I am reading Funding Beyond Discounting: Collateral Agreements and Derivatives Pricing by V. Piterbarg.
Now I have a question about the relation of the different funding rates in the paper.
- $r_C$ is the short rate paid on collateral, e.g. Fed Funds (before the move to SOFR)
- $r_R$ is the short rate for a repo transaction with the underlying asset, e.g. a stock as collateral
- $r_F$ is the short rate for unsecured bank funding
In the paper, Piterbarg states that one expects that $r_C \leq r_R \leq r_F$ and that the existence of non-zero spreads between short rates based on different collateral can be recast in the language of credit risk.
It is clear that $r_R \leq r_F$ should hold since a repo corresponds to secured funding whereas $r_F$ is unsecured funding. However, there are two things I don't understand:
1.) Why should the collateral rate, which also corresponds to unsecured funding (one bank gives cash to another bank and gets it back the next day + interest) be lower than the repo rate, which corresponds to secured funding? For example, the Fed Funds rate is the interest rate banks charge each other for unsecured overnight funding.
2.) Why would the collateral rate and the funding rate be different ($r_C \leq r_F$)? Both should reflect the credit risk of the counterparty? I know that bank funding comes from various sources like deposits, issued bonds, etc. and that there is a spread for credit risk, but why would a counterparty not charge for this the same way for cash given to the bank via a collateral agreement.
I hope my question is clear.