A from-scratch PCA decomposition of the US Treasury yield curve into Level / Slope / Curvature, plus a candid backtest of a PC2 mean-reversion strategy.
This project reproduces Litterman & Scheinkman (1991) on the 2020-2026 sample, then pushes further by building and honestly evaluating a systematic strategy on top of the extracted PC2 (Slope) factor.
- The Question
- The Approach
- Key findings
- Demo figures
- Honest limitations
- Quick start
- Project structure
- Notebooks
- Glossary
- Notes
- 日本語版
Fixed income desks have talked about the yield curve in terms of three things — Level, Slope, Curvature — for as long as there has been a yield curve to talk about. The 1991 Litterman & Scheinkman paper showed those aren't just convenient labels: they fall out of PCA almost mechanically. The original spec for this project called for reproducing that finding on US Treasury data.
Three questions then become unavoidable:
- If the curve really decomposes into a few clean factors, can you trade them? The decomposition is the easy part — what does it actually buy you in execution?
- Where does the decomposition fail? PC1 ≈ 10Y is so tight that "trading PC1" is just trading 10Y. The interesting answer lives in what's not covered by the easy approximations.
- What happens if you flip the direction — use PCA for risk management instead of prediction? The same factor structure that fails to predict can succeed at hedging, and that asymmetry is the deepest finding here.
This project answers all three, on US Treasury data from 2020 to April 2026.
Three phases.
Phase 1 — Decomposition (Notebooks 01-05). Pull daily yields for ten US Treasury maturities (3M to 30Y) from FRED for 2020-01-02 → 2026-04-27 (1,580 business days). Clean and convert to bp changes, decompose via PCA, then interpret the resulting components against the traditional human-friendly factors (10Y level, 2s10s spread, butterfly). Verify the 96.2% explained-variance result and the textbook factor shapes; quantify exactly where the human indicators do and don't recover the PC structure.
Phase 2 — Strategy and diagnostics (Notebooks 06-09). Build a PC2-based mean-reversion strategy as the first development task. Run a baseline backtest, attribute the losses, layer in five candidate improvements, quantify the theoretical ceiling via perfect-foresight projection, and finally regress the PC scores against macro releases (CPI / NFP / FOMC) to see how much of the dynamics is explainable from public macro data. The honest takeaway: PCA can't be predicted off at meaningful Sharpe.
Phase 3 — Risk management (Notebooks 10-12). Take the same PCA model and use it the way Litterman & Scheinkman originally proposed — to hedge factor exposures rather than to predict them. Compute DV01 and PC factor exposures for a $100M long-30Y portfolio, solve a linear system for the hedge notionals, then backtest four variants (unhedged / PC1 only / PC1+PC2 / PC1+PC2+PC3) to quantify variance reduction. The result: 96.03% reduction in daily P&L variance with a 3-instrument hedge in-sample — exactly the explained-variance budget the PCA started with. Notebook 12 then re-runs the same hedge as a true walk-forward (refit the PCA each month on past data only, reapply for the next month) and finds the out-of-sample reduction holds at 93.27% — confirming the in-sample number isn't a look-ahead artifact. Same factors, opposite outcome from Phase 2.
The whole project is structured to be reproducible from scratch: every parquet artifact in data/processed/ and every figure in reports/figures/ is regenerated by running the notebooks in order against the installable yield_curve_pca package.
- PCA recovers the textbook factor structure on 2020-2026 data. Top three components explain 96.2% of variance (PC1 = 79.8%, PC2 = 11.8%, PC3 = 4.6%). The factor shapes match Litterman-Scheinkman exactly: PC1 loadings are uniformly positive (Level), PC2 flips sign across the belly (Slope), PC3 has positive wings vs. negative belly (Curvature).
- PC1 ≈ 10Y and PC2 ≈ 2s10s, but PC3 ≠ butterfly. Daily-change correlation of PC1 vs Δ10Y is 0.96, PC2 vs Δ2s10s is 0.91 — the spec targets are met. PC3 vs the spec's narrow 2-5-10 butterfly is only 0.26, and the best alternative wide butterfly tops out at 0.59. Curvature is where PCA's optimality actually buys you something a 3-point indicator cannot replicate.
