TCC / FIXED-INCOME RESEARCH
Term-structure laboratory
Close / Oct 28, 2026 PostgreSQL / linked Official observations
Research note · cross-sectional term structure

U.S. Treasury term-structure estimation and residual diagnostics

Coupon-consistent zero-curve bootstrap, Nelson–Siegel factor estimation, cross-sectional repricing errors, and DV01\mathrm{DV01}DV01-normalized butterfly signal evaluation.

valuation date
Oct 28, 2026
cross-section
n=10n=10n=10
estimator
tcc-ns-v1.0
history
T=956T=956T=956
source state
official / online
generated
18:58 UTC
Model state / ttt
Tt={3M, 6M, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 20Y, 30Y}\mathcal{T}_t=\{3\mathrm{M},\,6\mathrm{M},\,1\mathrm{Y},\,2\mathrm{Y},\,3\mathrm{Y},\,5\mathrm{Y},\,7\mathrm{Y},\,10\mathrm{Y},\,20\mathrm{Y},\,30\mathrm{Y}\}Tt​={3M,6M,1Y,2Y,3Y,5Y,7Y,10Y,20Y,30Y}
θ^t=(β^0,β^1,β^2,τ^)\widehat{\boldsymbol\theta}_t=(\widehat\beta_0,\widehat\beta_1,\widehat\beta_2,\widehat\tau)θt​=(β​0​,β​1​,β​2​,τ)
εi,t=104 ⁣[zi,t−z^NS(Ti;θ^t)]\varepsilon_{i,t}=10^4\!\left[z_{i,t}-\widehat z_{\mathrm{NS}}(T_i;\widehat{\boldsymbol\theta}_t)\right]εi,t​=104[zi,t​−zNS​(Ti​;θt​)]
st=max⁡i∈I∣εi,t∣s_t=\max_{i\in\mathcal I}|\varepsilon_{i,t}|st​=maxi∈I​∣εi,t​∣
Official par yields drive synthetic maturity nodes; auction CUSIPs are reference metadata only, and no displayed residual is an executable dealer quote.
Summary / t
Cross-sectional estimates
Units: percent, basis points
y(10Y)y(10\mathrm{Y})y(10Y) levelS.01
5.05%5.05\%5.05%
Official par-yield observation
slopeS.02
+38.0 bp+38.0\,\mathrm{bp}+38.0bp
y(10Y)−y(2Y)y(10\mathrm{Y})-y(2\mathrm{Y})y(10Y)−y(2Y)
discrete bowS.03
−2.0 bp-2.0\,\mathrm{bp}−2.0bp
2y(5Y)−y(2Y)−y(10Y)2y(5\mathrm{Y})-y(2\mathrm{Y})-y(10\mathrm{Y})2y(5Y)−y(2Y)−y(10Y)
fit RMSES.04
8.67 bp8.67\,\mathrm{bp}8.67bp
R2=0.9567R^2=0.9567R2=0.9567
threshold exceedancesS.05
999
∣εi∣≥1.5 bp|\varepsilon_i|\geq 1.5\,\mathrm{bp}∣εi​∣≥1.5bp
[01]
Figure 01 / estimated term structures

Zero-coupon term-structure estimation

Observed constant-maturity par yields, coupon-consistent zero nodes, and the four-parameter Nelson–Siegel projection.
RMSE⁡z=8.67 bp\operatorname{RMSE}_z=8.67\,\mathrm{bp}RMSEz​=8.67bp
zt(T), ytCMT(T)z_t(T),\ y_t^{\mathrm{CMT}}(T)zt​(T), ytCMT​(T)rate (%)\text{rate }(\%)rate (%)
Figure 01. Observed par yields, bootstrapped continuously compounded zero rates, and an analytically evaluated Nelson–Siegel fit at the Oct 28, 2026 close. Straight segments connect reported nodes and do not imply an auxiliary spline interpolation.
[T01]
Table 01 / coefficient estimates

Parametric factor decomposition

τ\tauτ profiled on a bounded grid; conditional coefficients solved by linear least squares.
parameterinterpretationestimate
β^0\widehat\beta_0β​0​
Long-rate asymptote
5.565%5.565\%5.565%
percent
β^1\widehat\beta_1β​1​
Short-minus-long limit
−1.378%-1.378\%−1.378%
percent
β^2\widehat\beta_2β​2​
Curvature-loading amplitude
0.000%0.000\%0.000%
percent
τ^\widehat\tauτ
Exponential decay scale
3.35 yr3.35\,\mathrm{yr}3.35yr
years
lim⁡T↓0zNS(T)=β0+β1,lim⁡T→∞zNS(T)=β0\lim_{T\downarrow0}z_{\mathrm{NS}}(T)=\beta_0+\beta_1,\qquad \lim_{T\to\infty}z_{\mathrm{NS}}(T)=\beta_0T↓0lim​zNS​(T)=β0​+β1​,T→∞lim​zNS​(T)=β0​
explained variation
Rz2=0.9567R_z^2=0.9567Rz2​=0.9567
1−∑jεj2 / ∑j(zj−zˉ)21-\sum_j\varepsilon_j^2\,/\,\sum_j(z_j-\bar z)^21−∑j​εj2​/∑j​(zj​−zˉ)2
zero-rate fit error
8.67 bp8.67\,\mathrm{bp}8.67bp
104N−1∑jεj210^4\sqrt{N^{-1}\sum_j\varepsilon_j^2}104N−1∑j​εj2​​
bootstrap repricing norm
9.6×10−129.6\times 10^{-12}9.6×10−12
∥ΔPclean∥∞\lVert\Delta P^{\mathrm{clean}}\rVert_\infty∥ΔPclean∥∞​
[02]
Figure 02 / residual operator

Cross-sectional specification errors

εt,jbp=104 ⁣[zt(Tj)−z^NS,t(Tj)]\varepsilon_{t,j}^{\mathrm{bp}}=10^4\!\left[z_t(T_j)-\widehat z_{\mathrm{NS},t}(T_j)\right]εt,jbp​=104[zt​(Tj​)−zNS,t​(Tj​)]
Ntail,t=#{j:∣εt,jbp∣≥1.5}=9N_{\mathrm{tail},t}=\#\{j:|\varepsilon_{t,j}^{\mathrm{bp}}|\geq1.5\}=9Ntail,t​=#{j:∣εt,jbp​∣≥1.5}=9
εt,jbp\varepsilon_{t,j}^{\mathrm{bp}}εt,jbp​basis points\text{basis points}basis points
Figure 02. Cross-sectional zero-rate specification errors at a single close. Dashed rules mark ±1.5 bp\pm1.5\,\mathrm{bp}±1.5bp. Positive residuals denote zero yields above the parametric fit; they are not bond-level cheapness estimates.
Research note / NS-4 estimator / TypeScript numerical kernel / Railway PostgreSQLIndicative analysis only / not executable pricing