# FRTB Delta CSR vs Delta GIRR

In Basel III, FRTB SA includes different market risk capital requirements for interest rate (GIRR §21.19) and credit spread risk (CSR §21.20) exposures. These are different risks, as credit spreads and risk-free rates can change independently.

As to the first approximation for those risks - delta - sensitivities are PV01 and CS01. PV01 is computed as the change in the instrument's value given by 1bp change in the risk-free rate for one of the prescribed tenors, while keeping the credit spread constant:

$$s_{k,r_{t}} = \frac{V_{i}(r_{t} + 0.0001, cs_{t}) - V_{i}(r_{t} , cs_{t})}{0.0001}$$

In the same fashion, CS01 results from 1bp change in credit spread for a specific tenor, keeping constant the risk-free rate.

$$s_{k,cs_{t}} = \frac{V_{i}(r_{t}, cs_{t} + 0.0001) - V_{i}(r_{t} , cs_{t})}{0.0001}$$

There's a discussion about credit spreads on this book, where CS01 is computed by changing z-spreads. Z-spread $$\textbf{z}$$ is obtained from the quoted price of a bond and the risk-free rate. The bond's cash flows are discounted by spot rates which consists of the sum of the risk-free rate and the credit spreads for each tenor:

$$P_{mkt} = ce^{-(r_{i}+\textbf{z}_{i})}+ e^{-(r_{t}+\textbf{z}_{t})}$$

If the spot rate for tenor $$i$$ equals $$r_{i}+{z}_{i}$$, how are PV01 and CS01 for tenor $$i$$ supposed to be numerically different sensitivies, as it doesn't matter if one's giving a shock to $$r_{i}$$ or $${z}_{i}$$?

• Examples that come to mind are: different curve pillars and curve interpolation techniques. Commented May 2, 2023 at 6:54
• For fixed coupon bonds indeed the measures would be similar. But many instruments it may not be the case. Eg A floating rate bond would have very little rates sensitivity but a large CS01
– dm63
Commented Jul 7, 2023 at 15:08
• That makes sense, but none of the books I could find use a floating rate bond to exemplify CS01. Commented Jul 7, 2023 at 19:26

## 1 Answer

I don't know whether the following is allowed for FRTB. I would appreciate if someone who knows whether this was explicitly allowed or disallowed tells everyone.

Make some assumption about the loss given default. Calculate the CDS spread from the bond price, the risk-free interest rate, and the LGD. Perturb the CDS spread, and back out the new bond price. Scale the bond price change to 100% shock to get CS01.

If you take this approach for a fixed-coupon bond, then perturbing the CDS spread is close to, but not exactly the same as perturning the risk-free rate or the yield.

Also, not very material, when calculating the PV01, you may ptefer to keep the CDS spreads constant and recalculate the PD from the perturbed interest rate, and reprice the bond from the perturbed interest rate and the recalculated PD.

• FRTB (CSR §21.9) sets as risk factors both the Bond and CDS credit spread curves for the same issuer. So I thought these would be different risk sensitivities. Commented Jun 7, 2023 at 13:47
• I'm not sure if a CDS spread derived from bond price, or the formula in Malz, etc would be "modelable" in FRTB sense? Commented Jun 7, 2023 at 16:06