CR 20 – CVA

Credit value adjustment, usually shortened to CVA, is the amount by which the value of a derivative portfolio falls once the possibility of the counterparty defaulting is taken into account. A clean way to define it is as the gap between two valuations of the same trades: one computed as if the counterparty could never fail, and one that allows for its default. CVA is the first member of a wider family of valuation adjustments, often labelled xVA, and debt value adjustment (DVA) is its mirror image, reflecting the default of the party doing the valuation. Under fairly standard assumptions both can be built from just three ingredients: credit exposure, default probability and loss given default (LGD).

Banks have paid close attention to CVA because credit spreads became volatile and because accounting rules (IFRS 13) and capital rules (Basel III) both require it. The topic is not confined to banks, though. Any financial institution or corporation that hedges its economic risks with a sizeable book of over-the-counter (OTC) derivatives faces the same kind of exposure to the dealers on the other side.

From a reserve to an exit price

CVA began life as a credit charge in pricing and as a reserve or provision in the financial statements, and it was often estimated with historical volatilities and historical default rates. Accounting standards have since moved it towards an exit price concept: the adjustment should reflect what another market participant would require in order to take over the position, which means computing it with market-implied, risk-neutral parameters wherever that is practical. For the exposure side this is fairly natural, since simulating exposure resembles option pricing and many inputs, such as forward rates and implied volatilities, can be read from market prices. Some inputs, correlations in particular, are not observed at all, and long-dated volatilities may need interpolation or extrapolation. Risk-neutral volatilities tend to be higher than historical estimates, although not always.

The default probability is the more contentious input. A market-implied default probability normally requires a proxy credit spread curve, because few counterparties have quotes of their own, and several objections can be raised. Market-implied default probabilities sit well above real-world default frequencies. Most counterparties have no liquid single-name credit default swap (CDS), so their default cannot really be hedged. And a bank whose business model is to warehouse credit risk is, in the end, exposed only to real-world defaults. These objections are largely academic, since most banks and many other institutions are required by accounting standards to use credit spreads when they report CVA. A few smaller regional banks with modest derivative books still use historical default probabilities, on the argument that their exit price would be set by a local competitor using the same method, and in those increasingly rare cases CVA behaves like an actuarial reserve rather than a market-implied exit price.

Why a swap is harder than a bond

Pricing the credit risk of a bond is comparatively simple because every payment flows one way: discount each cash flow allowing for default and add the value of any recovery. If the issuer defaults, a given coupon is fully at risk apart from recovery. A swap is different because payments run in both directions and can be fixed, floating or contingent. A payment due from the counterparty is only partly at risk, since it is partly cancelled by payments the surviving party would itself have owed. So a swap carries less credit risk than a comparable bond (it also lacks a principal repayment), yet the fraction of its cash flows that is genuinely at risk is hard to pin down, depending on the shape of the yield curve, forward rates and volatilities. That two-way, state-dependent character is what makes counterparty risk difficult to quantify.

Until the sections on wrong-way risk (WWR), exposure, default probability and recovery are treated as independent, and CVA and DVA are considered apart from the other xVA terms, although in practice those terms overlap.

CVA answers three questions at once: will the counterparty default, what will the exposure be if it does, and how much of that exposure will be lost? Written as a single expectation, these give the standard unilateral CVA (UCVA), so called because it assumes the party doing the calculation cannot itself default:

Here t is today, T is the final maturity of the portfolio and τ is the counterparty’s default time. The indicator 1(τ ≤ T) equals one if default happens before maturity and zero otherwise, so its expectation is the cumulative default probability over the interval from t to T. The term V(t, τ)⁺ = max(V(t, τ), 0) keeps only the positive part of the portfolio value at the default date, discounted back to today; this is the exposure, and its expected value at a given default time is the expected positive exposure (EPE). LGD is the share of the exposure expected to be lost, equal to 100% minus the recovery rate R. The leading minus sign is a reporting convention: CVA reduces value, so it is shown as a negative number.

A simulation recipe and the direct calculation

Read literally, the expectation suggests a Monte Carlo scheme. Draw a default time consistent with the counterparty’s credit spread curve; value the portfolio at that date and keep only its positive part; draw an LGD; multiply the three; then repeat many times and average. When default time, exposure and LGD are independent (no WWR), the scheme becomes much simpler and is called a direct CVA calculation. A single average LGD is enough, the portfolio only has to be valued at one future date in each simulation, and it is efficient to draw default times only inside [t, T] and then scale the result by the counterparty’s probability of defaulting in that window.

The integral and the discrete sum

Independence also lets CVA be written as a one-dimensional integral over time, using the instantaneous default intensity λ_C of the counterparty and a risky discount factor:

In practice the integral is approximated on a grid of future dates t₁, t₂, up to t_m:

Each term weights the discounted EPE at date t_i by the probability that the counterparty defaults between t_i-1 and t_i. LGD carries no time behaviour, so it sits outside the sum as a plain multiplier. Risk-free discounting is normally applied while the EPE is being computed rather than through separate discount factors. Pulling the discount factors out explicitly would need care in an interest rate product, where high rates come with small discount factors; handling that convexity properly means measuring exposure under the T-forward measure (Jamshidian 1989), after which discount factors depend only on expected future rates.

