})[(\theta_{1}^{\mathbb{Q}}\theta_{2}^{\mathbb{Q}}\theta_{3}\mathbb{Q})-(\begin{array}{l}X_{1,t}X_{2,t}X_{3,t}\end{array})]dt+\Sigma d(W_{1t}^{\mathbb{Q}}W_{2}^{\bigotimes_{t}}W_{3}^{\mathbb{Q}^{t}})$(3.1.4)This model is just a class of 3-factor Gaussian model. In this paper, we modify this model bywith a CIR process. In Proposition 1 below, we derive the zeroreplacing the first factor
coupon yield curves of our model.
$X_{1}$

Proposition 1. (MAFNS)
Assume the shon rate is given as in (3.1.3), and $X_{t}=(X_{1,t}, X_{2,t}, X_{3,t})$ solve the following SDEs:$d(\begin{array}{l}X_{1,t}X_{2,t}X_{3,t}\end{array})=(\begin{array}{lll}\kappa_{1}^{\mathbb{Q}} 0 00 \lambda -\lambda 0 0 \lambda\end{array})[(\begin{array}{l}\theta_{l}^{\mathbb{Q}}00\end{array})-(\begin{array}{l}X_{1,t}X_{2,t}X_{3,t}\end{array})]dt$,(3.1.5)$=\exp\{A(T-t)+B_{1}(T-t)X_{1,t}+B_{2}(T-t)X_{2,t}+B_{3}(T-t)X_{3,t}\}$ ,(3.1.6)$+(\begin{array}{lll}\sigma_{11} 0 0\sigma_{21} \sigma_{22} \sigma_{23}\sigma_{3i} \sigma_{32} \sigma_{33}\end{array}) (\sqrt{X_{1,t}}00 001 001)d(W_{1}^{\mathbb{Q}}W_{2}^{\mathbb{Q}^{t}}W_{3,t}^{\copyright^{t}})$where$\kappa_{1}^{\mathbb{Q}}>0,$$\theta_{1}^{\mathbb{Q}}>0,$$\lambda>0$. Then the zero-coupon bond $pri$ ces are given by$P(t, T)=\mathbb{E}^{\mathbb{Q}}[e^{-\int_{t}^{T}r_{u}du}|\mathcal{F}_{t}]$where$B_{1}( \tau)=-\frac{2(1-e^{-\eta\tau})}{\eta+\kappa_{1}^{\mathbb{Q}}+(\eta-\kappa_{1}^{\mathbb{Q}})e^{-\eta\tau}}$$B_{2}( \tau)=-\frac{1-e^{-\lambda\tau}}{\lambda}$(3.1.7),(3.1.8),$B_{3}( \tau)=-\frac{1-e^{-\lambda\tau}}{\lambda}+\tau e^{-\lambda\tau}$(3.1.9),$A( \tau)=(K^{\mathbb{Q}}\theta^{\mathbb{Q}})’\int_{0}^{\tau}B(t)dt+\frac{1}{2}\sum_{j=2}^{3}\int_{0}^{\tau}(\Sigma’B(t)B(t)’\Sigma)_{jj}dt$,(3.1.10)and the zero-coupon yield curves are$y(t, T)= \frac{2(1-e^{-\eta(T-t)})}{(T-t)(\eta+\kappa_{1}^{\mathbb{Q}}+(\eta-\kappa_{1}^{\mathbb{Q}})e^{-\eta(T-t)})}X_{1,t}+\frac{1-e^{-\lambda(T-t)}}{(T-t)\lambda}X_{2,t}$$+[ \frac{1-e^{-\lambda(T-t)}}{\lambda}-e^{-\lambda(T-t)}]X_{3,t}-\frac{A(T-t)}{T-t}$,(3.1.11)

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with

$\eta=\sqrt{(\kappa_{1}^{\mathbb{Q}})^{2}+2\sigma_{11}^{2}}.$

Proof. see Appendix A. 1.
As we can see, the factor loadings on the 2nd and 3rd factors are identical to those of the
AFNS model, while the loading on the lst factor is no longer 1. It is equal to$\frac{2(1-e^{-\eta(T-t)})}{(T-t)(\eta+\kappa_{1}^{\mathbb{Q}}+(\eta-\kappa_{1}^{\mathbb{Q}})e^{-\eta(T-t)})}$.

(3.1.12)

However, when the time-to-maturity $T-t$ or is small, (3.1.12) can be approximated by 1.
Figure 1 plots the factor loadings of MAFNS for the value of parameter estimates of $US$ treasury
and JGB zero-coupon yield data. As we can see, for both sets of data, the lst factor loading
decays slightly from 1. Therefore, MAFNS may no longer be an $NS$ -type model, rather it is anapproximation of the $NS$ -type model. Two things can be said about this. First, since it is anapproximation, it may still preserve the property that the lst factor is the long-term yield, the
2nd and 3rd are respectively the slope and the curvature of yield curve. Second, since (3.1.12)
is more flexible than 1, it may help improve fitting performance.
$\eta$

Figure 1: Factor Loadings of MAFNS(a) $US$ TREASURY(b) JGB$\tau$$\tau$

Also note that there is one factor that enters the volatility structure of (3.1.5). This meansthat MAFNS is in fact a subclass of theclass of affine DTSMs. In the same mannerthat AFNS is the $NS$ -specifiedclass of affine DTSMs, MAFNS can also be considered as
the $NS$ -specifiedclass of affine DTSMs. In Appendix A.2, we show using an invariant
transformation that imposing some parameter restrictions on the canonical representation ofwill give us the MAFNS model.
$\mathbb{A}_{1}(3)$$\mathbb{A}_{0}(3)$$\mathbb{A}_{1}(3)$$\mathbb{A}_{1}(3)$3.2

The change of measure

To estimate the $mo$ del, we also need to know the distribution of the state variables under thereal world probability measure . The link between theand themeasures is given by themeasure change$\mathbb{P}$$\mathbb{P}-$$diW_{t}^{\mathbb{Q}}=dW_{t}^{\mathbb{P}}+\Lambda_{t}dt,$$\mathbb{Q}-$128
represents the risk premium. Following Duffee (2002), we consider the essentially risk
wherepremium specification of the form$\Lambda_{t}$$\Lambda_{t}=\sqrt{S_{t}}\hat{\Gamma}_{1}+\sqrt{S_{\overline{t}}}\hat{\Gamma}_{2}X_{t},$where$\hat{\Gamma}_{1}$is a $3-\dim$ vector,$\hat{\Gamma}_{2}$is a$3\cross 3$

matrix, and$S_{\overline{t}}=(\begin{array}{lll}0 0 00 l 00 0 1\end{array}).$Thus, we have$\Sigma\sqrt{S_{t}}\Lambda_{t}=(\begin{array}{lll}\sigma_{11}X_{1,t} 0 0\sigma_{21}X_{1,t} \sigma_{22} \sigma_{23}\sigma_{31}X_{1,t} \sigma_{32} \sigma_{33}\end{array})\hat{\Gamma}_{1}+(\begin{array}{lll}0 0 00 \sigma_{22} \sigma_{23}0 \sigma_{32} \sigma_{33}\end{array})\hat{\Gamma}_{2}X_{t}$$=\Gamma_{0}+\Gamma_{1}X_{t},$where$\Gamma_{0}$is a $3-\dim$ vector, and$\Gamma_{1}$is a$3\cross 3$

matrix.

