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Sizing a Rate Butterfly to Neutralize Level and Slope PCs

Article Quant Q&A · Author: Qing
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Summary

The note shows how to choose the 5-year and 10-year weights in a 5s/7s/10s interest-rate butterfly so its exposure to the first two principal components is zero. It represents the portfolio’s key-rate sensitivities in PV01 terms, sets the 7-year position as the fixed center leg, and projects the three-leg risk vector onto the level and slope component loadings. Setting both projected exposures to zero produces a two-equation linear system for the outer-leg weights.

A closed-form solution is given when the matrix formed from the 5-year and 10-year loadings is invertible. If it is singular, the proposed weights do not exist uniquely under this setup. The note warns that even when a solution exists, small changes in correlations or covariances can make the weights unstable. It also mentions using the third principal component as an alternative weighting idea, but provides little detail about how to implement or evaluate that approach.

Key ideas

  • Express each maturity’s risk in PV01 terms before calculating portfolio exposure to principal components.
  • Set the butterfly’s projected exposure to each of the first two components to zero.
  • Solve the resulting two-equation system for the 5-year and 10-year weights around the fixed 7-year leg.
  • A unique closed-form solution requires an invertible matrix of outer-maturity component loadings.
  • Small changes in the covariance structure can make the calculated hedge weights unstable.

Tags

Fixed incomePortfolio constructionRisk managementStatistics
Full text
# Construct a butterfly interest rate portfolio to eliminate PCA exposures


# Construct a butterfly interest rate portfolio to eliminate PCA exposures












I have data from 2012 to 2016 for interest rates whose term range from 2 month to 30 years, a total of 10 Principal components can be calculated. Then I want to construct a portfolio, $$WFLY = w_1 *5Y - 7Y +w_2 * 10Y$$, this portfolio will have no exposure to the first two principle components, level and slope. How to determine the coefficients $w_1$ and $w_2$?

## Answer by Attack68 (score 6, accepted)

https://quant.stackexchange.com/a/42673

Let $S$ be your risk sarray, expressed in pv01, for each of your (implied) 10 instruments. You restrict the array to all zeroes except those corresponding to the 5Y, 7Y and 10Y risks, e.g. if 1Y:10Y were your instruments you would have:

$$S = [0, 0, 0, 0, w_1, 0, -1, 0, 0, w_2]^T $$

You seek the solution of $w_1$, $w_2$ such that your risk expressed in PCs is zero relative to the first two components, i.e. where $E_{1,2}$ are the two columnwise PCs:

$$ S^T E_{1,2} = [0, 0] $$

This reduces to:

$$[w_1, -1, w_2] \begin{bmatrix} e_{5y,1} & e_{5y,2} \\ e_{7y,1} & e_{7y,2} \\ e_{10y,1} & e_{10y,2} \\ \end{bmatrix} = [0, 0]$$

Taking the constant to the right to form a linear system (Ax = b) is:

$$ [w_1, w_2] \begin{bmatrix} e_{5y,1} & e_{5y,2} \\ e_{10y,1} & e_{10y,2} \\ \end{bmatrix} = [e_{7y,1}, e_{7y,2}] $$

The solution is (if it exists):

$$ [w_1, w_2] = [e_{7y,1}, e_{7y,2}] \begin{bmatrix} e_{10y,2} & -e_{5y,2} \\ -e_{10y,1} & e_{5y,1} \\ \end{bmatrix} \frac{1}{e_{5y,1}e_{10y,2}-e_{5y,2}e_{10y,1}} $$

A word of warning: these weights might be very unstable depending upon small correlation/covariance changes.

> You may also be interested in How to adjust butterfly 2s5s10s swaps trade for directionality?

## Answer by advouk7 (score 0)

https://quant.stackexchange.com/a/51335

Just implement PCA and use the 3rd PC (curvature) as weights

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