CR 19 – Future Value and Exposure

Counterparty risk exists because a default affects the two parties to a derivative very differently, and the difference depends entirely on which way the contract value points at the moment of default. When a counterparty fails, the surviving party is allowed to terminate the affected contracts, stop making further contractual payments, work out the single net amount that is now owed between the two firms, and then adjust that figure for any margin that has changed hands. What remains is either a number in the survivor’s favour or a number against it, and the two cases lead to very different outcomes.

When the net value is negative

Suppose the net amount, after margin, is against the surviving party. The survivor then owes money to the defaulted firm, and the default does not cancel that debt. The administrators of the defaulted counterparty will expect payment, and apart from a few special legal cases the survivor cannot simply walk away. From a valuation point of view, very little has changed: the survivor owed the amount before the default and still owes it afterwards. So, a party with a negative position neither gains nor loses because its counterparty failed.

When the net value is positive

Now suppose the net amount is in the survivor’s favour. The defaulted firm can no longer honour its future obligations, so the survivor is left with a claim for the positive value, usually ranking as an unsecured creditor in the insolvency. It will eventually recover some portion of that claim, in the same way that bondholders recover part of the face value of a defaulted bond. That eventual recovery is uncertain, and by convention it is kept outside the definition of exposure, which measures the amount at risk before any recovery.

Putting the two cases together gives the core asymmetry of counterparty risk: the survivor loses when the value is positive and gains nothing when it is negative. This one-sided payoff is what drives credit value adjustment (CVA), and it is captured by a simple formula.

Two practical points follow from the formula. First, exposure is a function of value, so estimating exposure at any future date requires an estimate of what the portfolio will be worth on that date. Second, the value that goes into the formula must already reflect the netting rules that apply across transactions and any margin held, because both change the amount that would actually be lost.

Why negative exposure also matters

Counterparty risk runs in both directions. Either firm can default, and either can suffer the loss. Seen from a party’s own side, its own default would harm any counterparty to which it owes money. The mirror image of positive exposure measures this, and it is called negative exposure.

A negative exposure translates into a gain for the defaulting party, since its counterparty bears the corresponding loss. This is often called own counterparty risk, and it is priced through the debt value adjustment (DVA). Some readers find it uncomfortable to describe a firm’s own default as a source of gain, but the logic is purely symmetric: one side’s loss is the other side’s gain. Throughout this lesson, the word exposure on its own means positive exposure, and negative exposure is always named explicitly.

The word value in Equations 19.1 and 19.2 hides a real difficulty. It refers to the worth of every transaction in the portfolio at each date on which the counterparty, or the party itself, might default, after allowing for netting and margin. There are two candidate definitions. The base value is a clean model price, the kind produced by a standard valuation system with no adjustments for the identity of the parties. The actual value is the amount that is finally agreed with the administrators of the defaulted firm once the close-out process has run its course.

The base value is far easier to work with, but it is not what derivative documentation usually refers to. Ideally, a firm would want the amount agreed bilaterally after a default to match its own unilateral valuation from just before the default, because any mismatch creates a jump in value at the moment of default. In practice the actual value does not line up neatly with a single objective number, since it is shaped by valuation adjustments and by the mechanics of close-out.

Why the actual value creates a recursive problem

The close-out amount defined in International Swaps and Derivatives Association (ISDA) documentation may take account of the creditworthiness of the surviving party. In practical terms, a surviving bank may be able to reduce what it owes to a defaulted counterparty, or increase its claim against it, to reflect the CVA and similar charges it would incur when replacing the trades with a new counterparty. Those replacement charges themselves depend on exposure. So, to know today’s exposure one would need to know future xVA, and to compute future xVA one needs exposure: the definition feeds back on itself.

Market practice cuts through this loop by building exposure on the base value. It is straightforward to define and to model, and the errors introduced by ignoring documentation detail, jurisdiction and market conditions at the time of default are generally believed to be modest. The theoretical value used in exposure models cannot capture those aspects, so the working assumption is that they add only small uncertainty.

Delay in agreeing the close-out

Even once the definition is settled, there is a timing problem. Until the surviving firm and the administrators agree on the close-out amount, the survivor does not know precisely what it owes or what its unsecured claim is worth. Where a default involves a very large number of contracts, as with the over-the-counter derivative book in the Lehman bankruptcy, the operational volume alone can stretch this process out for a long time. In a bank, the risk carried during this period often sits with the xVA desk.