The formula is intuitive because it splits CVA into a market risk component (EPE) and a credit risk component (PD), with default entering only through its probability. A simulation is still needed to produce the EPE profile, but default events themselves never have to be simulated, and for independent inputs the discrete sum and the direct calculation give the same answer.

The path-wise formula looks like the obvious choice, since it presents CVA as a sum over the EPE profile, which is how exposure is naturally examined. Yet direct calculations can be considerably faster. Every path-wise simulation requires the portfolio to be valued at each of the m grid dates, and refining the grid improves the approximation of the integral only at the price of more valuations, which are usually the bottleneck.

A comparison for an uncollateralised 10-year receiver interest rate swap makes the point. With a quarterly grid of 40 time steps, a budget of 40,000 valuations buys 1,000 path-wise simulations, whereas the direct approach uses the same budget on 40,000 separate default times. As the number of valuation calls grows, the direct estimate settles down far sooner. Repeating each calculation 20 times shows the standard deviation of the direct estimate to be about six times smaller. Because Monte Carlo error falls with the square root of the number of simulations (a consequence of the central limit theorem), a sixfold cut in error is worth 6 × 6 = 36 times as many valuations; the direct method reaches a given accuracy with 36 times fewer.

The reason is autocorrelation along a path. Portfolio values two years ahead and two years and three months ahead are very strongly positively correlated, so successive grid points in a path-wise simulation add little new information, and convergence measured per valuation is slow. Direct simulation spends every valuation on an independent draw.

Amdahl’s law: how much of the gain survives

The speed-up from faster valuation only matters to the extent that valuation dominates run time. Amdahl’s law gives the overall speed-up when a fraction P of a calculation is accelerated by a factor S:

The direct method also brings overheads of its own, such as generating the whole path up to the default time, which some models and every path-dependent case (collateralised portfolios, some exotics) require. So the direct method may well be faster, depending on where the Monte Carlo engine actually spends its time; it is less naturally suited to path dependence, but it may be helpful when modelling wrong-way risk.

CVA is computed as an upfront amount, but for pricing it is often more useful as a running spread, a charge per annum that can be added to a rate the client pays or deducted from a rate the client receives. The simplest conversion divides the upfront CVA by an annuity or duration for the trade’s maturity. That shortcut misses a feedback loop sometimes called the CVA of the CVA: charging CVA raises the value of the trade to the bank, which raises the exposure and with it the total CVA. The exact answer solves recursively for the spread at which the trade, including its CVA, is worth zero. Vrins and Gregory (2011) show that the size of this effect can be bracketed by dividing by a risky annuity and by a risk-free annuity.

A second shortcut assumes the EPE stays constant over the life of the trade, which gives

CVA comes out in the same units as the credit spread, and the spread should be the one for the maturity of the trade. Average EPE is the simple average of the EPE profile when time steps are equal (a weighted average otherwise), and strictly it should use undiscounted EPE, though with low interest rates the difference is small, especially for short maturities. The approximation works reasonably well when the EPE profile, or the default probability profile, is fairly flat. It is rarely used for production numbers, but it is a useful way to think about CVA because it separates the credit component (the spread) from the market risk component (average EPE), and it has no LGD input at all.

For a small charge like this one the recursive effect is minor, but it grows for riskier counterparties and longer-dated trades. It is also systematic rather than random: a desk that charges only the first-order CVA will lose money, all else equal, the moment the trade is booked, because the extra CVA on the CVA has not been collected.

A special case: deep in-the-money trades

When a trade or portfolio is far in the money (ITM), with a large positive value, it is very unlikely to turn negative, so EPE is close to the expected future value (EFV). CVA can then be computed by putting the EFV into the CVA formula, or by a discounting approach that allows for LGD. Plotting CVA for payer and receiver swaps against the fixed rate shows the swap results converging on this simpler answer as the trade moves deep ITM, which is the swap behaving more and more like a bond, with cash flows that are effectively one-way.

The level of the credit spread matters, but so does its term structure. Consider three counterparty credit curves, upwards-sloping, flat and inverted, each with a five-year spread of 150 bps. All three imply the same cumulative default probability at five years, yet they distribute that probability very differently over time. With a flat curve, marginal default probabilities are roughly even across years. An upwards-sloping curve pushes defaults towards later years (back-loaded), and an inverted curve concentrates them in the early years (front-loaded).

The upwards-sloping curve produces the largest CVA, mainly because extrapolating it gives the highest 10-year spread, while the inverted curve gives the smallest for the opposite reason. So two counterparties quoted at the same five-year spread can carry CVAs that differ by more than half of the smaller figure once the trade runs beyond five years.

