4.7.3 CVA & DVA

Basic Prerequisites: PFE and Credit Risk & the Merton Model (PD, LGD, hazard rates).

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1. Intuition & Practical Objective

CVA is the number that prices a defaultable promise. The practical objective is to turn an exposure profile and a credit curve into one spread, in basis points, that a trader charges or hedges. The formula has a physical reading: walk along the survival curve; at each date, if the counterparty dies there, you lose the LGD fraction of what you are owed; discount and average.

Two implementations reach the same number:

• Direct (default-time) method - simulate a default time, value the portfolio once at that date, multiply by the probability the default happened in the interval.
• Path-wise (EPE) method - sum, over a grid, $\text{EPE}(t_i)\times \text{PD}(t_{i-1},t_i)$.

They are equal (with no wrong-way risk), but the direct method converges far faster: for a 10-year swap Gregory’s Spreadsheets 17.1–17.2 show a 6× smaller standard deviation for the same number of valuations - a 36× speed advantage (Monte Carlo error $\propto 1/\sqrt{n}$).

The LGD cancellation - the most useful practical fact. In the discrete formula the actual LGD appears in the numerator and the market LGD (from the CDS calibration) appears in the denominator of the hazard. Where seniority matches, they cancel to first order: moving LGD from 60% to 50% changes CVA by <2% (Gregory Fig. 17.9). So do not agonise over LGD for a name where the CDS seniority matches your exposure.

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2. Mathematical Ground Truth & Derivations

2.1 From first principle to the three CVA formulas

Let $\tau$ be the default time, $\text{LGD}=1-R$, and $V(\tau {)}^{+}$ the exposure at default. The direct (unilateral) CVA is

$$\text{UCVA}(t)=-\mathbb{E}\left[1_{\{\tau \le T\}}V(t,\tau {)}^{+}\text{LGD}\right].\text{\hspace{1em}(Gregory 17.1)}$$

Writing the expectation as an integral over the default-intensity (survival) curve gives the path-wise integral form:

$$\boxed{\text{UCVA}(t)=-\text{LGD}{\int }_t^{\infty }{\lambda }_C{D}_{r+{\lambda }_C}(t,u)\text{EPE}(t,u)du}\text{\hspace{1em}(17.2)}$$

with $D_{r+{\lambda }_C}(t,u)=\exp (-{\int }_t^u(r+{\lambda }_C)ds)$ the risky discount factor and $\text{EPE}(t,u)=\mathbb{E}[V(t,u{)}^{+}]$. Discretising the integral over a grid:

$$\text{UCVA}(t)\approx -\text{LGD}\sum _{i=1}^m\text{EPE}(t,t_i)\text{PD}(t_{i-1},t_i).\text{\hspace{1em}(17.3)}$$

This is the product-of-market-and-credit identity. Exposure (market risk) multiplies default probability (credit risk); with independence, everything else is bookkeeping.

2.2 The hazard comes from the CDS spread

For a flat spread $s$ and LGD, the risk-neutral hazard is (Hull, Eq 24.2)

$$\lambda =\frac{s}{\text{LGD}},\text{PD}(t_{i-1},t_i)=e^{-\lambda t_{i-1}}-e^{-\lambda t_i}.$$

Substituting into (17.3) and using $e^{-x}\approx 1-x$ gives the spread form

$$\text{UCVA}\approx -\overline{\text{EPE}}\times s\text{\hspace{1em}(17.4)}$$

• CVA quoted as a spread in basis points, the desk’s natural unit. (Gregory Table 17.1: for a 10-year swap the recursive “CVA of the CVA” spread is $-1.96$bp, the risky-annuity estimate $-1.92$bp, and the EPE approximation $-2.01$bp.)

2.3 The LGD-adjusted form

Keeping the two LGDs separate:

$$\text{UCVA}(t)=-{\text{LGD}}_{\text{actual}}\sum _{i=1}^m\text{EPE}(t,t_i)\left[e^{-s_{i-1}{t}_{i-1}/{\text{LGD}}_{\text{mkt}}}-e^{-s_i{t}_i/{\text{LGD}}_{\text{mkt}}}\right].\text{\hspace{1em}(17.5)}$$

When ${\text{LGD}}_{\text{actual}}={\text{LGD}}_{\text{mkt}}$ the LGD factors cancel - the practical reason (17.4) contains no LGD.