A valuation of today’s trades and today’s margin gives the current exposure, subject only to the close-out uncertainty described above. Today’s figure, and every figure in the past, is known. What matters far more for counterparty risk is the exposure that might exist at some later date when a default could actually occur. That future quantity cannot be known in advance: it depends on how market variables move, on contractual events in the trades, on the netting rules and on the way margin is called and posted. So, describing future exposure means describing two things at once, its likely level and the uncertainty around that level.

A useful picture is a fan of possible value paths that starts from a single known point today and spreads out as the horizon lengthens. At any future date the paths form a distribution of values. The part of that distribution above zero is positive exposure, and the part below zero is negative exposure. The same picture applies whether the portfolio is a single trade or thousands of trades netted together under margin terms.

Exposure as a short option position

Because exposure is max(value, 0), it has the shape of an option payoff. From the surviving party’s side, the defaulting counterparty effectively holds an option not to pay, and the survivor is short that option. The word short reflects the fact that exposure is a loss to the survivor. Two consequences follow directly from option pricing intuition. Volatility of the portfolio value, including the value of margin, becomes a central input, just as volatility drives an option premium. And because options are harder to price than their underlying instruments, quantifying exposure can be demanding even when the underlying trade is plain.

The analogy extends further. A portfolio of trades with one counterparty behaves like a basket option on all of them together, and a margin agreement shifts the effective strike. By the same symmetry, a firm is long an option on its own default, and that is the intuition behind DVA.

Where the analogy stops

Treating xVA as one enormous exotic option is correct in principle but can mislead in practice. The payoff itself, the exposure, cannot be written down exactly, because it depends on netting, margin and close-out terms that are partly subjective. xVA also contains other inputs, such as credit curves, funding curves and wrong-way risk, which are not option-pricing questions at all. An options-pricing problem sits at the centre of exposure calculation, but the sheer complexity of the embedded options means they cannot be handled with the precision normally applied to options, and a purely theoretical approach tends to neglect important qualitative judgements.

Value-at-risk (VaR) has been the standard way to describe market risk for roughly two decades, and the same style of thinking now underpins many initial margin models. VaR and exposure both ask what the distribution of a portfolio’s value looks like at some future date, so at first sight credit exposure is just a market risk measure. The similarity is real, but exposure carries three extra layers of difficulty.

Time horizon

VaR looks a short distance ahead. Traditional market risk capital models use a 10-day horizon, which is close to the horizon used for bilateral initial margin, while central counterparty initial margin for over-the-counter derivatives commonly uses five days. The Fundamental Review of the Trading Book extends some horizons to as long as 120 days depending on asset class. Exposure, by contrast, must be estimated at many dates stretching out to the final maturity of the longest trade, often many years away. That longer view brings two complications that a 10-day model can ignore.

• Ageing of the portfolio. Over years, trades pay cash flows, mature, hit termination triggers, have options exercised and generate margin calls. Each of these changes the portfolio, and some create path dependency, where the exposure on one date depends on what happened on an earlier date.
• Drift. Over a short horizon the expected change in a market variable is tiny compared with its volatility, so VaR models routinely set the trend to zero. Over many years, the trend (or drift) of market variables matters alongside their volatility and their joint dependence.

Risk mitigants

Exposure is normally reduced by netting and margin, and an exposure model must reproduce their effect at every future date. Applying netting correctly can require knowing the legal agreements and how courts in the relevant jurisdiction would interpret them. Future margin is harder still, especially dynamic initial margin, because the type of collateral and the precise timing of its receipt are uncertain. Termination clauses and other contractual features add further judgement, and every one of these elements has to be modelled.

Application

Market risk and initial margin models serve risk management alone. Exposure serves two masters: risk management and the pricing of xVA. These two uses can call for different probability assumptions, so a firm may end up running two sets of exposure calculations, a real-world set for risk limits and a risk-neutral set for pricing.

Several summary numbers are drawn from the distribution of future value, and each suits a different purpose. The definitions below apply to any group of trades. In practice the relevant group, or aggregation level, is usually all the trades covered by one legal netting agreement with a counterparty, known as a netting set, and the word portfolio is used here for that group together with its margin. Naming conventions vary, and regulatory texts in particular sometimes use the same label for a different quantity, so the alternative names are given alongside each metric.

Expected future value (EFV)

EFV is the mean of the portfolio value at a future date, taking both positive and negative outcomes at face value. It is also called the expected mark-to-market (EMTM), and some texts use the label expected exposure for it. At time zero EFV is simply today’s value. At later dates it can move well away from today’s value for three related reasons.