Under this specification, the dynamic underform$\mathbb{P}$of the state variables$X_{t}$is still affine of the(3.2.1),$dX_{t}=K^{\mathbb{P}}(\theta^{\mathbb{P}}-X_{t})dt+\Sigma\sqrt{S_{t}}dW_{t}^{P}$

with

3.3

$K^{P}=K^{\mathbb{Q}}-\Gamma_{1}$, and$K^{\mathbb{P}}\theta^{P}=K^{\mathbb{Q}}\theta^{\mathbb{Q}}+\Gamma_{0}.$The independent-factor model

In this study, we focus on the independent-factor models because they are easy to estimate andare proven to have outperformed correlated-factor models. In analogy with the independentfactor DNS and AFNS models, we specify the MAFNS model so that the factors are independent, andare specified as a diagonal matrix as follow:
under -measure, that is,$K^{\mathbb{P}}$$\mathbb{P}$$\Sigma$$K^{P}=(\begin{array}{lll}\kappa_{1}^{\mathbb{P}} 0 00 \kappa_{2}^{F} 00 0 \kappa_{3}^{\mathbb{P}}\end{array}), \Sigma=(\begin{array}{lll}\sigma_{1} 0 00 \sigma_{2} 00 0 \sigma_{3}\end{array}),$where we assume $K^{P}>0$ , so that the process is stationary.
Under this specification, the deterministic term is given by$A( \tau)=\sigma_{2}^{2}[\frac{1}{2\lambda^{2}}-\frac{1}{\lambda^{3}}\frac{1-e^{-\lambda\tau}}{\tau}+\frac{1}{4\lambda^{3}}\frac{1-e^{-2\lambda\tau}}{\tau}]$$+ \sigma_{3}^{2}[\frac{1}{2\lambda^{2}}+\frac{1}{\lambda^{2}}-\frac{1}{4\lambda}\tau e^{-2\lambda\tau}-\frac{3}{4\lambda^{2}}e^{-2\lambda\tau}-\frac{2}{\lambda^{3}}(1-e^{-\lambda\tau})+\frac{5}{8\lambda^{2}}\frac{1-e^{-2\lambda\tau}}{\tau}]$$- \kappa_{1}^{\mathbb{Q}}\theta_{1}^{\mathbb{Q}}[\frac{2}{\tau\sigma_{1}^{2}}\log(\frac{\kappa_{1}^{\mathbb{Q}}+\eta+(\eta-\kappa_{1}^{\mathbb{Q}})e^{-\eta\tau}}{2\eta})+\frac{2}{\eta+\kappa_{1}^{\mathbb{Q}}}]$.

(3.3.1)

Unless mentioning, from now on we will speak of AFNS and MAFNS when referring totheir independent-factor specifications. Since the conditional and unconditional expectationand variance of the state variables under -measure are useful for the estimation process, westate these in Appendix A.3 for easy reference.3$\mathbb{P}$amore general affine process, see for example, Duan and Simonato (1999) on how to calculate conditional$\overline{3For}$mean and variance.

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4

Empirical analysis

4.1

Data

In this paper, we estimate the models using interest rates data of the $US$ and Japan. For $US$
data, we use monthly Treasury zero-coupon yield covering the period from January 1987 to
December 2002. This data is constructed by Gurkayna, Sack and Wright (2006) based on the
Svensson $(1995)$ ’s model, and is obtained from the Federal Reserve Board website.4 The zerocoupon yields to be included in our estimation are those at maturity spectrum of 3/12, 6/12,
9/12, 1, 2, 3, 4, 5, 7, 8, 9, 10, 15, 20, 30 years. The data contains zero-coupon yields withtime to maturity ranging from 1 year to 30 year as well as the Svensson parameter estimates.
Using these parameter estimates, we are able to calculate the 3-month, 6-month, and 9-monthzero-coupon yields. The data is available in daily frequency. Choosing estimates on the last
day of month as the monthly estimates, we obtain a total of 192 observations for monthly data
sample. We also note that the sample period we use is the same as in the previous studies.
For Japan data, we use JGB zero-coupon yields estimated using the method proposed by
Steeley (1991). Kikuchi and Shintani (2012) show that the Steeley’s method is the best among
popular yield-curve estimation methods based on some considered criteria.5 The daily data
based on this method is downloaded from Bank of Japan $(BoJ)$ ’s website. We convert daily
data to weekly data by choosing estimates on Thursday (if available, and on Wednesday ifnot) as the weekly estimates. We skip those weeks where both the estimates on Thursday andWednesday are not available. The time-to-maturity spectrum included in our study are 0.5, 1,
1.5, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 15, 18, and 20 year, and the sample consists of 692 weekly
observations covering the period from January 1999 to December 2012.
Figure 2 are plots of term structure of $US$ treasury and JGB zero-coupon yields. Movementof the term structure of JGB yields is subject to zero-interest rate policy (ZIRP) started fromFebruary 1999. It was lifted temporarily during the ICT bubble in 2000, and restarted after
the burst of the ICT bubble in 2001. Under this monetary easing, the long-term rate began todecrease to 0.43%, the lowest rate ever recorded in the history of the world, in 2003. The ZIRP
was then lifted again in 2006 at the time of economic recovery, and restarted in December 2008
at the time it was introduced in the $US$ , during the world financial crisis.

4.2

Estimation method

In this paper, the models are estimated using maximum likeliho $od$ estimation method based on
the Kalman filter. The MAFNS model can be formulated in a state space form as follow.
System Equations:$X_{t}=(I-e^{-K^{P}\Delta t})\theta^{\mathbb{P}}+e^{-K^{1P}\Delta t}X_{t-\triangle t}+u_{t}$.

(4.2.1)

Measurement Equations:$\{\begin{array}{l}y_{t}(\tau_{1})y_{t}(\tau_{N})\end{array}\}=\{\begin{array}{l}-\frac{A(\tau_{1})}{\mathcal{T}1}-\frac{A(\tau_{N})}{\tau_{N}}\end{array}\}+\{\begin{array}{lll}-\frac{B_{1}(\tau_{1})}{\tau_{1}} -\frac{B_{2}(\tau_{1})}{\mathcal{T}1} -\frac{B_{3}(\tau_{1})}{\tau_{1}}-\frac{B_{1}(\tau_{N})}{\tau_{N}} -\frac{B_{2}(\tau_{N})}{\tau_{N}} -\frac{B_{3}(\tau_{N})}{\tau_{N}}\end{array}\}X_{t}+\epsilon_{t}$.

