Banking / credit

CVA hedge and internal-model capital

A counterparty exposure carries two different numbers: the CVA the trading book marks, and the capital the internal model demands. This prices both from one exposure, sizes the CDS hedge that neutralises the CVA spread sensitivity, and shows what a parallel spread move leaves behind.

Hedge effectiveness97.1%CS01$7,845SegmentCorporate

CVA

$1,033,459

9.9% implied default over 5y

CDS hedge notional

$17,000,000

At 125 bp, 100% hedged

IRB risk weight

91.1%

PD 0.80% · LGD 45%

Capital requirement

$1,822,875

RWA $22,785,943

CVA, hedge and capital

The same exposure seen through the trading book and the regulatory capital model.

$1,033,459

CVA, hedge and capital
Measure	Value	Note
CVA	$1,033,459	λ 2.08% from the CDS spread
CS01	$7,845	CVA change for a one basis point widening
Hedge notional	$17,000,000	CDS protection, rounded to the trade step
Hedge P&L (10 bp)	-$80,723	Protection gains as spreads widen
CVA move (10 bp)	$78,445	Mark-to-market loss before hedging
Residual CVA	$2,277	Unhedged CVA plus the net hedge error
Expected loss	$90,000	PD × LGD × EAD
Risk-weighted assets	$22,785,943	Correlation 0.200
Capital	$1,822,875	Unexpected loss held against the exposure
Provision gap	$0.00	Expected loss not covered by provisions

CVA uses the market-implied default probability; IRB capital uses the internal PD. They are different questions about the same loan.

CVA against the CDS spread

The hedge is sized at the design spread; the gap between the lines is what a parallel move leaves behind.

Loss and capital stack

Expected loss, capital held and the provision gap.

Canonical equations

The exact relationships implemented by this engine.

λ = s / (1 − R) · From the CDS spread and market recovery.

LGD · EPE · (1 − e^{−λT}) · DF̄ · Market-implied default probability.

∂CVA/∂s · 1bp · Sets the hedge notional.

[LGD·N(·) − PD·LGD] · MA · Basel internal-ratings-based formula.

Show model notation and computationOpen the symbolic form, the variable definitions, and the step-by-step derivation when you need the detail.

A clear path from inputs to results.

The same counterparty exposure carries a mark and a capital charge. The mark is CVA, priced off the CDS curve and hedged with protection; the charge is internal-model capital, priced off the bank's own probability of default. This calculator computes both, sizes the CDS hedge that neutralises the CVA spread sensitivity, and shows what a parallel move leaves behind.

Step-by-step methodWorked exampleAssumptions explained

How to use

Enter the exposure, its tenor and the loss given default.

Set the counterparty CDS spread and the market recovery rate.

Choose how much of the CVA to hedge and how finely to round the notional.

Set the internal PD and maturity for the IRB capital calculation.

Read the CVA, the hedge notional, the risk weight and the capital.

The formula

\mathrm{CVA} \approx \mathrm{LGD}\cdot\mathrm{EPE}\cdot(1-e^{-\lambda T})\cdot\overline{DF} \;;\; K = \left[\mathrm{LGD}\,N(\cdot) - \mathrm{PD}\,\mathrm{LGD}\right] MA

Reference: Basel II/III IRB corporate formula (BCBS 128, ¶272); CVA as the risk-neutral expected loss on expected positive exposure.

Worked example

$25m exposure, 125 bp CDS, 5 years, PD 0.8%

Inputs

Exposure $25,000,000, LGD 45%, tenor 5 years.
CDS spread 125 bp, market recovery 40%.
Hedge 100% of the CVA, rounded to $1m, 10 bp spread shock.
Internal PD 0.8%, maturity 3 years, corporate.

Calculation path

1
λ = 1.25% / 0.6 = 2.08%, so the five-year default probability is 9.9%.
2
CVA = 0.45 × 25M × 0.099 × 0.93 = $1.03m.
3
CS01 = $7,845 per basis point, which a $17m CDS notional neutralises.
4
A 10 bp widening costs $78,445 of CVA and earns $80,723 on the hedge: net −$2,277, 97.1% effective.
5
IRB: correlation 0.200, maturity adjustment 1.376, risk weight 91.1%, capital $1.82m.

Result

CVA $1.03m · hedge $17m · 97.1% effective · IRB capital $1.82m.

Interpreting the result

The hedge is sized on CS01, not on the CVA level, so it neutralises a small parallel move almost exactly and leaves the level of CVA untouched. Rounding to $1m is what produces the 2.9% residual. The capital number is a different question entirely: it uses the internal PD, ignores the hedge, and does not fall when the CVA is hedged.

Assumptions & limitations

Flat expected positive exposure over the tenor; no profile or collateral modelling.

CVA default probability taken from the CDS spread, as an XVA desk marks it.

CDS CS01 approximated by the annuity of the survival curve.

Basel IRB corporate formula with the supervisory maturity adjustment; no double-default or securitisation treatment.

Frequently asked questions

Why does hedging CVA not reduce capital?

CVA is a valuation adjustment; IRB capital is a regulatory charge on the exposure. A CDS hedge transfers spread risk but the loan still sits on the balance sheet until the hedge is recognised under the framework.

What is CS01?

The change in CVA for a one basis point move in the counterparty's spread. It is the number the hedge is sized against.

Why is the hedge notional larger than the CVA?

The notional is set to match CS01, and the CVA is a small fraction of the exposure. Matching a spread sensitivity needs protection on a notional close to the exposure itself.

Why is the IRB risk weight so high at 0.8% PD?

The formula is calibrated to a 99.9% downturn loss. Even a low default probability produces a large capital requirement once loss given default and correlation are applied.