- A naive mean-reversion strategy on PC2 loses money (Sharpe -0.38 over 2020-2026), and the loss is statistically distinguishable from random-strategy baselines. The strategy bleeds during regime shifts; loss attribution shows |z-score| has correlation -0.55 with daily P&L, meaning bigger signals correlate with worse trades — the opposite of the textbook expectation.
- Adding a regime classifier flips the sign of the Sharpe. A simple "ranging vs trending" filter (|60-day cumulative PC2| ≤ 30bp = ranging) restricts trading to 17% of days but takes Sharpe to +0.06. This is still noise-level, but it isolates the one filter that matters: when to trade is a bigger lever than how.
- The theoretical Sharpe ceiling is ~16 (perfect-foresight projection onto PC2). Industry-grade systematic strategies typically realize 0.5-1.5; the mapping from hit rate to Sharpe shows that just 52-54% directional accuracy on a daily PC2 signal would already produce 0.5-1.0.
- PC1 reacts to macro releases by ~1.5x. PC1 std on FOMC / CPI / NFP release days is 22.9 bp vs 14.9 bp on non-release days. In joint regression, the FOMC dummy is the only feature that survives (β = −7.24 bp on PC1, p < 0.001) — confirming PC1 as the macro-policy factor. CPI/NFP proxy surprises don't reach significance because the free-data substitute (deviation from rolling mean) doesn't capture true consensus surprise; comparable studies with Bloomberg consensus typically reach R² of 5-20%.
- PCA-based immunization eliminates 96% of P&L variance. A $100M long-30Y portfolio has daily P&L std of $1.66M unhedged. Hedging PC1 with a 10Y instrument cuts that 85%; adding a 2Y to hedge PC2 gets to 90%; adding 3M and 20Y to hedge PC3 reaches 96.03% variance reduction — exactly the 96.2% explained-variance budget the PCA started with. The same PCA model that couldn't be predicted off in Notebooks 06-08 can be hedged off cleanly, which is the asymmetry the project is built around.
- Walk-forward validation: 93.27% out-of-sample. Refitting the PCA on a rolling 252-day window with monthly rebalances (no future information at any decision point) preserves 93.27% of the in-sample 96.53% — a look-ahead premium of just 3.26 points. The economic mechanisms behind Level / Slope / Curvature are stable enough that PCA loadings barely drift across the 2020-2026 sample, which is why the immunization holds up out-of-sample.
Each line is a month-end snapshot, colored by date. The arc from low/flat (2020 COVID) → high/inverted (2023) → re-steepened (2026) makes the Level and Slope dynamics visible without any analysis.
These three shapes — emitted by a procedure that knows nothing about finance — match the qualitative factors fixed-income desks have used for decades.
Cumulative PC1 tracks the 10Y level almost perfectly. Cumulative PC2 tracks 2s10s. PC3 tracks butterfly only loosely — see Notebook 04 for the full discussion.
Things to be aware of when reading the results:
- Walk-forward only on the immunization side. Notebook 12 adds a true walk-forward to the immunization track (refit PCA each month on past data, hedge OOS, ≈93% variance reduction). The strategy backtests in Notebooks 06-08 still use the full 2020-2026 sample for fitting; the 6×6 sensitivity grid and random-strategy baseline there are partial robustness checks, not a walk-forward in the strict sense.
- No transaction costs in the backtest. Adding 0.25-0.5 bp per trade leg would shift the regime-strategy Sharpe from +0.06 to negative. The +0.06 should be read as "the strategy is not a money pit before frictions," not "the strategy makes money in production."
- Proxy macro surprises only. Notebook 09 uses
(actual MoM) − (6-month rolling average MoM)as a stand-in for true consensus surprise. TheR² < 1%regression result is a data-quality limitation; with Bloomberg/Refinitiv consensus data, comparable studies typically reach R² of 5-20%. - Single market. US Treasuries only. The factor structure is known to look qualitatively similar across major sovereigns (Bunds, JGBs, Gilts), but cross-country comparison is not in scope for this iteration.
