# Attributing the change in NII to Shift, Twist and Butterfly

The movement of the zero rate curves can be decomposed into a shift movement (the level of interest rates) and a twist movement (the slope of the curve) and butterfly (the curvature of the curve). If we want to stress the net interest income (NII) of a bank by shocking these three factors, how do we attribute the change in NII of each of these factors? So far, I have identified following steps. Please correct me if I am going wrong or missing something somewhere:

1. Find the Shift, Twist and Butterfly (STB) - Which one is industry practice to arrive at the shift, twist and butterfly factors from the current term structure- PCA or factor model?
2. Shock S,T,B - Apply the required shock to the Eigen vectors of the three factors
3. Find the new yield curve - Add the new Eigen vectors weighted by their Eigen values to arrive at the new yield curve.
4. Recalculate NII based on the new yield curve - The thing that is not clear to me is how do I attribute the change in NII to the shocks given to shift, twist and butterfly.

Thanks in advance.

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## 1 Answer

Your steps 1. to 3. sound reasonable. I am not sure about industry practice (what industry?) I always do step 1. using PCA on historical correlations. If you plan to do a regulatory exercise better check with your regulator what he prefers.

Most interesting to me is step 4. which - I think - is in general impossible to do. This can be achieved only in very special cases. From what you describe, you model single discrete stress scenarios such as +/-100bps flat, which is good since it is a very special case. Assuming one stress each for S,T,B you have in total 7 = 2^3-1 different scenarios (3 single stresses of S,T,B , 3 where you have two stresses S&T, S&B and T&B and one where you stress S&T&B). If your portfolio reacts in a "simple" fashion (technically: The stresses do not interact) then Delta(S&T) = Delta(S) + Delta(T) and so on and you can decompose in the obvious way by reporting the marginal changes, i.e. the differences from each single stress. If your portfolio is more complicated, you are in big trouble and you should seriously consider whether such an attribution makes sense at all. If your stress factors are random variables you need to apply the thinking above in a regression context.

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Thanks for the response. I actually referred to 1. Barra Risk Model Handbook 2. papers.ssrn.com/sol3/papers.cfm?abstract_id=1601518 where there is a description of attributing returns to these factors. Although I didn't get all of what was given in these two documents, I thought that it might be possible for NII also. – user3212376 Feb 17 '14 at 8:59