Spread widening and CVA gamma

CVA generally becomes larger in magnitude as the counterparty’s credit spread widens, and for small changes the relationship is close to linear. It cannot grow without limit, however. As spreads become extreme, default becomes almost certain and imminent, and CVA converges to the loss on the exposure at that moment, which for a trade currently at par may be zero. The result is a hump: CVA first grows with the spread and then shrinks back for very large widenings. That curvature means a large gamma with respect to big spread moves, which matters for understanding jump-to-default risk.

LGD, or equivalently the recovery rate, enters CVA twice. Once default probabilities are implied from credit spreads and substituted into the discrete formula, the expression becomes

Both LGDs should be expected values. LGD_actual is the loss the bank actually expects if this counterparty defaults on this claim. LGD_mkt is the value used to back out market-implied default probabilities from spreads, and it should match the seniority of the instrument behind the spread, usually senior unsecured debt for most CDS contracts. The Basel Committee (BCBS 2011b) stresses that LGD_mkt in the regulatory CVA charge is a market assessment and differs from the internal LGD estimate used for the internal ratings-based default charge.

Why equal LGDs largely cancel

Derivative claims typically rank alongside the senior unsecured debt referenced by CDS, so the natural assumption is LGD_actual = LGD_mkt, and the regulatory capital formula requires it because only LGD_mkt appears there. With equal LGDs the two effects offset each other to first order: a higher LGD lowers the default probability implied by a given spread but raises the loss when default happens. The offset works because exp(−x) ≈ 1 − x for small x, so it is most accurate at small spreads. What remains is a second-order effect, and in the swap example moving LGD from 60% to 50% changes CVA by less than 2%. This is also why the average-EPE approximation needs no LGD input.

When a different actual LGD is justified

Institutions sometimes use a different LGD_actual. In rough order of how easily it can be defended, the reasons are a different seniority in the exposure waterfall (trades with securitisation special purpose vehicles, for example) or a structurally different LGD (project finance); credit enhancements such as margin; and a belief that the institution’s own work-out experience earns it higher recoveries. Seniority is the most common and most defensible reason. J.P. Morgan’s 2015 annual report describes adjusting CDS-implied recoveries for the difference between a derivatives creditor and a senior unsecured bondholder, and BCBS (2015a) allows regulatory LGDs to differ only where that seniority difference can be shown.

Project finance is the standard case for a lower LGD. These are infrastructure and industrial projects funded off-balance sheet against the project’s own projected cash flows, mostly with senior debt that is structured to have priority over all material assets. Recoveries average around 75% in project finance against about 40% for corporates, and Standard & Poor’s (2016) puts the project finance average at 77%. With LGD_mkt held at 60% and only LGD_actual varied, CVA moves roughly in proportion to LGD_actual, a far bigger effect than when both move together.

One safeguard is needed: the LGD must not also have shaped the credit curve. If a proxy spread was assigned from a rating that already rewards strong recovery, a better rating has already produced a lower spread, and cutting LGD as well counts the same benefit twice. Ratings used for spread mapping should reflect default probability only, not expected loss. Standard & Poor’s definitions are default-probability based, whereas Moody’s definitions suggest recovery forms part of the rating. Finally, the xVA desk commonly bears LGD risk on actual defaults: the difference between LGD_actual and the loss finally realised after the work-out.

Debt value adjustment (DVA) arises once the party making the calculation is allowed to default as well. Assuming it cannot fails sounds harmless and fits the going-concern principle of accounting, under which a business is presumed to continue indefinitely. International accounting standards nevertheless allow, and arguably require, an entity to reflect its own default in the value of its liabilities through an own-credit adjustment. DVA can also be seen as part of the exit price, since any counterparty taking over the position would charge its own CVA. DVA relates to the liability side through the expected negative exposure (ENE), just as CVA relates to the asset side through EPE, and using the two together is called bilateral CVA (BCVA).

The accounting push

The accounting move began in 2006, when FAS 157 required banks to reflect their own credit standing in the fair value of liabilities measured at fair value, since non-performance risk includes the reporting entity’s own credit risk. Several large US banks, and some Canadian ones, started reporting DVA, with BCVA treated as an exit price rather than a reserve. Most other banks still treated CVA as a reserve, often with historical default probabilities. IFRS 13, effective from January 2013, changed that: fair value of a liability reflects non-performance risk, which includes but may not be limited to own credit risk, and auditors have generally read it as requiring market-implied default probabilities and both CVA and DVA. Practice has largely converged, though methods are still not standardised (ESMA 2017). The arrival of funding value adjustment (FVA) has partly undercut DVA, because in pricing many banks set DVA aside and replace it with funding considerations.

Price symmetry

Two parties valuing the same portfolio with the same base method should agree, so V_A = −V_B; otherwise value would be created or destroyed. Adding CVA alone breaks this, because each side makes a negative adjustment. Adding DVA restores it:

Historically, a bank charged CVA to an end user such as a corporate, based on the client’s credit quality and the exposure, and the client could hardly argue back, since bank default looked remote and bank spreads were tiny. The global financial crisis made bank spreads material and raised two awkward questions: should banks themselves be charged CVA, and how can two banks trade with each other at all? DVA answers both by creating price symmetry, so that in principle both sides can agree on a price.