• Cash flow differential. Some trades have lopsided payment schedules. Early in an interest rate swap on an upward-sloping curve, the fixed payments usually exceed the floating ones; in a cross-currency swap the two legs can differ by several percent a year because the interest rates differ. A party that pays more now can expect the trade to be worth more later, and vice versa. In a portfolio, a trade that matures with a large final payment changes the value of everything left behind.
• Forward rates. Forward rates can sit far from spot rates, which implies a trend in the underlying variables. That trend alone pushes future value up or down before any volatility is considered. This is really the same effect as the cash flow differential, viewed through forward curves.
• Asymmetric margin terms. Under a one-way margin agreement, the future value tends to be higher where the terms are unfavourable to the party and lower where they are favourable.

Because exposure horizons are long, EFV can be a significant component of exposure, whereas in short-horizon market risk it hardly matters.

Potential future exposure (PFE)

PFE answers the risk manager’s question of how bad the exposure could become. It is a high quantile of the value distribution at a chosen confidence level. A 99% PFE is a level that the exposure exceeds with a probability of no more than 1%. In this sense PFE plays the same role for credit exposure that VaR plays for market risk. A PFE can also be read at a low confidence level, on the left side of the distribution, which gives the corresponding worst case from the counterparty’s point of view.

Expected positive exposure (EPE) and expected negative exposure (ENE)

EPE is the average of the positive exposure from Equation 19.1, and ENE is the average of the negative exposure from Equation 19.2. The averaging runs over every outcome, not just the favourable ones: when computing EPE, each negative outcome contributes zero but still counts in the probability weighting. EPE is also called expected exposure (EE), and ENE is also called negative expected exposure (NEE). ENE is the positive exposure as seen by the counterparty. Like PFE, both depend heavily on the spread of the distribution, so higher volatility raises them. Pricing of some xVA terms needs both, EPE for CVA and ENE for DVA.

The four metrics are easiest to understand on a tiny discrete distribution, where every average can be checked by hand, and then on a continuous one, where the effect of volatility becomes visible.

The identity in the final step is worth remembering. Since max(value, 0) + min(value, 0) = value for every outcome, EPE + ENE = EFV at every date, which gives a quick check on any table of results.

The same metrics for a normal distribution

When future value is assumed to follow a normal distribution, closed-form expressions give EPE and ENE directly, and PFE is the mean plus a multiple of the standard deviation. At 99% confidence that multiple is the one-tailed 99% point of the standard normal distribution, 2.326 to three decimals (2.32635 more precisely).

The second example shows why EPE, ENE and PFE all depend on volatility while EFV depends only on the centre of the distribution. It also shows that the out-of-the-money side of the distribution is far more volatile in relative terms than the in-the-money side. Source tables sometimes print the ENE values here as magnitudes (0.17 and 0.79); with the sign convention of Equation 19.2 they are negative.

The metrics in the previous sections describe one horizon. Repeating the calculation at a grid of future dates produces exposure profiles: an EPE profile, a PFE profile, and so on. Two summary numbers then compress a whole profile into a single figure.

Average EPE

Average EPE is the time-weighted mean of the EPE profile over the life of the portfolio. Each EPE point is weighted by the length of the time interval it represents, so when the dates are evenly spaced the weighted mean reduces to a simple average of the EPE values. Some texts call this quantity simply EPE, which can be confused with the single-date EPE defined earlier, and the definitions used in the Basel framework differ again, so it pays to check which convention a document is using.

The single average EPE number is often described as a loan equivalent, because it represents the average amount that has effectively been lent to the counterparty. Expressing an uncertain, time-varying exposure as one loan-like figure is clearly a rough approximation, since it averages away both the randomness of the market and the shape of the profile through time. Even so, average EPE turns out to have a strong theoretical basis for regulatory capital and for xVA calculations, which is why it is widely used.

Maximum PFE

The maximum PFE is the peak of the PFE profile across all future dates. It answers the question of the worst exposure, at the chosen confidence level, that the counterparty relationship could produce at any point in its life, and it is the natural number to compare with a credit limit.

The shape of an exposure profile is set by a handful of drivers: maturity, the frequency of payments, option exercise, the roll-off of cash flows over time, and default events in the underlying. In the illustrations that follow, exposure is expressed as a percentage of a unit notional, the current value is assumed to be zero unless stated otherwise, the profile shown is EPE, the current time is t and the maturity is T. PFE profiles usually have the same shape as EPE profiles, only scaled up.