- The 2020-2026 window is unusual. It contains the COVID emergency easing, the fastest US hiking cycle in 40+ years, and the longest yield-curve inversion on record. The factor decomposition (and any strategy fit to it) may not generalize to calmer regimes — PC1 explained variance, for instance, lands at 79.8% here vs the 85-90% typical of pre-COVID samples.
- Sharpe +0.06 is not investable. The regime-classifier strategy turns positive but stays well inside noise. Treat it as a quantification of "where mean reversion can and can't work," not as a deployable signal.
Each of these is a deliberate scope choice, not an oversight. They map onto natural next steps listed in reports/findings.md.
Requires Python 3.11+. Tested on Python 3.12.10 (Windows / Git Bash).
# Clone and enter the project
git clone <YOUR_FORK_URL> yield-curve-pca
cd yield-curve-pca
# Create a virtual environment with uv (or use python -m venv)
python -m pip install uv # if uv isn't installed
python -m uv venv
# Activate the venv
source .venv/Scripts/activate # Windows (Git Bash)
# source .venv/bin/activate # macOS / Linux
# Install the package and dev dependencies in editable mode
uv pip install -e ".[dev]"
# Run the test suite
pytest
# Run the notebooks (or just open them in JupyterLab)
PYTHONUTF8=1 jupyter labThe notebooks under notebooks/ are numbered and meant to run in order. Each one loads its inputs from the parquet files left behind by the previous notebook, so re-running notebook 03 is enough if you only want to re-fit the PCA. To regenerate everything from scratch, delete data/processed/ and reports/figures/ first, then run notebooks 01 → 12 in sequence (or run make notebooks if a Makefile is present).
yield-curve-pca/
├── src/yield_curve_pca/ # Installable package (`pip install -e .`)
│ ├── data/
│ │ ├── loader.py # FRED CSV fetching
│ │ └── preprocessor.py # Clean & convert to bp changes
│ ├── analysis/
│ │ ├── pca_analyzer.py # YieldCurvePCA class
│ │ └── backtest.py # Strategy backtest engine
│ ├── visualization/ # Reserved for Phase 2 expansion
│ └── utils/
│ └── config.py # Constants (maturities, FRED endpoint, …)
│
├── notebooks/ # Phase 1 (01-05) + development tasks (06-09)
├── tests/ # 50 unit tests (pytest)
│
├── data/
│ ├── raw/ # Raw FRED CSV (gitignored, regenerable)
│ └── processed/ # Cleaned parquet artifacts
│
├── reports/
│ ├── figures/ # Publication-quality PNGs
│ └── findings.md # Written analysis summary
│
├── pyproject.toml
├── README.md
└── yield_curve_pca_spec.md # Original project specification
| # | File | Purpose |
|---|---|---|
| 01 | 01_data_exploration.ipynb |
Fetch yields from FRED, clean, plot, save |
| 02 | 02_curve_dynamics.ipynb |
Daily bp changes, slope indicators, Bull/Bear × Steep/Flat regimes |
| 03 | 03_pca_basics.ipynb |
Apply PCA, explained variance, loadings interpretation |
| 04 | 04_pca_interpretation.ipynb |
Compare PCs against 10Y / 2s10s / butterfly |
| 05 | 05_event_study.ipynb |
FOMC days vs ordinary days; three case studies; generates findings.md |
| 06 | 06_pc2_mean_reversion.ipynb |
First systematic strategy attempt (negative result, with diagnosis) |
| 07 | 07_loss_analysis_and_improvements.ipynb |
Loss attribution + five improvement variants |
| 08 | 08_strategy_ceiling_and_walkthrough.ipynb |
Theoretical Sharpe ceiling + entry/exit walkthrough |
| 09 | 09_macro_factor_regression.ipynb |
Regress PC scores on CPI/NFP/FOMC |
| 10 | 10_pca_immunization.ipynb |
DV01, portfolio, PC factor exposures, P&L decomposition (Phase 1) |
| 11 | 11_immunization_backtest.ipynb |
Hedge construction + 4-variant backtest, 96% variance reduction (Phase 2) |
| 12 | 12_walk_forward_immunization.ipynb |
Walk-forward (rolling-window) validation — 93% out-of-sample reduction |
For a project-specific reference of the financial and statistical terms used here (yield, CMT, bp, butterfly, DV01, look-ahead bias, walk-forward, regime classifier, …), see docs/glossary.md.
