Basic Prerequisites: 03 · CVA & DVA and 05 · Failure Modes & Practice.
1. Intuition & Practical Objective
Counterparty risk is only the first cost of trading derivatives. A firm also funds the position, posts initial margin, and holds regulatory capital against it. Each of these has a price, and each gets its own adjustment. Together they are the xVA family:
The practical objective is to see that all the xVAs are the same integral shape - an expected usage profile against a cost curve - so CVA is a template, not a special case:
| xVA | Usage profile (market component) | Cost curve (cost component) |
|---|---|---|
| CVA / DVA | EPE / ENE | counterparty / own default probability |
| FVA | EFV (expected future value) | own funding spread |
| MVA | EIM (expected initial margin) | IM funding spread |
| ColVA | expected collateral balance | collateral remuneration gap |
| KVA | expected capital (ECP) | cost of capital |
The two halves of funding. FVA is the cost of being under-collateralised (you must fund the variation-margin shortfall). MVA is the cost of being over-collateralised (you must fund the posted initial margin). FVA can be a benefit; MVA essentially cannot.
2. Mathematical Ground Truth & Derivations
2.1 FVA = FCA + FBA
Funding value adjustment mirrors CVA but replaces the counterparty’s credit spread with your own funding spread , and the exposure with the expected future value EFV (Gregory §18.2):
Positive value (an asset) must be funded → FCA cost; negative value (a liability) is a funding source → FBA benefit. The Burgard–Kjær integral form makes the analogy exact:
2.2 The DVA/FBA double-count - the one thing to get right
DVA and FBA are two names for the same benefit of a negative exposure. DVA books it as avoided payment on own default; FBA books it as funding relief. Including both discounts the same cash flow twice (§18.2.5). When the funding spread equals the own-credit LGD-weighted hazard (), the two are identical (Burgard–Kjær). So a firm must choose exactly one of:
CVA + FCA + FBA- “CVA + symmetric funding” (consistent with Basel III, which has no DVA), orCVA + DVA + FCA- “bilateral CVA + asymmetric funding”.
Never CVA + DVA + FCA + FBA.
The FVA debate (Gregory §18.2.6). Hull–White (2012a, 2014) argue funding costs should not enter a risk-neutral value: including them breaks the law of one price and creates arbitrage (buy from a low-funding bank, sell to a high-funding bank). The counter-view (Castagna, Carver, Kenyon–Green) is that there is no market for uncollateralised derivatives, so the “fair value” argument is void and FVA is real. Current settlement: FVA belongs in pricing (entry price maximising shareholder value); the debate persists on valuation. Only the non-default (liquidity) part of a funding spread is universally accepted as a true FVA - estimable from the CDS–bond basis. Hull–White’s own decomposition reads (DVA₂ = benefit of defaulting on funding liabilities), which nets back to CVA+DVA at the firm level.
2.3 MVA
Initial margin must be posted, segregated (non-rehypothecable), and funded. Its cost is
Key facts (§20.2–20.4): MVA is asymmetric (segregation means it is a cost only, never a benefit); the EIM profile can be non-monotonic (a large offsetting trade ageing out can raise future IM); and MVA is not trivially computed alongside CVA - future SIMM sensitivities are needed. MVA and KVA are not mutually exclusive (§20.4): posting more IM raises MVA but can lower KVA (less capital), and the optimum bilateral IM posting is generally below the full regulatory requirement.
2.4 Modelling WWR (the rigorous route)
Beyond the correlation trick of 05, two families (Gregory §17.6.3):
- Intensity approach - a stochastic credit-spread intensity correlated with the exposure drivers; default drawn from the intensity. Tractable, but even correlation tends to understate the true effect.
- Structural approach - map exposure and default distributions onto a bivariate distribution (e.g. a Gaussian copula); no revaluation needed, stronger effect, but the dependence is opaque and hard to calibrate.
- Jump approach - the strongest empirical support: allow the exposure to jump at default. The only family that makes collateral genuinely ineffective.
3. Computational Implementation - FVA, the DVA/FBA overlap, and MVA
Build an ITM forward’s EFV/EPE/ENE profile, compute FCA/FBA/FVA, show numerically that DVA ≈ FBA (so adding both double-counts), and price MVA from an initial-margin profile. Standard library only.
Read the block: the ITM forward is an asset (EPE avg 27.2 vs ENE avg −7.6), so funding it costs FCA = −1.25; the small negative-exposure tail gives back FBA = +0.34, leaving FVA = −0.90. The critical line is DVA (0.33) ≈ FBA (0.34) - literally the same benefit computed two ways, which is why they must never be summed. And MVA (0.46) is the price of funding a constant ~$11.6 IM; if the IM profile is assumed to decay linearly, MVA halves to 0.22 - a reminder that the EIM shape is where MVA model risk lives.
4. Failure Modes & First-Principles Breakdowns
- Summing DVA and FBA. The single most common xVA accounting error - it double-counts the negative-exposure benefit (§2.2). Pick one framework and stick to it.
- Hedging the wrong FVA. Only the non-default (liquidity) component of a funding spread is a genuine economic cost; the credit-default component overlaps with DVA. Using the whole bond spread inflates FVA (Morini–Prampolini; Hull–White 2014).
- MVA EIM oversimplification. Forward-IM or amortisation proxies are decent for linear swap portfolios but bad for options and CCP-style methodologies, where IM is convex (§20.2.3). Understating EIM understates MVA.
- MVA/KVA asymmetry ignored. Treating MVA and KVA inconsistently (pricing one, not the other) incentivises sub-optimal structures - ‘backloading’ to a CCP or discretionary IM that looks cheap on one adjustment but costs more on the other (§20.4).
- WWR model choice is not neutral. Intensity models understate the effect; structural models are hard to calibrate; only jump models reproduce the empirically observed magnitudes. Choosing the intensity approach “because it is tractable” systematically understates CVA.
- The recursive valuation problem. The true value is both an input to and an output of xVA (close-out references a value that already contains xVA). Practice uses linear superposition on a base value - an approximation that fails exactly when xVA is large (§3.3.2).
5. References
- Gregory, Jon: The xVA Challenge (5th ed., 2025)
- Burgard & Kjær (2011a,b, 2013): the funding-cost derivation and the conditions under which DVA ≡ FBA.
- Hull & White (2012a, 2014): the case against FVA in valuation; Andersen, Duffie & Song (2016): the shareholder-value view that FVA belongs in entry prices.
- Brigo, Morini & Pallavicini (2013): the rigorous CVA/FVA-with-collateral pricing framework.