Linear Risk Transformations for Floating-Rate Bond Basis Risk

# How to calculate basis risk for a floating rate bond?


# How to calculate basis risk for a floating rate bond?












Right now the bank, I work in, calculates basis risk (BR01) for floating rate bonds (FRNs) in the following way.

In the post below Dimitri Vulis and Kermittfrog suggested using Jacobian matrices for translation of DV01s into "the basis risk space".

DV01 on LIBOR vs. SOFR basis Swaps

I would really appreciate if someone could show how to apply the approach to my example.

Personally, I could not figure out how to make a Jacobian matrix from two absolutely different curves:

- The discounting curve is built from government bonds (the FRN was issued by government);

- The coupon curve is constructed from OIS swaps.

Edited:

What perplexes me in the current answer however is the fact that spread widening risks (BR01) of FRN and inverse FRN are the same, whereas intuitively they should be different.

May I suggest changing the transformation matrix to A2. This is equivalent to assuming equal distribution of spread widening/tightening between curves.

Is A2 a valid transformation matrix or am I missing something crucial?

P.S.: The transformation matrix approach is hard to extended to multiple curves. I experimented with Euclidean distance, but eventually reverted to the formula:

$\max(|\max(X,0)|, |\min(X,0)|) = X + \min(|\max(X,0)|, |\min(X,0)|),$ where $X$ is DV01, $\min(..)$ is BR01, and $\max(..)$ is the maximum possible risk.

## Answer by Attack68 (score 2, accepted)

https://quant.stackexchange.com/a/85232

If you want a really simple transformation, since your are working in Excel, I would do the following:

A express your input risk as vector, S:

Then define the matrix transformation, A, as follows:

And finally obtain your desired risk with $$ S_{new} = A S_{old} $$

If you are trying to something more complicated than this I would suggest you describe your target result with further details.

> If we change -383 in my example to -500, your method gives BR01 735, whereas the bank's calculation gives 698 due to taking "min of abs". Which approach is better?

This approach is better. Delta risk is linear, meaning if you define risks for elements A, B and C when you combine these in a portfolio you get A + B + C, you do not get some discontinuous combination of risks using an arbitrary min and abs function.

Furthermore, the entire function (and definition) of risk is the financial sensitivity to the movement in the instrument rates/price, so that you should be able to use it to estimate PnL.

If you have market moves measured against the curve instruments in the old coordinate system you can derive market movements in the new coordinate system.

Note that,

$$ PnL = \Delta_{old}^T \mathbf{S_{old}} $$

which is the same as,

$$ PnL = \Delta_{new}^T \mathbf{S_{new}} = (\mathbf{A^{-T}} \Delta_{old})^T \mathbf{A S_{old}} $$