- PCA in practice: standardization is not free — it changes how loadings are interpreted. For same-unit inputs, skipping it preserves direct bp interpretability of the factors.
- Reproducibility discipline: every parquet is the output of an upstream notebook; no silent edits. This took deliberate engineering once notebooks started multiplying.
- Backtest correctness: implementing
position[t] = f(z_score[t-1])correctly via.shift(1)is non-negotiable. Verifying it with a deliberate "smash the future" perturbation test confirms the modular implementation has no look-ahead. - Statistical hygiene: comparing a strategy's Sharpe to a random-strategy distribution is what made "+0.06" legible as "still noise" rather than "small but real." Without that benchmark you'd be tempted to keep tuning a fundamentally negative strategy.
- Module extraction: the rewrite from notebooks →
src/yield_curve_pca/reproduced the original results bit-for-bit, which is the only meaningful test that the refactor didn't break anything.
- Yield, not coupon: the number on a screen is the implied yield from today's price, not the bond's coupon — they only coincide at issuance. Price ↑ → yield ↓ is the same arbitrage statement as duration sensitivity.
- Inversion ≠ recession in 2022-2024: the longest US inversion in 40+ years was driven by the Fed pre-emptively hiking against inflation, not by the market pricing recession. The textbook signal-from-history broke because the cause was unusual.
- It's the surprise, not the action: the 75bp hike on 2022-06-15 produced a Bull-Steepener, not the textbook Bear-Flattener, because the WSJ leak two days earlier had moved the priced-in expectation. The market reacts to (actual − expected), not to actual.
- PCA reveals what curves move on, but it doesn't trade itself: PC1 ≈ 10Y is so tight that a PCA-based "Level alpha" is just a 10Y trade. The interesting use cases are PC2 (Slope) regime detection and PC3 (Curvature) for trades a 3-point butterfly literally cannot represent.
イールドカーブは何で動いているのか — それを 3 つの因子(Level / Slope / Curvature)で説明できる、というのが Litterman & Scheinkman (1991) の古典的発見。本プロジェクトはまずこれを 2020-2026 年の米国債データで再現し、その上で 3 つの問い に答えた:
- 3 因子に分解できるなら、それを使ってトレードできるか?(分解は易しい、で実装は?)
- どこで分解は破綻するか?(PC1 ≈ 10Y は強すぎる近似、本当に面白いのは「人間指標で捕捉できない部分」)
- 予測ではなくリスク管理に使うとどうなるか?(同じ PCA を逆方向に — ヘッジに — 使った場合の効果は?)