(4.2.2)

4The data is updated periodically, and is available at http: $//www.$ federalreserve. $gov/pubs/feds/2006.$
5The criteria considered are 1) Non-negativity of interest rates 2) Ruling out abnormal values, 3) Goodness-of-fit to the market prices, and 4) Little unevenness in zero-coupon yield curves.

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Figure 2: Term Structure of Zero-coupon Yields(b) JGB(a) $US$ TREASURY$\frac{\tau}{\frac{Q)}{\succ{\}}}$$\frac{\tau}{\succ\underline{q)}}$denotes infinitesimal time
are defined as in $(3.1.7)-(3.1.10),$
whereinterval, and the variance of the measurement errors $Var(\epsilon_{t})=$ H. In this paper, we assume
that $H$ is a diagonal matrix, and the filtering algorithm is initialized with the unconditional
is approximated by a Gaussian process
expectation and variance. Note that the CIR processduring the estimation process.
negativetobecould turn out
as in (4.2.1). This means that
When that happens, we put it equal to zero, and let the filtering algorithm continues.
$\triangle t$$B_{1}(\tau),$ $B_{2}(\tau),$ $B_{3}(\tau),$ $A(\tau)$$X_{1}$$X_{1}$

4.3

Empirical results

We argue that our proposed model is not only closed related to DNS and AFNS, but can also
be used as an altemative model. To justify this argument, we conduct empirical analysis by
estimating these models with the term structure of $US$ Treasury and JGB zero-coupon yield
data, and then compare, in terms of performance and properties, our MAFNS model with the
DNS and AFNS models. As mentioned earlier, we only focus on the independent-factor models.
In-sample and out-of-sample performances are examined, and then to see whether modifying thefirst factor as in MAFNS would change the economic interpretation of the model’s state variables,
we calculate correlations of the estimates of these factors with long-term rates, empirical slopes,
and empirical curvatures. First, the results for $US$ treasury yields, and then the results for JGB
yields are presented subsequently.

4.3.1

Results for $US$ treasury yield data

likelihood values are reported in Table 1 for the three models.
The parameter estimates and
The “Estimates” columns show parameter estimates, and the “Std. Err.” columns show standard
errors. The estimated value of the decay parameter is 0.4378 for DNS, and 0.361 for AFNS
model. These values are lower than values estimated in previous studies, which range between
0.5 and 1; however, it is not surprising because in our study, Svesson’s method is used as yield
curve fitting method, whereas in previous studies Unsmoothed Fama Bliss method is used. We
may expect that smaller value for the decay rate results in better fit for long-term yield thanshort-term yield. For MAFNS model, is estimated to be 0.6282.
$\log$$\lambda$$\lambda$

131
Table 1: Parameter Estimates and Standard Errors ( $US$ TREASURY)

Notes: The
likelihood value of each model is respectively 16778, 16576, and 17279, and the BIC is-33425,-33022, and-34421 respectively. Since lower value of BIC suggests better fit, based on this criterion, MAFNS has
the best fit to the data.
$\log$The estimate of the long run mean for DNS’s level factor is 0.0758, which is slightly higher
than the sample average of long-term yields: 0.069, 0.0715, 0.0723, and 0.0715 for the 10, 15, 20,
and 30-year yields respectively. On the other hand, the estimate of the long-run mean for AFNS’sfactor is 0.0816, a magnitude higher than the sample average. This result may look skepticalat first; however, recall that unlike in DNS where the level factor
alone represents the longterm yield, in AFNS, the long-term yield is represented byplus a deterministic adjustment
term. This deterministic adjustment term is a downward function of time-to-maturity so thatthe average of the long-term yield estimated by the model is about the sample average. Thisis confirmed in Figure 3 (a) which shows a plot of AFNS’s deterministic yield adjustment term
as a downward function of time-to maturity. The same thing can be said about the result of
MAFNS with the estimate of the long-run mean ofof 0.0517. In MAFNS, the long-term yield
is a linear transformation of the factor, with the loading as coefficient and the deterministic
adjustment term as the intercept. Figure 3 (b) shows plot of MAFNS’s deterministic adjustment
term as upward function of time-t maturity. In fact, the factor
in both AFNS and MAFNSmodels can be considered as the long-term factor because as will be seen later, it captures the
movement of long-term yields.
It is a stylized fact for yield curve dynamics that long-term yields are more persistent thenshort-term yields. To see how persistent ur considered models’ factors are, we examine the parametersfor DNS, and the parameters
for AFNS and MAFNS. For DNS, theseparameters are estimated to be [0.975, 0.9836, 0.8218], suggesting that the most persistent is theslope factor, then the level factor, and the curvature is the least persistent. For AFNS, the estimates ofis diag([0.9900, 0.9797, 0.9009]),6 which means that the long-term factor is themost persistent, while the slope is less persistent. Finally, in MAFNS the long-term yield has themost persistent dynamic since estimates ofin our $mo$ del is diag([0.9943, 0.9806, 0.8941]).
this result, we see that our model does capture this stylized fact of the term structure ofinterest rates.
$X_{1}$$X_{1}$$X_{1}$$X_{1}$$X_{1}$$(\succ$$X_{1}$$0$$[a_{1}, a_{2}, a_{3}]$$e^{-\kappa^{P}\triangle t}$$e^{-\kappa^{P}\triangle t}$$e^{-\kappa^{P}\triangle t}$$\mathbb{R}om$

In-sample fitting performance
6Where diag$(\nu)$

denotes a diagonal matrix with a diagonal vector$\nu.$

132
Figure 3: Deterministic Adjustment Term ( $US$ TREASURY)(a) AFNS(b) MAFNS$\tau$$\tau$

likelihood value of DNS is 16778 versus 16576 of AFNS. Since both models have theThesame number of parameters, this suggests that DNS better fitted to the data than AFNS. This
is in line with Christensen et al.(2011). On the other hand, MAFNS has the highest likelihood
value of all, but it has one parameter more than the other two models. However, the value of
BIC, which places penalty on model complexity, suggests that MAFNS has the best fit to thedata among the three models. That the overall performance of MAFNS is better than DNS, and
AFNS can also be confirmed in Table 2, which reports value of RMSE in basis points for different
time-to-maturities and the ratios of each model’s RMSE to AFNS’s RMSE, and in Figure 4,
which shows plots of RMSE versus time-to-maturity for the three models. From this table, we
also notice that MAFNS appears to fit better than AFNS for short- and medium-term yields.
In addition, Figure 5 shows plots of sample average yield curve, and the models’ fit average
yield curves. As we can see, on average, these models appear to provide good fit especially forshort and medium term. However, DNS does seem to have relatively hard time fitting the very
long-term yield compared to AFNS and MAFNS. This may be because DNS does not have yield
adjustment term. On the hand, MAFNS has the best fit for long-term yields on average.
$\log$

Figure 4: RMSE ( $US$ TREASURY)

Figure 5: Average Yield Curve ( $US$ TREASURY)$\tau$$\tau$

133
Table 2: RMSE ( $US$ TREASURY)

Notes: ‘RMSE” columns report root mean square errors (RMSE) in basis points for different time-to-maturity foreach model. “DNS/AFNS”, “AFNS/AFNS”, and “MAFNS/AFNS” columns respectively report ratios of DNS’s,
AFNS’s, and MAFNS’s RMSE to AFNS’s RMSE.

