Using PCA Eigenvectors to Hedge Parallel Shifts and Yield-Curve Slope

Summary

The document addresses how to construct a hedge across two-, five-, and ten-year Treasury futures that is neutral to parallel shifts and changes in yield-curve slope. It describes running principal component analysis on a returns matrix and interpreting the first component as a level shift, the second as a slope change, and the third as curvature. In the example, the third eigenvector is orthogonal to the first two, so holdings proportional to that vector have no exposure to those components in the sample PCA representation.

The response gives illustrative eigenvector values and notes that estimates from more data may be more reliable. This is a component-based hedge construction, not a full implementation recipe: it does not discuss scaling positions by contract risk, conversion factors, estimation windows, or out-of-sample hedge performance. The resulting neutrality depends on the PCA inputs and on the components capturing the relevant movements in the curve.

Key ideas

• PCA can summarize Treasury futures returns in components interpreted as level, slope, and curvature movements.
• The third eigenvector is orthogonal to the first two and can represent a portfolio neutral to their components.
• Holdings proportional to that eigenvector provide the described hedge in the PCA representation.
• Position scaling and hedge effectiveness depend on data and implementation details not covered in the example.

Tags

# Calculating PCA hedge ratio for 3-leg spread


# Calculating PCA hedge ratio for 3-leg spread












I'm wondering how can I find PCA hedge ratio for a 3-leg spread? I've taken the simple steps laid out in here.

I've taken some treasury futures data for 2yr,5yr,10yr and ran the PCA. The first eigenvector correspond to parallel shift hedge, the second to slope hedge, the third to curvature hedge. I'm wondering how would I use the PCA to calculate a hedge ratio that is hedged to parallel shift and slope?

Thanks.

## Answer by Chris Taylor (score 3, accepted)

https://quant.stackexchange.com/a/25357

Let's use the following returns matrix, X

```
 2Y        5Y        10Y
 --------------------------
 0.0143    0.0910    0.1451
 0.1791    0.3505    0.4588
 0.0572    0.1358    0.0120
 0.0357    0.1809    0.2884
-0.0571   -0.1096   -0.0719
 0.0286    0.0710    0.1319
 0.0429    0.1806    0.2754
-0.0357   -0.0579   -0.1075
 0.0714    0.2513    0.4304
-0.0214   -0.0771   -0.1667
```

The first PCA eigenvector is (0.2, 0.55, 0.8) corresponding to a shift, and the second eigenvector is (0.55, 0.62, -0.56) corresponding to a change in the slope.

The third eigenvector is (0.81, -0.55, 0.17) which by construction is orthogonal to the first two, and is therefore hedged against changes in both the level and slope of the curve - any portfolio with holdings proportional to this eigenvector will be hedged in the way you describe.

Obviously if you use more data, you will get a more reliable estimate of the PCA eigenvectors and a more effective hedge.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.