# 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. May 2 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
Jul 7 at 15:08
• That makes sense, but none of the books I could find use a floating rate bond to exemplify CS01. Jul 7 at 19:26