Accounting, economic and regulatory value

Whether DVA belongs in a price is another matter. Including it causes no accounting loss if DVA is also in the valuation, but its economic substance is doubtful: losses when a counterparty defaults are real, whereas a gain from one’s own default is harder to accept, and while CVA can be linked to buying CDS protection on the counterparty, hedging DVA would mean selling protection on oneself. A party that sees no economic value in DVA will be reluctant to price it, and markets do not need symmetry to function, since banks can quote prices that end users accept or decline. Regulation takes the same line by requiring DVA gains to be derecognised from equity.

The split between shareholders and bondholders reconciles these views. Suppose a firm enters an out-of-the-money (OTM) trade, receiving an upfront payment, with a large DVA. If the firm defaults, its other creditors share in that amount and enjoy a higher recovery, so for bondholders the DVA is real (assuming the trade’s creditors have no seniority over other creditors). Shareholders receive nothing in default and see no value in it. Capital rules set the required shareholder equity, so derecognising DVA is consistent; financial reporting that aims at the value of the whole firm supports showing DVA as a bondholder benefit; and pricing from a shareholder perspective would leave DVA out.

When both the party making the calculation (P) and its counterparty (C) can default, BCVA is the sum of a CVA term and a DVA term:

The CVA term resembles the unilateral integral, except that the risky discount factor now contains both default intensities, D = exp(−∫(r + λ_C + λ_P)ds), which is the probability that neither party has defaulted. This is the first-to-default effect: a default by one party only counts if the other has not defaulted first. DVA is the mirror image, built from ENE, the party’s own default intensity and its own LGD. Because ENE is negative, DVA is positive and offsets CVA as a benefit, matching the idea that a party which defaults while owing money makes a gain equal to the loss its surviving counterparty suffers. Since one party’s ENE equals the other’s EPE with opposite sign, one party’s DVA should in theory equal the other party’s CVA with opposite sign, provided both compute them the same way.

On a time grid, the integrals become

The bracketed survival terms are what separate these bilateral CVA and DVA terms from unilateral CVA (UCVA) and unilateral DVA (UDVA), which are the same sums without them. So bilateral CVA is never larger in magnitude than UCVA, and bilateral DVA is never larger than UDVA.

The ITM and OTM trades use a fixed rate of 1% instead of the market rate of 1.49%. Some sums differ from their components by 0.1 because the components are rounded.

The survival-adjusted (contingent) CVA and DVA are each smaller than their unilateral counterparts, but because they have opposite signs the net effect on BCVA can go either way. The striking feature is the asymmetry between paying and receiving fixed, caused by the skew in the exposure profiles: the receive-fixed swap has a large ENE and so a large DVA, leaving its BCVA close to zero. Off-market trades exaggerate this, and the OTM receive-fixed swap even has a positive BCVA of 9.4. Whether a bank would really charge so little for the receive-fixed swap, or pay through mid to enter the OTM one, is doubtful.

BCVA as a running spread

Extending the average-EPE approximation gives an intuitive spread version of BCVA:

When average EPE and average ENE are mirror images, which is often roughly true for collateralised relationships, a party simply charges its counterparty the difference in their credit spreads, and pays the counterparty if the difference is negative. Weaker credits would pay stronger ones to trade, which in theory allows agreement even when both have poor credit quality, although the economic doubts about DVA still apply.

The bilateral formulas simplify three linked issues. The first is the survival adjustment. UCVA and UDVA ignore whether the other party survives, whereas the bilateral CVA and DVA terms include it. The first-to-default logic, that contracts end when the first party defaults so a second default is irrelevant, seems to justify including survival probabilities, but doing so makes CVA sensitive to the bank’s own credit quality, which looks unnatural and which regulation may forbid. The second is default dependency: the formulas assume the two parties default independently, and a strong dependency would change both terms. The third is close-out. EPE and ENE are normally computed from base valuations, assuming trades settle at those values at default, whereas real close-out may use actual values; the 2002 ISDA documentation lets the claim take into account the creditworthiness of the determining party, which suggests CVA and DVA could themselves feed into the close-out amount.

Gregory (2009a) quantified how survival probabilities and default correlation alter BCVA, leaving close-out aside, and Brigo and Morini (2010) studied close-out for one-sided exposure. Modelling close-out properly is hard because each future default requires the future BCVA, which makes the problem recursive. For a loan, Brigo and Morini found that a risky close-out, in which DVA forms part of the close-out value, cancels the survival probability of the calculating party, so the CVA formula without that survival term is correct in the one-sided case. Gregory and German (2013) found no simple result for the two-sided case but concluded that the bilateral formulas without survival probabilities are probably the best approximation short of a far more sophisticated model. Practice remains mixed: in a 2012 Ernst & Young survey, six of 19 respondents reported contingent (survival-adjusted) CVA and DVA and seven reported non-contingent figures, the rest not reporting DVA at the time.

Can DVA actually be monetised?

