Credit Valuation Adjustment

Two banks sign a five-year interest rate swap. Standard pricing tells you exactly what that swap is worth, discounting every future cash flow at the risk-free rate. There is just one problem: that price assumes both banks stay solvent on every payment date. One of them might not. If your counterparty goes bankrupt owing you money, you do not collect the risk-free price — you collect whatever the bankruptcy court hands out, usually far less. Credit valuation adjustment, CVA, is the number you subtract from the textbook price to account for that.

Lending to a friend versus lending to a bank

If a friend asks to borrow $1,000 and promises to pay it back with interest, you would not charge them the same rate a AAA-rated bank pays on a deposit. Your friend might lose their job, move away, or simply not pay you back, so you charge more — enough extra interest to compensate you, on average, for the times it goes wrong. That extra charge is not a guess about your friend's character; it is arithmetic, once you know roughly how likely default is and how much you would recover if it happened.

A derivatives desk faces the identical problem every time it trades with anyone other than a central bank. Whatever the "true" risk-free price of a swap or an option is, the desk shaves it down by exactly the expected cost of the counterparty failing to pay. That shaved-down amount is CVA, and unlike the friend example, a bank can actually compute it, because default probabilities and recovery rates are quoted in the market every day via credit default swaps.

The three ingredients

CVA is built from three pieces, each of which answers one question in plain words:

• Expected exposure, $EE(t)$ — if the counterparty defaulted at time $t$, how much would they typically owe you? This is not the notional of the trade; it is the trade's expected mark-to-market value, and only when that value is positive (a value that's negative to you is money you owe them, and their default does not erase your debt).
• Probability of default, $PD(t_{i-1}, t_i)$ — the chance the counterparty defaults during a specific future time slice, not before and not after.
• Recovery rate, $R$ — the fraction of what they owe you that you actually collect from the bankruptcy estate. $(1-R)$, the loss given default, is the fraction you lose.

Put them together, summed over every time slice out to the trade's maturity, and discounted back to today with a discount factor $DF(t_i)$:

$$\text{CVA} \;=\; (1-R) \sum_{i=1}^{n} EE(t_i)\, PD(t_{i-1}, t_i)\, DF(t_i) .$$

In plain English: for every future date, work out how much you stand to lose if default happens exactly then — expected exposure times loss-given-default — weight it by how likely default is in that particular window, discount it back to today's dollars, and add up all the dates. The result is a single number, in currency, that gets subtracted from the risk-free price.

Worked example 1: a three-year interest rate swap

You hold an uncollateralized swap with a bank counterparty. A risk-free pricing model says the swap is worth $50,000 to you today. Its expected exposure profile, in annual buckets, along with the counterparty's marginal default probability in each year and the relevant discount factor, is:

Year	$EE(t_i)$	$PD$ in that year	$DF(t_i)$
1	$1,000,000	1.0%	0.97
2	$1,500,000	1.5%	0.94
3	$800,000	1.0%	0.91

Assume a recovery rate $R = 40\%$, so $(1-R) = 0.6$.

Step 1 — loss-weighted exposure per year. Multiply exposure by that year's default probability and discount factor:

• Year 1: $1{,}000{,}000 \times 0.01 \times 0.97 = 9{,}700$
• Year 2: $1{,}500{,}000 \times 0.015 \times 0.94 = 21{,}150$
• Year 3: $800{,}000 \times 0.01 \times 0.91 = 7{,}280$

Step 2 — sum and apply loss-given-default. Add the three: $9{,}700 + 21{,}150 + 7{,}280 = 38{,}130$. Multiply by $(1-R) = 0.6$: $\text{CVA} = 22{,}878$.

Step 3 — adjust the price. The CVA-adjusted value of the swap is $50{,}000 - 22{,}878 = 27{,}122$. Nearly half the risk-free value has been eaten by counterparty risk, because the exposure happens to peak right in the year where default risk is also highest — a coincidence you should never assume away, since real desks price it in explicitly (Wrong-Way Risk).

Worked example 2: a purchased option

You buy a one-year call option from a dealer for a risk-free price of $50,000 (10,000 shares at $5.00 each). Because you paid up front and the dealer owes you the payoff at expiry, your exposure is simply the option's own value — there is nothing complicated about a profile here, because a bought option is a one-way bet: if the dealer defaults, you lose whatever the option was worth, and never less than zero.

Suppose the dealer's hazard rate implies a 3% chance of default over the year, and the average value of the option between now and expiry (accounting for time decay) is estimated at $48,000. Recovery is 40%, and the one-year discount factor is 0.97.

$$\text{CVA} = (1-R)\, PD \, EE \, DF = 0.6 \times 0.03 \times 48{,}000 \times 0.97 = 838.08 .$$

The dealer will not sell you that option for $50,000; they will quote $50,000 minus $838, or $49,162 — roughly $4.92 per contract instead of $5.00. That eighteen-cent-per-contract haircut is the entire reason CVA desks exist: someone has to price, hedge, and hold capital against that number across every trade on the bank's book.

What this means in practice

CVA is not a footnote — since the 2008 crisis, most of the derivatives losses banks reported came from CVA moving against them (counterparties' credit spreads widening) rather than from the underlying market risk of the trades themselves. That is why every major bank runs a dedicated CVA desk that prices this adjustment into every quote, hedges its sensitivity to counterparty credit spreads with credit default swaps, and holds regulatory capital against it under Basel III. It is also why collateral agreements (CSAs) exist: posting collateral against a trade's mark-to-market shrinks expected exposure toward zero, which shrinks CVA toward zero along with it — the whole reason cleared and collateralized trades carry far less counterparty charge than a bilateral, uncollateralized swap (Netting and CSA Agreements).

Key terms

• Expected exposure $EE(t)$ — the expected positive mark-to-market value of the trade at a future date if the counterparty defaulted then.
• Recovery rate $R$ — the fraction of exposure recovered from the bankruptcy estate; $(1-R)$ is loss given default.
• Marginal default probability — the chance of default in one specific future window, derived from the counterparty's credit default swap curve via Hazard Rates And Survival Probabilities.
• Wrong-way risk — exposure and default probability rising together, which the basic CVA formula does not capture.
• DVA — the mirror-image adjustment for your own default risk, which is why CVA is only half of the full XVA picture (XVA: FVA, DVA And MVA).