2 Ways to Define/Calculate "FVA"? - Same or Different? (Simple XVA Question)

Question

I've got a very simple question on 2 different ways of defining or calculating the FVA of an uncollateralized swap.

One definition I've often seen is that the FVA is the difference in the net present value of the swap discounted using the risk free rate (e.g., OIS) and that of the same swap discounted using the bank's funding rate (e.g., LIBOR). So, basically a difference in discounting.

The other thing I've seen often is that it requires more than that, as the swap's EPE and ENE over its lifetime need to be simulated. Based upon the EPEs and ENEs simulated, the FVA is calculated using the spread between the actual funding rate and the risk-free rate. This indeed makes sense.

I am just wondering if these two are totally different methods, e.g., if the first one is considered a "simplified method" and the second one is a "simulated method". I don't see much connection between the two, but I may be wrong.

Are these two pretty much the same, or totally different definitions as I suspect?

Thanks!

Answer 1

Can I add my 2c:

We agree that in the case of a receivable, there is a funding cost C, and in the case of a payable, there is a funding benefit B. The cost C is essentially unsecured rate minus risk free rate, often calculated as Libor minus Fed Funds. The benefit B may or may not be Libor minus Fed Funds. I elaborate slightly with 2 statements:

a) if B=C, the 2 methods described in the OP will give you the same answer. This is because if C=B, the EPE and ENE are using the same discount rate inside the simulation. That's the same as method 1 (discounting all the flows at 2 different rates and taking the difference).

b) if B=0 (no benefit is being given for funding, for some reason), then the 2 methods will give different answers, with the simulation method giving a higher charge for FVA.

You may ask then which of these is correct? I would say usually a) is correct, because a funding benefit is typically monetizable, in that it reduces the overall funding need of the bank. Sometimes b) is appropriate (for example, if the counterparty is a mutual fund in the US, the incoming collateral must be segregated in a separate account and therefore does not reduce the funding need of the bank)

Answer 2

Imagine a derivatives desk at a large bank gets a call from a client (a non-financial corporation, say a medium-sized or a small-sized enterprise, that is not big enough to have a sophisticated treasury operation). Imagine that this client doesn't have a CSA signed, so they don't need to post collateral. The client executes a derivative transaction (say an IRS), to hedge something.

The desk charges the client a small spread (say 1 bps) and decides to hedge this trade by entering into an offsetting IRS with another big bank (with which it has a CSA, so a collateral is posted on a daily basis).

• Scenario 1 (funding cost): If the IRS executed with the client starts to be in the money for the bank, it will receive no collateral on this "in-the-moneyness" from the client (remember, no CSA), but it will have to post collateral to the other bank with which it hedged the client IRS: where will they get this cash to post this collateral? They will need to raise this from the treasury at the funding rate. This collateral will then be remunerated at risk-free rate whilst being posted with the counterparty.

• Scenario 2 (funding benefit): If, on the other hand, the client position is in the money, the bank would need to post collateral to the client, but because there is no CSA, the bank doesn't need to post anything to the client, whilst receiving collateral from the hedging counterparty. This collateral received will be remunerated at the risk-free rate (i.e. cost to the bank), but this collateral would normally be deposited somewhere by the treasury at around the funding rate.

The FVA is trying to capture the costs arising from scenario 1 or the benefit arising from scenario 2 at inception of the trade, so that it can be added to (or subtracted from) the client charge.

The only way to capture this is the second method you describe: scenario 1 is basically: $$\int_{h=0}^{h=t}DF(h)ENE(h)*(FundingRate(h)- RiskFreeRrate(h))dh=\int_{h=0}^{h=t}\mathbb{E}^Q\left[DF(h)\left(R(h)-k_0\right)^{-} (r_{funding}-r_{riskFree})\right]dh$$

Above, $R(h)$ is the value of the fixed swap rate at time $h$ that sets the MTM of the IRS executed against the client to zero as of time $h$, whilst $k_0$ is the strike that set the MTM to zero at inception. $DF(h)$ is the discount factor.

Scenario 2 is the same type of calculation, but with the EPE instead of ENE.

The difference between the two integrals (i.e. one with EPE and the other one with ENE) will then tell us if the transaction against the client will generate a funding cost or a funding benefit (the above formulas are somewhat simplified, I didn't add conditioning on the client and the bank surviving, i.e. not defaulting prior to time $h$).

I don't see how a one-off snapshot of the net present value of a swap "as of today" multiplied by the difference between the funding rate and the risk-free rate could give you FVA: that will just tell you what the funding is costing right now at this point in time for the current day, but it won't reflect how the funding profile will evolve going into the future (unless by "net present value" we mean the expected NPV of the swap between inception until maturity: i.e. the whole MTM profile at various time points, not just a snapshot).

The whole point of an FVA is to capture the funding cost for the duration of the whole trade and reflect this in the pricing of the transaction at inception. A one-off NPV snapshot won't achieve this.