DVA on derivatives is part of a wider question about own-credit in liability valuation. FASB issued SFAS 157 in 2006 (effective 2007), the IASB amended IAS 39 in 2005, and IFRS 13 reinforced the position from 2013. After the crisis, volatile bank spreads produced large accounting gains as banks’ own credit worsened, and equity analysts came to strip DVA out as an accounting effect of no interest to shareholders. The main criticism is that DVA is hard to realise; Kenyon (2010) adds that if DVA is recognised, goodwill, which is worthless at default, should also depend on own credit quality, so goodwill losses would offset DVA gains as spreads widen. The proposed routes to monetisation are weak:

Defaulting realises DVA, but it is about as sensible a strategy as cashing in one’s own life insurance. Unwinds and novations can crystallise it: MBIA paid Morgan Stanley $1.1bn in 2011 against an exposure of several billion, the difference being in effect a DVA gain. But banks only unwound because the monolines were near default, and a replacement trade would attract a new CVA charge cancelling the benefit. The close-out process after a counterparty default is another route, and claims including DVA were common in the Lehman Brothers bankruptcy, though courts did not always accept them. Hedging is the most obvious route, and the weakest. CVA can be hedged by shorting the counterparty’s credit, buying CDS protection or shorting its bonds. Hedging DVA would require going long one’s own credit, either by buying back one’s own bonds, which needs funding, or by selling CDS protection on oneself, which is impossible. Selling protection on correlated names instead is inefficient and dangerous; some banks sold protection on Lehman Brothers before its default as a supposed DVA hedge.

Unsurprisingly, the Basel Committee (BCBS 2011d) derecognised DVA from the CVA capital charge, so that a riskier bank could not lower its capital through DVA. FASB has required DVA gains and losses to go to other comprehensive income. In pricing, practice is divided: many banks pass on only part of the DVA benefit, and Totem consensus submissions appear to include FVA but no DVA. The debate has largely been settled by treating DVA as a funding benefit: buying back one’s own debt presumes that debt was issued, which ties DVA to funding and to FVA.

Netting and margin reduce CVA, but their effect can only be measured on the whole netting set, the group of trades with one counterparty covered by a single netting agreement. Pricing and valuation still need numbers per trade, so CVA has to be allocated. Because netting can only lower exposure, the CVA of a netting set is never more negative than the sum of the standalone CVAs of its trades:

The netting benefit, CVA becoming less negative, is a portfolio effect: strong for balanced, offsetting books and weak for directional ones.

The incremental CVA formula

The most natural allocation charges each change in the portfolio, most often a new trade, with the change it causes in netting set CVA:

A trade’s charge is then its contribution to total CVA at the moment it is executed, which makes this the right number to charge to salespeople, traders, business lines and ultimately clients. The result depends on the order in which trades arrive but is not revised by later trades, and a desk charging it exactly offsets the immediate change in its total CVA. The same calculation applies to unwinds, restructurings and changes in contractual terms such as a new margin agreement, all occasions on which a counterparty may need to be charged or refunded. If the change does not alter the counterparty’s PD or LGD, the result is simply the usual formula with incremental EPE in place of standalone EPE:

The PD assumption can be argued over (a counterparty that agrees to post more margin might be more likely to default), although any such effect should show in its spread. Incremental EPE needs exposure at the netting set level, where the netting set value on a simulation path s at time t is the sum of the trade values:

For a new trade this means comparing the set with and without it; a complex restructuring needs a full recalculation. Incremental EPE can be negative when the trade reduces risk, giving a positive incremental CVA, a benefit rather than a cost, as when unwinding a trade. And incremental CVA with netting is never more negative than standalone CVA. In practice this lets a bank that already trades with a client offer better terms on a new trade, as Cooper and Mello (1991) showed for forward contracts, and clients in jurisdictions where netting enforceability is doubtful may trade only with banks that treat netting as enforceable. Accurate netting usually requires a full Monte Carlo exposure simulation, run close to real time for pricing.

A directional existing book barely reduces the new trade’s CVA, while an offsetting one roughly halves it. DVA actually changes sign in the offsetting case, which deserves the label offsetting, whereas CVA shrinks without changing sign, which is better called neutral. BCVA moves much less than either component because the changes partly cancel; with equal spreads for both parties BCVA would have no incremental effect at all, much like a comparable result for FVA.

The netting benefit also depends on the size of the new trade. A small trade gets most of the benefit, and its incremental CVA, quoted as a running spread, approaches a lower limit of -6.2 bps; as the trade grows relative to the existing book, the benefit fades and incremental CVA approaches the standalone value (-17.9 bps).

Marginal CVA replaces EPE in the CVA formula with marginal EPE, each trade’s share of the netting set EPE. Its key property is additivity: the marginal CVAs of all trades in a netting set add up to its total CVA, whatever the order in which they were dealt. It is not suited to pricing new trades one at a time, because every new trade changes the marginal CVA of every existing one, which would imply revaluing trades already done. It does fit three situations: pricing several trades executed together, perhaps as one deal; a policy of computing CVA charges only periodically, after several trades have accumulated with a counterparty; and allocating CVA to trades at a point in time for accounting, which also highlights trades worth restructuring, novating or unwinding.

