Choosing Regression Residuals for Treasury Yield-Curve Fly Trades
Summary
The document compares two ways to construct a residual for a relative-value trade in a three-bond yield-curve fly. One approach regresses a fly measure on a belly yield and a wing-spread measure, aiming to remove level and curve exposures. The other regresses the middle yield directly on the two outer yields and uses the residual as the dislocation measure.
The discussion asks whether the second regression’s hedge ratios minimize variance and whether that makes its residual more suitable for trading or potentially higher in Sharpe ratio. It also raises a concern that collinearity between the outer yields could leave curve exposure embedded in the residual. The author notes that the regression coefficients can be mapped between formulations, but the resulting residual series differ. No answer resolves which residual is preferable, so the document frames a model-selection question rather than providing trading evidence or a validated hedge rule.
Key ideas
- A yield-curve fly can be residualized against level and curve measures to define a relative-value signal.
- An alternative construction regresses the belly yield on the two wing yields.
- The residual series from the two formulations need not be identical even when their coefficients are related.
- Minimum-variance hedge ratios may be relevant to trading objectives, but the document does not establish which residual performs better.
- Potential hidden curve exposure is raised as a concern rather than resolved.
Tags
Full text
# Which residual to trade for fly RV? # Which residual to trade for fly RV? Starting with a "dislocation" measure that says whether bond is cheap wrt two others, I want to understand what the correct betas are for a fly that captures this dislocation. The way I initially learned this was the following: $y_1 - 2y_2 + y_3 = \beta_1y_2 + \beta_2(y_3-y_1) + \varepsilon \rightarrow \varepsilon = (1+\beta_2)y_1 - (2+\beta_1)y_2 + (1-\beta_2)y_3$ To me, this makes sense because it reminds me of PCA (although I am aware the two variables are not orthogonal) in the sense that the fly is driven by a level component (the belly) and a curve component (the wings), both of which we want to hedge; this results in a clean exposure to the residual. However, someone told me to think about the regression $y_2=\gamma_1y_1 + \gamma_2y_3 + \nu \rightarrow \nu = -\gamma_1y_1+y_2-\gamma_2y_3$ as a better way to find hedge ratios for the fly. I didn't quite understand some of the points they made, specifically with regards to the unconditional nature of the second regression vs the conditional one of the other. They also claimed something about the second regression being the minimum variance one which may be more suitable for trading since it may have a higher SR. The way I am thinking about it for now is that the latter regression would ignore the curve component since both variables are collinear, thus including the "PC2" information in $\nu$, which would result in trading a fly with hidden curve exposure. Any help would be appreciated. Thank you! EDIT: As @dm63 has mentioned below, it is true that there exists a mapping between $\beta_i$ and $\gamma_i$. However, the two residual time series $\varepsilon$ and $\nu$ will not be the same. Which of the two is the correct one to trade?
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.