# How to measure Steepener/Flattener/Butterfly sensitivity? (in 01)

This seems like a simple concept but I'm a bit lost. How can I calculate the dollar value sensitivity for a yield curve slope or butterfly position? I understand how DV01 can be calculated, but it seems that it's only applicable to a parallel shift in the yield curve.

For example, if I have a 2s10s steepener position, is there a formula to calculate how much I will make in $if the 2s10s increases by 1bp? Wondering about this for butterflies as well, e.g. how can I calculate this for a 1bp increase in the 2s5s10s? ## 2 Answers As you said, dv01 is the P&L from a 1 basis point parallel shift of the interest rate curve - i.e., all the instruments used to build the curve simultaneously move 1 bp. This is the most basic risk measure that everyone understands and uses. It has some obvious limitations: • the exposure is likely not to be linear, i.e. $$n \times$$ dv01 is not a great estimate of the P&L from interest rates moving $$n$$ bps for large $$n$$ and/or non-linear products like swaptions. • the dv01 does not tell you what happens if different curve-building instruments move by a different amount, rather than all in parallel, as you said, i.e. your sensitivity to the shape of the interest rate curve. • if you're working with different currencies and curves - 1bp move is a bigger deal when an interest rate is close to 0 than when an interest rate is close to 10%. It would be nice to have some kind of historical context for your risk. Some of the ways you can address these limitations inclide: • The most obvious calculuation that everyone with a sensitivity to the curve shape should be doing, in addition to perturbing all the curve-building instruments in parallel, is to see your interest rate sensitivities by tenor bucket. For this, pick some standard set of tenors for your risk reporting, e.g. 6m, 1y, 2y, ... 5y, 7y, 10y, 15, 20y, 20y, 25y, 30y - you decide what makes the most sense for your book, but your'll want this set of tenors not to change across time and various books. Then, for each tenor bucket, calculate the P&L impact of this rate only changing 1bp, while everything else stays constant. (Note that if you're using ED futures to build your IR curve, but you're looking for the impact of 1y, 2y... swap rates changing, this will require a little work, but can be done with inverse Jacobian). In most situations, these sensitivities will add up to your dv01 (up to some noise, which can grow materially large if the shape of the curve is unusually strange). (Some people prefer to calculate sentivities to forward rates. There are some advantages to that, but I feel that sensitivities to maket-observable rates are easier to understand and make more transparent P&L Explain.) So right away, you can see the sensitivity to 2s5s10s as a linear combination of the sensitivities in these tenor buckets. • Furthermore, this is more work, but knowing the history of your interest rates curves, you can run principal components analysis on each curve, and report sensitivities to a 1 historical stanard deviation (rather than fixed number of basis points) movement in the first three principal components. The PCs have intuitive geometric interpretation: parallel shift, slope, and curvature. • you can perform 'reverse stress tests', i.e. look for curve shocks that are plausible (in terms of the historical principal components) and cause the most adverse P&L. For example, you can run a Monte Carlo simulation petrurbing the first few principal components and see which Monte Carlo scenarios cause the most damage. (Note that running MC on curve-building instruments generally does not work very well because it leads to too many curve scenarios that just aren't plausible.) You can also run stress tests where you make up shocks (that don't necessarily look plausible based on the historical principal components) manually or based on historical events. • Thanks for the reply Dimitri. Oct 20, 2021 at 9:13 • ....but I don't quite understand how you can get from calculating bucket sensitivities to being able to "see the sensitivity to 2s5s10s as a linear combination of the sensitivities in these tenor buckets." For example, by using key rate durations to determine the sensitivities of each bucket, I have -$10 in 2y, $20 in 5y, and -$30 in the 10y bucket. How can I use this to determine the sensitivity to the 2s5s10s? Oct 20, 2021 at 9:30
• In this case, if the 2s5s10s increases by 1bp, I could gain 10 dollars if the 2s and 10s stays flat, and 5s increases 0.5bps. However, I could also gain 20 dollars – if the 2s and 10s each decrease 0.5bp and the 5s stays flat. Under both of these scenarios, the 2s5s10s is increasing by 1bp. Oct 20, 2021 at 9:31
• "Duration" (including key rate durations) is less intuitive than looking at the P&L from key rate moves (IR delta by tenor bucket). Suppose, you've calculated for each tenor bucket n that if nS moved 1bp, then your P&L is Pn . Then if you're concerned about a scenario that 2S and 10S go up 2bp while 5S goes down 3bp (is it plausible given your historical PCA analysis?), and are OK to ignore the convexity (IR gammas, cross gammas between tenors) then the P&L under this scenario is close to the sum 2P2-3P5+2P10. Or, fully reprice under your stress scenario for better accuracy. Oct 20, 2021 at 15:14

I think this question may be referring to something like what's done for curve gamma - a measure that's pretty important to spread options portfolios, for example. Leaving the PCA route aside, one simple way is to fix an anchor tenor (say 5y), and do a -/+ 0.5bp twist of the wings (of say 2y,30y) - with the intervening tenor shifts appropriately interpolated. Then $$\Delta PnL$$ will give the steepener/flattner position as a single number, while $$\Delta dv01$$ gives the curve gamma (i.e. how the portfolio's steepener/flattener delta position changes with a twisting of the curve). So for instance, if u had a duration flat portfolio with a 2s30s position of +100k dv01 at the moment and the curve gamma computed earlier was -5k, then you'll know that if 2s30s inverted by 10bps tomorrow then you'd be left with a +150k steepener (and your loss will be lot more than 1mm). I assume this concept can be extended to flys though I haven't come across it in practice.