The IRS is the swap from the incremental example. In the IRS-first row the components add to −37.0 rather than −36.9 because of rounding.

Whichever trade comes first under incremental allocation carries its full standalone CVA, and the second gets the whole netting benefit, so the amount charged depends heavily on timing. The marginal split is the fairer one. Timing dependence could invite gaming, but in practice it rarely causes trouble for two reasons: a client is usually covered by one trader or salesperson who sees the total charge across the client’s portfolio, even if particular trades look favourable at a given moment, and most clients trade fairly directionally, for example to hedge, so netting effects are modest anyway.

Margin affects CVA only through exposure. The CVA formula is unchanged; the EPE fed into it is simply computed under the margin agreement. Strong collateralisation might suggest CVA becomes negligible, yet it has become common to model the effect of variation margin even with zero thresholds, because exposure remains over the margin period of risk (MPoR), the time between the last successful margin call and the close-out of a defaulted counterparty. The 10-day minimum MPoR is strictly a regulatory capital rule, but it has become standard in other CVA calculations, including accounting CVA. The reduction is real but smaller than intuition suggests, and because an institution may have many collateralised counterparties with large portfolios, their combined CVA can be material. Charging it is harder: end users posting margin may pay some CVA, but interbank counterparties cannot be charged, so there is less incentive to price collateralised CVA.

Collateralised swaps in numbers

With a classical MPoR model of 10 business days, the collateralised EPE of a pay-fixed swap is a thin, nearly flat profile compared with the large hump of the uncollateralised EPE, and because the MPoR assumptions are symmetric the receive-fixed swap is its mirror image.

Collateralised values are much more symmetric between paying and receiving, although more detailed modelling of cash flows within the MPoR could change that. A simple rule of thumb puts the reduction in CVA (through EPE) at about 8.4 times:

The pay-fixed CVA (and the receive-fixed DVA) fall by more than this because the positive skew of the uncollateralised exposure inflates the starting point, while the receive-fixed CVA falls by less because of the negative skew. Small collateralised BCVA is some comfort, since such costs are hard to charge to interbank counterparties. DVA for a collateralised counterparty is even more questionable than usual: it rests on the assumption that the bank stops posting margin in the 10 days before its own default, and any resulting benefit accrues only to bondholders, whose recovery is boosted by the margin not returned.

Initial margin and thresholds

Initial margin drives exposure, and with it CVA and DVA, towards zero. A threshold works like negative initial margin and the reverse is also true: as the threshold rises, CVA tends to its uncollateralised value, and as initial margin rises, CVA tends to zero. Modelling dynamic initial margin is complex, and given how small CVA should be once initial margin is held, the effort may not be warranted; for accounting, a party might argue CVA is zero where initial margin coverage is high, or ignore the initial margin benefit altogether and accept a material CVA in exchange for simplicity. Initial margin a party posts does not appear in these calculations if it is segregated, but it carries a cost captured by margin value adjustment (MVA).

CVA to central counterparties

Central counterparties (CCPs) are not risk-free, as a recent Nasdaq clearing member default showed: clearing members can lose their default fund contributions even if the CCP stays solvent, and default fund capital charges are relatively high for that reason. CVA to CCPs is not yet standard, but it is likely to become routine. Every input is hard. The default probability is that of one or more defaults among the other clearing members, which requires many credit curves and their dependence, and the LGD used to imply it will not cancel as it does elsewhere. The total exposure is the loss beyond the defaulters’ own resources, which depends on unknown individual margin and default fund amounts and on how well portfolios can be auctioned in a crisis. The allocation of that loss to a given member may be pro rata, but auction incentive pools or variation margin gains haircutting can make it uneven. Recovery is also very uncertain.

There is a conceptual objection as well: CVA is an expected loss, meaningful when exposures are diversified or hedgeable, and a member’s exposure to one of a handful of CCPs is neither. As a practical method, Arnsdorf (2019) proposes basing the calculation on the member’s own posted initial margin, valid if default fund contributions track initial margin and losses are shared homogeneously (both roughly true), with other members’ spreads for default probabilities and extreme value theory for the loss beyond the defaulter-pays resources.

Wrong-way risk describes an unfavourable dependence between exposure and the counterparty’s credit quality: exposure tends to be high exactly when the counterparty is more likely to default. It obviously raises CVA (and affects DVA), and similar dependencies through collateral or funding can reach other xVA terms. Right-way risk is the favourable case, where exposure tends to be low when default is likely, and it reduces CVA. Ignoring WWR is often reasonable, but when it does bite the effect can be dramatic, and it is hard to spot, model and hedge because it stems from subtle macroeconomic and structural links. Dealers lost heavily in the Asian crisis of 1997/1998, when defaults coincided with collapsing local currencies, and from 2007 on protection bought from monoline insurers.

