South Korea 5-Year Zero-Coupon Bond AFNS Risk-Neutral Rate

The AFNS risk-neutral yield is the yield obtained from the Arbitrage-Free Nelson-Siegel model by projecting the yield curve factors under the risk-neutral measure (Q), reflecting the average of expected future short-term rates with risk compensation removed (Christensen, Diebold, and Rudebusch 2011). The model parameterizes the yield curve with 3 interpretable Nelson-Siegel factors (level, slope, curvature) and an explicit yield-adjustment term $A(\tau)/\tau$ that enforces no-arbitrage.

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

where $L$, $S$, $C$ are the level, slope, and curvature factors, $\lambda$ is the Nelson-Siegel decay parameter, and $A(\tau)/\tau$ is the yield-adjustment term that arises from the no-arbitrage constraint.

A decline in the AFNS risk-neutral yield suggests the market anticipates future rate cuts, while an increase signals tightening expectations. Its trend reveals the future policy rate path that the market prices in.

Estimated using the independent-factor Arbitrage-Free Nelson-Siegel (AFNS) model of Christensen, Diebold, and Rudebusch (2011), following the Christensen-Rudebusch (2012) specification used in production at the Federal Reserve Bank of San Francisco.

(1) Model Structure. The AFNS model parameterizes the yield curve through three latent factors

with Nelson-Siegel factor loadings. The Q-measure dynamics are:

In the independent-factor specification, the Q-measure mean-reversion matrix $K^Q$ and long-run mean $\theta^Q$ are restricted so that each factor evolves independently under Q. The P-measure dynamics, which govern observed factor behavior, are unrestricted:

The yield-adjustment term $A(\tau)/\tau$ is a deterministic function of $\Sigma$, $K^Q$, $\theta^Q$, and $\lambda$, computed in closed form from the Riccati equations of the affine framework.

(2) Estimation. The model is cast in state-space form. The measurement equation maps the latent factors to observed zero-coupon yields at multiple maturities, while the transition equation encodes the VAR(1) discretization of the P-measure factor dynamics. Parameters are estimated by maximum likelihood via the Kalman filter, which simultaneously filters the latent states and evaluates the log-likelihood. The RTS (Rauch-Tung-Striebel) smoother is applied post-estimation to obtain smoothed factor paths.

(3) Risk-Neutral Yield Computation. The risk-neutral yield at maturity $\tau$ is computed using the Q-measure factor dynamics:

where $A^Q(\tau)$ and $B^Q(\tau)$ are the Q-measure affine coefficients derived from the model's no-arbitrage restrictions. The Nelson-Siegel factor loadings under Q are

and $A^Q(\tau)$ is the yield-adjustment term that ensures absence of arbitrage.

The AFNS risk-neutral yield provides an independent, structurally different estimate of the bond market's embedded expectations for the future path of monetary policy. The AFNS model employs 3 interpretable Nelson-Siegel factors (level, slope, curvature) rather than the 5 statistical principal components used in the ACM model, and is estimated by maximum likelihood via the Kalman filter. Because its factor representation and estimation strategy differ fundamentally from the ACM, which relies on OLS-based multi-step regression, it provides a foundation for cross-validating expected rate paths. Diebold and Li (2006) demonstrated that the Nelson-Siegel three-factor structure maps directly onto Litterman and Scheinkman (1991)'s level, slope, and curvature decomposition of yield curve variation, providing an economically interpretable foundation for term structure analysis.

Agreement between the ACM and AFNS risk-neutral rates strengthens confidence that the estimated expected policy path is robust to model specification choices. Kim and Wright (2005) noted that model uncertainty is a first-order concern in term structure decompositions, and that comparing estimates across structurally different models is essential for robust inference. When the two models agree that, for example, the 3-year risk-neutral rate has declined, this convergent evidence strongly suggests that market participants have genuinely revised down their expectations for the average policy rate over the next three years. Persistent divergence between the two estimates, by contrast, signals sensitivity to modeling assumptions, a finding that is itself informative for monetary policy analysis.

The three-factor structure also permits an intuitive decomposition of risk-neutral yield movements following Diebold, Rudebusch, and Aruoba (2006). Changes in the level factor ($L$) shift the entire risk-neutral curve, reflecting revisions to the long-run equilibrium rate that Laubach and Williams (2003) associate with the natural rate of interest. Changes in the slope factor ($S$) tilt the curve, capturing near-term policy expectations relative to the long run, as emphasized by Gürkaynak, Sack, and Swanson (2005) in their decomposition of monetary policy surprises into target and path factors. Changes in the curvature factor ($C$) affect the belly of the curve, reflecting medium-term rate expectations and policy cycle turning points.

The Korean risk-neutral rate is exposed not only to expectations for Bank of Korea policy but also to spillovers from U.S. monetary policy. Obstfeld (2015) documented that U.S. monetary policy shocks transmit to emerging market yield curves through the expectations channel, shifting risk-neutral rates even absent domestic policy changes. The AFNS model, with its parsimonious 3-factor structure, can track these cross-border spillovers in a more interpretable framework than the 5-factor ACM, enabling researchers to decompose international monetary policy transmission into level (global neutral rate), slope (relative policy stance), and curvature (cycle divergence) channels.

The AFNS risk-neutral yield provides fixed income practitioners with an interpretable factor framework for yield curve positioning and risk management. Duffee (2002) demonstrated that affine term structure models produce economically meaningful yield forecasts, and the AFNS specification preserves this property while adding the interpretability of the Nelson-Siegel factor structure.

For relative value analysis, comparing the ACM and AFNS risk-neutral rates at the same maturity reveals model-dependent variation in the expectations-risk premium decomposition. Kim and Orphanides (2012) demonstrated that risk-neutral rate estimates are sensitive to the factor structure and estimation method, and that multi-model comparison is essential for identifying robust signals. When the two models produce similar risk-neutral rates but different term premiums, it suggests that the level of expected rates is well-identified but the risk compensation estimate is model-sensitive.

For liability-driven investors, the AFNS model provides a benchmark for 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 a factor model with no-arbitrage restrictions supplies an internally consistent curve for asset-liability matching. Ilmanen (2011) emphasized that long-horizon institutional investors should account for the full distribution of interest rate scenarios, not merely the central forecast, when constructing duration-matching strategies.

This is a model-derived series, the output of the Christensen-Rudebusch arbitrage-free 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 2000-12-31 to 2026-06-09.

The five-year AFNS 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 = 0.8556, the Phillips and Perron (1988) test concurring at p = 0.896, 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 = 139.59 and Q = 145.37 at p = 0.000 and p = 0.000, and the automatic portmanteau of Escanciano and Lobato (2009) concurs with a statistic of 77.27 at p = 0.000. The Bai and Perron (1998, 2003) procedure finds a single break in the mean at 2020-08-31, 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).