Out-of-sample forecasting performance
To examine the out-of-sample forecasting performance of our model as compared to therelated models, we calculate l-month-, 3-month-, 6-month-ahead forecasts. The forecasting
window is the period from Sept. 1999 to Dec. 2002 having respectively 42, 40, and 37 observations for l-month-, 3-month-, and -month-ahead forecasts. The estimation window starts from
Jan. 1987, and we adopt the expanding estimation window approach, that is, to make forecasts
at a time point, we use data starting from Jan. 1987 up to that time. Table 3 reports root
mean squared forecast errors (RMSFE) of AFNS and ratios of DNS’s and MAFNS’s RMSFE to
AFNS’s RMSFE. Figure 6 are plots of RMSFE for different forecasting horizons. The result isobvious. For short and medium terms, the three models’ performance is not much different fromeach other. However, it turns out that AFNS has the best performance among the three models
for forecasting long-term yields. The result for DNS and AFNS is consistent with Christensent
et al. (2011) who show that the independent-factor AFNS outperforms the independent-factor
DNS for out-of-sample forecasts. That our model performs the worst for forecasting long-termyields might have been due to over-fitting as a result of introducing flexibility to the loading
factor ofand the deterministic adjustment term. As we can see in Figure 3, the magnitudeof the MAFNS’s deterministic adjustment term is so high compared to the AFNS’s.
$6$$X_{1}$

Economic interpretation of the latent variables
Finally, we want to know how modifying the loading factor of AFNS affects the economic
interpretation of the state variables, namely level, slope, and curvature. To see this, we calculate
correlations between the estimates of state variables and the empirical levels, slopes, and curvatures. Table 4 reports this result for the three models. Panel A reports correlations between

134

Table 3: RMSFE ( $US$ TREASURY)

Notes: “i-m” is short form for -month. Column 5 to 7 are l-month-, 3-month-, and 6-month-ahead root meansquared forecast errors (RMSFE) of AFNS in basis points. Column 2 to 4 are ratios of DNS’s RMSFE to AFNS’s
RMSFE. Column 8 to 10 are ratios of MAFNS’s RMSFE to AFNS’s RMSFE.
$i$

Figure 6: Out-of-Sample Forecasting Performance ( $US$ TREASURY)(a) l-month-ahead Forecasts(b) 3-month-ahead Forecasts(c) 6-month-ahead Forecasts$\tau \tau \tau$135
the estimates ofof each model and long-term (10-year, 15-year, 20-year, and 30-year) yields.
Panel reports correlations between the estimates ofof each model and negative empiricalslopes defined as the difference between a short and a long-term (10-year, 15-year, 20-year, and30-year) yield with 0.25-year yield chosen as the short-term yield. Finally, Panel reports correlations between the estimates ofand empirical curvatures defined as 2 times medium-termyield (4-year yield) minus a short-term (1-year) yield and minus a long-term (10-year, 15-year,
20-year, and 30-year) yield. As expected, for DNS and AFNS,, and
are respectively
highly correlated with empirical levels, slopes, and curvatures. On the other hand, for MAFNS,
andrespectively have high correlations with long-term yields and empirical slopes, while
appears to have weak correlations with empirical curvatures; the highest correlation is only
about 73%. This suggests that the factor
in the MAFNS may lose its interpretation ascurvature.
To sum up, in this section we have examined the performance and properties of our proposed$mo$ del relative to the related models when used as a term
stmcture of $US$ treasury yields. Ourmodel appeared to have better fit to the data than the related models, but this was offset bypoorer forecasting performance especially for very long-term yields as compared to the related$mo$ dels. However, our model to some extent
inherits some go $od$ properties from its original AFNS
model. This includes the fact that its long-term yield factor has strong persistent dynamic, andtwo of its factors can be interpreted as level, and slope. In Section 5 below, we will show byusing an example that our model can be used in place of AFNS for modeling purpose. For now,
we examine our model’s performance and properties when used to describe the term structureof JGB yields.
$X_{1}$$B$$X_{2}$$C$$X_{3}$$X_{1},$$X_{1}$$X_{2}$$X_{3}$$X_{2}$$X_{3}$$X_{3}$

Table 4: Correlations between the state variables and economic variables ( $US$ TREASURY)$\frac{PanelA:CorrelationsbetweenX_{1}andlong-termyields}{10y15y20y30y}$$X_{1}^{DNS}$$X_{1}^{AFNS}$$X_{1}^{MAFNS}$

0.90200.88870.9384

0.94230.93440.96690.9512
0.94490.97199297
0.91550.9557$0$$\frac{}{}\frac{Pane1B:CorrelationsbetweenX_{2}andnegativeempirica1.s1opes}{X_{2}^{DNS}0.96490.9792098050.9657[0.25y-10y][0.25y-15y][0.25y-20y][0.25y-30y]}$$X_{2}^{AFNS}$$X_{2}^{MAFNS}$

0.94380.98580.96550.98780.96990.98390.95410.9703$\frac{PanelC:Corre1ationsbetweenX_{3}andempirica1curvatures}{[2^{*}(4y)-1y-10y][2^{*}(4y)-1y-15y][2^{*}(4y)-1y-20y][2^{*}(4y)-1y-30y]}$$X_{3}^{DNS}$$X_{3}^{AFNS}$$X_{3}^{MAFNS}$

0.83250.92790.45050.96880.97070.71410.93950.91510.73640.77500.69070.6701

Notes: Panel A reports correlations between the estimated path of the factorof each model and long-termyields (10-year, 15-year, 20-year, and 30-year yields). Panel reports correlations between the estimated path ofof each model and negative empirical slopes defined as the difference between a short and a long-term (10-year,
15-year, 20-year, and 30-year) yield with 0.25-year yield chosen as the short-term yield. Finally, Panel reports
correlations between the estimates ofand empirical curvatures defined as 2 times medium-term yield (4-yearyield) minus a short-term (1-year) yield and a long-term yield.
$X_{1}$$B$$X_{2}$$C$$X_{3}$

