Why Perfect Collateralization Removes Funding Valuation Adjustment

# FVA for a perfectly collateralised trade


# FVA for a perfectly collateralised trade












Consider a perfectly collateralised swap.

Numerous sources discuss how FVA arises from banks having to fund collateral at a spread to the CSA rate. One example here:

> The asymmetric nature of this cost of collateral adds additional costs to transacting the swap. The size of this cost relates to the difference between the bank’s unsecured borrowing rate and the CSA rate. In this sense the FVA is related to the DVA which is a reflection of the bank’s own likelihood of default.

This is intuitive - if we have to fund a margin call at some rate $r_F$, but only receive $r_C$ on this margin, then we expect to be losing money on a net basis.

Yet Funding beyond discounting: collateral agreements and derivatives pricing by Piterbarg states that the value of a trade under (potentially imperfect/no) collateralisation is given by:

$$ V_t = E_t \left[ e^{-\int_t^Tr_C(u)du}V_T\right]-E_t \left[\int_t^Te^{-\int_t^ur_C(v)dv}\left( r_F(u)-r_C(u)\right) \left(V_u-C_u\right)du \right]$$

With perfect collateralisation, we have $C(t) == V(t)$, and the equation becomes:

$$ V_t = E_t \left[ e^{-\int_t^Tr_C(u)du}V_T\right]$$

Which is just $V_T$ discounted at the collateral rate $r_c(t)$. Notice how the funding rate $r_F(t)$ does not appear here. This is consistent with numerous sources that state that FVA is only applied for imperfectly (potentially uncollateralised) swaps.

I can't fault either of these conclusions. What is the intuitive explanation on how FVA is not applicable to a perfectly collateralised derivative, even though the collateral will be funded at a rate $r_F > r_C$.

## Answer by Attack68 (score 4, accepted)

https://quant.stackexchange.com/a/76672

See what you think of this intuitive argument.

Let $V_T$ be a cashflow, say 1mm USD, payable today (the derivative expires today). Do we agree that the PV of this is -1mm USD? This is becuase there is no time left to maturity ($T-t=0$) and therefore no time for any interest / funding to accrue. This PV does not take into account the funding rate of the bank or its future expected cost, it is simply a settlement amount payable of 1mm USD.

Now suppose instead that $V_T$ is payable tomorrow, AND that it is collateralised. Thus the bank needs to post something today. How should the PV be assessed? If the bank posts 999,900 USD in collateral then after 1 day, with collateral interest rates at 3.60%, the bank will have 999,900 + 100 = 1mm USD which will settle the liability as above. In this scenario the funding rate of the bank is also not used to derive the PV becuase it is not relevant to amount of money that is posted as collateral to settle the 1mm USD as it falls due. The economic value of the future liability is 999,900, which is 1mm discounted by the collateral rate.

In the case where the payment is not collateralised the bank does not have to post the collateral. That amount of collateral (999,900) could be put elsewhere, reducing the funding requirement elsewhere. This means that the bank does not have to fund something else at its funding rate of 3.60% + funding spread. Therefore the liability of 1mm USD payable tomorrow can be discounted at 3.60% + funding rate. Suppose the funding spread is also 360 bps then the value of tomorrow's cashflow is 999,800. I.e. it is a smaller liability.

You can reverse these arguments in the case of an asset and the same is true.

$$ V_t = \underbrace{E_t \left[ e^{-\int_t^Tr_C(u)du}V_T\right]}_{\text{value discounted at collateral}}-\underbrace{E_t \left[\int_t^Te^{-\int_t^ur_C(v)dv}\left( r_F(u)-r_C(u)\right) \left(V_u-C_u\right)du \right]}_{\text{adjustement for how much of value is not collateralised}}$$