XVA Explained: CVA, DVA and FVA

XVA is the family of valuation adjustments that turns a risk-free derivative price into a price reflecting counterparty default, own default, funding, collateral, and capital economics. It is portfolio- and agreement-specific: summing trade-level “CVA charges” generally fails because netting and collateral are nonlinear.

Core definitions

For counterparty \(C\), unilateral CVA is the discounted expected loss on positive exposure:

CVA = (1 - R_C) ∫ D(0,t) EE(t) dPD_C(t)
DVA = (1 - R_B) ∫ D(0,t) ENE(t) dPD_B(t)

EE(t) is expected positive exposure after netting and collateral; ENE(t) is expected negative exposure. Bilateral valuation requires a first-to-default treatment, so the compact formulas are intuition rather than a complete legal valuation.

Adjustment	Economic source	Sign intuition
CVA	counterparty may default	lowers asset value
DVA	bank may default	raises own liability value
FVA	unsecured funding spread	often lowers value
MVA	initial-margin funding	lowers value
KVA	capital cost	lowers value

Exposure is a simulation problem

A swap’s future mark-to-market changes with rates; a cross-currency swap adds FX and basis; an option is nonlinear. Simulate market paths, revalue a netting set at exposure dates, apply the CSA, then average positive and negative values. Collateral threshold, minimum transfer amount, margin period of risk, and disputes can dominate a supposedly “small” counterparty spread.

def collateralized_exposure(values, threshold=0.0, mta=0.0):
    collateral = np.maximum(values - threshold, 0.0)
    collateral[np.abs(collateral) < mta] = 0.0
    return np.maximum(values - collateral, 0.0)

def expected_exposure(path_values):
    return np.mean(np.maximum(path_values, 0.0), axis=0)

The toy function deliberately omits calls, lag, independent amount, and wrong-way risk. It establishes the sequencing: value first, collateralize second, aggregate only within legal netting sets.

FVA and its controversies

FVA measures the cost or benefit of funding exposure not remunerated at the collateral rate. It cannot be derived from a universal “funding curve” independent of a dealer’s treasury policy. Funding benefits on own default may overlap DVA, creating double-counting unless the framework is internally consistent. Desk pricing often uses FTP curves and explicit governance rather than pretending these are market-observable derivatives.

Wrong-way risk and hedging

Wrong-way risk occurs when exposure rises as counterparty quality deteriorates: selling protection to a bank while holding a positive-rate exposure to it is a familiar pattern. Independent market and credit simulations understate this. Add stressed dependencies, sector scenarios, and explicit limits.

CVA risk itself is usually hedged with CDS indices or single-name CDS, but a hedge does not eliminate exposure-profile risk, basis, jump risk, or liquidity. Netting diversification reduces expected exposure, which is why CVA is measured at agreement level.

Governance

Reconcile clean PV, collateral, exposure, and each adjustment. Validate simulation against analytic cases, backtest realized margin behavior, and treat model uncertainty as a reserve. XVA introduces a new risk factor into portfolio optimization: carry that ignores funding and tail exposure is overstated.

Key takeaways

• CVA is loss on simulated positive exposure, not simply notional times spread.
• Netting, CSA terms, and margin period determine exposure economics.
• FVA/DVA require consistent funding policy and careful avoidance of double counting.

Numerical controls

Exposure simulation needs convergence diagnostics at each future date, not only a single portfolio PV tolerance. Use common random numbers for sensitivities, validate trade-level profiles against analytic vanilla cases, and explain changes caused by market data, credit curves, CSA amendments, and model releases separately. Wrong-way risk scenarios should be transparent overlays when calibration data is weak.

The legal data is as important as the stochastic model. Link each trade to its correct netting set, CSA, collateral currency, and margining frequency; missing or duplicate legal mappings can dominate a sophisticated Monte Carlo improvement. A robust XVA process also preserves historical agreement versions, because today’s terms cannot reconstruct yesterday’s exposure.

Finally, reconcile pricing XVA with capital and limits. A desk hedge may reduce market-sensitive CVA without reducing stressed exposure or regulatory capital. Present these measures side by side and prevent a reported valuation benefit from obscuring a concentration that remains economically material.