Term structure forecasting in affine framework with time-varying volatility

Abstract

This study extends the affine Nelson–Siegel model by introducing the time-varying volatility component in the observation equation of yield curve, modeled as a standard EGARCH process. The model is illustrated in state-space framework and empirically compared to the standard affine and dynamic Nelson–Siegel model in terms of in-sample fit and out-of-sample forecast accuracy. The affine based extended model that accounts for time-varying volatility outpaces the other models for fitting the yield curve and produces relatively more accurate 6- and 12-month ahead forecasts, while the standard affine model comes with more precise forecasts for the very short forecast horizons. The study concludes that the standard and affine Nelson–Siegel models have higher forecasting capability than their counterpart EGARCH based models for the short forecast horizons, i.e., 1 month. The EGARCH based extended models have excellent performance for the medium and longer forecast horizons.

Appendices

Appendix 1

The yield-adjustment term A in the AFNS and AFNS-EGARCH models

The appendix reports the analytical solution for the yield-adjustment term A as derived in the Christensen et al. (2011). For a general volatility matrix,

$$\begin{aligned} {\Sigma }=\left[ {\begin{array}{ccc} \sigma _{11} &{}\quad \sigma _{12} &{}\quad \sigma _{13}\\ \sigma _{21} &{}\quad \sigma _{22} &{} \quad \sigma _{23}\\ \sigma _{31} &{}\quad \sigma _{32} &{}\quad \sigma _{33}\\ \end{array} } \right] \end{aligned}$$

the analytical solution for is: ${A}(t,T)/(T-t)$ is:

$$\begin{aligned} -\frac{A(t,T)}{m}= & {} -\sigma _{11}^{2}\frac{(T-t)^{2}}{6}-\left( \sigma _{21}^{2}+\sigma _{22}^{2} \right) \\&\left[ \frac{1}{{2\lambda }^{2}}-\frac{1}{\lambda ^{3}}\frac{1-e^{-\lambda (T-t)}}{T-t}+\frac{1}{4\lambda ^{3}}\frac{1-e^{-2\lambda (T-t)}}{T-t} \right] \\&-\left( \sigma _{31}^{2}+\sigma _{32}^{2}+\sigma _{33}^{2} \right) \left[ \frac{1}{{2\lambda }^{2}}-\frac{1}{\lambda ^{2}}e^{-\lambda (T-t)}-\frac{1}{4\lambda }(T-t)e^{-2\lambda (T-t)}\right. \\&\left. -\frac{3}{4\lambda ^{2}}e^{-2\lambda (T-t)}-\frac{2}{\lambda ^{3}}\frac{1-e^{-\lambda (T-t)}}{T-t}+\frac{5}{{8\lambda }^{3}}\frac{1-e^{-2\lambda (T-t)}}{T-t} \right] \\&-\sigma _{11}\sigma _{21}\left[ \frac{1}{2\lambda }(T-t)+\frac{1}{\lambda ^{2}}e^{-\lambda (T-t)}-\frac{1}{\lambda ^{3}}\frac{1-e^{-\lambda (T-t)}}{T-t} \right] \\&-\sigma _{11}\sigma _{31}\left[ \frac{3}{\lambda ^{2}}e^{-\lambda (T-t)}+\frac{1}{2\lambda }(T-t)\right. \\&\left. +\frac{1}{\lambda }(T-t)e^{-\lambda (T-t)}-\frac{3}{\lambda ^{3}}\frac{1-e^{-\lambda (T-t)}}{T-t} \right] \\&-\left( \sigma _{21}\sigma _{31}+\sigma _{22}\sigma _{23} \right) \left[ \frac{1}{\lambda ^{2}}+\frac{1}{\lambda ^{2}}e^{-\lambda (T-t)}-\frac{1}{2\lambda ^{2}}e^{-2\lambda (T-t)}\right. \\&\left. -\frac{3}{\lambda ^{3}}\frac{1-e^{-\lambda (T-t)}}{T-t}+\frac{3}{{4\lambda }^{3}}\frac{1-e^{-2\lambda (T-t)}}{T-t} \right] \end{aligned}$$

Since, the A(t, T) term is a complex function of the volatility matrix $\Sigma$, decay parameter $\uplambda$ and maturity $(T-t)$. In both affine based models, i.e., AFNS and AFNS-EGARCH, we use the above analytical solution to estimate the yield curve.

Appendix 2

Coefficients and latent variable in the general state-space form

In the statistical formulation of the models in Sect. 2.3, the matrices and vectors for the state and observations equations should be considered as follows. The matrices and vectors in state-space system in (12–14) for the simple AFNS model should be defined as:

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} B={\Lambda }\left( \lambda \right) : (\hbox {N}\times 3)&{} X_{t}=\left( X_{1t}, X_{2t}, X_{3t} \right) ^{'}: (3\times 1)&{} w_{t}=\varepsilon _{t}: (\hbox {N}\times 1) \\ C=\left( I_{3}-e^{-K{\Delta }t} \right) \theta : (3\times 1)&{} F=e^{-K{\Delta }t}: (3\times 3)&{} u_{t}=v_{t}: (3\times 1) \\ \Omega =\Omega : (\hbox {N}\times \hbox {N})&{} Q_{t}={\Sigma }_{v}: (3\times 3)&{} {\tilde{A}}=-A: (\hbox {N}\times 1) \\ \end{array} \end{aligned}$$

while the system in (12–14), for the AFNS-EGARCH can be specified as:

