South Korea Cochrane–Piazzesi Bond Return-Forecasting Factor

This indicator is the single return-forecasting factor that combines the forward rates of the Korean government bond curve into one object to forecast one-year-ahead bond excess returns (Cochrane and Piazzesi 2005). The original paper showed that one linear combination of the 1-year yield and the 2–5-year annual forwards, $x_t = \gamma^{\top} f_t$, forecasts the one-year-ahead excess returns on bonds of all maturities simultaneously, and the headline forecasting factor of this indicator is that fitted combination evaluated on the daily curve (percent). The per-maturity expected excess returns $b_n \cdot x_t$ ($n = 2..5$ years) are exposed as hidden toggles.

A rising factor means the curve is pricing higher expected one-year-ahead excess returns on holding intermediate and long bonds, and this indicator is a descriptive return-forecasting gauge, not a guaranteed forecast. The repo's zero curve observes only the 1, 3, 5, and 10-year tenors, so the 2-year and 4-year yields are interpolated from the PCHIP curve through the {0.25, 1, 3, 5, 10}-year knots (the same object the ACM model consumes), which makes the factor a summary of effectively lower-dimensional curve information rather than five independent maturity signals. The honest verdict is that this grid cannot resolve whether the tent-shaped coefficient pattern exists in Korea, not that the tent is absent.

Estimated using the two-stage regression of Cochrane and Piazzesi (2005). With $y^{(n)}_t$ the $n$-year zero yield,

the log price, and

the annual forward, the one-year holding excess return is

Step 1 regresses the average excess return on all forwards,

and the fitted value $x_t = \gamma^{\top} f_t$ is the factor. Step 2 regresses each maturity's excess return on the single factor to estimate the loadings $b_n$, whose mean equals one by construction because $x_t$ is the fitted value of the average.

The grid is annual 1–5y, with the 2y and 4y sampled from the PCHIP curve through the {0.25, 1, 3, 5, 10}-year knots. The 10y knot anchors the long end, so the 4–5y region is interpolation rather than extrapolation. Estimation runs on the month-end panel, and the MA(11) errors induced by 12-month overlapping returns are handled with Newey-West HAC standard errors at lag 18. The estimated $\gamma$ and $b_n$ are applied to the daily curve's forwards to produce the daily forecasting factor and expected excess returns (monthly fit, daily display). Coefficients refit on the full sample every run, so historical fitted values revise as the sample grows.

The validation file records the condition number of the z-scored regressor matrix, the spanning $R^2$ from regressing the factor on the first three principal components of the 1–5y yield levels, the joint Wald, the tent-shape verdict, the $b_n$ monotonicity, and the crisis-window means.

Cochrane and Piazzesi (2005) show on U.S. data that one tent-shaped combination of forwards forecasts one-year excess returns on all maturities with $R^2$ of 0.30–0.44, strongly rejecting the unpredictability the expectations hypothesis implies. The finding builds on the forward-rate excess-return predictability documented by Fama and Bliss (1987) and Campbell and Shiller (1991), and has two prongs. First, return-forecasting information concentrates in a single common factor rather than living maturity-by-maturity, against the level-slope-curvature three-factor benchmark of Litterman and Scheinkman (1991). Second, the factor is not fully spanned by the first three principal components of yields, which makes it a candidate carrier of macro information that yields alone do not encode.

The wider literature surrounding the CP factor is divided over whether yields carry unspanned information that does not surface in their principal components. On the side that supports unspanning, macro factors extracted from large panels carry independent return-forecasting power (Ludvigson and Ng 2009), expected returns decompose into a trend-inflation cycle and an interest-rate cycle that form two components sharing information with the CP factor (Cieslak and Povala 2015), and yields encode hidden state variables not visible in PCA (Duffee 2011). These findings are all consistent with the unspanned-component claim. Once standard small-sample corrections are applied, by contrast, the unspanned-component claim is harder to sustain, and the residual return predictability often stays within the range of sampling variation (Bauer and Hamilton 2018). The CP factor therefore sits in a literature that is empirically rich and methodologically contested, so the honest reading of any country-level replication has to acknowledge both sides.

The Korean sample replicates only half of the original finding. What replicates is the joint predictability and the loading structure. The joint Wald is significant at $p < 0.0001$ under NW-HAC(18), supporting that the five forwards jointly forecast one-year-ahead excess returns, and the per-maturity loadings

rise monotonically with maturity exactly as in the original and satisfy the mean-one restriction. The Step-1 $R^2$ of 0.28 sits below the U.S. but within the international range that Sekkel (2010) reports for non-U.S. sovereign bond markets, so the joint-predictability prong reads as a genuine cross-country regularity rather than a U.S.-only stylized fact.