136
4.3.2

Results for JGB yield data

In this section, we examine the performance and properties of these $mo$ dels for the case of JGB
zero-coupon yield data. It is important to note that, to our knowledge, our study is the firstto estimate JGB zero-coupon yields using the AFNS model. Though DNS is sometimes seenused as a term structure model of JGB zero-coupon yields in previous literatures, this type ofmodels is not popular among academic literatures because of at least two basic reasons. First,
they are Gaussian models which do not rule out the negativity of the interest rates. This is ofgreat concem especially for Japan whose standard short-term rate is near zero, as a result of
which the estimates from these models are likely to be negative. The second reason may be thefact that these models do not incorporate regime switch which is often observed in JGB interest
rates subject to ZIRP. Nevertheless, for the sake of reference, we notice the need for writing
down a few pages in the literature regarding the results of using AFNS and the related models
to describe the term structure of JGB zero-coupon yields.
The parameter estimates and standard errors are reported in Table 5. For DNS, the decayrate parameter is estimated to be equal to 0.3523, a magnitude that is often seen adopted inusual practice.7 Meanwhile, the estimates of for AFNS and MAFNS are 0.3072 and 0.2878
of DNS are equal torespectively. Furthermore, the estimates of the parameters0.9887, , 0.9807])
diag([0.9881,
of AFNS is[0.9912, 0.9882, 0.9828], and the estimates ofand of MAFNS is diag([0.9995, , 0.9914, 0.9825]). This suggests that the dynamics of the threefactors in the three models are highly persistent.
Figure 7 plots the deterministic adjustment term of AFNS and MAFNS. Just as in the case$US$treasury data, the AFNS’s deterministic adjustment term is a downward function of timeof
to-maturity, whereas the MAFNS’s is an upward function of time-to-maturity. However, unlikein the case of $US$ treasury data, the magnitude of the MAFNS’s adjustment term is not muchdifferent from the AFNS’s. This could have important consequence. As we have seen in the case
$US$ treasury, MAFNS have poor forecasting performance for long-term yields relative to AFNS.
This may be due to large adjustment term estimated by MAFNS. With small adjustment justas in JGB case, we may expect that the MAFNS’s performance and properties do not deviate
much from the AFNS’s. In fact, this is the case as we examine the results subsequently below.
$\lambda$$\lambda$$(a_{1}, a_{2}, a_{3})$$exp(-\kappa^{P}\Delta t)$

Figure 7: Deterministic Adjustment Term (JGB)$\tau$$\tau$

In-sample fitting performance
7As usual practice, the decay rate parameter$\lambda$

is sometimes assumed constant and fixed at about 0.36.

137
Table 5: Parameter Estimates and Standard Errors (JGB)

Notes: The
likelihood value of each model is respectively 67903, 68055, and 68133, and the BIC is-135636,-135940, and-136091 respectively. Since lower value of BIC suggests better fit, based on this criterion, MAFNS
has the best fit to the data.
$\log$The MAFNS model has the smallest BIC, -136091 versus-135940 of the AFNS, and-135636
of the DNS. Thus, MAFNS fits the best to the data. This is consistent with RMSE values
reported in Table 6, and plots of RMSE in Figure 8. The differences between the three though
are not so obvious. Moreover, the three models fit well for medium and long-term yields, buthappen to have hard time fitting 0.5-year yield. This may be because of the typical -shape of
JGB yield curves. The sample mean of 0.5-year yields is 0.32% larger than 0.28% and 0.31%,
the sample mean of 1-year and 1.5-year yields respectively. Average yield curves of sample data,and of the models’ fitted values are plotted in Figure 9. Although during the sample period,
JGB yield curves may assume many different shapes, we can see the sample average yield curve
takes an -shape, and we see that the three models are not able to fit the very short end.
$S$

$S$

Figure 8: RMSE (JGB)$\tau$

Figure 9: Average Yield Curve (JGB)$\tau$

Out-of-sample forecasting performance
To make out-of-sample comparison, we calculate forecasts for 4-week, 12-week, and 26-

138
Table 6: RMSE (JGB)

Notes: “RMSE” columns report root mean square errors (RMSE) in basis points for different time-to-maturity foreach model. “DNS/AFNS”, “AFNS/AFNS”, and “MAFNS/AFNS” columns respectively report ratios of DNS’s,
AFNS’s, and MAFNS’s RMSE to AFNS’s RMSE.

week forecasting horizons. The forecasting window is the period from week 3 of May 2010
to week 4 of Dec. 2011 having respectively 88, 80, and 66 observations for 4-week-, 12-week-,
and 26-week-ahead forecasts. The estimation window, which starts from the lst week of Jan.
1999, is expanded during the forecasting process. The RMSFE of AFNS and ratios of DNS’s
and MAFNS’s RMSFE to AFNS’s RMSFE are reported in Table 7. Figure 10 are plots of
RMSFE for different forecasting horizons. From Figure 10, it looks obvious that DNS has thebest forecasting performance for 12-week, and 26-week. For the 4-week-ahead forecasts, thedifferences between the three models are not obvious. Also, comparing MFANS with AFNS
across different forecasting horizons, we see that the difference between them is not so obvious.
This is in line with our expectation that MAFNS do not deviate much from AFNS because themagnitudes of the adjustment terms of both models are not much different.

Economic interpretation of the latent state variables
Same as above, to see whether the factors in our model can be interpreted as level, slope,and curvature, we calculate correlations between estimates of these factor and empirical levels,
slopes, and curvatures. Table 8 Panel A reports correlations between the estimates of the factorof each model and long-term (13-year, 15-year, 18-year, and 20-year) yields. Panel reportsof each model and negative empirical slopes definedcorrelations between the estimates of(13-year, 15-year, 18-year, and 20-year) yieldlong-term
a
and
a
short
as the difference between
Finally, Panel reports correlations between
yield.
with 0.5-year yield chosen as the short-termand empirical curvatures defined as 2 times medium-term yield -year
the estimates ofyield) minus a short-term (0.5-year) yield and a long-term (13-year, 15-year, 1-year, and 20and negative empirical slopes, andyear) yield. As we can see, high correlations between$B$$X_{1}$$X_{2}$$C$$($$X_{3}$$X_{2}$

139

Table 7: RMSFE (JGB)

Notes: Column 5 to 7 are 4-week-, 12-week-, and 26-week-ahead RMSFE of AFNS in basis points. Column 2 to
4 are ratios of DNS’s RMSFE to AFNS’s RMSFE. Column 8 to 10 are ratios of MAFNS’s RMSFE to AFNS’s
RMSFE.

