CVA: The Price of Default

You’ve done the hard half of this course already. You know how to build an exposure profile — the expected positive exposure (EPE) curve that says “if the counterparty vanishes at time $t$, how much would they owe me?” You know that netting and collateral squash that curve down toward zero. Good. But knowing the size of the residual risk isn’t the same as knowing its price. A derivative dealer can’t just feel uneasy about a shaky counterparty — they have to put a number on it, charge it into the trade, book it, and hedge it. That number is CVA, the credit valuation adjustment, and it’s the single most important price in this entire course. This lesson turns your exposure profile into dollars.

CVA: the discount for dealing with a defaultable counterparty

Analogy. Imagine two friends each owe you $100, payable in a year. One is your reliably-employed accountant cousin; the other is your charming brother-in-law who’s between jobs, again. Both promised $100. But if you had to sell those two IOUs today, you would not get the same price for them — the brother-in-law’s IOU trades at a discount, because there’s a real chance you collect nothing. That discount, the haircut the market puts on a promise from someone who might not pay, is exactly what CVA is for a derivative.

The definition. The credit valuation adjustment (CVA) is the difference between the risk-free value of a trade — what it would be worth if your counterparty literally could not default — and its actual, risky value once you account for the chance they do:

$\text{CVA} = V_{\text{risk-free}} - V_{\text{risky}}$

So the risky value you actually hold is $V_{\text{risky}} = V_{\text{risk-free}} - \text{CVA}$. CVA is subtracted — it’s a cost, a reduction in value. And crucially, it is an expected loss on the derivative: not the worst case, not a capital buffer, but the probability-weighted average of how much you lose to default, today’s-dollars.

That framing matters. You already met expected loss in credit risk: $\text{EL} = \text{PD} \times \text{LGD} \times \text{EAD}$. CVA is that same expected-loss idea, but applied to a derivative whose exposure (the EAD) isn’t a fixed loan balance — it wanders up and down over the life of the trade, which is exactly why you spent earlier lessons building the exposure profile.

The three ingredients

CVA isn’t a single formula you plug numbers into once — it’s a sum over time slices of the trade’s life. Chop the life of the deal into intervals $[t_{i-1}, t_i]$. In each slice you ask: if they default in this window, how much do I expect to lose, and how likely is that? Multiply, discount, add it all up:

$\text{CVA} \approx \text{LGD} \times \sum_{i} \text{EE}(t_i)\, \cdot\, \text{DF}(t_i)\, \cdot\, \text{PD}(t_{i-1}, t_i)$

Four ingredients, each one familiar:

• EE — expected exposure. $\text{EE}(t_i)$ is the expected positive exposure at time $t_i$ — the EE/EPE curve you built in the profile lessons. It answers “if they default at $t_i$, how much do they owe me on average?” This is the exposure profile doing its job: CVA integrates the price of default against your EE curve, slice by slice.
• PD — marginal default probability. $\text{PD}(t_{i-1}, t_i)$ is the probability the counterparty defaults in that specific window, given they survived to its start. You read it off their credit curve, which the market hands you through their CDS spread — recall the credit triangle, $\lambda \approx \text{spread} / (1 - R)$.
• LGD — loss given default. $\text{LGD} = 1 - R$, the fraction of exposure you don’t recover when they fail. A 40% recovery means a 60% LGD.
• DF — discount factor. $\text{DF}(t_i)$ pulls each future expected loss back to today’s dollars. A loss four years out is worth less today than the same loss next quarter.

The per-period building block is just the expected-loss triangle you already know — exposure, default odds, and severity, multiplied together. Here it is, live: drag the CDS spread and recovery to watch the implied default probability (and therefore each slice’s contribution to CVA) move.

A worked CVA, end to end

Numbers make it real. Take a clean, deliberately round example you can carry in your head:

• Trade: a 5-year interest-rate swap with a flat expected positive exposure of $1,000,000 (in reality EPE has a humped shape; we flatten it so the arithmetic is transparent).
• Counterparty credit: a CDS-implied hazard rate of ~2% per year ($\lambda = 0.02$).
• Severity: LGD = 60% (a 40% recovery).
• Discounting: ignore it for the headline number, to keep the spotlight on the credit math.

Step 1 — cumulative default probability over 5 years. Survival compounds exponentially, so:

$\text{PD}(0, 5) = 1 - e^{-\lambda \cdot 5} = 1 - e^{-0.02 \times 5} = 1 - e^{-0.1} \approx 1 - 0.905 = 0.095$

About a 9.5% chance they default somewhere in the five years.

Step 2 — the headline CVA. Expected loss = severity × exposure × default probability:

$\text{CVA} \approx \text{LGD} \times \text{EPE} \times \text{PD}(0,5) = 0.60 \times 1{,}000{,}000 \times 0.095 \approx 57{,}000$

So the desk must charge roughly $57,000 into this trade to cover the expected cost of the counterparty defaulting. That’s not a fee they pocket — it’s the fair price of the credit risk they’re absorbing.

Step 3 — see it accrue year by year. CVA is a sum over slices, so let’s break the $57k into its yearly pieces. The marginal default probability in each year is $\text{PD}(t_{i-1}, t_i) = e^{-\lambda t_{i-1}} - e^{-\lambda t_i}$ — survive to the start of the year, then fail during it. Multiply each by LGD × EPE = $600,000:

Each year chips in roughly $11–12k, sliding gently downward because surviving to later years gets less likely. Add the five slices and you land back on $57,000 — the same headline, now shown as the accrual it really is. (Switch discounting back on and each later slice shrinks a bit more, nudging the total down.)

