U.S. Kim-Wright 10-Year Term Premium

USKWTP10 is the published ten-year term premium recorded as-is without any transformation, expressed in percent per annum as the part of the fitted ten-year zero-coupon yield that a three-factor arbitrage-free model of the term structure does not attribute to the expected average future short rate (Kim and Wright 2005). KRED serves the value exactly as recorded, so the object on this page is a model-identified split produced by its authors, not a KRED estimate and not a replication or a validation of any term premium KRED computes itself.

The premium is a model quantity rather than an observed price. What markets quote is the yield, and the division of that yield into an expectations part and a compensation part exists only relative to a stated model, which is why the term-structure tradition writes the long rate as the expected path of short rates plus a risk premium and treats the premium as the residual of that identity (Hicks 1939). The yield being decomposed is itself a construction, a point on a zero-coupon curve fitted to coupon-bond prices by spline or parametric methods and read off daily (Macaulay 1938; McCulloch 1971; Nelson and Siegel 1987; Svensson 1994; Gürkaynak, Sack, and Wright 2007).

Inside the model three latent factors are updated as each new day of yields arrives, while the parameters governing their dynamics are held fixed over long stretches and re-estimated only occasionally. Separating the expectations part from the premium in such a model is the hard identification problem of this class, which is why survey information on future short rates has been brought into term-structure estimation (Kim and Orphanides 2012). Because a new parameter vintage rewrites the whole history, a value read today for a past date can differ from the value read for that same date earlier, and nothing here is claimed about the series as first published.

Term premiums of this kind are estimand-specific. A decomposition built from a filtered factor model and a decomposition built from excess-return regressions on the same curve answer the same question with different machinery and do not deliver the same number (Adrian, Crump, and Moench 2013; Cochrane and Piazzesi 2005), so this series is never called the term premium, and its level is never read as independent evidence for or against a term premium KRED estimates on its own. Small-sample bias in the estimated persistence of the factors moves the split between expectations and premium directly, which is why that bias is treated as a first-order design question in the estimation literature (Bauer, Rudebusch, and Wu 2012; Bauer and Rudebusch 2020).

KRED applies no transformation to USKWTP10 and stores the published premium exactly as recorded, neither rescaling, deflating, smoothing, nor annualizing it, and it neither re-estimates the model nor re-derives the split. The methodology is therefore confined to the measurement basis by which such a premium comes to exist.

That basis is a Gaussian three-factor arbitrage-free model of the term structure, in which three latent factors follow a first-order autoregression, a stochastic discount factor prices every maturity without admitting arbitrage, and the fitted yield at each maturity separates into an expected-average-short-rate part and a residual premium (Kim and Wright 2005). The state is recovered from the observed cross-section of yields by filtering. Pinning the expectations part down from yields alone is the identification problem this class of model faces, and the standard response in the literature has been to bring survey information on future short rates into the estimation (Kim and Orphanides 2012).

The yields the model is fitted to are themselves measured constructions. A zero-coupon curve is recovered from coupon-bond prices by cubic or exponential splines, by parsimonious or extended parametric forms, and by the daily curve construction that is now standard, and those methods have been compared against one another (McCulloch 1971, 1975; Vasicek and Fong 1982; Nelson and Siegel 1987; Svensson 1994; Gürkaynak, Sack, and Wright 2007; Bliss 1997). The convention behind any single maturity point remains the yield-to-maturity identity that equates a bond's price to its discounted cash flows (Macaulay 1938).

Two properties of the construction bear on how the number is read. The persistence of the fitted factors is estimated with a downward bias in samples of this length, and because the premium is what the expectations part leaves over, that bias transmits directly into the level of the split (Bauer, Rudebusch, and Wu 2012; Bauer and Rudebusch 2020). The parameters are also held fixed between infrequent re-estimations while the factors are filtered forward, so the published history is the latest parameter vintage and revises with it, and no dated vintage panel is kept from which the size of such a revision could be quantified (Orphanides and van Norden 2002).

None of that construction is carried out by KRED, which keeps the published premium exactly as recorded and never recomputes it from a curve of its own.

A term premium is the compensation a holder of a long bond earns for bearing the risk that short rates turn out differently than expected, so reading a long yield without splitting it confuses a change in what markets expect with a change in what they charge (Hicks 1939; Kim and Wright 2005).

The split matters for policy analysis because the stance is set at the shortest maturity while transmission runs through long rates (Bindseil 2004; Borio 1997), so a fall in a ten-year yield means one thing when the expectations part drives it and another when the premium does. The forward-rate structure that makes the decomposition meaningful is the same one under which a long rate is an average of expected short rates plus a premium (Hicks 1939), and it is read off curves built by the standard daily construction (Nelson and Siegel 1987; Svensson 1994; Gürkaynak, Sack, and Wright 2007).

That premiums move rather than sitting at a constant is what the return-forecasting literature established, since excess bond returns are predictable from the shape of the curve and that predictability is the empirical footprint of a time-varying premium (Cochrane and Piazzesi 2005; Hansen and Hodrick 1980). Slope-based recession readings rest on the same logic, since a compressed slope mixes an expectations path with a premium (Estrella and Mishkin 1998; Engstrom and Sharpe 2018).