Figure 10: Out-of-Sample Forecasting Performance (JGB)(a) 4-week-ahead Forecasts(b) 12-week-ahead Forecasts(c) 26-week-ahead Forecasts$0 5 10 15 20$$\tau$$\tau$$\tau$

140of theand empirical curvatures, are observed for the three models. However,between
MAFNS
ofthree models has weaker correlation with long-term yields than expected, andcorrelations.
theweakesthas$X_{1}$$X_{3}$$X_{1}$

Table 8: Correlations between the state variables and economic variables (JGB DATA)$\frac{PanelA:CorrelationsbetweenX_{1}and1ong-termyields}{13y15y18y20y}$0.75970.80790.8310.8424$\overline{X_{1}^{DN6}}$$X_{1}^{AFNS}$$X_{1}^{MAFNS}$

0.71320.66690.6416
0.59390.75810.71320.77090.7290$\frac{PanelB:Correlations.betweenX_{2}andnegaveempirica1s1opes}{[05y-13y][0.5y-5y][0.5y-18y][0.5y-20y]}$$X_{2}^{DNS}$$X_{2}^{AFNS}$$X_{2}^{MAFNS}$

0.92380.89520.87020.92380.87380.83810.91180.90250.88560.92570.91930.9031$\frac{PanelC:Corre1ationsbetweenX_{3}andempirica1curvatures}{[2^{*}(5y)-0.5y-13y][2^{*}(5y)-0.5y-15y][2^{*}(5y)-0.5y-18y][2^{*}(5y)-0.5y-20y]}$$X_{3}^{DNS}$$X_{3}^{AFNS}$$X_{3}^{MAFNS}$

0.96610.9766
0.96810.95050.9507
0.94190.95790.98140.96940.94450.96530.9533of each model and the long-term
Notes: Panel A reports correlations between the estimated path of the factor
reports correlations between the estimated path of(13-year, 15-year, 18-year, and 20-year) yields. Panelof each model and negative empirical slopes defined as the difference between a short and a long-term (13-year,
15-year, 18-year, and 20-year) yield with 0.5-year yield chosen as the short-term yield. Finally, Panel reports
and empirical curvatures defined as the 2 times medium-term yieldcorrelations between the estimated path of(5-year yield) minus a short-term (0.5-year) yield and a long-term (13-year, 15-year, 18-year, and 20-year) yield.
$X_{1}$$X_{2}$$B$$C$$X_{3}$

5

An example of using MAFNS in credit risk pricing model

After checking the performance and properties of our model relative the existing related models,
we can say that even though our model does not outperform the existing $NS$ -type models, tosome extent, it preserves the properties and performances of these models. We argue that formodeling purpose, our model can be used as an alternative model to AFNS model. In thissection we illustrate this point by giving an example of using our model in credit risk modelingwithin the reduced-form framework.
In credit risk modeling, there are some styhzed facts that are difficult to model. One of thesestylize facts that most studies tend to ignore due to modeling and implementation difficulty is
the fact that recovery rate is not constant, but time-varying and negatively correlated with default intensity. Most previous studies assume constant recovery rates. Studies that incorporate
stochastic recovery rates include Christensen (2007), Hotch and Zagst (2010), Doshi (2011),
Jaskowski (2011), among others. One other stylized fact is that default intensity is negatively
correlated with interest rates. In most practices, interest rates are assumed constant, or independent of default intensity. Studies that attempt to capture this stylized fact are for instant,
Doshi (2011), and Hotch and Zagst (2010).
Our attempt is to capture such difficult-to-model stylized facts using MAFNS as an interest

141
rate model. The reason for using $NS$ -type model is two folds. First, since one factor of suchmodels can be interpreted as slope, which is known to be an indicator of future economic growth,using $NS$ -type models, we are able to incorporate such economic indicator into our credit pricing
model. This is crucial because sub-prime loan crisis witnesses the need to incorporate future
economic indicator into credit pricing models. Ignoring such important information may have
led to underpricing of credit risk; as a consequence, the financial crisis may have occurred.
Second, since the other two factors can be interpreted as level and curvature, we are able toexplore the dynamic interaction between such economic factors and credit spread and $/or$ default
probability and $/or$ recovery rates).
In addition, the reason for preferring MAFNS to AFNS is that we are able to model correlation between interest rate factors and default intensity, and at the same time taking int
account the positivity of default intensity. For example, we may consider modeling default
intensity process as follow:
$($

$0$

$h$

$h_{t}=\rho X_{1,t}+X_{4,t};-1\leq\rho\leq 0,$$dX_{4,t}=\kappa_{4}^{Q}(\theta_{4}^{Q}-X_{4,t})dt+\sigma_{4}\sqrt{X_{4,t}}dW_{4,t}^{Q}.$Since$X_{1,t}$is a CIR process as in MAFNS, imposing the following parameter restrictions,$\kappa_{4}^{Q}=\kappa_{1}^{Q},$$\rho\theta_{1}^{Q}+\theta_{4}^{Q}>0$, and$\sigma_{4}=\sigma_{1}\sqrt{|\rho|},$it can be easily shown that the default intensity is a positive process. The negative correlation
between default intensity and interest rate is captured through the negative parameter
Also, we may consider stochastic recovery rate
model as follow:$h$$\rho.$$\varphi(.)$,$\varphi(\mathbb{X}_{t})=exp(\beta 0+\beta_{1}X_{1,t}+\beta_{2}X_{2,t}+\beta_{3}X_{3,t}+X_{5,t})$where $\mathbb{X}_{t}=(X_{1,t}, X_{2,t}, X_{3,t}, X_{4,t}, X_{5,t})$ , andcan be a Gaussian process independent of otherfactors, and can be considered as recovery rate idiosyncratic factor.
Under this model specification, first note that the default intensity and recovery rate share
a common factor, so its correlation is captured by the parameter . Moreover, correlations
between the recovery rate and interest rate factors depend on$i=1,2,3,4$ . Imposing zero
restriction on these parameters, the model reduces to a constant recovery rate model. Since,
all factors are modeled as affine processes, explicit formulas for corporate bond, and CDS pricecan be obtained by applying the transform result of Duffie, Pan, and Singleton (2000). Using
corporate bond price data and $/or$ CDS spread data, we are able to estimate the joint modelusing Markov Chain Monte Carlo (MCMC) method or Monte Carlo Particle filtering method.
Significances of these parameters can then be tested to see whether interest rate factors interact
with default intensity as well as recovery rate processes. Furthermore, some stylized facts may
also be drawn by investigating the $sign$ of these parameters.
Nonetheless, it should be noted that in the above model, recovery rate does not take thevalue between and 1. An alternative model as in Chen and Joslin (2012) for recovery rate may
take the following form in order to ensure it takes the value in $[0,1].$$X_{5}$$X_{1}$$\beta_{1}$$\beta_{i},$$0$$\varphi(\mathbb{X}_{t})=\frac{1}{1+e^{-\beta_{0}+\beta_{1}.X_{t}}}.$

Since are affine processes, theoretical corporate bond price and CDS price may be calculated byapplying the generalized transform result of Chen and Joslin (2012). However, explicit solution
may not be guaranteed in this case.
Therefore, we have shown that using MAFNS $mo$ del, we were able to model credit risk pricing
model that captured some important stylized facts as mentioned above while guaranteeing thepositivity of default intensity.
$\mathbb{X}$142
6Conclusion

