South Korea 3-Year Zero-Coupon Bond AFGNS Risk-Neutral Rate

The AFGNS risk-neutral yield is the market's expected average path of future short-term rates, stripped of risk compensation, derived from a 5-factor generalized Nelson-Siegel yield curve. It is estimated using the Arbitrage-Free Generalized Nelson-Siegel model of Christensen, Diebold, and Rudebusch (2009), which extends the 3-factor AFNS by adding a second slope-curvature pair $(S^2, C^2)$ governed by a slower decay parameter $\lambda_2 < \lambda_1$, capturing richer cross-sectional dynamics at long maturities while preserving no-arbitrage consistency.

The zero-coupon yield at maturity $\tau$ is:

where $L$ is the level factor, the first pair $(S^1, C^1, \lambda_1)$ captures short-to-medium dynamics identically to the AFNS, and the second pair $(S^2, C^2, \lambda_2)$ captures medium-to-long dynamics. The term $\bar{A}(\tau)/\tau$ is the yield-adjustment term derived from the no-arbitrage restriction.

A decline in the AFGNS risk-neutral yield indicates that the market anticipates future policy rate cuts, while an increase signals tightening expectations. The AFGNS derives risk-neutral yields from a factor structure different from those of the ACM and AFNS, but all three models are fitted to the same Korean government bond yield curve, so comparing them shows whether the expected policy path is consistent across specifications rather than confirming it independently.

Estimated using the independent-factor Arbitrage-Free Generalized Nelson-Siegel (AFGNS) model of Christensen, Diebold, and Rudebusch (2009). The estimation methodology extends the same Kalman filter maximum likelihood (MLE) framework as the 3-factor AFNS to 5 factors, and the full model specification is given in the AFNS risk-neutral rate series (KRRNAFNS).

(1) Model Structure. The model uses five latent factors

and two decay parameters $\lambda_1$ and $\lambda_2$, where $\lambda_1$ is fixed identically to the AFNS and $\lambda_2$ is estimated under the constraint $\lambda_2 < \lambda_1$. The Q-measure dynamics follow the canonical AFGNS structure with $K^Q$ encoding both $\lambda_1$ and $\lambda_2$, while the P-measure dynamics use independent mean-reversion:

(2) Yield-Adjustment Term. The no-arbitrage yield-adjustment generalizes the AFNS formula with additional terms for the second slope-curvature pair:

where $f_S$ and $f_C$ are the slope and curvature adjustment functions from the AFNS model, applied to each $(\sigma, \lambda)$ pair.

(3) Estimation. 17 parameters are estimated via L-BFGS-B, comprising $\kappa^P$ (5), $\theta^P$ (5), $\sigma$ (5), $\sigma_\varepsilon$ (1), and $\lambda_2$ (1). The search starts from the best point of a likelihood profile over a grid of fixed $\lambda_2$ values and then continues as a global search over all parameters. As in the AFNS model, the published yields are computed from the one-sided filtered factors at the full-sample maximum-likelihood parameters, re-estimated at every update, with the filter started from the unconditional mean and covariance of the factors, so the first observations lean on that start value, no smoother enters the published series, and the series is not a real-time estimate (Orphanides and van Norden 2002). The model runs on month-end curves, so each month is represented by its month-end curve, and the current month's point is the latest trading day so far, recomputed at every update until the month ends.

The AFGNS is a third model specification of the risk-neutral rate, alongside the ACM and AFNS. Christensen, Diebold, and Rudebusch (2009) showed that the Svensson (1995) extension of the standard 3-factor Nelson-Siegel model cannot satisfy the no-arbitrage condition, whereas a 5-factor generalization adding a second slope factor can. The additional factors governed by $\lambda_2$ provide expressive power at long maturities, where the 3-factor model's single curvature factor has decayed to near zero.

Comparing the AFGNS alongside the ACM and AFNS provides a model-robustness check on estimates of the expected policy path. Model uncertainty is a first-order concern in term premium decompositions, and comparing estimates across structurally different models is essential for robust inference (Kim and Wright 2005). The ACM uses 5 principal components, the AFNS uses 3 Nelson-Siegel factors, and the AFGNS uses 5 generalized Nelson-Siegel factors, but all three are fitted to the same Korean government bond yield curve and estimate the same risk-neutral path, so they are different specifications of one object rather than independent measurements. Agreement among the three on the direction of change in risk-neutral rates therefore shows that the result is consistent across these specifications, not that independent evidence corroborates a shift in market expectations, and divergence identifies the segments where the specification choice affects the result.

The two additional factors give the AFGNS the capacity to absorb cross-sectional variation that a three-factor model leaves to the observation noise $\sigma_\varepsilon$. Duffee (2011) showed that almost half of the variation in bond risk premia cannot be detected from the yield cross-section, so standard term structure models may confound systematic yield curve variation with measurement error, and this hidden-information problem is what the added capacity is aimed at.