Unilateral vs. bilateral CVA, and DVA

Everything so far has been unilateral CVA — it counts only their default, on the exposure where they owe you. But a derivative is symmetric: there are scenarios where the position flips and you owe them. And here’s the uncomfortable truth of credit pricing — you might default too, and if you do, you might not pay what you owe. From your own narrow, self-interested point of view, that possibility is a benefit.

Price up that benefit and you get DVA, the debit (or debt) valuation adjustment. It’s the exact mirror of CVA, with every term flipped to your side:

• where CVA used their default probability, DVA uses yours;
• where CVA used your positive exposure (they owe you), DVA uses your negative exposure (you owe them);
• where CVA used their LGD, DVA uses yours.

$\text{DVA} \approx \text{LGD}_{\text{you}} \times \sum_{i} \text{NEE}(t_i)\, \cdot\, \text{DF}(t_i)\, \cdot\, \text{PD}_{\text{you}}(t_{i-1}, t_i)$

Put the two together and you get bilateral CVA (BCVA) — the net credit adjustment that recognises both parties can fail:

$\text{BCVA} = \text{CVA} - \text{DVA}$

CVA is the cost of their default; DVA is the “benefit” of yours; BCVA is the net. Two counterparties of equal credit quality, perfectly symmetric, would net to roughly zero — which is the whole appeal of the bilateral framing.

The DVA paradox

Now stare at what DVA actually implies, because it’s genuinely strange. DVA goes up when your own default probability goes up. And since DVA is added back (it improves the value of your book), a rising DVA books a profit. Read that again: as your own credit deteriorates, you record a gain on your derivatives.

The logic isn’t a glitch — it’s internally consistent and deeply weird. If you’re more likely to default, your liabilities are more likely to go unpaid, so a market-value-of-liabilities accounting says those liabilities are “worth less,” which shows up as a gain. The catch: to actually realise that gain, you’d have to default — monetising your DVA means going bankrupt, which is not a strategy any treasurer puts in the annual plan.

This isn’t hypothetical. In 2011, as the European debt crisis widened bank credit spreads dramatically, major banks reported enormous DVA gains — billions in some cases — purely because their own spreads had blown out. The headlines wrote themselves: “Bank posts profit because markets fear it might collapse.” Accounting-real, economically dubious. Regulators eventually agreed it was nonsense for solvency purposes and filtered DVA out of regulatory capital — you don’t get to count “I might go bust” as capital strength.

CVA is a TRADED risk: the CVA desk

Here’s the shift that elevates CVA from a pricing formula to a trading business: CVA is not static. It’s built from live market inputs — the counterparty’s credit spread, interest rates, FX rates — and every one of those moves daily. So CVA has Greeks, just like an option:

• Credit sensitivity (CS01). Widen the counterparty’s CDS spread and their PD rises, so CVA increases — your book takes a mark-to-market hit. CVA is fundamentally a short credit position on every counterparty you face.
• Rates and FX sensitivity. Rates and FX reshape the EE profile itself (a swap’s expected exposure depends on the rate path; a cross-currency trade’s on FX). Move those, and the exposure CVA is priced against moves too.

Because these sensitivities are real P&L, banks run a dedicated CVA desk that centralises the counterparty risk of the whole institution and hedges it — buying single-name CDS or credit-index protection to offset the short-credit exposure, and rates/FX hedges for the profile sensitivities. The trading desks pay the CVA desk a charge at inception; the CVA desk then manages the aggregate risk as a portfolio.

And this isn’t an academic nicety. During the 2007–09 crisis, an estimated two-thirds of counterparty-credit-related losses came not from actual defaults but from CVA mark-to-market swings — counterparty spreads widening violently, repricing CVA, gutting bank P&L before anyone actually defaulted. That “CVA volatility” story is precisely why Basel III introduced a dedicated CVA capital charge: regulators decided the volatility of CVA, not just realised defaults, was a risk that had to be capitalised.

Putting it together

CVA is the market price of counterparty credit risk — the expected loss on your derivative from the chance the counterparty defaults, equal to the gap between the trade’s risk-free value and its actual risky value. You compute it by integrating the expected-loss triangle against your exposure profile: $\text{CVA} \approx \text{LGD} \times \sum_i \text{EE}(t_i)\,\text{DF}(t_i)\,\text{PD}(t_{i-1},t_i)$, where EE is your EPE curve, PD comes from the counterparty’s CDS-implied credit curve, LGD $= 1 - R$, and DF discounts. On a 5-year swap with a flat $1m EPE, a 2%/yr hazard ($\text{PD}(0,5) \approx 9.5\%$), and 60% LGD, the headline CVA is $0.60 \times 1{,}000{,}000 \times 0.095 \approx 57{,}000$ (i.e. ~$57,000) — accruing ~$11–12k a year. DVA mirrors CVA on your own default (a benefit you add back), and BCVA $= \text{CVA} - \text{DVA}$ nets the two — but DVA carries the paradox that your deteriorating credit books a paper gain you can only collect by failing, which is why regulators stripped it from capital. Finally, CVA is a traded risk with Greeks, centralised and hedged on a CVA desk, and its mark-to-market volatility — two-thirds of crisis-era counterparty losses — is why Basel III demanded a CVA capital charge.

Next — The XVA Family — we zoom out from CVA and DVA to the full alphabet soup: FVA for the cost of funding uncollateralised exposure, MVA for the cost of posting initial margin, and KVA for the cost of holding the very capital Basel III just demanded. Each is the same move you’ve now mastered — a real economic cost, priced into the trade as an adjustment to value.