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.

  • $\begingroup$ He writes that $r_C$ is the rate corresponding to "the safest possible collateral" (cash) and $C$ stands for CSA. So here the collateral is safer than the asset applied in repo. $\endgroup$
    – fes
    Jul 5 at 14:36
  • $\begingroup$ Fed funds rate is an unsecured rate so it should be closer to $r_F$ than $r_C$ $\endgroup$
    – fes
    Jul 5 at 14:37
  • $\begingroup$ @fes Thanks for your comment. Of course, cash is the safest form of collateral. But the collateralization essentially means that if a derivative has positive value $V$ for the bank, then the counterparty will post cash and charge the collateral rate, e.g. fed funds for it. Isn't this like the counterparty providing a loan of $V$ at the rate $r_C$? Why would the counterparty not charge a rate associated with the credit risk of the bank, i.e. $r_F$? $\endgroup$ Jul 5 at 17:44
  • 1
    $\begingroup$ Yeah technically a repo still has some risk and by $r_C$ he means a rate that is perfectly risk-free. Now a deal fully collateralised with cash would make the borrowing risk-free, though I don't know how this would be implemented in practice. $\endgroup$
    – fes
    Jul 5 at 18:03
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    $\begingroup$ @fes Obviously, a risk-free rate is a theoretical concept and the fed funds rate (or SOFR now) would be only a proxy for it. Actually SOFR is even closer to risk-free since it is essentially a repo rate. $\endgroup$ Jul 5 at 18:15


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