What does not replicate is the shape and the spanning. The estimated $\gamma$ slopes (0.62, −1.97, −0.28, +6.89, −4.66) do not trace a tent. The large opposite-signed pair on adjacent maturities (+6.89/−4.66) is the classic footprint of the collinearity (condition number about 87) created by the interpolated 2y and 4y yields, so the correct reading is that the grid cannot resolve the tent rather than that Korea lacks one. The spanning $R^2$ against the three Litterman-Scheinkman principal components is 0.88, so most of the factor is explained by level, slope, and curvature, and the Bauer and Hamilton (2018) caution against over-asserting an unspanned component applies a fortiori here. The crisis-window mean of the factor, covering the GFC and COVID, is indistinguishable from the full-sample mean, so a countercyclical-return-premium reading is not confirmed in this sample.

The Korean KTB benchmark structure runs on the 1/3/5/10y tenors that the Bank of Korea publishes, and the 2y and 4y yields that enter the CP regression are PCHIP interpolations on this knot grid. The grid constraint is not a modelling choice but a data fact, and it makes the per-maturity allocation of the $\gamma$ coefficients weakly identified even while the joint Wald and the loading structure remain sound. The Ludvigson and Ng (2009) and Cieslak and Povala (2015) channels are visible in the Korean data only to the extent the four-tenor grid can carry their information, which is the structural limit of this gauge on this market.

For the fixed-income desk the factor reads as an auxiliary duration-positioning signal. A high factor means the curve is pricing expected one-year excess returns that favour holding intermediate and long bonds, and since $b_n$ rises with maturity the signal amplifies toward the long end. Given that the 0.28 $R^2$ is an in-sample fit with no out-of-sample validation, and that the daily fitted value inherits the small movements of an interpolated curve, the factor is best used as a companion read alongside the ACM term-premium (KRTP) and risk-neutral (KRRN) decomposition rather than as a standalone trading signal. The forward-rate predictability tradition that runs from Fama and Bliss (1987) through Campbell and Shiller (1991) to Cochrane and Piazzesi (2005) is the conceptual backbone here.

The factor is most useful as a complementary signal when cross-checked against the ACM term-premium decomposition. ACM separates expected short rates from premia inside a no-arbitrage model structure, while the CP factor extracts return-forecasting information by regression alone with no model structure, which is the Duffee (2011) hidden-state argument applied at the regression layer. When the two approaches point the same way the signal is more credible, and when they diverge the divergence flags whether a model assumption or a sample feature is the source, while how aggressively to bet on small divergences is governed by awareness of the sampling variation that Bauer and Hamilton (2018) caution about.

This indicator follows the same descriptive-gauge framing as KRGAR and KRFXAR. Full-sample re-estimation each run revises history, the interpolated grid constrains per-maturity coefficient interpretation, and the unresolved tent plus high spanning ($R^2 = 0.88$ against level-slope-curvature) do not permit a claim that the U.S. result replicates in Korea. The trust this indicator carries extends exactly as far as the parts that do replicate, namely the joint predictability that Sekkel (2010) documents for international sovereign markets and the rising-with-maturity loading structure, and no further.

The 1/3/5/10y KTB benchmark grid that the Bank of Korea publishes is the structural limit of this gauge, and the 2y and 4y forwards are sampled from a PCHIP curve through the published knots. Until the Korean curve carries a richer set of liquid annual tenors, the CP gauge will remain the joint-predictability and loading-structure object that the data can support, and the more granular Cieslak and Povala (2015) decomposition or the macro-factor extension of Ludvigson and Ng (2009) is not implementable on this knot set without grid enrichment.

Over 6307 observations from 2000-12-18 to 2026-06-09, the unit-root reading of the Cochrane-Piazzesi single forward-rate factor is ambiguous, since the Dickey and Fuller (1979) and Phillips and Perron (1988) tests reject a unit root at p = 0.0053 and p = 0.0005 while the Kwiatkowski et al. (1992) test also rejects stationarity at p < 0.01, the near-unit-root disagreement of a persistent bounded signal, so no clean order is assigned. On the level the first-order autocorrelation is 0.993 with an implied half-life of about 94.4 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 = 58219.00 and Q = 111474.00 at p = 0.000 and p = 0.000, and the Bai and Perron (1998) procedure, by the Bai and Perron (2003) algorithm, finds 4 breaks in the mean at 2007-05-28, 2011-03-10, 2014-12-30, 2021-09-27 (Andrews 1993; Perron 1989).

This signal, the Cochrane-Piazzesi single forward-rate factor, 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).