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.