In this paper, we proposed a modified arbitrage-free Nelson-Siegel, a 3-factor arbitrage-freeaffine DTSM. We compared our model with the related models (DNS, and AFNS), in termsof empirical performance and properties using 2 sets of sample data; one is the $US$ treasury
zero-coupon yields, and the other one is the Japan govemment bond zero-coupon yields. For
the $US$ treasury data, we found that our model outperforms AFNS in terms of in-sample fit, but
this is offset by poorer forecasting performance. However, our model still maintains some good
properties, which include the fact that one of its factors can be interpreted as long-term yields,
and one as slope, and that the long-term yield factor has the most persistent dynamic. For JGB
data, we did not find significant difference between MAFNS and AFNS, which means that bothmodels have similar properties and performance. Giving an example of using MAFNS in joint
credit risk pricing model within the reduced-from framework, we conclude that our model canbe used as an alternative to AFNS. Using MAFNS, we showed that not only is the joint model
able to capture some important stylized facts, positivity of default intensity is also ensured.
However, it should be noted that there are some important facts that are not addressed in ourstudy.
ne is the positivity of interest rates, and another is the regime which is observed in
JGB yields. Furthermore, even though our proposed joint model of credit risk seems to work
find, it still awaits empirical analysis to evaluate its practicality. Such problem is thus left for
future research.
$O$

143
A

AppendicesA. 1

Proof of Proposition 1

Proof. With the represent as in (3.1.1) and (3.1.2), Duffie and Kan (1996) prove that the zerocoupon prices are given by$P(t, T)=exp\{B(T-t)’X_{t}+A(T-t)\},$

where $B(t)$ and $A(t)$ satisfy the following ODEs$\dot{B}(t)=-\rho_{1}-K^{\mathbb{Q}}B(t)+\frac{1}{2}\sum_{j=1}^{n}(\Sigma’B(t)B(t)’\Sigma)_{jj}\delta_{1}^{j}$( . 1.1),$A$$A(t)=- \rho_{0}+(K^{\mathbb{Q}}\theta^{\mathbb{Q}})’B(t)+\frac{1}{2}\sum_{j=1}^{n}(\Sigma’B(t)B(t)’\Sigma)_{jj}\delta_{0}^{j}$with the boundary conditions: $B(O)=0,$ $A(O)=0$ . The dot onderivative w.r.
With the specification as in Proposition 1, we then have
$t.$

$B$

,

and

( .1.2)$A$$A$

denotes partial$t.$$\dot{B}(t)=-(\begin{array}{l}110\end{array})-(\begin{array}{lll}\kappa_{1}^{\mathbb{Q}} 0 00 \lambda -\lambda 0 0 \lambda\end{array}) B(t)+\frac{1}{2}(\begin{array}{l}(\Sigma’B(t)B(t)’\Sigma)_{11}00\end{array}),$$A(t)=( K^{\mathbb{Q}}\theta^{\mathbb{Q}})’B(t)+\frac{1}{2}\sum_{j=2}^{3}(\Sigma’B(t)B(t)’\Sigma)_{jj}.$

Since

A.2

$(\Sigma’B(t)B(t)’\Sigma)_{11}=\sigma_{11}^{2}B_{1}^{2}(t)$, solving these ODEs is straightforward.

Parameter restrictions on the canonical representation of theof affine term structure models$\square$$\mathbb{A}_{1}(3)$class

Dai and Singleton (2000) classify the -factor affine term structure models into $n+1$ classes
denoted by, where $m$ in
denotes the number of factorsthat enter into the volatility structure of the affine diffusion process. The canonical representation of theclass is admissible (in the sense that the model is well-defined) and maximally
flexible among the models within this class. In this appendix, using an invariant transformation
we look for the parameter restrictions to be imposed on the canonical representation ofclass so as to arrive at the independent-factor MAFNS model.
$n$$\mathbb{A}_{0}(n),$ $\mathbb{A}_{1}(n),$$\ldots,\mathbb{A}_{m}(n),$$\mathbb{A}_{n}(n)$$\ldots,$$\mathbb{A}_{m}(n)$$\mathbb{A}_{m}(n)$$\mathbb{A}_{1}(3)$The canonical representation of
The instantaneous short rate:$\mathbb{A}_{1}(3)$:$r_{t}=\delta_{0}+\delta_{1}’Y_{t}=\delta_{0}+\delta_{11}Y_{1,t}+\delta_{12}Y_{2,t}+\delta_{13}Y_{3,t}$

The dynamic under$\mathbb{Q}$( .2.1)$A$:$d(\begin{array}{l}Y_{1,t}Y_{2,t}Y_{3,t}\end{array})= (\kappa_{31}^{Q,Y}\kappa_{21}^{Q,Y}\kappa^{Q,Y}11 \kappa_{32}^{Q,Y}\kappa_{22}^{Q,Y}0 \kappa_{33}^{Q,Y}\kappa_{23}^{Q,Y}0)[(\begin{array}{l}\theta_{1}^{\mathbb{Q}}00\end{array})-(\begin{array}{l}Y_{1,t}Y_{2,t}Y_{3,t}\end{array})]dt$$+$$(\sqrt{Y_{1,t}}00$$\sqrt{1+\beta_{2}Y_{1,t}}00$$\sqrt{1+\beta_{3}Y_{1,t}}00)d(W_{3,t}^{\otimes^{W_{1}^{\mathbb{Q}}}}W_{2t}^{\copyright^{t}})$( .2.2)$A$

144
The dynamic under :$\mathbb{P}$$d(\begin{array}{l}Y_{1,t}Y_{2,t}Y_{3,t}\end{array})= (\kappa_{2}^{PY}i_{Y}\kappa_{3}^{P}i1 \kappa_{32}\kappa_{\mathscr{C}_{Y}}^{P,Y}0 \kappa_{23}^{P,Y}\kappa_{33}^{P,Y}0)[(\theta_{1}^{\mathbb{P}}\theta_{2}^{P}\theta_{3}^{P})-(\begin{array}{l}Y_{1,t}Y_{2,t}Y_{3,t}\end{array})]dt$$+$$(0\sqrt{Y_{1,t}}0$$\sqrt{1+\beta_{2}Y_{1,t}}00$$\sqrt{1+\beta_{3}Y_{1,t}}00)d(W_{1}^{P}W_{2t}^{\beta^{t}}W_{3,t}^{I}\emptyset)$.