Loans and bonds as a reference point

Debt instruments are not usually discussed as counterparty risk, but their exposure is a helpful benchmark. A loan or bond exposes the lender to an amount that is close to deterministic and roughly equal to the notional. A fixed-rate bond carries a little extra uncertainty, since its value, and so the exposure, rises when interest rates fall and falls when they rise. Loans are mostly floating rate, and their exposure can shrink over time if the borrower prepays. Derivatives are very different: their exposure can be large or small, can switch sign, and is shaped by margin and other mitigants.

Future uncertainty and the square root of time

The simplest derivative profile comes from a contract with a single exchange at maturity, such as a forward. Nothing is paid before the end, so the only thing that changes over time is the uncertainty about the final payment, and that uncertainty keeps growing as the horizon lengthens. If returns on the underlying market variable are independent and identically distributed, the standard deviation of value grows with the square root of the horizon, so exposure does too.

A foreign exchange (FX) forward is the typical example. With a volatility of 15% and a normal distribution of value, the EPE climbs steeply at first and then more gently, approaching roughly 19% of notional by year 10. One consequence of the formula is that the maturity of the forward plays no part in the exposure before maturity: a five-year forward and a ten-year forward have the same exposure at year 3, and the only role of maturity is that exposure drops to zero once the contract has settled. Vanilla options with the premium paid upfront follow much the same rising shape, while exotic options can behave quite differently.

Many over-the-counter derivatives exchange cash flows periodically over their life, and each payment removes some of the value that could otherwise be lost. An interest rate swap is the standard case. Two forces act against each other. Uncertainty about future rates grows with time, as for a forward, which pushes exposure up. At the same time, every coupon date retires a fixed payment and a floating payment, so the amount of remaining business keeps shrinking, which pulls exposure down. The result is a hump-shaped profile.

Early on the square root term dominates and the curve rises. Later the remaining maturity term dominates and drags exposure to zero at the final payment date. With a 1% interest rate volatility and a normal distribution, a 10-year swap peaks at a little under 5% of notional, while 3-year and 5-year swaps peak much lower and earlier. Longer swaps carry far more risk because they live longer and have more payments still to exchange. With semiannual payments, a 3-year swap has 6 payment dates, a 5-year swap 10, and a 10-year swap 20.

Payment frequency mismatches

The precise timing of cash flows can reshape the profile considerably. In a basis swap, or any swap where one side pays more often than it receives, the profile shifts. Consider a swap in which a party receives quarterly but pays only semiannually. Between payment dates it has collected a quarter’s flow without yet paying its own, so value moves against it and exposure is lower than for an equivalent swap with matched frequencies. The profile then shows a sawtooth pattern, with exposure dropping after each unmatched receipt. Reversing the arrangement, paying more often than receiving, raises exposure for the same reason.

Two swaps that are exact opposites of each other do not have mirror-image exposure profiles once the yield curve slopes. In a payer swap the party pays a fixed swap rate on each date and receives a floating rate that is not known until its fixing date. A receiver swap is the reverse. At inception the swap rate is set so that the risk-neutral discounted expected values of the two legs are equal, which makes it a par swap with zero value. Equal value in total does not mean equal payments on each date, though: the projected floating payments, meaning the risk-neutral expected value of each floating cash flow, follow the shape of the yield curve.

An upward-sloping curve

When the yield curve slopes upward, as it usually does, the early projected floating payments are below the fixed rate and the later ones are above it. Consider the receiver swap, which receives fixed and pays floating. In the first periods it expects to collect net payments, since it receives the higher fixed rate against a lower floating rate. In the later periods it expects to pay net amounts. Having collected early, the receiver swap is expected to be worth less in the future, so its EFV is negative. Its EPE is lower than that of a swap with no cash flow differential and its ENE is more negative. For the payer swap all three effects reverse: higher EPE, less negative ENE and a positive EFV.

This single effect can be described in three linked ways, and all three say the same thing:

• it is a cash flow differential between the fixed and floating legs;
• it reflects an upward-sloping curve, which embeds a market-implied expectation that rates will rise; and
• it means forward swap rates exceed the current swap rate, so a forward starting swap that receives today’s swap rate already has negative value.

