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

The AFNS instantaneous forward rate is the risk-neutral rate of interest that the market expects to apply over an infinitesimally short interval beginning at a future date. It is derived analytically from the bond price function of the arbitrage-free Nelson-Siegel model of Christensen, Diebold, and Rudebusch (2011).

The instantaneous forward rate at tenor $\tau$ has the closed-form expression:

The forward-rate loadings

are the derivatives of the Nelson-Siegel yield loadings. Unlike yields, which average expected future short rates over the entire maturity, forward rates provide point estimates of the expected short rate at each specific horizon.

The AFNS instantaneous forward rate complements the existing ACM instantaneous forward rates (KRIFR series), which are derived from the 5-factor PCA representation of Adrian, Crump, and Moench (2013). Together with the AFGNS forward rates, three independent forward rate estimates are available for cross-validation of the expected policy path.

The AFNS forward rate is computed analytically from the Christensen, Diebold, and Rudebusch (2011) model by differentiating the bond price function. See the AFNS risk-neutral yield series (KRRNAFNS) for the full model specification.

(1) Forward-Rate Loadings. The instantaneous forward rate at tenor $\tau$ uses the derivative of $\tau \cdot b_i(\tau)$ for each Nelson-Siegel factor, giving the loading matrix

. Svensson (1995) documented that forward-rate loadings are more peaked than yield loadings, providing sharper identification of factor movements at specific tenors. This property is intrinsic to the Nelson-Siegel class, where the slope factor's forward loading $e^{-\lambda\tau}$ is a pure exponential decay (no averaging), and the curvature loading $\lambda\tau e^{-\lambda\tau}$ is a peaked hump that resolves medium-term dynamics more sharply than its yield counterpart.

(2) Forward-Rate Yield-Adjustment. The adjustment $d\bar{A}(\tau)/d\tau$ is the analytical derivative of the yield-adjustment term from the no-arbitrage restriction (Christensen, Diebold, and Rudebusch 2011). At long tenors the level contribution $-\sigma_1^2 \tau^2 / 2$ dominates, which is three times larger in absolute value than the yield adjustment $-\sigma_1^2 \tau^2 / 6$. Because of this amplification, noted by Piazzesi (2010), forward rates are more sensitive to volatility misspecification than yields, making accurate $\sigma$ estimation all the more important.

(3) Historical Series. Computed from RTS-smoothed factors using the forward-rate loadings and adjustment, evaluated at tenors 365d (1Y), 1095d (3Y), 1825d (5Y), and 3650d (10Y). The Rauch-Tung-Striebel smoother (Rauch, Tung, and Striebel 1965) provides the optimal linear estimate of the latent factors given the full sample.

Risk-neutral forward rates represent the market-implied future path of the policy rate after removing risk compensation. Because forward rates encode the local slope information of the yield curve, they are more sensitive to localized changes than yields, which are averaged over the maturity (Heath, Jarrow, and Morton 1992). Gürkaynak, Sack, and Swanson (2005) used high-frequency movements in forward rates around FOMC announcements to decompose monetary policy surprises into a 'target' factor (unexpected changes in the current rate) and a 'path' factor (revisions in the expected future path), demonstrating that forward rates at specific horizons are more informative about policy expectations than yields.

Fama and Bliss (1987) provided early evidence that forward-spot spreads predict 1-year excess bond returns, demonstrating that forward rates contain term premium variation. Cochrane and Piazzesi (2005) extended this by showing that a tent-shaped linear combination of forward rates predicts excess returns with $R^2$ up to 0.44, capturing information beyond the standard three Nelson-Siegel factors. The AFNS forward rate, derived from an explicit no-arbitrage model, provides a structural interpretation of this predictive content by decomposing forward rates into risk-neutral expectations and term premium components.

Campbell and Shiller (1991) tested the expectations hypothesis using U.S. term structure data, finding that yield spreads forecast long-run changes in short rates but not short-run changes in long rates. This asymmetry implies that forward rates at long horizons are primarily informative about long-run rate expectations, while short-horizon forward rates contain substantial term premium variation.

At longer horizons (5–10 years), the risk-neutral forward rate converges toward the market's estimate of the long-run neutral nominal interest rate, $r^* + \pi^*$. Gürkaynak, Sack, and Wright (2007) showed that far-forward rates in countries with explicit inflation targets exhibit less volatility, suggesting that credible monetary policy regimes help anchor long-horizon expectations. For Korea, where the BOK has maintained an explicit inflation target since 1998, the AFNS far-forward rate provides a market-based measure of the credibility of the inflation targeting framework.

From a macroeconomic modeling perspective, Rudebusch and Wu (2008) combined a no-arbitrage affine term structure model with macro equations for output and inflation, showing that forward rate factors have macroeconomic underpinnings. Ang, Piazzesi, and Wei (2006) found that the short rate extracted from no-arbitrage models has more predictive power for GDP than any term spread, reinforcing the value of model-based forward rate decompositions for macroeconomic surveillance.

Forward rates are foundational instruments in fixed income derivatives pricing. They are used to price forward rate agreements (FRAs), interest rate swaps, caps, floors, and swaptions under the risk-neutral valuation framework. Duffie and Kan (1996) established that the no-arbitrage forward rate ensures consistency between spot and derivative markets within the affine class.

For bond portfolio management, instantaneous forward rates serve as the basis for estimating roll-down returns. Litterman and Scheinkman (1991) showed that three factors explain over 99% of yield curve variation, and forward rates provide higher-resolution information about curvature dynamics than par or zero-coupon yields. The AFNS forward rate preserves this factor structure while adding the no-arbitrage restriction, ensuring that the forward curve, yield curve, and discount factor curve are mutually consistent.

For monetary policy surveillance, comparing three independent forward rate estimates (ACM from KRIFR, AFNS from KRIFRAFNS, AFGNS from KRIFRAFGNS) at the same tenor reveals which aspects of the forward curve are robust to model specification. Kim and Orphanides (2012) demonstrated that term premium estimates are sensitive to the factor structure and estimation method. When all three models agree on the implied policy rate path, this provides strong evidence for the market consensus on future BOK actions; persistent disagreement identifies where model uncertainty is economically meaningful.

In the Korean market, the risk-neutral forward rate curve provides a benchmark for evaluating whether the Korean IRS curve embeds consistent rate expectations or is distorted by supply-demand imbalances. Borio et al. (2016) documented that FX hedging demand by foreign investors creates persistent cross-currency basis, which can distort the IRS curve relative to the government bond forward curve. The AFNS forward rate, derived entirely from KTB yields, is not subject to these swap market distortions.

For liability-driven investors, the no-arbitrage forward curve provides a benchmark for future reinvestment rates consistent with the current term structure. Ilmanen (2011) emphasized that long-horizon institutional investors should account for the full distribution of forward rate scenarios when constructing asset-liability matching strategies, and the AFNS forward rate provides curve information with the structural guarantee of no-arbitrage consistency.

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 2000-12-18 to 2026-06-09.

The three-year AFNS 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 = 0.7395, the Phillips and Perron (1988) test concurring at p = 0.5771, 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 = 43.57 and Q = 81.09 at p = 0.000 and p = 0.000, and the automatic portmanteau of Escanciano and Lobato (2009) concurs with a statistic of 6.71 at p = 0.010. 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).