3 フェーズ構成:
- Phase 1(Notebooks 01-05): FRED から 10 年限の日次利回りを取得 → クリーニング → bp 変化に変換 → PCA → 人間指標(10Y、2s10s、バタフライ)との対応関係を定量化。
- Phase 2(Notebooks 06-09): PC2 を使った平均回帰戦略を構築 → 失敗 → 損失帰属で原因特定 → 5 つの改良案 → 理論上限の計算 → マクロ要因との回帰。「PCA を予測に使うと負ける」という結論を honest に出す。
- Phase 3(Notebooks 10-12): 同じ PCA を リスク管理ツールとして 使う。仮想ポートフォリオの PC factor exposures を計算 → 連立方程式でヘッジ比率を解く → 4 variant のバックテストで 96.03% の分散削減 を確認(残差 3.97% は PCA の未説明分散 3.8% とほぼ完全に一致)。さらに walk-forward(月次リバランス、252日ローリング窓で PCA 再 fit)で out-of-sample 検証も実施し、削減率 93.27% を確認(look-ahead premium はわずか 3.26 ポイント)。Phase 2 と同じ PCA、逆方向の結果。
src/yield_curve_pca/ に installable package、50 件の pytest、CI/CD まで揃えた reproducible な構成。
- PCA が金融用語を再発見: PC1 ≈ Level、PC2 ≈ Slope、PC3 ≈ Curvature。Litterman-Scheinkman そのまま。
- PC1 と 10Y は相関 0.96、PC2 と 2s10s は相関 0.91、PC3 と3点バタフライは 0.26(narrow)〜 0.59(best wide) → Curvature は PCA の最適性が必要な唯一の領域。
- 素朴な平均回帰戦略は Sharpe -0.38 で負け。失敗の原因を損失帰属分析で定量化(|z-score| が P&L と相関 -0.55、つまり 強い信号ほど負ける)。
- Regime classifier だけが Sharpe をプラス(+0.06)に。「いつトレードするか」が「どうトレードするか」より重要。
- Sharpe の理論上限は約 16(perfect foresight)。Production レベルは 0.5-1.0、これに必要な hit rate は約 52-54%。
- PC1 はマクロ発表に 1.5 倍反応: PC1 std がリリース日 22.9 vs 通常日 14.9 bp。Joint regression で FOMC ダミーのみ有意(β = -7.24 bp、p<0.001)。CPI/NFP の代理サプライズは無料データの限界で R² < 1%(本物のコンセンサス予想なら 5-20% の見込み)。
- PCA イミュナイゼーションで P&L 分散を 96% 削減: $100M 30Y ロングのポートフォリオに対して、PC1+PC2+PC3 を {3M, 5Y, 20Y} でヘッジ → 日次 std $1.66M → $0.33M(96.03% 削減)。PCA で予測しようとして負けた(Sharpe -0.38)同じ3因子が、ヘッジに使うと劇的に効くという非対称性が本プロジェクトの中核メッセージ。
- Walk-forward で OOS 検証も完了(93.27%): 月次リバランス、252日ローリング窓で PCA を再 fit する真の out-of-sample バックテストでも 93.27% の削減を達成(in-sample 96.53% との差は 3.26 ポイント)。PCA loadings が時代を超えて安定なため(Litterman & Scheinkman の核心命題)、look-ahead を取り除いても結果が崩れない。
- Walk-forward は immunization のみ実施(strategy 側は未実施) — Notebook 12 でリスク管理側に walk-forward を実装(月次リバランス、93% OOS 削減)。一方で Notebook 06-08 の戦略バックテストは全期間 fit のまま。感度グリッドとランダム比較で部分的な robustness は確認済み。
- 取引コストなし — Sharpe +0.06 は「コスト前で赤字ではない」レベル。コスト含めると簡単に負ける。
- マクロサプライズは代理変数 — 真のコンセンサス予想(Bloomberg)が無料で取れないため、6ヶ月平均からの乖離で代用。R² < 1% はデータの限界。
- 米国債のみ — Bunds、JGBs への展開は scope 外(発展課題で残置)。
- 2020-2026 は異常な期間 — COVID、過去最速の利上げ、過去最長の逆イールドを含む。他の局面に外挿できる保証はない。
- Sharpe +0.06 は実運用不可 — production には Sharpe 0.5+ が必要、HMM・クロスアセット features 等の追加が必要。
詳細は reports/findings.md の「Limitations and what I'd do next」を参照。
詳細は上の英語セクションを参照。要点だけ:
src/yield_curve_pca/— installable パッケージ。loader / preprocessor / pca_analyzer / backtest / immunization(walk-forward 含む)notebooks/01-12— Phase 1(基礎)+ 発展課題(戦略バックテスト + マクロ回帰 + リスク管理 + walk-forward 検証)tests/— pytest ユニットテスト 50 件reports/findings.md— プロジェクトの分析サマリー
MIT — see LICENSE.
- Federal Reserve Economic Data (FRED) for the raw yield series.
- Litterman, R. and Scheinkman, J. (1991). "Common Factors Affecting Bond Returns." The Journal of Fixed Income, 1(1), 54-61.