WWR is often a natural feature of markets. A mortgage lender in a recession faces falling house prices at the same time as rising defaults. In derivatives, classic cases appear in every asset class:

A put option on a stock or index is wrong-way when the underlying’s fortunes track the counterparty’s; buying a put on one bank’s shares from another bank is the textbook example, and equity call options are correspondingly right-way. Foreign exchange (FX) forwards and cross-currency swaps carry WWR when the currency the bank pays could weaken just as the counterparty weakens, most obviously when paying a sovereign’s own currency to that sovereign, or to a bank in the same region. A cross-currency swap is like a loan secured by the other currency, so a currency collapse destroys the value of the security. The link can run both ways: a weak currency may signal a weak economy and a struggling counterparty, while the default of a sovereign, bank or large corporate can itself trigger devaluation. With interest rate products, a corporate paying fixed in a strong economy is wrong-way for the bank receiving fixed, because rates tend to be cut in a recession; paying fixed into a recovery, when rates rise, can be right-way. A commodity producer, such as a mining company, hedging its output price creates right-way risk, since it owes money only when prices are high and its business is doing well, a feature of hedging rather than speculation. Finally, buying CDS protection creates exposure when the reference entity’s spread widens, so a strong link between reference entity and protection seller means extreme WWR, a bank selling protection on its own sovereign being the clearest case, while selling protection in the same situation is right-way.

Evidence

Duffee (1998) found corporate defaults clustering in periods of falling interest rates, which points to recessions bringing both rate cuts and high default rates. Banks have seen the same on uncollateralised receiver swaps that moved in the money as their counterparties weakened, a cross-gamma effect between spreads and rates that operates even without defaults. For FX, Levy and Levin (1999) measured average residual currency values after sovereign defaults ranging from 17% for triple-A to 62% for triple-C issuers, indicating how far an exchange rate can jump at the moment of default.

General and specific wrong-way risk

Regulators distinguish general WWR, driven by macroeconomic relationships, from specific WWR, driven by a causal link between the exposure or collateral and the counterparty’s default.

Specific WWR is especially hard because it may not show in historical or market data at all. In the European sovereign debt crisis of 2010, many sovereigns weakened together with the euro, yet past data gave no hint of the relationship because neither those sovereigns nor the currency had ever been under serious credit stress.

The most direct way to bring WWR into CVA is to replace unconditional EPE with exposure conditional on default occurring at that date:

This also allows a heuristic treatment, judging how far conditional exposure might exceed unconditional exposure. Regulatory capital does something similar: a conservative alpha multiplier plus stressed exposure data partly capitalise general WWR.

A one-parameter illustration

A simple model links EPE and default with a single correlation. The parameter is abstract and lacks economic intuition, but it makes WWR easy to see. With a correlation of 50%, conditional EPE roughly doubles; with −50%, right-way risk cuts it by at least half. More surprisingly, under WWR the exposure is larger for a better-rated counterparty, because for a strong credit default is a rarer event and therefore signals a bigger shock when it happens. The general lesson is that WWR increases with the credit quality of the counterparty.

Richer models run into three problems. Historical data may be uninformative, because WWR can be subtle and absent from any time series of correlations. The relationship may be misspecified: dependence may be causal rather than a correlation, and zero measured correlation does not prove independence (if X is normal and Y = X², the two are uncorrelated but plainly dependent). And the direction may be unclear: low rates usually accompany recessions with wide spreads, yet a credit crisis with high rates is not impossible.

Hazard rate (intensity) approach

Here the counterparty’s credit spread, or default intensity, follows its own stochastic process that is correlated with the processes driving exposure. Defaults are generated from the intensity, and conditional EPE is computed in the usual direct or path-wise way, but only on paths with a default. It is tractable, since spread paths come first and exposure is simulated only where default occurs, and the correlations can be estimated from historical data. For a swap paying fixed, with lognormal spreads correlated to rates, positive correlation means rates tend to be high on default paths, raising positive exposure (WWR), while negative correlation enlarges the negative exposure (right-way). Empirically the link between rate changes and default rates is generally negative. The catch is that even correlations near ±90% produce only weak dependence between exposure and default, so this approach may understate WWR.

Structural approach

An even simpler route specifies dependence directly between the default time and the exposure distribution. The two marginal distributions are mapped onto a bivariate distribution, for example with a Gaussian copula, so that with positive dependence an early default is paired with a high exposure. Pre-computed, unconditional exposures are simply resampled, which bolts WWR onto an existing system, but that is also its weakness, since the unconditional distribution may not contain the information that defines the WWR. Compared at the same 50% correlation, the structural model gives a much larger EPE uplift than the intensity model, and across correlations both produce larger CVA as correlation rises, the structural effect being far stronger. Its correlation parameter is opaque and hard to calibrate; richer versions correlate default more directly with the variables behind exposure, which makes estimation more achievable.

Parametric approach

Hull and White (2011) link the default probability to exposure through a simple functional relationship, calibrated either intuitively from a what-if scenario or from history by revaluing the current portfolio on past dates and relating its value to the counterparty’s spread. If high portfolio values have coincided with wider than average spreads, WWR is present. This requires the current portfolio to resemble the historical one and the history to show a meaningful relationship.