Cross-currency swaps magnify the drift

The effect is stronger still in a cross-currency swap that pays a high-interest-rate currency and receives a low-interest-rate one. The payer of the high rate makes larger interest payments throughout, and in risk-neutral terms these are expected to be compensated by a gain when the notionals are exchanged at maturity, because the high-rate currency is expected to depreciate. That expected gain at the final exchange gives the payer of the high rate a large exposure. In the reverse position, paying the low rate, the swap is increasingly likely to have negative value over time; the negative drift keeps exposure much lower.

The drift created by cash flow differentials is one of the most important features of exposure for xVA, because it shifts the whole distribution and therefore every metric derived from it.

Every profile so far has started from a par trade with zero value. Real trades are often far from par. A trade with positive current value is in-the-money (ITM) and one with negative value is out-of-the-money (OTM), and this moneyness changes the profile substantially.

An in-the-money receiver swap

Take a receiver interest rate swap that is already well in the money. Its future value is very likely to stay positive, so the chance of slipping below zero is small and its ENE is correspondingly small. Its EPE is large and lies close to its EFV, because almost every outcome counts in full. As a result, the EPE profile becomes much more predictable. The option analogy explains why: a deep ITM option behaves almost like the underlying, with a nearly deterministic payoff and low sensitivity to volatility. An OTM trade is the opposite, with small EPE that is highly sensitive to volatility, just as the ENE in the normal distribution example responded most to a doubling of volatility.

Combination profiles: the cross-currency swap

Some products combine more than one exposure driver. A cross-currency swap exchanges interest payments in two currencies and then exchanges the notionals at the end, so it behaves like an interest rate swap combined with an FX forward. Its EPE profile therefore blends the rising square root of time shape of the forward with the hump of the swap. The FX component usually dominates, because FX volatility is high, the maturities are long and the final notional exchange is large; the interest rate component adds comparatively little.

How closely the two interest rates move with the FX rate also shapes the combined profile. In practice that correlation is often fairly low, and a low correlation tends to increase the exposure of the cross-currency swap because the two components do not offset each other.

Comparing cross-currency swaps of 3, 5 and 10 years shows profiles that rise similarly at first, driven by the FX forward element, before each drops to zero at its own maturity. The longer swaps carry slightly more risk at any given date because they still have more interest payments to exchange.

Simple options with an upfront premium have the rising profile of a forward. Exercise decisions complicate matters, because once an exercise date has passed the underlying trade is either alive or not, depending on the path the market took. This matters most when settlement is physical, so that exercise creates a new trade that lives on.

Consider a European interest rate swaption with one year to expiry that, if exercised, delivers a five-year swap, and compare it with the equivalent forward starting swap, which is an obligation to enter the same swap. Because the underlying swap has unequal payment frequencies on its two legs, both profiles show a sawtooth pattern after the start date.

Before exercise

Up to the exercise date, the swaption always has at least as much exposure as the forward starting swap. The right to enter a contract can never be worth less than the obligation to enter the same contract, since the holder can always decline when entering would be unattractive.

After exercise

After the exercise date the ordering reverses. There are paths on which the swap later ends up with positive value, yet at the exercise date it had negative value, so the swaption holder rationally let the option lapse. On such a path the forward starting swap, which had to be entered, now has exposure, but the swaption has none because no swap exists. On a path where the swap was valuable at the exercise date, the swaption is exercised and its exposure from then on matches the swap. So, after exercise the swaption’s exposure is at most that of the forward swap.

A cash-settled swaption has the same exposure as the physically settled one up to the exercise date and zero exposure afterwards, since exercise produces a single payment rather than a live swap. Physical settlement is standard in some interest rate markets, and whether cash or physical settlement is more common depends on the currency.

Exercise and the recursive problem

The analysis above uses the base value of the underlying swap, so the valuation paths are the same for cash and physical settlement and exposure can be read straight from future swaption values. Using actual value instead would link future value, exposure and the exercise decision, because a rational holder would take the xVA of the resulting swap into account when deciding whether to exercise. That produces another recursive problem, similar to the one in the definition of value, and where several exercise decisions fall at the same time every combination would need to be considered.

Credit derivatives are awkward for exposure assessment on two counts. The first is wrong-way risk, the link between the counterparty’s default and the value of the trade, which is especially strong when the counterparty is selling credit protection. The second applies even without wrong-way risk: the payoffs are discrete, so exposure can jump.

Consider a long protection position in a single-name credit default swap (CDS). If the reference entity does not default, the CDS behaves much like a swap whose value drifts with credit spreads, and its EPE shows a familiar low, swap-like profile. If the reference entity defaults, the protection buyer is owed one minus the recovery rate on the notional, a large and sudden amount. With an assumed recovery rate of 40%, that payout is 60% of notional.

