Credit Valuation Adjustment

Collateralized Debt Obligations showed how tranching prices default risk on a whole pool of loans. But default risk isn't confined to CDS and CDOs — it lurks inside every over-the-counter derivative you've met in this course. An interest-rate swap, an FX forward, an exotic option bought from a dealer bank: whenever one side owes the other a positive mark-to-market value, that side is exposed to the other's default, exactly like a bondholder. Credit valuation adjustment (CVA) is the market's answer to a simple question: how much should that risk shave off the price of the trade?

What CVA measures

Every derivative price you've computed so far in this course implicitly assumed both counterparties are default-free. CVA is the correction: the amount a derivative's value should be reduced because your counterparty might not be able to pay what it owes you at exactly the moments the trade is in your favor.

\text{price with counterparty risk} = \text{default-free price} - \text{CVA}.

The key idea that makes this harder than pricing a bond is exposure: a bond's exposure to its issuer is simply its face value, fixed and known in advance. A derivative's exposure is its replacement cost — how much it would cost to re-enter an equivalent trade with someone else — and that number moves with the market every single day, sometimes swinging from positive to negative and back. You only lose money to counterparty default at moments when the trade is in your favor (positive exposure); if it's underwater, your counterparty's default barely matters to you at all — you owed them.

For a single exposure period, a common simplified approximation is:

\text{CVA} \approx (1-R) \times EE \times PD,

where EE is the expected (positive) exposure at that point in the trade's life, PD is the counterparty's default probability over the relevant period, and R is the recovery rate on the counterparty's obligations. Over a multi-period trade, a full CVA calculation sums this expression, period by period, discounted back to today — exactly the same machinery Credit Default Swaps used to value the protection leg of a CDS.

A swap's exposure profile

Unlike a bond, a swap's expected exposure typically rises then falls over its life — a characteristic "hump" shape. Early on there's a long time left for rates to drift far from where they started, building up potential replacement cost; late in the trade's life there's less time left for that drift and fewer remaining cash flows at stake, so exposure fades back toward zero as maturity approaches. Explore that shape below.

Worked example

A 3-year interest-rate swap with a counterparty bank has the following expected exposure profile, and the bank's hazard rate is \lambda = 1.5\% per year, recovery rate R = 40\%, and the risk-free rate is r = 3\%.

Year	EE ($M)	Marginal PD	DF(t)	Contribution ($)
1	2.0	0.01489	0.9704	28,904
2	3.0	0.01466	0.9418	41,430
3	1.5	0.01445	0.9139	19,811
Sum				90,145

Each marginal probability of default is S(t-1) - S(t) = e^{-\lambda(t-1)} - e^{-\lambda t}, and each contribution is EE(t) \times PD(t) \times DF(t). Summing and multiplying by (1-R):

\text{CVA} = (1-R) \times \$90{,}145 = 0.6 \times \$90{,}145 \approx \$54{,}000.

A roughly \$54{,}000 charge on a swap whose exposure peaks around \$3 million — a modest but very real haircut, and one the dealer bank will build directly into the price it quotes the client, exactly as it would price in a swap's credit risk more generally.

The rest of the "xVA" family

CVA is the best known member of a whole family of valuation adjustments desks now compute for every OTC trade:

DVA (debit valuation adjustment) — the mirror image of CVA: an adjustment reflecting your own default risk. If your firm might default before paying what it owes, that liability is worth slightly less to you — so DVA adds value back.
FVA (funding valuation adjustment) — the cost of funding an uncollateralized derivative position at your bank's own (above risk-free) borrowing rate, rather than at the idealized risk-free rate every pricing formula in this course has assumed so far.

Together these are often just called xVA, and most large banks run a dedicated "xVA desk" whose sole job is computing and hedging these adjustments across the entire firm's derivatives book.

Before the crisis, many banks priced OTC derivatives close to the "default-free" formula, treating counterparty risk as a footnote handled separately by credit officers rather than built into the price itself. Lehman Brothers' collapse in September 2008 ended that complacency overnight: banks holding derivatives with Lehman as counterparty discovered that "too big to fail" wasn't a pricing input they could rely on, and losses from unwound Lehman trades ran into the billions. Dedicated CVA desks — and, soon after, a Basel III regulatory capital charge specifically for CVA volatility risk — became standard practice almost immediately afterward. Today, pricing a swap or an option without CVA baked in isn't considered incomplete pricing; it's considered a mispriced trade.

DVA is philosophically awkward. Under DVA accounting, a bank's own derivatives book can show an accounting gain when the bank's own credit worsens — you're "profiting" from becoming more likely to stiff your counterparties. Regulators and accountants remain uneasy about this, and DVA is treated very cautiously (often excluded from regulatory capital) precisely because it can never be realized as real cash unless the bank actually defaults.
The simple CVA formula assumes exposure and default are independent — they often aren't. When a counterparty's default probability rises exactly when your exposure to them is largest, that's wrong-way risk, and it means the simplified (1-R) \times EE \times PD formula understates the true CVA. A classic example: an oil producer's swap counterparty that itself depends heavily on oil prices, so a crash that boosts your swap's value also raises your counterparty's default risk at the same moment.