ACMTP10 Meaning and Computation: Understanding the Adrian-Crump-Moench (2013) Model

ACMTP10 is not an analyst opinion but the output of a five-factor regression published by the New York Fed in 2013, reproducible with standard statistical software and published every trading day by the New York Fed.

Three economists, Adrian, Crump and Moench, operationalized in 2013 what the affine term-structure theory had promised since Vasicek (1977): decomposing the Treasury yield into economically interpretable components.

Origin, Publication and Institutional Status of the ACM 2013 Model

The Adrian-Crump-Moench model appeared in the New York Fed Staff Reports series as number 340, Pricing the Term Structure with Linear Regressions, in August 2008, then in the April 2013 revision everyone cites. The paper was subsequently published in the Journal of Financial Economics, volume 110, number 1, pages 110 to 138. The three authors then held research positions at the Federal Reserve Bank of New York: Tobias Adrian led the Capital Markets Function group, before joining the International Monetary Fund in 2017 as Director of the Monetary and Capital Markets Department; Richard K. Crump and Emanuel Moench belonged to the same research team, Moench later moving to the Bundesbank.

The model’s institutional status matters. Unlike many academic contributions on the term structure, ACM 2013 was produced in the explicit context of the FOMC’s diagnostic work on the effects of QE programs running since 2008. The model was immediately integrated into the Fed's analytical toolkit: the New York Fed has published the model's estimates for maturities 1 to 10 years since 2013, in an ACMTermPremium.xls workbook carrying one monthly sheet and one daily sheet. That file is the source to cite: the ACM series are not in FRED, whose only published term premia are those of the competing Kim-Wright model (THREEFYTP10). For the 10-year, three columns matter: ACMY10, the model-fitted yield, ACMTP10, the term premium, and ACMRNY10, the risk-neutral yield, that is the expected-short-rate component.

Beyond the Fed, ACM 2013 adoption has been gradual but is now established. The European Central Bank publishes its own Bund 10-year term-premium estimate using an ACM-inspired methodology adapted to the euro curve structure. The Bank of England has provided since 2014 a Gilt 10-year estimate following the same class of multi-factor affine models. The Bank for International Settlements (BIS) regularly references ACM estimates in its Quarterly Review for comparative analyses of global financial conditions. Central-bank steering of the term structure thus rests on a common decomposition framework, without making it an explicit communication object toward the broader public.

The Five Latent Factors and What They Capture

The heart of the ACM 2013 model is the extraction of five latent factors via principal component analysis applied to the historical matrix of Treasury yields across maturities. The choice of five factors is not arbitrary: it extends the Litterman-Scheinkman (1991) tradition that had shown three factors, level, slope and curvature, already capture more than 99% of yield variance, adding two additional factors to capture residual movements the term premium needs isolated. A related fiscal angle is developed in the mechanics tying the federal deficit to gold reserves.

The first factor captures the level component of the curve. It represents parallel movements affecting the entire term structure in the same direction. Empirically, this factor explains 94.3% of the variance of Treasury constant-maturity yields from 1 to 30 years, and 98.5% of that of the model-fitted curve since 1961. When the FOMC revises aggregated expectations of policy rates over the visible horizon, or when an inflationary shock shifts the entire curve upward, it is primarily the level factor that moves.

The second factor captures slope. It represents the difference between the long end and the short end of the curve: when it rises, the curve steepens; when it falls, the curve flattens or inverts. This factor explains 5.4% of the total variance of constant-maturity yields. It moves when the market revises its read of the monetary cycle: slowdown expectations (flattening), return of growth (steepening).

The third factor captures curvature. It describes the hump or trough in the intermediate zone of the curve (typically 5 to 7 years) relative to a simple linear interpolation between the short and long ends. This factor explains only 0.26% of the variance of constant-maturity yields, and 0.07% of that of the model curve: the 2% order of magnitude often quoted is Litterman and Scheinkman’s curvature share, measured on excess returns rather than on the variance of levels. It generally moves during technical adjustments tied to the composition of Treasury supply across intermediate maturities.

The fourth and fifth factors capture fine residuals: typically maturity-specific spreads, microstructure effects (segmentation among buyers seeking different duration profiles), inventory biases at primary dealers. Individually, each of these factors explains less than three hundredths of a percent of variance. Collectively, they are indispensable for a clean term-premium estimate: these are precisely the residuals that carry information about the economic compensation demanded beyond what level, slope and curvature already determine.

Adrian, Crump and Moench argue in their original publication that using five factors rather than three significantly improves fit quality to the data and the stability of term-premium estimates. Their argument is comparative rather than consensual: the paper shows that the five-factor specification outperforms out of sample the four-factor Cochrane and Piazzesi specification, while producing similar in-sample term-premium dynamics.

Three-Step Regression: Why Linear and Not Maximum Likelihood

The methodological specificity of ACM 2013 versus prior affine models (Duffie-Kan 1996, Dai-Singleton 2000) lies in the estimation procedure. Classical affine models are estimated by joint maximum likelihood over all dynamic and pricing parameters, a numerically heavy procedure, sensitive to initial conditions, that can converge to local optima depending on starting specifications. Adrian, Crump and Moench propose instead, in the words of their own abstract, “a three-step ordinary least squares estimator”: three successive regressions, each of them ordinary least squares.