Why the PFE jumps

PFE is a quantile. At the 99% level it looks at the worst 1% of outcomes. While the cumulative probability that the reference entity has defaulted by a given date is below 1%, the default outcomes are too rare to reach the 99% quantile, so the PFE reflects only spread moves and stays moderate. Once that cumulative default probability passes 1%, the default payout falls inside the worst 1% of outcomes and the PFE leaps to 60%. The result is confusing: the PFE may or may not reflect the credit event, and its level depends sharply on the confidence level chosen.

Part of this behaviour comes from common modelling choices that treat default as a sudden, unanticipated jump with a known recovery value. Modelling default more realistically, with an unknown recovery, makes the profile more continuous. Using expected shortfall instead of a quantile partly solves the problem too, since expected shortfall averages over the tail rather than picking one point in it; this is one reason the Fundamental Review of the Trading Book prefers expected shortfall to VaR. EPE is itself an average, similar in spirit to expected shortfall, so it does not show the jump. For CDS indices the effect fades, because each single default among many reference names has only a small impact.

Exposure is rarely measured trade by trade. Close-out netting is a contractual arrangement that lets the values of all trades under one netting agreement offset each other in a default, so all those trades must be added together before exposure is calculated. A counterparty may have more than one netting agreement with the same firm, in which case each netting set is treated separately. Netting set aggregation is the relevant level for counterparty-specific quantities such as CVA, while funding and capital metrics such as FVA and KVA may need aggregation at other levels.

Why aggregation can only reduce exposure

Picture two trades whose values move in opposite directions over time, one positive while the other is negative. Added together, they partly cancel. The positive exposure of the combined position is smaller than the sum of the two positive exposures, and in a clean case the combined negative exposure can disappear altogether. This reduction is the whole rationale for netting. Mathematically, it rests on a simple inequality: the positive part of a sum can never exceed the sum of the positive parts.

EFV behaves differently. It is a plain average of values, so it is additive across trades: the aggregated EFV is exactly the sum of the individual EFVs. EPE and ENE are not additive, because each throws away part of the distribution before averaging.

Positive correlation: little benefit

The table below uses five equally likely scenarios at one date. The two trades move together, so their values usually have the same sign.

EFV adds exactly, since 10 – 5 = 5. The aggregated EPE of 21 is only slightly below the 22 obtained by adding 18 and 4, and the aggregated ENE of -16 is only slightly smaller in size than -8 plus -9, which is -17. Netting helps only in scenarios where the two values have opposite signs, and that happens in scenario 3 alone, where 10 and -5 offset to 5. Interestingly, the values themselves are perfectly correlated (each step down is the same), yet the exposures, being only the positive or negative parts, are less than perfectly correlated, and that is where the small benefit comes from. Such a pattern arises with otherwise identical trades struck at different rates, so that their current values differ.

Negative correlation: strong benefit

Transaction 2 now moves against Transaction 1, so the two values have opposite signs in every scenario. The individual metrics are unchanged, yet the aggregated EPE falls to 9, from 22 on a stand-alone basis, and the aggregated ENE shrinks to -4 from -17. EFV is still 5. The general lesson is that the benefit of aggregation grows as the correlation between trades falls, and it also grows with the number of trades in the portfolio, because more trades offer more chances to offset.

Offsetting within a netting set does not come only from the structural correlation between trades. The current value of the existing portfolio matters just as much, so relative moneyness can produce large aggregation effects even when the trades are highly correlated or all point the same way.

Adding a trade to an out-of-the-money portfolio

Suppose the existing portfolio with a counterparty has a significantly negative value. For the netting set to show positive exposure in the future, the portfolio value would have to rise by enough to cross zero, which is unlikely. When a new trade is added, much of its positive exposure is absorbed by the negative value already in the set. So, the new trade adds far less exposure than it would on its own, and its incremental EPE is damped. Another way to see this is that the exposures are less correlated than the underlying future values, which is the same mechanism seen in the positively correlated aggregation table.

Adding a trade to an in-the-money portfolio

The reverse case also helps. If the existing portfolio has a significantly positive value, it carries large positive exposure. A new trade that tends to have negative value offsets part of that positive exposure. Put another way, the ENE of the new trade is absorbed by the EPE of the portfolio, and the net position benefits.

These effects show that even strongly directional portfolios can produce large netting effects, and they help explain why the xVA of a new trade depends heavily on the netting set it joins.