2.4 Bilateral CVA and DVA

Accounting (FAS 157 / IFRS 13) requires own credit risk in the value of liabilities. That is DVA, and the bilateral pair is

$$\text{BCVA}=\text{CVA}+\text{DVA}\text{\hspace{1em}(17.7a)}$$ $$\text{CVA}(t)=-{\text{LGD}}_C{\int }_t^{\infty }{\lambda }_C{D}_{r+{\lambda }_C+{\lambda }_P}(t,u)\text{EPE}(t,u)du,\text{DVA}(t)=-{\text{LGD}}_P{\int }_t^{\infty }{\lambda }_P{D}_{r+{\lambda }_C+{\lambda }_P}(t,u)\text{ENE}(t,u)du.$$

The joint discount factor $D_{r+{\lambda }_C+{\lambda }_P}$ is the first-to-default survival of both parties. In discrete form (17.8a/b) each term carries the other party’s survival probability $[1-{\text{PD}}_{\text{other}}(0,t_{i-1})]$. Because $\text{ENE}\le 0$, DVA $\ge 0$ is a benefit opposing CVA - “my CVA is your DVA”. And when EPE ≈ −ENE,

$$\text{BCVA}\approx -\overline{\text{EPE}}\times (s_C-s_P)\text{: the weaker credit pays the stronger.}$$

DVA is derecognised from regulatory capital (BCBS 2011d) even though accounting demands it - a genuine conflict discussed in §4.

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3. Computational Implementation - CVA, DVA and BCVA on a simulated profile

Reuse the ATM-forward exposure engine of 02, discount the EE to today, evaluate the discrete sum (17.3), and cross-check it against the integral (17.2) on a fine grid. Standard library only.

Read the block: CVA is −1.06 on a zero-value ATM forward - the price of the promise. The discrete sum and the fine-grid integral agree to ~3% (the residual is Monte Carlo noise plus the crude 20-step grid), which is the practical check that the two identities (17.2) and (17.3) are the same object. DVA (+0.73) uses the weaker own spread; it is smaller than CVA because our own credit is better, and BCVA lands at −0.327. The spread ladder shows CVA rising monotonically with spread - roughly linearly at moderate spreads, with the curvature that creates the large CVA gamma (and jump-to-default risk) discussed in 05 · Failure Modes.

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4. Failure Modes & First-Principles Breakdowns

1. Real-world vs risk-neutral PD. CVA must use market-implied (CDS) default probabilities, not historical default rates - the exit-price/hedging standard (Gregory §3.1.7). Substituting a real-world PD breaks the hedge and the accounting value.
2. The DVA moral hazard. DVA increases in value when your own credit deteriorates - “book a profit by becoming less creditworthy”. Bondholders gain in firm default; shareholders do not, which is exactly why regulators derecognise DVA from capital (Gregory §17.3.5). Treat DVA as a funding benefit, not distributable income.
3. DVA double-counts FBA. DVA and the funding benefit adjustment FBA are two names for the same negative-exposure benefit; adding both discounts it twice (§18.2.5). Choose CVA + FCA + FBA or CVA + DVA + FCA, never all four.
4. Survival-adjustment and close-out ambiguity. Whether to include the other party’s survival probability (contingent vs non-contingent CVA), and whether close-out references base or actual (risky) value, changes the number and is not standardised across firms (Gregory §17.3.4).
5. Spread-shape blindness. The same 5-year spread with an up-sloping, flat, or inverted curve gives materially different 10-year CVA ($-24.6/-20.0/-15.7$bp in Gregory Table 17.2). Using a single flat spread discards the term structure that drives the integral.
6. “CVA of the CVA” forgetting. The exact spread that zeroes a CVA-inclusive value solves a recursive equation; the naive EPE approximation misses ~5% (Gregory Table 17.1).

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5. References

• Gregory, Jon: The xVA Challenge (5th ed., 2025)
• Brigo, Morini & Pallavicini: Counterparty Credit Risk, Collateral and Funding (2013)
• Hull, Options, Futures, and Other Derivatives
• BCBS (2011d): DVA derecognition; BCBS (2017) Basel III CVA Risk Framework for the capital treatment.

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6. Connected Graph Bridges

• Back: PFE
• Forward: 04 · Collateral, Netting & SA-CCR · Index Hub
• Base: Credit Risk & the Merton Model · Value at Risk & Expected Shortfall