Yield Curve Decomposition and Parallel Curvature Risk

# Term structure decomposition of a yield curve


# Term structure decomposition of a yield curve












The BCBS gives the following definition for Curvature GIRR risk factors (21.8):

> Curvature GIRR: (a) The GIRR curvature risk factors are defined along only one dimension: the constructed risk-free yield curve per currency with no term structure decomposition. For example, the euro, Eonia, three-month Euribor and six-month Euribor curves must be shifted at the same time in order to compute the euro-relevant risk-free yield curve curvature risk capital requirement. For the calculation of sensitivities, all tenors (as defined for delta GIRR) are to be shifted in parallel.

What does it mean to have a yield curve without term structure decomposition? I thought ‘yield curve’ and ‘term structure of interest rates’ were the same thing. Also, isn’t any yield curve just a series of dates with corresponding interest rates?

## Answer by Dimitri Vulis (score 2, accepted)

https://quant.stackexchange.com/a/81091

There are many ways to calculate interest rate delta and gamma.

For example, for interest rate delta, you can pick a set of standard tenors (e.g. 3M, 6M, ..18M, 2Y, ... 30Y), and perturb the interest rate only at this tenor - ceteris paribus, reprice, and get the P&L impact / sensitivity for this tenor. If you add up the resulting sensitivities, the sum will be close to the P&L impact of simultaneously perturbing the rates at all the tenors in parallel. This is what most people do these days. Many have additional ways of looking at the 1st order sensitivities, e.g. in terms of historical principal components.

For the interest rate gamma, there are many more possible approaches, all of which may be reasonable depending on the needs. For example, a few people risk-managing complicated payoffs compute a tenor-x-tenor square matrix. But for most others, this matrix would have just immaterial noise off its diagonal, i.e. the cross-gammas between different tenors, like the cross-gamma of the 2Y and 10Y rate. And if you have calculated delta by tenor from the previous paragraphs, you could get "for free" the gamma by tenor, i.e. the matrix diagonal. In practice, this is still too much information for linear positions. For example, for a P&L attribution of a 30-year interest rate swap, you do want one number for the gamma to avoid too much unexplained P&L, but having a gamma for each tenor may be overkill.

You could calculate the one number for the gamma by simply adding up the gammas at each tenor if you have them; or by using as weights the interest rate deltas by tenor; or in lots of other ways.

However the FRTB spec says that you don't need gamma broken down by tenor, unless you want it. Just perturb all the rates in parallel up and down and get the one interest rate gamma number.