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. Because the AFGNS produces risk-neutral yields from a factor structure independent of those underlying the ACM and AFNS, comparing the three models enables robust cross-validation of the expected policy path.

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). A profile likelihood approach over $\lambda_2$ ensures robust convergence.

The AFGNS provides a third independent estimate 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, so the three models employ mutually independent representations. When all three agree on the direction of change in risk-neutral rates, this constitutes strong evidence of a genuine shift in market expectations; when they diverge, it identifies precisely the segments where specification choice affects the result.

The AFGNS's improved cross-sectional fit strengthens the reliability of the risk-neutral rate estimates. Because the observation noise $\sigma_\varepsilon$ is roughly half that of the AFNS, the 5-factor structure reclassifies as factor dynamics the variation that the 3-factor model attributes to noise. 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 the AFGNS's improved fit directly mitigates this hidden-information problem.

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 output of the Christensen-Diebold-Rudebusch arbitrage-free generalized Nelson-Siegel five-factor model, so the unit-root reading describes the fitted curve rather than a directly observed price, and the serial-correlation and break diagnostics run on the first difference. The series is tested over its own span from 2000-12-31 to 2026-06-09.

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 at p = 0.6737, the Phillips and Perron (1988) test concurring at p = 0.6576, and the Kwiatkowski et al. (1992) test rejecting stationarity. On the first difference the Ljung and Box (1978) portmanteau rejects white noise at lags 10 and 20, Q = 15.12 and Q = 24.04 at p = 0.128 and p = 0.241, and the automatic portmanteau of Escanciano and Lobato (2009) detects no further dependence with a statistic of 0.69 at p = 0.405. The Bai and Perron (1998, 2003) procedure finds no break in the mean, a reading consistent with the parameter-instability inference of Andrews (1993) on the differenced object (Perron 1989).

This is a daily model output with no low-integer seasonal period, so the seasonal-unit-root and seasonal-stationarity machinery of Hylleberg et al. (1990) and Canova and Hansen (1995) is deliberately not run (Beaulieu and Miron 1992; Ghysels and Osborn 2001).