Three limits belong with the reading. The premium is model-identified, so a different arbitrage-free specification on the same curve returns a different level (Adrian, Crump, and Moench 2013; Bauer, Rudebusch, and Wu 2012), the level is revised whenever the model is re-estimated (Orphanides and van Norden 2002), and a sample containing a long low-rate stretch makes the persistence of the factors, and therefore the split, harder to pin down (Bauer and Rudebusch 2020). For the same reason the series is read on its own terms rather than as agreement or disagreement with a term premium KRED estimates by another method.

For a bond desk the premium is the part of a long yield that carries expected excess return, so it enters relative-value work differently from the expectations part. Where the premium is wide, holding duration is being paid for; where it is narrow, the same duration is held for a smaller expected reward, which is the practical content of predictable excess returns (Cochrane and Piazzesi 2005; Hansen and Hodrick 1980).

A position taken on such a reading is executed through the curve rather than through the premium itself, since the premium is not a traded instrument. Exposure is measured by the duration that maps yield changes into price changes and managed by immunization defined on the whole term structure (Macaulay 1938; Fisher and Weil 1971), and the discounting and hedging curves come from spline recovery, parametric fitting, and the standard daily construction, whose differences have been compared (McCulloch 1971, 1975; Vasicek and Fong 1982; Nelson and Siegel 1987; Svensson 1994; Bliss 1997; Gürkaynak, Sack, and Wright 2007).

Three cautions bound the use. The number is model output rather than a price, so its level carries the specification's assumptions and a different arbitrage-free decomposition of the same curve returns a different answer (Kim and Wright 2005; Adrian, Crump, and Moench 2013). The history revises when the model's parameters are re-estimated, so a backtest treating today's series as what was visible at each past date overstates what a desk could have acted on (Orphanides and van Norden 2002; Kim and Orphanides 2012). And the estimated persistence of the factors, biased downward in samples of this length, moves the premium level directly (Bauer, Rudebusch, and Wu 2012; Bauer and Rudebusch 2020).

USKWTP10 is a published model-identified split of the fitted Treasury curve into an expected-average-short-rate part and a residual premium, served here as recorded, so the statistical object is that published premium rather than a price observed in a market or a quantity KRED estimates (Kim and Wright 2005). Because a risk premium is theoretically mean-reverting, the deterministic term is pinned to a constant with no trend, since fitting a trend to a premium of this kind is a specification error and not a discovery. Over 9126 daily observations spanning 1990-01-02 to 2026-07-24 the three procedures agree on the opposite of that prior. The augmented Dickey-Fuller test of Dickey and Fuller (1979), in the lag-augmented form of Said and Dickey (1984) with lag length chosen by an information criterion as in Ng and Perron (2001), does not reject a unit root with a statistic of −2.3885 at p = 0.145008, the nonparametric Phillips and Perron (1988) test concurs with a statistic of −2.15666 at p = 0.22235, and the KPSS test rejects stationarity with a statistic of 12.7239 at p < 0.01 (Kwiatkowski et al. 1992), an honest agreement of all three that carries a clean I(1) verdict. This is not a lone-dissent reading, so the GLS-detrended escalation of Elliott, Rothenberg, and Stock (1996) is not required. The verdict is reported exactly as the battery returned it, and what it says is that over this sample the premium is not distinguishable from a series carrying a stochastic trend, which is weaker than the mean reversion its economic interpretation would suggest.

On the level the first-order autoregressive coefficient is 0.999394 and the implied half-life is 1142.71 observations, a descriptive persistence summary and not an integration claim, read with the small-sample downward bias of Andrews (1993) in mind. The Ljung and Box (1978) portmanteau on the level, refining the original Box and Pierce (1970) form, rejects the no-autocorrelation null at lags 10 and 20, with Q = 90500.3 and Q = 179444 and p = 0.000 in both cases. Because the triplet returned a clean integrated reading, the multiple-break search of Bai and Perron (1998), computed by the dynamic-programming algorithm of Bai and Perron (2003), moves from the level to the first difference, where a break search in the mean is well defined, and it finds no break in the mean of the differenced series (Perron 1989).

Three limits are stated rather than implied. The model's parameters are re-estimated only occasionally while the factors are filtered forward, so the whole history is rewritten at a new parameter vintage, no dated vintage panel is stored from which the size of such a revision could be quantified, and the series therefore carries no claim about its values as first published (Orphanides and van Norden 2002; Kim and Orphanides 2012). Nothing here is a statement about the premium's sign or about its economic level, since the only persistence figures reported are the ones the battery produced. And the verdict is a property of this decomposition alone, since a term premium KRED estimates from the same curve by a different method is a different estimand whose reading would be neither independent evidence nor corroboration (Adrian, Crump, and Moench 2013; Bauer and Rudebusch 2020).

This series is daily and has no low-integer seasonal period, so the seasonal-unit-root machinery of Hylleberg et al. (1990) and the Canova and Hansen (1995) seasonal-stationarity test are not applicable and are deliberately not run, the seasonal auxiliary regression being degenerate at a daily period (Beaulieu and Miron 1992; Ghysels and Osborn 2001).