Credit valuation adjustment is three columns multiplied and added. Done by hand on a corporate swap, it tells you how much of the credit charge in the rate is credit.
Multiply three columns and add: for each period, the discounted expected positive exposure, times the probability the counterparty defaults in that period, times the loss given default. On an illustrative 120M five-year swap with 100bp of rate volatility and a 200bp credit spread, CVA is 143,510, about 2.7bp a year. The bank in this case embedded 11bp in the rate, roughly four times as much.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The swap is Kilmartin Industrial's, from Derivatives: 120,000,000 notional, five years, Kilmartin pays fixed and the bank receives it. The book's case states the credit charge the bank built into the fixed rate, 11bp, or 132,000 a year on the swap, and observes that the credit charge, carried in the swap and the currency forwards alike, is the largest of the hedge's real costs and the only one never invoiced. The case takes the 11bp as given rather than computing what it should be. To do it here, three inputs are added and are illustrative: the volatility of swap rates, Kilmartin's credit spread and its recovery rate. The curve is held flat at the five-year par rate for simplicity.
| Input | Value |
|---|---|
| Notional | 120,000,000 |
| Tenor, annual fixed payments | 5 years |
| Discount rate, flat | 4.36% |
| Normal volatility of swap rates (illustrative) | 100bp |
| Kilmartin credit spread (illustrative) | 200bp |
| Recovery rate (illustrative) | 40% |
| Credit charge in the swap rate | 11bp |
A swap starts at zero value. Its value to the bank at a future date is the change in the swap rate since inception times the remaining annuity. If rates fall the bank's receive-fixed position is in the money and the bank is exposed to Kilmartin; if they rise, it is not. With rate changes normally distributed with volatility σ, the expected positive part of a zero-mean normal is σ√t times 0.3989 (the normal density at zero), so:
EPE(t) = Notional × σ × √t × 0.3989 × remaining annuity at t
Year 2, at its midpoint: 120,000,000 × 1.00% × 1.2247 × 0.3989 × 3.4489 = 2,022,190
Excel: =N*Sigma*SQRT(t)*NORM.S.DIST(0,FALSE)*RemAnnuity
The remaining annuity is the sum of the discount factors on the payments still to come, already discounted to today, so EPE here is in present value.
The credit triangle turns a spread into a hazard rate: 200bp divided by a 60 per cent loss given default is 3.333 per cent a year. Survival to time t is e−3.333% × t, and the probability of default in a year is the fall in survival across it. Five-year survival is 84.65 per cent, a cumulative default probability of 15.35 per cent.
PD(year i) = e−λ(i−1) − e−λi, with λ = spread / LGD
CVA = LGD × Σ EPE(ti) × PD(year i)
| Year | √t | Remaining annuity | Expected positive exposure | Default probability in year | LGD × EPE × PD |
|---|---|---|---|---|---|
| 1 | 0.7071 | 4.4072 | 1,491,883 | 3.278% | 29,346 |
| 2 | 1.2247 | 3.4489 | 2,022,190 | 3.171% | 38,473 |
| 3 | 1.5811 | 2.5307 | 1,915,622 | 3.067% | 35,251 |
| 4 | 1.8708 | 1.6509 | 1,478,601 | 2.966% | 26,317 |
| 5 | 2.1213 | 0.8078 | 820,403 | 2.869% | 14,123 |
| CVA | 143,510 |
The exposure profile is the familiar hump: uncertainty grows with √t while the remaining annuity shrinks, so exposure peaks in year two at 1.69 per cent of notional. A monthly grid instead of annual buckets gives 141,457, 1.4 per cent lower, so the hand version is good enough to reason with.
143,510 is about 0.120 per cent of notional. Spread over the life of the swap, divide by notional times the annuity of 4.4072: 2.7bp a year. The bank's charge was 11bp, 132,000 a year or 581,745 in present value, which is 4.05 times the CVA on these inputs. Run the other way, the 11bp would be a fair CVA only if Kilmartin's credit spread were around 1,111bp at the same volatility.
The gap is not proof of overcharging. A bank's credit charge usually also covers the capital held against the exposure and the cost of funding an uncollateralised trade, and its own spread. What the calculation gives the treasurer is a number to put the question with: of 11bp, which part is default risk and what is the rest?
| Credit spread | Vol 75bp | Vol 100bp | Vol 125bp |
|---|---|---|---|
| 100bp | 55,839 (1.1bp) | 74,452 (1.4bp) | 93,065 (1.8bp) |
| 200bp | 107,632 (2.0bp) | 143,510 (2.7bp) | 179,387 (3.4bp) |
| 300bp | 155,671 (2.9bp) | 207,561 (3.9bp) | 259,451 (4.9bp) |
CVA is proportional to volatility in this model and close to proportional to the spread. Even the top-right cell, a weak credit in a volatile market, reaches 4.9bp, under half the 11bp charged.
The usual error is to estimate exposure as a fixed percentage of notional for the whole life, or as the current mark-to-market, which is zero on day one and gives a CVA of zero. Exposure is an expectation over future rate paths and it has a shape. The second error is using the cumulative default probability on every bucket rather than the marginal one, which multiplies CVA several times over. The third is ignoring the direction of the curve: on an upward-sloping curve the fixed payer's swap is expected to drift in its favour, which lowers the bank's exposure further than the flat-curve figure here.
A collateral agreement turns this credit exposure into a liquidity one: see how much more margin a five-year hedge calls than a rolling one.
The three real cost lines of Kilmartin's hedge, with the credit charge set beside the two spreads, are in the free workbook for this case.
CVA equals loss given default times the sum, over time buckets, of discounted expected positive exposure times the probability of default in that bucket. On a 120M five-year swap with 60 per cent loss given default and a 3.333 per cent hazard rate, five annual buckets give 143,510, about 0.120 per cent of notional.
Use the credit triangle: hazard rate equals spread divided by loss given default. A 200bp spread with 40 per cent recovery gives 2.0 over 0.60, a 3.333 per cent hazard rate. Survival to year t is e to the minus hazard times t, so five-year survival is 84.65 per cent and the marginal default probability in year one is 3.278 per cent.
Two forces pull in opposite directions. Rates have more time to move, so uncertainty grows with the square root of time; but fewer payments remain, so each basis point is worth less. On a five-year swap the expected exposure peaks around year two, here at 2,022,190 or 1.69 per cent of notional, then falls to 820,403 in year five.
This article is one calculation from Derivatives. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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