3.10.4 FVA & MVA

Basic Prerequisites: 03 · CVA & DVA.

1. Intuition & Practical Objective

Credit risk (CVA/DVA) is about default. Funding is about cash. This page prices the two funding adjustments: FVA (Funding Value Adjustment, the cost/benefit of financing the derivative’s value and margin) and MVA (Margin Value Adjustment, the cost of posting initial margin). The practical objective: compute each, understand the crucial asymmetry - FVA can be a cost or a benefit, MVA is almost always a cost - and avoid the double-counting that plagues naive implementations.

The one-line distinction (Gregory §16.3.2):

FVA = cost of being under-collateralised (variation margin, running value).
MVA = cost of being over-collateralised (initial margin, segregated and non-rehypothecable).

Where does funding come from? It is literally value − margin (Gregory Eq 18.1). A derivative with positive value must be funded - you borrow to finance the asset - a cost. A negative value gives you a funding benefit (the “borrowing” is a liability others fund you for). Under perfect collateralisation value − margin ≈ 0 and there is no FVA; under- or partial collateralisation leaves a residual to fund.

2. Mathematical Ground Truth & Derivations

2.1 FVA (Gregory Ch 18)

The funding profile is the EFV (expected future value) of the transaction. Symmetric FVA (Eq 18.3):

F V A = - i = 1 \sum m EF V (t_{i}) \times FS (t_{i - 1}, t_{i}) \times (t_{i} - t_{i - 1}),

where $FS (t_{i - 1}, t_{i}) \approx \frac{FS ( 0 , t _{i} ) t _{i} - FS ( 0 , t _{i - 1} ) t _{i - 1}}{t _{i} - t _{i - 1}}$ is the forward funding spread over the valuation (OIS) rate. Because $EPE + ENE = EF V$ , FVA splits into

F V A = FC A + FB A, FC A = - i \sum EPE (t_{i}) FS Δ t, FB A = - i \sum ENE (t_{i}) FS Δ t .

FCA (Funding Cost Adjustment): cost of funding positive exposure.
FBA (Funding Benefit Adjustment): benefit of negative exposure.
Compared to CVA (Eq 17.3), the bank’s funding spread replaces the counterparty credit spread, and EFV replaces EPE (Gregory §18.2.4).

The FVA debate. Hull & White (2012) argued FVA should not be in pricing/valuation at all: including a party’s own funding cost breaks price symmetry and the law of one price, and creates arbitrage (buy options from a low-funding-cost bank, sell to a high-cost one). The counter (Kenyon & Green 2014; shareholder view): there is no market for uncollateralised derivatives, so incremental FVA belongs in entry prices when maximising shareholder value. The current consensus: FVA in pricing under the shareholder view is broadly accepted; FVA in accounting valuation is contested (Gregory §18.2.6).

The double-counting trap. DVA (own default avoids paying) and FBA (negative exposure gives a funding benefit) price the same benefit twice. Two consistent frameworks (Gregory §18.2.5):

CVA + symmetric funding = CVA + FCA + FBA (Basel III consistent, no DVA; the majority market choice for pricing), or
Bilateral CVA + asymmetric funding = CVA + DVA + FCA (IFRS 13 consistent). Never both DVA and FBA.

2.2 MVA (Gregory Ch 20)

Initial margin is segregated, non-rehypothecable, custodian-held, and (for cash IM) remunerated at or below OIS - so it is a liability, not an asset (Gregory Table 20.3: cost only, asymmetric, sub-OIS reference). MVA (Eq 20.1):

M V A \approx - i = 1 \sum m E I M (t_{i}) \times FS (t_{i - 1}, t_{i}) \times (t_{i} - t_{i - 1}),

where $E I M$ is the discounted expected initial-margin profile. IM evolves through portfolio ageing (it can increase as offsetting long-dated trades mature), look-back-window roll, and annual SIMM recalibration (Gregory §20.2.3) - which is why MVA is not trivially computable alongside CVA.

MVA vs KVA trade-off (Gregory §20.4). Posting more IM raises MVA (more funding) but lowers KVA (less capital, since IM covers exposure). The optimal posted IM is generally below the full regulatory requirement, because KVA relief under SA-CCR has diminishing returns (conservative add-ons + the 5% floor). KVA and MVA must be treated equivalently in pricing, or the bank incentivises sub-optimal structures (backloading to a CCP, discretionary IM).

3. Computational Implementation - FVA decomposition and MVA

Stdlib only. The FVA example uses consistent profiles (EPE + ENE = EFV), so FCA + FBA must equal FVA exactly - a built-in sanity check.

The payer swap is a net asset (EFV > 0 throughout), so FVA is a cost; but notice FBA is positive because the swap occasionally has negative value. MVA is larger than FVA here because initial margin (100→30) dwarfs the variation-margin funding profile - typical of a bilaterally-margined trade.

4. Failure Modes & First-Principles Breakdowns

Inconsistent profiles. FVA is built on EPE + ENE = EFV. If you feed EPE and ENE from one model and EFV from another, FCA + FBA ≠ FVA - a silent, hard-to-catch inconsistency. The check is built into the code above.
Double-counting DVA with FBA. Counting the benefit of a negative exposure twice (own-default and funding) double-discounts the cash flow (Gregory §18.2.5). Pick CVA + FCA + FBA or CVA + DVA + FCA and stick to it.
Symmetric FVA where funding is asymmetric. Symmetric FVA assumes you borrow and lend at the same unsecured spread; real funding is asymmetric (borrow unsecured, lend at OIS) and must be computed at the funding-set level, not trade level (Gregory §18.3, Eqs 18.8–18.9). This is inconsistent with the NSFR.
MVA as a benefit. Unlike FVA, segregated IM can never be a funding benefit - posting IM is a liability with no rehypothecation. Pricing MVA as if it could net against FBA is wrong.

5. References

Gregory, The xVA Challenge
Piterbarg, Vladimir (2010): Funding beyond discounting (Risk)
Burgard, Christoph & Martin Kjaer (2011a,b): Partial differential equation representations of derivatives with bilateral counterparty risk and funding costs
Hull & White (2012, 2014) and Kenyon & Green (2014): the FVA debate (against / for).

6. Connected Graph Bridges

Back: 03 · CVA & DVA
Forward: 05 · Failure Modes · 06 · Advanced Extensions · Index Hub
Base: Interest Rate & Term Structure Models (the discounting/FVA connection)

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