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

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)

• If B=C, and if ENE is not the same as EPE, then still I don't think the the two methods are the same. Say we are short a Put and long a Call (i.e. "long forward"). The EPE will be the expected MTM of the Call, and the ENE will be the expected MTM of the Put, at each point in time (t): those two are not equal, unless rates are zero. So the time integrals of over the EPE and ENE will not be equal, even if B=C. Aug 20, 2022 at 18:17
• To clarify: I don't think the initial MTM times the funding charge will always necessarily be equal to the time integral of the expected future MTM, times the funding charge, discounted to "today". (expected future MTM is essentially the EPE minus ENE). Aug 20, 2022 at 20:39
• But in your words method 1 is the time integral of the PV and method 2 is the time integral of the PE minus the time integral of the NE. By your last comment these are equal , assuming the discount rates are the same for PE and NE.
– dm63
Aug 21, 2022 at 6:38
• Yes, I think we’re on the same page now. I assumed that method 1 meant the time integral.
– dm63
Aug 21, 2022 at 11:19
• Could it be that the 'simplified' method is just an application of equation (3.2) in Piterbarg for an uncollateralised trade? $V(t) = \mathbb{E}_t\left[e^{-\int_t^Tr_F(u)du}V(T)+\int_t^Te^{-\int_t^ur_F(v)dv}(r_F(u)-r_C(u))C(u)du \right]$, so given it is uncollateralised we have $C(t) = 0$ giving $V(t) = \mathbb{E}_t\left[e^{-\int_t^Tr_F(u)du}V(T) \right]$, i.e: $V(T)$ discounted using the funding rate. Sep 11 at 14:04

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.

• I am still confused as I find the following explanations elsewhere, for example: "Use the OIS curve for the base valuation and mark the difference between 3M Libor discounting and OIS discounting as FVA." (Quant StackExchage Q&A Link) "[FVA] is the difference between the NDV obtained when the risk-free rate is used for discounting and the NDV based on discounting at the dealer’s cost of funds. (Hull and White (2014) p.4 [Link](www-2.rotman.utoronto.ca/~hull/downloadablepublications/… Aug 20, 2022 at 2:43
• Hull and other authors are wonderful mathematicians and are certainly smarter than me, but they are not practitioners. Their definitions are vague and don't paint the whole picture. The only author I would recommend on XVA is John Gregory. Aug 20, 2022 at 7:15
• So, during your career, may I assume that you have never encountered (or heard of) a case where just the simple PV differential based on the different discount rates (e.g. OIS vs. LIBOR) is booked as an FVA? (not even during the early strage of the FVA adoption?) Aug 20, 2022 at 10:47
• @Curiosity: have a look at the comments underneath dm63's answer. It depends what you meant: if by PV you mean "today's PV" only, then method 1 doesn't compute FVA. If by PV you mean "PV profile between now and maturity", then method 1 and method 2 are equivalent (as long as funding benefit interest rate and funding cost interest rate are the same). Aug 21, 2022 at 11:54