Jump approaches

Jumps suit specific WWR, especially in FX, where the CDS market reveals the effect. CDS on the same sovereign quoted in two currencies trigger on identical credit events and differ only in the currency of the payout, so their price gap, the quanto basis, measures the expected currency move on default. Ehlers and Schönbucher (2006) showed that intensity models cannot reproduce such data even at ±100% correlation. Levy and Levin (1999) instead let the FX rate jump at the default time, finding implied jumps larger for better-rated sovereigns, 83% for AAA and 27% for BBB, because their default reflects or causes a more severe shock. Jumps implied from CDS for the euro against the US dollar during the European sovereign crisis were 9%, 17%, 20% and 25% for Greece, Italy, Spain and Germany respectively, again largest for the strongest and most systemic credit. Chung and Gregory (2019), using Japanese CDS, found average implied jumps of 13.5% for financial institutions, 8% for non-financial corporations and 38.4% for the sovereign.

Euro-denominated protection is about 30% cheaper at most maturities (the two-year point is nearer 22%), because a euro payout would be received in a currency expected to fall if Italy defaulted. Dollar and yen quotes on Japan show a similar persistent basis, so the CDS market both reveals currency WWR and offers a hedge for it.

Credit derivatives are a special case because WWR is built in: buying protection on one entity from another always links the payout to two credits, and the link can be specific, as when buying protection on a sovereign from a bank based there. Because CDS risks are contagious and systemic, margin may do little to help.

Consider the fair premium for buying protection from a risky seller as a function of the correlation between the reference entity and the seller. The reference entity’s CDS spread is 250 bps, the counterparty’s is 500 bps, and both LGDs are 60%. A buyer should pay less than the risk-free premium, and the reduction is the CVA. At 60% correlation the fair premium is only about 200 bps against 250 bps with a risk-free seller. At 100% correlation the seller defaults whenever the reference entity does, so the protection can never pay out and the premium falls to just above 100 bps, the value of recovery alone. Selling protection is the opposite story: the seller needs a higher premium, but the CVA effect is much smaller and shrinks as correlation rises, because at high correlation the protection is almost certainly a liability for the buyer when the buyer defaults, and that amount must still be paid.

Does collateral protect against wrong-way risk?

Because WWR can make exposure surge, margin’s ability to keep up matters, and the answer depends on timing. If exposure builds gradually before default, margin can be called and received; if it jumps at default, margin is useless. Chung and Gregory (2019) show margin has very little effect on an FX portfolio under a jump model, whereas continuous models such as the intensity and structural approaches suggest margin works well. Reality lies in between and depends on the trade and the counterparty. Pykhtin and Sokol (2013) argue that the benefit of margin under WWR must allow for jumps and for higher volatility during the MPoR, that WWR is stronger for systemic parties such as banks, and that WWR reduces the value of collateralisation overall. Ironically, the parties that post margin, banks, are the systemic ones most prone to severe WWR, while non-systemic corporates often post none.

Wrong-way collateral

WWR can also sit between the collateral and the exposure. A payer swap collateralised with a high-quality government bond is general wrong-way collateral: rising rates increase the swap’s value while lowering the bond’s, though with low rate volatility the problem is usually minor. A receiver swap with the same collateral is right-way. A cross-currency swap collateralised with cash in one of its two currencies is a more serious example: if margin is held in the currency the bank pays, an FX move raises exposure and lowers collateral value at once. Specific wrong-way collateral involves a direct link to the poster’s credit: a counterparty posting its own bonds gives almost no protection against its own default, though it may help with FVA, and a bank posting its own sovereign’s bonds is similarly problematic.

Wrong-way risk at central counterparties

CCPs rely on variation and initial margin as their main defence, which makes them especially prone to WWR, above all when clearing products like CDS. Their aim is a defaulter-pays outcome in which a defaulting member’s losses are covered by that member’s own resources; if those prove insufficient, the CCP passes losses to surviving members and could itself become insolvent. CCPs tend to separate credit quality from exposure: members must meet a credit standard set by the CCP, but margin and default fund contributions are driven mainly by the market risk of their portfolios, so WWR can be implicitly ignored. For CDS, WWR must somehow be built into initial margin and default fund sizing, which is difficult. Since WWR increases with credit quality, and Pykhtin and Sokol (2012) argue a large dealer brings more WWR than a smaller or weaker one, the uncomfortable implication is that better-rated, more systemic members should post more, because although they default less often, their default would do more damage.

Several authors (Murphy 2013, Pirrong 2013, Gregory 2014) have noted that a CCP waterfall behaves like a collateralised debt obligation (CDO). The terms of this hypothetical CDO keep changing, but the second-loss position should be hit only in an unlikely event, or initial margin would be too thin. That makes members’ exposure economically similar to a senior tranche, and senior tranches are known to be concentrated in systemic risk. CCPs also face WWR on the collateral they accept. Pressure to take a wide range of securities as initial margin adds risk and leans heavily on haircuts that may be underestimated, and it invites adverse selection, since members and clients will post the collateral that is riskiest relative to its haircut and possibly most wrong-way for the CCP, such as a European bank posting European sovereign debt.