Margin generally reduces exposure, and as a first approximation it can simply be deducted from the portfolio value before the positive and negative parts are taken. Equations 19.1 and 19.2 then become:

Received margin lowers positive exposure and pushes negative exposure further below zero. Posted margin does the reverse: it raises positive exposure, because the posted assets could be lost if the counterparty defaults, unless they are offset by the value owed. This matches intuition, since receiving margin protects against counterparty risk while posting margin can create it. Segregation, covered later, complicates this simple picture.

Example with two-way margin

The table below assumes, loosely, two-way variation margin with no initial margin. It shows the value, the margin held (negative where margin has been posted) and the value net of margin for five equally likely scenarios, with an extra column for the reduction in positive exposure.

In scenarios 1 to 3 the portfolio is positive and margin has been received, so most of the exposure is removed, but not all of it. In practice collateralisation is imperfect, because value can rise faster than margin arrives and because thresholds and minimum transfer amounts hold back part of the call. In scenario 4 the value is negative and the party has posted 3; the posting does not create positive exposure, since the value still exceeds the posted amount in size. In scenario 5 the party has posted 16 against a value of -15, which overshoots by 1 and turns into positive exposure. Such over-posting arises in practice when margin posted earlier has not yet been returned. This last effect is small compared with the benefits elsewhere, but it shows that margin can raise exposure as well as lower it.

Margin usually reduces exposure, but measuring how much it reduces exposure requires care and a number of partly subjective assumptions. Thresholds and minimum transfer amounts must be modelled faithfully, and the margin period of risk (MPoR) must be analysed to establish how long the firm is really exposed before margin arrives. Three separate effects leave a collateralised exposure above zero.

• Granularity. A firm cannot always call for the full margin it is owed. A threshold is an amount of exposure left uncollateralised by agreement, and a minimum transfer amount suppresses calls that would be too small. Margin therefore moves in discrete steps. Occasionally this produces a brief overcollateralisation, where margin held exceeds the exposure, which helps the holder. Exposure analysis must also allow for margin the firm has posted, not only margin received.
• Delay. Margin arrives late. The operational chain of calling, agreeing and receiving margin takes time, and disputes over valuations can stretch it further. These delays are captured in the MPoR.
• Margin value risk. Unless the margin is cash in the same currency as the exposure, its own value can move, so the protection it provides can shrink just when it is needed.

Margin is also path dependent. The amount called at any date depends on how much has already been called or posted, so the margin balance carries the history of the portfolio with it. This matters most under two-way agreements, where margin can flow in either direction.

Levels of collateralisation

The EPE of a five-year interest rate swap under different margin terms illustrates the range of outcomes:

• Uncollateralised. The full swap hump, peaking a little above 4% of notional.
• Partially collateralised. A material threshold means the reduction is incomplete. Exposure rises until it reaches the threshold and then stays roughly flat, near 3%, so the threshold acts approximately as a cap.
• Strongly collateralised. Thresholds are zero but there is no initial margin, so the residual exposure comes from the MPoR. EPE stays near 1% and falls over the life of the swap.
• Overcollateralised. Initial margin is also held, so exposure drops further, close to zero, and to zero entirely if the initial margin is large enough.

Market practice now accepts that uncollateralised derivative positions that carry counterparty risk also have to be funded, and the resulting cost is priced through the funding value adjustment (FVA). The logic is simple. A derivative with positive value is an asset, and like any asset it has to be financed. One with negative value is a liability, which acts as a source of funding, a funding benefit. Costs and benefits partly cancel, because funding is assessed for the portfolio as a whole rather than trade by trade.

Not every asset creates a funding cost. A liquid, high-quality bond such as a Treasury can be financed by repo, and the cash raised in the repo largely covers the purchase. Derivatives cannot be repoed in that way, so they carry a funding cost. In the same spirit, margin received against a derivative asset can offset its funding cost, provided the margin can be reused, and margin posted against a derivative liability removes its funding benefit. The funding position is therefore the value net of margin.

Where funding and credit exposure line up

The parallel with exposure is close. A positive exposure is what is lost if the counterparty defaults, and it is also what has to be funded if the counterparty survives. A negative exposure relates to the firm’s own default and is also a funding benefit, and these two are generally seen as overlapping. Margin received against a positive value reduces both the counterparty risk and the funding cost; margin posted against a negative value reduces both own counterparty risk and the funding benefit.