At long maturities, the second slope-curvature pair $(S^2, C^2)$ captures systematic variation in the 5-to-10-year segment, which Gürkaynak, Sack, and Wright (2007) documented as empirically important for pricing long-dated government bonds. Cochrane and Piazzesi (2005) showed that a single tent-shaped return-forecasting factor, capturing information beyond the standard three Nelson-Siegel factors, predicts excess bond returns with $R^2$ up to 0.44, suggesting that the additional factors carry economically meaningful risk premium information.

The AFGNS's ability to identify long-end yield curve dynamics is useful for tracking trend changes in the equilibrium rate. Accounting for time variation in the equilibrium real rate and trend inflation is crucial for understanding yield curve dynamics and extracting unbiased term premium estimates (Bauer and Rudebusch 2020). Because the AFGNS model separately identifies long-run dynamics through $\lambda_2$, the second factor pair tracks how the secular decline in $r^*$ and $\pi^*$, the so-called 'falling stars', shifts the long end of the curve. Christensen and Rudebusch (2019) estimated, using a related AFNS framework, that the longer-run equilibrium real rate has fallen approximately 2 percentage points, and identifying this result requires the model's capacity to capture long-maturity variation.

The AFGNS model's improved cross-sectional fit makes it particularly suited for relative value analysis at long maturities. Litterman and Scheinkman (1991) established the factor-based approach to fixed income by showing that three principal components, namely level, slope, and curvature, explain the vast majority of bond return variation, and Christensen, Diebold, and Rudebusch (2009) extended this to five arbitrage-free factors. The AFGNS decomposition separates the medium-term and long-term curvature signals that the three-factor structure conflates. A change in $C^1$ (governed by $\lambda_1$) indicates movement in the 1Y–5Y segment, while a change in $C^2$ (governed by $\lambda_2$) indicates movement in the 5Y–10Y segment, so bond traders can execute more targeted butterfly trades.

The AFGNS's 5-factor structure spans a wider range of yield curve variation than the 3-factor AFNS, so relative value traders can identify long-maturity mispricings that the simpler model cannot detect. In Gaussian dynamic term structure models, the pricing factors are expressed as observable portfolios of yields, and the number of factors determines the span of yield curve variation that the model can capture (Joslin, Singleton, and Zhu 2011).

In extrapolating the yield curve beyond observed maturities, the AFGNS's second decay parameter $\lambda_2$ governs the rate at which the curve converges to its long-run asymptote, providing a structural basis for extrapolation that purely statistical approaches lack. Nelson-Siegel-type models produce extrapolated long-maturity yields with biases of only a few basis points, negligible for risk management (Christensen, Lopez, and Mussche 2022).

For liability-driven investors, the AFGNS model provides a benchmark for long-maturity discount rates consistent with the current term structure. Under K-ICS and K-IFRS 17, Korean insurers must discount insurance liabilities using a risk-free curve, and variation in 10-year discount rates directly affects reserve adequacy. Ilmanen (2011) emphasized that long-horizon institutional investors should account for the full distribution of interest rate scenarios when constructing duration-matching strategies, and the AFGNS model's richer factor structure provides a more complete characterization of long-end uncertainty than the 3-factor alternative.

This is a model-derived series, the three-year risk-neutral yield of the Christensen-Diebold-Rudebusch arbitrage-free generalized Nelson-Siegel five-factor model computed from one-sided Kalman-filtered factors at the full-sample maximum-likelihood parameters, so the unit-root reading describes the fitted curve rather than a directly observed price, and the curve is a quasi-final estimate filtered at parameters estimated on the full sample, not a real-time one (Orphanides and van Norden 2002). It is a monthly series of 307 observations from 2000-12-31 to 2026-06-09, the last being the latest day of the running month, and the serial-correlation and break diagnostics run on the first difference.

The three-year AFGNS risk-neutral yield is integrated of order one on the level, with the Dickey and Fuller (1979) test in the Said and Dickey (1984) form not rejecting a unit root at p = 0.649, the Phillips and Perron (1988) test concurring at p = 0.644, and the Kwiatkowski et al. (1992) test rejecting stationarity at p < 0.01, so the three tests agree. The GLS-detrended test of Elliott, Rothenberg, and Stock (1996), reserved for ambiguous outcomes, was not needed on this agreeing reading. On the first difference the Ljung and Box (1978) portmanteau does not reject white noise at either monthly lag, Q = 18.59 at p = 0.099 at lag 12 and Q = 26.70 at p = 0.319 at lag 24. The automatic portmanteau of Escanciano and Lobato (2009) is not run, because the matrix reserves it for daily members. The Bai and Perron (1998) procedure, computed by the algorithm of Bai and Perron (2003), finds no break in the mean of the differenced series, a reading consistent with the parameter-instability inference of Andrews (1993) on that object (Perron 1989).

This is a monthly model-derived series with no posited seasonal quantity, whose inputs are deseasonalized where relevant, so the seasonal-unit-root and seasonal-stationarity machinery of Hylleberg et al. (1990) and Canova and Hansen (1995) is not applicable and is not run (Beaulieu and Miron 1992; Ghysels and Osborn 2001).