What are the level, slope, and curvature factors of the yield curve?

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AcadiFi · Accepted Answer

Principal Components Analysis (PCA) applied to historical yield curve changes decomposes movements into three orthogonal factors that explain ~99% of curve variation. **Factor 1: Level (~80–85% of variation)** — Parallel shift of the entire curve up or down. All yields move in the same direction by roughly the same amount. Driven by: Fed policy expectations, inflation, risk-off/risk-on regime. **Factor 2: Slope (~10–12% of variation)** — Twist of the curve around a pivot point. Short rates move one direction, long rates the other. Driven by: growth expectations (steepening in recovery), central bank near-term policy divergent from long-term. **Factor 3: Curvature (~3–5% of variation)** — Butterfly movement. Belly moves one direction, wings move the other. Driven by: liquidity preferences at specific maturities, relative value supply/demand (Treasury auctions). **Combined**: ~98–99% of daily curve variation explained by three factors. **Practical implications**: 1. Most single-factor duration hedging (hedging against Level) captures ~85% of rate risk. 2. Adding key-rate duration or slope hedges captures ~95%. 3. Adding curvature hedge captures ~99%. 4. Portfolio risk models use these three factors as primary rate risk drivers. Worked example: Aetherline Risk Analytics decomposes $10B Treasury portfolio daily P&L using PCA: - Level factor contributed 82% of variance in 2024. - Slope contributed 13%. - Curvature contributed 4%. - Residual (idiosyncratic points) contributed 1%. Risk budget: 60bp level risk (1σ), 20bp slope risk, 8bp curvature risk, aggregated via factor covariance. ```mermaid graph LR A[Curve Change] --> B[Level 85%
parallel shift] A --> C[Slope 12%
twist] A --> D[Curvature 3%
butterfly] B --> E[Duration
hedge] C --> F[Key-rate
duration] D --> G[Butterfly
hedge] ``` **CFA test**: Know the approximate % contributions (80 / 10 / 5 roughly) and that these three factors together capture nearly all curve variation.

What are the level, slope, and curvature factors of the yield curve?

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