# 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.

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.

• Thank you Chris. This makes total sense. I also have a related question: should we use PCA on the difference in price or on the price itself? – A1122 Apr 12 '16 at 2:38
• Always on the difference in price. If you are using data at a frequency of >= 1 day then it is better to compute relative differences, i.e $(p(t+1) - p(t)) / p(t)$. – Chris Taylor Apr 12 '16 at 7:33
• Hi Chris, I'm revisiting this post. What is your thought on the data frequency for intraday trading? Is doing PCA on 5-minute data better than daily data for calculating PCA hedge ratios in the sense that it provides a more precise hedge? Is there an industry standard data frequency for this? – A1122 Aug 27 '16 at 10:38
• There is no industry standard. The data frequency should correspond to the time period over which you are trying to mitigate risk. The Epps effect says that correlations will be lower at higher sampling frequencies, therefore you can expect the same position to be a less effective hedge over short time periods than over longer periods. The advantage of short periods is that you can use more independent samples, so you will get a more accurate approximation to the hedge ratios. This implies that over short time horizons you should worry more about... – Chris Taylor Aug 27 '16 at 11:47
• ...the idiosyncratic risk of each asset (i.e. the risk that is not explained by factors) and expend hedging effort trying to reduce this (by trading in the same asset, or very highly correlated assets). Over longer horizons you worry more about your factor risk (particularly the level factor, which dominates the other two). – Chris Taylor Aug 27 '16 at 11:49