South Korea AFGNS Risk-Neutral Instantaneous Forward Rate 3 Years Hence

The AFGNS instantaneous forward rate is the risk-neutral interest rate expected to apply over an infinitesimally short interval at a future time $\tau$, derived from the 5-factor Arbitrage-Free Generalized Nelson-Siegel model of Christensen, Diebold, and Rudebusch (2009). It extends the 3-factor AFNS forward rate with a second slope-curvature pair governed by a slower decay parameter $\lambda_2$, and the closed-form expression is:

The added second factor pair provides richer forward rate dynamics at long tenors (5Y–10Y), where the 3-factor AFNS forward loadings have decayed to near-zero. The 5-factor structure captures independent medium-term and long-term curvature dynamics and thus expresses a wider variety of forward curve shapes than the 3-factor AFNS, providing another independent estimate of forward rates for the expected policy path.

Same methodology as the AFNS forward rate series (KRIFRAFNS), extended to 5 factors with two decay parameters $\lambda_1$ and $\lambda_2$. See the AFGNS risk-neutral yield series (KRRNAFGNS) for the full 5-factor model specification.

(1) Forward-Rate Loadings. The 5-factor forward-rate loading matrix is:

The second pair's loadings

decay more slowly than the first pair's, providing additional sensitivity at long tenors. At $\tau = 10$ years with $\lambda_2 \approx 0.025$ months$^{-1}$, the second slope loading $e^{-\lambda_2\tau} \approx 0.05$ still carries meaningful signal, while the first slope loading $e^{-\lambda_1\tau} \approx 0.001$ has effectively vanished.

(2) Forward-Rate Yield-Adjustment. The derivative $d\bar{A}(\tau)/d\tau$ uses the same functional forms as the AFNS forward adjustment (Section 2C.8 in THEORY.EN.md), applied independently to each $(\sigma, \lambda)$ pair for the two slope and two curvature factors.

The AFGNS forward rate captures long-tenor dynamics that the 3-factor AFNS cannot separately identify. At 5Y and 10Y tenors, the second slope-curvature pair $(S^2, C^2)$ governed by $\lambda_2$ provides additional sensitivity to shifts in the long-run equilibrium rate and is well suited to capturing these slowly evolving components. Time variation in the equilibrium real rate and trend inflation, the so-called 'falling stars', is crucial for understanding forward rate dynamics at long horizons (Bauer and Rudebusch 2020).

The AFGNS model's 5-factor structure provides a richer decomposition of the equilibrium-rate information embedded in long-horizon forward rates, enabling separate identification of medium-term ($\lambda_1$) and long-term ($\lambda_2$) components of the expected policy path. Christensen and Rudebusch (2019) used a related AFNS framework to show that forward rates at long maturities contain information about the secular decline in the equilibrium real rate. Del Negro et al. (2017) attributed this secular decline to rising demand for safe assets, a structural shift that manifests precisely in the long-horizon forward rates that the AFGNS is designed to capture.

The improved cross-sectional fit of the AFGNS has direct implications for forward rate estimation, where observation noise $\sigma_\varepsilon$ is roughly half that of the AFNS. Standard term structure models may confound systematic forward rate variation with measurement error (Duffee 2011). The AFGNS's 5-factor structure reclassifies variation that the 3-factor model attributes to noise as genuine factor dynamics. Forward-spot spreads carry economically meaningful predictive content for future bond returns (Fama and Bliss 1987), and the AFGNS forward rate preserves this content more completely by reducing the information loss from model misspecification.

A credible monetary policy framework stabilizes long-horizon expectations, and far-forward rates in inflation-targeting countries exhibit lower volatility (Gürkaynak, Sack, and Wright 2007). The AFGNS forward rate's ability to separately track medium-term and long-term forward rate dynamics distinguishes revisions in near-term policy expectations (the $\lambda_1$ factors) from shifts in long-run equilibrium perceptions (the $\lambda_2$ factors), providing a more granular decomposition of monetary policy credibility than the 3-factor model can offer.

Comparing forward rates across models with different factor structures identifies which features of the expected policy path are robust to specification choices (Piazzesi 2010). The AFGNS supports this cross-validation by providing a third independent forward rate estimate alongside the ACM-based KRIFR and the AFNS-based KRIFRAFNS.

The AFGNS forward rate's advantage over the AFNS forward rate is concentrated at long tenors (5Y–10Y), where the additional curvature factor $C^2$ provides sharper identification of forward rate dynamics. Three PCA factors explain most bond return variation (Litterman and Scheinkman 1991), but a fourth and fifth factor carry economically significant return-forecasting information (Cochrane and Piazzesi 2005). The AFGNS forward rate captures this additional information through its second slope-curvature pair.

For curve trading strategies, the AFGNS decomposition separates medium-term from long-term forward curve signals. A change in $C^1$ (governed by $\lambda_1$) indicates shifts in the 1Y–5Y forward curve segment, while a change in $C^2$ (governed by $\lambda_2$) signals movement in the 5Y–10Y segment. This two-curvature structure supports more targeted butterfly trades that separately exploit medium-term and long-term forward curve shape changes. Capturing this dual curvature in the affine class requires at least 4–5 factors (Dai and Singleton 2000).

The AFGNS's 5-factor structure captures a wider range of forward curve variation than the 3-factor AFNS, enabling identification of long-maturity forward rate mispricings that the simpler model cannot detect. In Gaussian dynamic term structure models, the pricing factors can be expressed as observable portfolios of yields, and the number of factors determines the span of forward curve variation the model can capture (Joslin, Singleton, and Zhu 2011).

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

For K-ICS and K-IFRS 17 liability discounting, forward rate dynamics at long horizons are directly relevant. The tail risk in 10-year forward rates affects insurance reserve adequacy. Institutional investors should account for the full distribution of forward rate scenarios when constructing duration-matching strategies (Ilmanen 2011). The AFGNS model's richer factor structure provides a more complete characterization of long-end forward rate uncertainty than the 3-factor alternative.

This is a model-derived series, the output of the Christensen-Diebold-Rudebusch generalized Nelson-Siegel 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 2000-12-18 to 2026-06-09.

The three-year AFGNS instantaneous forward rate 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 = 0.4677, the Phillips and Perron (1988) test concurring at p = 0.3281, 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 = 47.93 and Q = 73.78 at p = 0.000 and p = 0.000, and the automatic portmanteau of Escanciano and Lobato (2009) concurs with a statistic of 5.06 at p = 0.024. 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).