How Synthetic Tenors Affect Yield Curve PCA
Summary
Yield curve PCA needs observations on a consistent tenor grid, but some markets have quotes at only a few maturities. The note compares running PCA on the sparse observed points with interpolating or extrapolating rates to create a fuller grid. The choice of curve construction method affects the principal components and the risk factors they represent.
The discussion highlights two risks of adding synthetic tenors. With very few observed rates, a curvature component can be driven almost entirely by the interpolation method: a flat-forward assumption may suppress curvature, while a spline may imply unsupported curvature. Also, interpolated points are highly correlated with nearby points, which can inflate the variance explained by PCA. The document gives conceptual examples rather than empirical comparisons, and does not recommend a particular interpolation method. Its central caveat is that synthetic data can add apparent detail without adding independent market information.
Key ideas
- PCA on yield curves generally requires a consistent set of tenors across observations.
- Adding interpolated tenors can create a fuller grid when market quotes are sparse.
- With few observed rates, the interpolation method can determine the shape of the curvature component.
- Highly correlated synthetic points can inflate the variance explained by PCA.
- PCA factors built from synthetic tenors should be interpreted in light of the curve construction assumptions.
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Full text
# Answer by David Duarte (score 2) # What are the pros/cons of adding synthetic tenors for PCA on a yield curve with with few observable rates, and using complicated interpolation? In emerging markets, some yield curves have observable rates only at a limited set of tenors, for example: 5Y and 10Y, and no quotes between them. But even in developed markets we observe 7Y and 8Y swap rates, and when we need a discount factor for a date with no directly observable quote, for example, 7.5 years, then we need to choose some interpolation/extrapolation method. Decades ago, everyone just assumed that forwards were constant between nodes, but now there are numerous complicated methodologies and papers, for example: - Patrick Hagan, Graeme West. Methods for Constructing a Yield Curve, Wilmott Magazine, May 2008, 70-81 - Patrick Hagan. Building Curves Using Area Preserving Quadratic Splines. Wilmott Magazine, May 2018 https://doi.org/10.1002/wilm.10676 and many others. In the context of principal component analysis on the yield curve, what are the advantages/disadvantages of adding synthetic quotes for more tenors? ## Answer by David Duarte (score 2) https://quant.stackexchange.com/a/85659 When you apply PCA to yield curves, the model needs a steady, evenly spaced set of tenors across your whole dataset to work its magic properly. Look at it this way: if you're stuck with just 5Y and 10Y data, you have two choices. You can either run the PCA on just those two points, which throws out nearly all the richness of the curve, or you can estimate the missing pieces via interpolation and extrapolation to build a full grid (like 1Y, 2Y, 3Y, etc.). How you choose to fill in those gaps isn't just a formatting choice, it fundamentally shapes what the PCA "sees" and dictates exactly what your final risk factors actually mean. To name a few disadvantages: the third component, normally curvature, will be totally defined by your choice of interpolation method. Think of flat foward, with zero curvature by definition, vs cubic spline which would be unreliable and potentially exhibit excess curvature, with only two points. Another one would be the unrealistically high explained variance introduced by the fact that the 6, 7, 8 and 9 tenors would be very highly correlated
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.