3.10.3 CVA & DVA

Basic Prerequisites: 02 · Exposure & Margin.

1. Intuition & Practical Objective

Credit Value Adjustment (CVA) is the price of counterparty risk - it internalises the expected loss from your counterparty’s default into the derivative’s value (Gregory Ch 17). Debt Value Adjustment (DVA) is its mirror image: the benefit of your own default (the liabilities you don’t pay). The practical objective: compute both, understand why they oppose each other, and know exactly which credit numbers feed in.

The three questions CVA answers (Gregory 17.2): Does the counterparty default? What is my exposure then? How much do I lose? In symbols,

C V A (t) \approx - L G D i = 1 \sum m EPE (t, t_{i}) \times P D (t_{i - 1}, t_{i}),

which is the discrete form of the continuous integral

C V A (t) = - L G D \int_{t}^{\infty} λ_{C} (u) D_{r + λ_{C}} (t, u) EPE (t, u) d u .

Three ideas matter:

It is a product of market risk and credit risk. EPE is the market component (how big is the exposure); PD × LGD is the credit component (how likely is default, how much is lost). This is the “market × credit” split that makes CVA both a derivatives problem and a credit problem.
Default probabilities are risk-neutral, market-implied. Since the accounting standards moved CVA to a market (hedging/exit-price) basis, PDs come from CDS credit spreads, not historical default rates (Gregory §3.1.7; Hull 24.5 - risk-neutral PDs exceed real-world ones).
DVA is the other half of the same coin. Your counterparty prices your default as a DVA against you; “my CVA is your DVA” is what makes bilateral pricing symmetric (Gregory 17.3).

2. Mathematical Ground Truth & Derivations

2.1 The CVA formula (Gregory Eqs 17.1–17.3)

Direct form - simulate the default time $τ$ , value the portfolio once at $τ$ :

U C V A (t) = - E [1_{τ \leq T} V (t, τ)^{+} L G D] .

Path-wise / discrete form - the workhorse (Eq 17.3):

U C V A (t) \approx - L G D i = 1 \sum m EPE (t, t_{i}) \times P D (t_{i - 1}, t_{i}),

where $EPE (t, t_{i}) = E [V (t, t_{i})^{+}]$ is the discounted expected positive exposure at date $t_{i}$ and $P D (t_{i - 1}, t_{i})$ the default probability over that interval (independence ⇒ default enters only via PD). With flat $λ$ , $P D (a, b) = e^{- λa} - e^{- λb}$ .

The credit inputs. From a CDS spread $s$ and $L G D = 1 - R$ , the hazard is $λ = s / L G D$ (Hull 24.2; BM 21.25), survival $Q (τ > t) = e^{- λ t}$ . The LGD-adjusted form (Eq 17.5) splits the market LGD (used to strip PDs from CDS, usually senior unsecured) from the actual expected LGD; when seniority matches they cancel to first order - changing LGD 60%→50% moves CVA <2% (Gregory §17.2.6).

CVA as a spread (Eq 17.4), valid for roughly-flat EPE/PD profiles: $U C V A \approx - \overline{EPE} \times s$ .

2.2 DVA and the bilateral framework (Gregory 17.3)

DVA is the own-default side, driven by the ENE (your counterparty’s positive exposure / your negative exposure):

D V A (t) = - L G D_{P} \int_{t}^{\infty} λ_{P} (u) D_{r + λ_{C} + λ_{P}} (t, u) ENE (t, u) d u .

The bilateral value is

BC V A = C V A + D V A,

with the discrete forms (Eqs 17.8a/b) carrying the survival probability of the other party (the “first-to-default” effect):

C V A (t) = - L G D_{C} i \sum EPE (t, t_{i}) P D_{C} (t_{i - 1}, t_{i}) [1 - P D_{P} (0, t_{i - 1})],

D V A (t) = - L G D_{P} i \sum ENE (t, t_{i}) P D_{P} (t_{i - 1}, t_{i}) [1 - P D_{C} (0, t_{i - 1})] .

Key facts:

ENE ≤ 0 ⇒ DVA ≥ 0: DVA is a benefit opposing CVA (your default wipes out the debt you owe).
Spread approximation (Eq 17.9): $BC V A \approx - \overline{EPE} s_{C} - \overline{ENE} s_{P}$ ; when $\overline{EPE} \approx - \overline{ENE}$ , $BC V A \approx - \overline{EPE} (s_{C} - s_{P})$ - the weaker (higher-spread) party pays the stronger. Gregory Table 17.3: 10y swaps, both LGD 60%, own spread < counterparty: $U C V A - 29.9/ U D V A + 8.1/ BC V A - 21.5$ (pay-fixed).
Accounting vs capital: IFRS 13 requires own-credit (DVA) in liability fair value; Basel III derecognises DVA from capital (Gregory §17.3.5, §13.3.1). Hence the debate in 05 · Failure Modes.

2.3 Wrong-way risk (Gregory 17.6)

The standard formulas assume independence. Wrong-way risk (WWR) = unfavourable dependence between exposure and counterparty credit quality (default is more likely when exposure is high). Replace $EPE (t, t_{i})$ with the conditional $EPE (t, t_{i} ∣ τ_{C} = t_{i})$ . Canonical WWR: buying a put on a name correlated to the counterparty; an FX forward with a sovereign paying local currency; a corporate paying fixed when rates fall in a recession. Gregory’s key empirical result: corr +50% roughly doubles EPE; −50% at least halves it - and WWR increases as the counterparty’s credit quality rises (a strong-name default is a bigger shock).

3. Computational Implementation - unilateral and bilateral CVA in full

A stylised 5-year payer swap with discounted EPE/ENE profiles and two credit curves. Stdlib only; reproduces the “market × credit × LGD” product and the unilateral→bilateral contrast.

Notice the structure: the counterparty is riskier than you (150bp vs 100bp), so CVA (−1.35) exceeds DVA (+0.47) and the swap is a cost to you (BCVA −0.89). The spread approximation (−0.40) has the same sign but is rough because the EPE/ENE profiles are not flat - exactly Gregory’s caveat on Eq 17.4.

4. Failure Modes & First-Principles Breakdowns

Using historical PDs. Real-world (historical) default rates are lower than risk-neutral (CDS-implied) ones; pricing CVA with historical PDs undercharges for risk and contradicts the market/exit-price basis required by accounting (Gregory §3.1.7, §17.2; Hull 24.5).
Ignoring wrong-way risk. Under the independence assumption, CVA is structurally underpriced when default and exposure co-move - corr +50% ≈ doubles EPE (Gregory §17.6). The assumption is no-WWR, not a neutral one.
Adding standalone CVAs. CVA is a netting-set quantity; the standalone sum over-tells the portfolio CVA because it ignores netting (Gregory Eq 17.10). Incremental/marginal CVA must be computed on the netting set, not trade-by-trade.
Mixing LGDs. Using one LGD to strip PDs from CDS and a different one as the loss severity - without the “actual vs market” separation of Eq 17.5 - distorts both inputs. When seniority matches, the LGDs cancel to first order; force them apart only when the waterfall justifies it (Gregory §17.2.6).

5. References

Gregory, The xVA Challenge
Hull, Options, Futures, and Other Derivatives
Brigo & Mercurio, Interest Rate Models
Pykhtin, Michael & Steven Zhu (2007): A Guide to Modeling Counterparty Credit Risk

6. Connected Graph Bridges

Back: 02 · Exposure & Margin
Forward: 04 · FVA & MVA · 05 · Failure Modes · Index Hub
Credit theory: Credit Risk & the Merton Structural Model · Probability & Measure

Kwant Atlas

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