First step: decompose the pricing factors into a predictable component and innovations, by regressing the factors on their own lagged levels. This is a standard VAR(1) on the principal components extracted from the curve, and it uses only classical econometric tools.

Second step: estimate the exposures of Treasury returns to the lagged levels of the factors and to the contemporaneous innovations of those same factors. This is a time-series regression of excess returns, delivering the sensitivities the third step needs.

Third step: obtain the market price of risk parameters from a cross-sectional regression of the exposures to lagged levels onto the exposures to contemporaneous innovations. That step turns the physical measure into the risk-neutral one and delivers the term premium directly. Worth flagging, because the confusion is common: the stagewise procedure in two parts that the paper describes is not its own, it is the one it attributes to Joslin, Singleton and Zhu.

The practical advantage is considerable: the full procedure can be coded in a few dozen lines in Matlab, R, Python or Stata. A public MATLAB implementation of the method is maintained by MathWorks, which allows any researcher to reproduce the estimates. The potential drawback is that stagewise estimation does not exploit all information from the joint structure: full likelihood would in principle have superior asymptotic efficiency properties, and the affine term-structure literature has documented small-sample biases that bootstrap corrections help attenuate.

What the Model Produces: Term Premium, Expectations, Decomposed Yield

The ACM 2013 model produces three families of outputs for each maturity from 1 to 10 years. The first is the term premium, ACMTP01 to ACMTP10. The second is the risk-neutral yield, ACMRNY01 to ACMRNY10, which carries the expected average short rate over the horizon. The third is the model-fitted yield, ACMY01 to ACMY10, whose gap with the market yield serves as a fit diagnostic.

The accounting identity the model produces reads: fitted yield = expected short rates + term premium. For the 10-year: ACMY10 = ACMRNY10 + ACMTP10, which holds to the sixteenth decimal across the last two thousand sessions. On 21 September 2026 it gives 4.92% = 4.34% + 0.58%. This is the framework for applying the decomposition to the 10-year yield that allows every Treasury market episode to be reread along two dimensions. The gap between ACMY10 and FRED’s DGS10 is 9 basis points at the median since 1990 and 6 since 2017; it reflects the difference in convention, zero-coupon on one side and par yield on the other, as much as fit quality, which makes it a diagnostic to handle with care.

Outputs are published in two formats inside a single workbook, ACMTermPremium.xls, downloadable from the New York Fed website. The ACM Monthly sheet carries month-end points back to June 1961; the ACM Daily sheet carries more than sixteen thousand sessions and ends two days before consultation. Publication is therefore daily, which is a stronger reproducibility argument than a periodic refresh: every session of the fitted curve, the premium and the expectations is separately available for all ten maturities, as summarised on the US term premium dataset reference page.

Limitations, Retrospective Revisions and Alternative Models

The ACM 2013 model has several limitations that its authors do not hide and that any operational user needs to keep in mind. The first concerns sensitivity to estimation choices: the historical window used, the assumptions on factor-innovation distributions, the exact number of factors retained: all these methodological decisions affect the final estimates at the margin. The NY Fed made a fixed specification choice and has maintained it since 2013, which guarantees the temporal consistency of the series but does not mean that other reasonable choices would produce identical numbers.

The second limitation concerns retrospective revisions. When the NY Fed re-estimates the model on the full history, an operation done periodically to incorporate new observations, past term-premium estimates can be revised. The June 2018 figure published in June 2018 is not exactly identical to the June 2018 figure read today on the dashboard; the gap typically remains modest (a few basis points) but can be larger on particularly hard-to-estimate points (March 2020, October 2023). This revisability property disqualifies ACMTP10 use for real-time automated decision rules but does not reduce its analytical value.

The third limitation concerns the epistemological status of the decomposition. The model assumes risk-neutral pricing for the mathematical derivation, and precisely the residual unexplained by that assumption constitutes the term premium. The approach is defensible but remains one modeling convention among others. Kim and Wright (2005), in an alternative model also published and maintained by the Fed, make different specification choices (three factors instead of five, distinct dynamic assumptions) and obtain qualitatively close but quantitatively distant estimates: the gap between the two ten-year premia is 47 basis points at the median across the 9,165 common sessions since 1990, 126 at the ninth decile and 236 at the maximum, and it changes sign across periods: the ACM premium exceeded Kim-Wright’s by 137 basis points at the median in 2010 and fell short of it by 89 basis points in 2022. The gap is less a margin of error than a measure of what the modeling convention decides. Joslin, Singleton and Zhu (2011) propose an approach with different canonical restrictions that also converges with ACM on directions but diverges at the margin on levels.

For broader context on the decomposition and its role in modern Treasury analysis, back to the decomposition identity.

Conclusion

The ACM 2013 model is neither a black box nor a truth. It is a reproducible empirical instrument, defended by its authors, adopted by the global central-banking community, and accompanied by alternative models published in parallel to allow methodological cross-checks. Its solidity rests on three properties: transparency of method, daily publication, and explicit acceptance of its own limitations. For anyone wishing to read the 10-year Treasury as more than a single number, ACMTP10 remains the most rigorous methodological entry point available.

Last updated — 6 October 2026