South Korea 10-Year Shadow Term Premium

The shadow term premium is the excess risk compensation that investors demand for holding a long-term bond to maturity, measured consistently with the zero lower bound (ZLB) constraint on nominal interest rates. It follows the same decomposition structure as the standard ACM term premium but assigns no probability to infeasible negative rate paths even when rates approach the lower bound.

The shadow term premium at maturity $n$ is defined by decomposing the ZLB-adjusted yield into a risk-neutral component and a residual risk compensation component:

where $y^{SR,P}_t(n)$ is the ZLB-adjusted model-implied yield computed using Q-measure dynamics with risk prices, and $y^{SR,Q}_t(n)$ is the ZLB-adjusted risk-neutral yield computed using P-measure dynamics with risk prices set to zero. Both yields are obtained by applying the Wu-Xia (2016) forward-rate-based option adjustment to the respective Gaussian affine coefficients.

When interest rates are well above the effective lower bound, the ZLB constraint is not binding and the shadow term premium converges to the standard ACM term premium. The two decompositions diverge meaningfully only when rates approach the lower bound, where the nonlinear option adjustment redistributes yield variation between the risk-neutral and term premium components. A positive shadow term premium indicates that investors require risk compensation beyond the expected short-rate path, and the size of the gap relative to the standard term premium reveals the degree to which the zero lower bound constraint distorts the yield curve decomposition.

Estimated using the Wu-Xia (2016) first-order ZLB approximation applied to the ACM model output. The shadow term premium requires two independent ZLB adjustments, one for each measure, because the conditional variance of the shadow rate differs under the physical and risk-neutral dynamics.

(1) Gaussian ACM Estimation. The standard ACM pipeline produces two sets of affine coefficients, one being the model-implied coefficients $(A^P_n, B^P_n)$ computed with risk prices $(\lambda_0, \Lambda_1)$ and the other being the risk-neutral coefficients $(A^{RN}_n, B^{RN}_n)$ computed with $\lambda = 0$. Each set is decomposed into one-period forward rates $f^G_j$ following the piecewise formula described in the KRSHR methodology.

(2) ZLB Adjustment (Model-Implied). Each Gaussian forward rate under the P-measure pricing is adjusted for the ZLB using the option formula

. The conditional variance $\sigma^2_j$ is computed under Q-measure dynamics using the transition matrix

, because bond pricing occurs under the Q-measure regardless of which set of affine coefficients is used. The adjusted yields $y^{SR,P}_t(n)$ respect the lower bound at all maturities.

(3) ZLB Adjustment (Risk-Neutral). Each Gaussian forward rate under the $\lambda = 0$ pricing is adjusted using $\sigma_j$ computed under P-measure dynamics ($\tilde{\Phi} = \Phi$), because the risk-neutral yield represents the expected path of short rates under the physical measure. The adjusted yields $y^{SR,Q}_t(n)$ similarly respect the lower bound.

(4) Shadow Term Premium. The shadow term premium is computed as the difference:

Both components are individually bounded below by $r_{lb}$, and their difference captures the risk compensation in a manner consistent with the ZLB constraint. When rates are far above the ZLB, all $z_j \to +\infty$ so $f^{SR}_j \to f^G_j$, and the shadow decomposition converges to the standard decomposition.

The shadow term premium provides a more accurate measure of risk compensation in the bond market during periods when the zero lower bound constraint significantly affects yield dynamics. When rates approach the lower bound, the underlying Gaussian model fails to respect the non-negativity of yields, so the standard term premium loses reliability near the bound (Kim and Singleton 2012).

The convergence property of the shadow and standard term premiums is particularly useful for policy analysis. During normal times, when rates are well above zero, the two measures agree closely, confirming that the ZLB adjustment introduces no artificial distortions. During low-rate episodes, the divergence between the two measures quantifies the degree to which the ZLB constraint affects the yield curve decomposition (Bauer and Rudebusch 2016).

From a monetary policy evaluation perspective, the shadow term premium isolates the risk compensation that investors demand in an environment where rates cannot fall below the effective lower bound. This is directly relevant for assessing the effectiveness of unconventional monetary policy. Quantitative easing compresses the term premium by absorbing duration supply (Gagnon et al. 2011), and the shadow term premium provides a cleaner estimate of this compression effect (Wu and Xia 2016).

The shadow term premium enables more accurate yield curve analysis and relative value trading when rates are near the zero lower bound (Wu and Xia 2016).

For curve positioning, the shadow term premium provides a better signal for assessing whether long-term bonds offer adequate risk compensation. Near the ZLB, the standard term premium may understate risk compensation because the Gaussian model assigns probability to negative rate paths that cannot materialize (Black 1995). By properly accounting for the truncated distribution of future rates, the shadow term premium provides a more reliable input for duration allocation decisions.

For bond investors, the difference between the standard and shadow term premiums at a given maturity serves as a gauge of how much the ZLB constraint is distorting the yield curve (Christensen and Rudebusch 2015).

Over 6307 observations from 2000-12-18 to 2026-06-09, the ten-year shadow term premium carries a unit root, with the Dickey and Fuller (1979) test not rejecting at p = 0.2883, the Phillips and Perron (1988) test concurring at p = 0.1756, and the Kwiatkowski et al. (1992) test rejecting stationarity, an agreed I(1). On the level the first-order autocorrelation is 0.999 with an implied half-life of about 498.9 observations, a descriptive persistence summary and not an integration claim, the Ljung and Box (1978) portmanteau on the level rejects white noise at lags 10 and 20, Q = 62130.90 and Q = 122592.00 at p = 0.000 and p = 0.000, and the Bai and Perron (1998) procedure, by the Bai and Perron (2003) algorithm, finds no break in the mean (Andrews 1993; Perron 1989).

This signal, the ten-year shadow term premium, has no daily 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).