( .2.3)$A$

There are 24 parameters in this maximally flexible model, and there are only 12 parametersin the MAFNS model. Thus, 12 parameter restrictions need to be imposed in order to obtainthe MAFNS model.
For the affine diffusion process $dY_{t}=K^{Y}(\theta^{Y}-Y_{t})dt+\Sigma^{Y}\sqrt{S_{t}^{Y}}dW_{t}$, we consider the affine
invariant transformation $X_{t}=T_{A}(Y_{t})=AY_{t}+\zeta$ , where A is a non-singular matrix, and isa vector having the same dimension as
From It\^o formula,$\zeta$$Y_{t}.$$diY_{t}=A^{-1}dX_{t}$$=A^{-1}[K^{X}(\theta^{X}-X_{t})dt+\Sigma^{X}\sqrt{S_{t}^{X}}dW_{t}]$$=A^{-1}K^{X}A(A^{-1}\theta^{X}-\zeta+\zeta-A^{-1}X_{t})dt+A^{-1}\Sigma^{X}\sqrt{S_{t}^{X}}dW_{t}$$=K^{Y}(\theta^{Y}-Y_{t})dt+\Sigma^{Y}\sqrt{S_{t}^{Y}}dW_{t}$( .2.4).

$A$

Then, we must have $K^{Y}=A^{-1}K^{X}A,$ $\theta^{Y}=A^{-1}\theta^{X}-\zeta$ , andThis is true both under and . It can be verified that$\mathbb{P}$$\Sigma^{Y}\sqrt{S_{t}^{Y}}=A^{-1}\Sigma^{X}\sqrt{S_{t}^{X}}.$$\mathbb{Q}$$A=(\begin{array}{lll}\sigma_{1}^{2} 0 00 \sigma_{2} 00 0 \sigma_{3}\end{array})$, and$\zeta=(\begin{array}{l}000\end{array})$

will transform the above canonical representation into the MAFNS model. Since$K^{Y,Q}=A^{-1}K^{X,Q}A= (\sigma_{1}^{-2}00 \sigma^{\frac{0}{02}1} \sigma^{\frac{00}{3}1})(\begin{array}{lll}\kappa_{1}^{\mathbb{Q}} 0 00 \lambda -\lambda 0 0 \lambda\end{array})(\begin{array}{lll}\sigma_{1}^{2} 0 00 \sigma_{2} 00 0 \sigma_{3}\end{array})$$=(\begin{array}{lll}\kappa_{1}^{\mathbb{Q}} 0 00 \lambda -\lambda_{\sigma_{2}}^{\Delta}\sigma 0 0 \lambda\end{array}),$four restrictions to be imposed on the mean reversion rate under$\mathbb{Q}$-measure are$\kappa_{2}^{Y}i^{Q_{=\kappa_{3}^{Y}}}i^{Q_{=\kappa_{32}^{Y,Q}=0,\kappa_{22}^{Y,Q}=\kappa_{33}^{Y,Q}}}.$Since$K^{Y,P}=A^{-1}K^{X,P}A= (\sigma_{1}^{-2}00 \sigma^{\frac{0}{20}1} \sigma^{\frac{00}{3}1})(\begin{array}{lll}\kappa_{1}^{P} 0 00 \kappa_{2}^{P} 00 0 \kappa g\end{array})(\begin{array}{lll}\sigma_{1}^{2} 0 00 \sigma_{2} 00 0 \sigma_{3}\end{array})$$=(\begin{array}{lll}\kappa_{1}^{P} 0 00 \kappa_{2}^{\mathbb{P}} 00 0 \kappa_{3}^{\mathbb{P}}\end{array}),$145four restrictions on the -dynamic mean reversion rate are$\mathbb{P}$$\kappa_{2}^{PY_{=\kappa_{23}^{P,Y}=\kappa_{3}^{PY}}}ii=\kappa_{32}^{P,Y}=0.$

Also,$\Sigma^{Y}\sqrt{S_{t}^{Y}}=A^{-1}\Sigma^{X}\sqrt{S_{t}^{X}}=$$(\sigma_{1}^{-2}00$$\sigma^{\frac{0}{02}1}$$\sigma^{\frac{00}{3}1})(\begin{array}{lll}\sigma_{1} 0 00 \sigma_{2} 00 0 \sigma_{3}\end{array})(\sqrt{X_{1,t}}00$$001$$001)$$=(\begin{array}{lll}\sigma_{1}^{-1} 0 00 1 00 0 1\end{array})(\sqrt{(100)AY_{t}}00 001 001)$$= (\sqrt{Y_{1,t}}00 001 001)$.

We obtain another 2 restrictions: $\beta_{2}=\beta_{3}=0.$
Finally, $r_{t}=X_{1,t}+X_{2,t}=$ 1 1 $0)X_{t}=(1 1 0)AY_{t}=(\sigma_{1}^{2} \sigma_{2} 0)Y_{t}$ . Thus,
the other 2 restrictions are$($$\delta_{0}=0,$ $\delta_{13}=0.$

A.3

Conditional and unconditional variance of state variables

Under the independent-factor assumption the conditional expectation and variance are explicitly
given as follow.
The conditional expectation and varvance:,( .3.1)$\mathbb{E}_{t-\triangle t}^{\mathbb{P}}[X_{t}]=(I-e^{-K^{||}’\triangle t})\theta^{P}+e^{-K^{JP}\Delta t}X_{r,t-\triangle t}$$Var_{t-\Delta t}^{\mathbb{P}}[X_{t}]=Q(X_{1,t-\triangle t}, \psi)$,( .3.2)$A$

where, and is the vector containing all the parameters, andmatrix with diagonal elements:$\mathbb{E}_{t}[.$$]\equiv \mathbb{E}[.

|\mathcal{F}_{t}]$$A$$\psi$$Q$

is a diagonal$Q_{11}= \frac{\sigma_{1}^{2}}{2\kappa_{1}^{JP}}[1-e^{-\kappa_{1}^{P}\triangle t}]^{2}+\frac{\sigma_{1}^{2}}{\kappa_{1}^{\mathbb{P}}}X_{1,t}e^{-\kappa_{1}^{F}\triangle t},$$Q_{ii}= \frac{\sigma_{i}^{2}}{2\kappa_{i}^{\mathbb{P}}}[1-e^{-2\kappa_{i}^{J1}\triangle t}], i=2,3.$

The unconditional expectation and vanance:$\lim_{tarrow\infty}\mathbb{E}^{\mathbb{P}}[X_{t}]=\theta^{\mathbb{P}}$$\lim_{tarrow\infty}Var^{\mathbb{P}}[X_{t}]=$,$(\overline{2}_{0}^{+_{\kappa_{1}}}\sigma_{0}^{2}$( .3.3)$A$$\ovalbox{\tt\small REJECT}^{\sigma^{2}}2_{0}0$$\ovalbox{\tt\small REJECT}_{2\kappa_{3}}^{\sigma^{2}}00)$.

( .3.4)$A$

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Graduate School of Economics
Osaka University, Toyonaka 560-0043, Japan-mail address: simdara@gmail.com$E$$\lambda\ovalbox{\tt\small REJECT} k\star^{\mapsto\backslash }\mp\cdot ffi’\backslash ;^{\backslash }g_{\mp WFa\ovalbox{\tt\small REJECT}_{\grave{t}}\}}^{\mapsto\backslash }$Dara Sim