Where they differ

The recursive problem in defining value still affects funding. The difference in margin delay explains why a collateralised derivative may have zero FVA even though its CVA is not zero: funding uses the short operational margin cycle, while credit exposure uses the longer period that follows a default. Despite the differences, CVA, DVA and FVA have much in common and are usually computed within one shared framework.

Margin serves two purposes. Its traditional role is to mitigate counterparty risk, but it also neutralises funding. Historically the first role dominated, which explains why counterparties of excellent credit quality were often given one-way margin agreements, under which they never posted. In a regime that is more sensitive to funding, such terms are expensive, because the funding cost does not depend on the counterparty’s credit quality at all; it depends on what it costs the bank to raise money.

Receiving margin against a positive value can deliver two benefits. In a default, the firm can keep the margin to cover close-out losses, which is counterparty risk reduction. In normal times, it can use the margin elsewhere, for example posting it against a negative value on another trade or on the hedge, which is a funding benefit. Each benefit needs a different property.

• Counterparty risk reduction needs no adverse link between the margin’s value and the counterparty’s credit quality. Margin with wrong-way risk still helps if it keeps some value after the default.
• Funding benefit needs the margin to be reusable: not segregated, and either transferred by title transfer or with rehypothecation allowed. Cash meets this automatically; non-cash margin needs rehypothecation rights so it can be reused or repoed.

Typical forms of margin

• Cash that need not be segregated gives both benefits.
• Securities that can be rehypothecated also give both, provided haircuts cover adverse price moves and also the haircuts that apply when the securities are reused, for example in repo or as margin elsewhere.
• Cash or securities that must be segregated or cannot be rehypothecated reduce counterparty risk, since they can be realised in a default, but give no funding benefit because they cannot be reused while the counterparty is alive. This case is used in the Totem xVA consensus pricing service as a way to separate counterparty risk from funding.
• A counterparty’s own bonds that can be rehypothecated give questionable counterparty protection, since they will themselves be in default when needed. They still help a little: more margin can be called if their price falls, and they retain some recovery value, although a sudden default with low recovery would leave them almost worthless. Provided they can be rehypothecated with adequate haircuts, they do give a funding benefit.

Sovereigns, supranationals and agencies

Sovereigns, supranationals and agencies (SSAs), usually rated triple-A, traditionally had one-way credit support annexes (CSAs) with banks and posted nothing. That arrangement leaves banks with a large funding obligation, which they pass back in the price of the swaps SSAs use to hedge their borrowing and lending. As funding costs rose and banks became more sensitive to them, some SSAs moved to two-way margin, sometimes posting their own bonds, to win better pricing. For a high-quality counterparty, posting own bonds is attractive because it avoids the liquidity strain of sourcing other collateral, and because the counterparty risk it represents matters less to the bank than the funding cost. In table terms, it targets the WWR-present, reusable cell: limited counterparty protection but a real funding benefit. Capital requirements for counterparty risk (KVA) may still be significant even when CVA is small.

Once segregation and rehypothecation are allowed for, received and posted margin no longer enter exposure and funding in the same way. For counterparty risk, all margin received can be used in a default, whatever its segregation status, while posted margin that is not segregated adds risk because it cannot be recovered if the counterparty defaults.

For funding, only received margin that can be reused helps, while every unit of posted margin has to be financed, whether segregated or not.

Applying the formulas to initial and variation margin

Initial margin. Under Equation 19.8, posted initial margin that is not segregated increases positive exposure, so it should be segregated. If both sides posted initial margin without segregation, the amounts would simply offset, and each side could in effect hand the other’s margin back. So two-way initial margin must be segregated, as bilateral margin rules require, and one-way initial margin should ideally be segregated too. Some parties outside the regulatory margin rules still hold non-segregated one-way initial margin, often called an independent amount, though this is becoming rare.

Variation margin. Under Equation 19.9, variation margin should be reusable to reduce funding costs. Posting it adds little counterparty risk, because it is posted against a negative value that is already owed. It is never segregated and can usually be rehypothecated, since it does not represent overcollateralisation.

With these conventions, writing VM for variation margin (positive when received, negative when posted), IM^R for initial margin received and IM^P for initial margin posted, the formulas become:

Variation margin offsets credit exposure in either direction, and initial margin received offsets it further. Funding is fully adjusted for variation margin and increased by initial margin posted. Without initial margin the exposure and funding formulas look alike, apart from the differences listed earlier. Received initial margin may even carry a funding cost, since it has to be held with a third-party custodian. It is convenient to split the funding effects into FVA for variation margin and margin value adjustment (MVA) for initial margin.