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} B=\left[ \varLambda (\tau ) \quad {\Gamma }_{\varepsilon } \right] : (\hbox {N}\times 4)&{} X_{t}=\left[ X_{1t}, X_{2t}, X_{3t},\varepsilon _{t}^{*} \right] ^{'}: (4\times 1)&{} w_{t}=\varepsilon _{t}^{+}: (\hbox {N}\times 1) \\ C=\left[ {\begin{array}{cc} \left( I_{3}-e^{-K\Delta t} \right) \theta \\ 0\\ \end{array} } \right] : (4\times 1)&{} F=\left[ {\begin{array}{cc} e^{-K\Delta t} &{} 0\\ 0 &{} 0\\ \end{array} } \right] : (4\times 4)&{} u_{t}=\left[ {\begin{array}{cc} v_{t+1}\\ \varepsilon _{t+1}^{*}\\ \end{array} } \right] : (4\times 1) \\ {\Omega =\Omega }: (\hbox {N}\times \hbox {N})&{} Q_{t}=\left[ {\begin{array}{cc} {\Sigma }_{v} &{} 0\\ 0 &{} h_{t+1}\\ \end{array} } \right] : (4\times 4)&{} {\tilde{A}}=-A: (\hbox {N}\times 1) \\ \end{array} \end{aligned}$$

where ${\Lambda }(\lambda )$ is $(\hbox {N}\times 3)$ matrix of loadings, ${\Gamma }_{\varepsilon }$ is ($\hbox {N}\times 1$) vector showing the sensitivity of various yields to common volatility component, $\theta$ is $(3\times 1)$ vectors of factors mean, and K is $(3\times 3)$ full-matrices of parameters. $\varepsilon _{t}$ and $v_{t}$ are $(\hbox {N}\times 1)$ and $(3\times 1)$ innovations vectors of the observation and state equations respectively, ${\Omega }$ is $(\hbox {N}\times \hbox {N})$ covariance matrix of the measurement equation innovations, and ${\Sigma }_{v}$ is $(3\times 3)$ lower triangular covariance matrix of the state innovations (assumed to be a full-matrix rather than a diagonal one). Moreover, $\varepsilon _{t}^{+}$ is $(\hbox {N}\times 1)$ errors vector of the observation equation in EGARCH based model.

Appendix 3

Data description

We consider JGB yields with maturities of 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, 120, 180, 240 and 300 months. The yields are derived from bid/ask average price quotes, from January 2000 through June 2013, using the Fama and Bliss (1987) methodology.

Table 8 Descriptive statistics of yields data across maturities

Fig. 4

The figure shows the yield curves, 2000:01–2013:06. The sample consists of monthly yield data from January 2000 to June 2013 (162 months) for maturities of 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, 120, 180, 240 and 300 months (20 maturities)

Table 8 provides summary statistics for the data-set. For each maturity, we report mean, standard deviation, minimum, maximum, skewness, kurtosis, and autocorrelation coefficients at various displacements. The summary statistics reveal that the average yield curve is upward sloping. Unconditional volatility increases with maturity and yields for all maturities are persistent, however, relatively short rates persistency is higher than those of the long rates.

In addition to the findings in Table 8, we see few interesting characteristics in Fig. 4, which plots cross-section of yields over time. The first noticeable fact is that long term yields vary significantly over time. Second, the short rates are almost zero during the prolonged period except with a little rise in late 2006 and early 2007 that causes a fall in the slope of the curves (apparent in figure). Moreover, when short rates are stuck at zero, the long end seem more volatile than the short end of the curves.

Appendix 4

1 month conditional estimates of mean-reversion matrix ${{\varvec{K}}}$

The state transition matrix K in AFNS and AFNS-EGARCH are modeled continuously and should be transformed in order to be comparable with the DNS based specifications. The continuous state transition matrix K is converted into 1-month conditional matrix by computing $\mathrm {exp}( -K{\Delta }t )$ with ${\Delta }t=1/12$. The resulting matrix for the AFNS is as:

$$\begin{aligned} {\tilde{F}}_{AFNS}=\mathrm {exp}\left( -K{\Delta }t \right) =\left[ {\begin{array}{ccc} {0.6178} &{} {-0.3883} &{} {-0.3459}\\ {-0.0132} &{} {0.8917} &{} {0.0062}\\ {-0.2649} &{} {0.0847} &{} {0.6967}\\ \end{array} } \right] \end{aligned}$$

while for the AFNS-EGARCH is as:

$$\begin{aligned} {\tilde{F}}_{AFNS-EGARCH}=\mathrm {exp}\left( -K{\Delta }t \right) =\left[ {\begin{array}{ccc} {0.9104} &{} {-0.0008} &{} {-0.0005}\\ {-0.0019} &{} {0.9207} &{} {0.0002}\\ {0.0584} &{} {-0.0288} &{} {0.9914}\\ \end{array} } \right] \end{aligned}$$

The 1 month conditional estimation results indicate that all the three factors are more persistent in the AFNS-EGARCH than the rest of three models, being very close to one. Cross-factors dynamics seems unimportant in the AFNS-EGARCH model, while reasonably stronger in AFNS framework than the rest of three models.