11
Future Value and Exposure

The concept of future value is a key determinant in xVA because it represents the potential value of a transaction or portfolio in the future. The utilisation components (Section 5.4.6) of all xVA terms are related – simply or otherwise – to the future value, and it is therefore a common component of all xVA adjustments. This chapter will be concerned with defining future value in more detail and explaining the key characteristics. We will focus initially on credit exposure – the core component of credit value adjustment (CVA) – including the important metrics used for its quantification. Typical credit exposure profiles for various products will be discussed, and we will explain the impact of netting and margin on credit exposure. We also describe the link between future value and funding costs, which are driven by similar components but have some distinct differences, especially when aspects such as segregation are involved. This leads us on to define the economic impact of funding, which is similar to the definition of exposure but has some distinct differences. The calculation of funding value adjustment (FVA) will then be similar to that of CVA.

11.1 CREDIT EXPOSURE

11.1.1 Positive and Negative Exposure

A defining feature of counterparty risk arises from the asymmetry of potential losses with respect to the value of the underlying transaction(s).¹ In the event that a counterparty has defaulted, a surviving party may close out the relevant contract(s) and cease any future contractual payments. Following this, the surviving party may determine the net amount owing between itself and its counterparty and take into account any margin that may have been posted or received. Note that margin may be held to reduce exposure, but any posted margin may have the effect of increasing exposure.

Once the above steps have been followed, there is a question of whether the net amount is positive or negative. The main defining characteristic of credit exposure (hereafter referred to just as exposure) is related to whether the net value of the contracts (including margin) is positive (in a party's favour) or negative (against it), as illustrated in Figure 11.1:

Negative value. In this case, the party is in debt to its counterparty and is still legally obliged to settle this amount (it cannot walk away from the transaction(s) except in specific cases (see Section 6.3.3)). Hence, from a valuation perspective, the position appears largely unchanged. A party does not generally gain or lose from its counterparty's default in this case.
Positive value. When a counterparty defaults, it will be unable to undertake future commitments, and a surviving party will have a claim on the (positive) value at the time of the default, typically as an unsecured creditor. The surviving party will then expect to recover some fraction of its claim, just as bondholders receive some recovery on the face value of a bond. This unknown recovery value is, by convention, not included in the definition of exposure.

Schematic illustration of the impact of a positive or negative contract value in the event of the default of a counterparty. — **Figure 11.1** Illustration of the impact of a positive or negative contract value in the event of the default of a counterparty.

The above feature – whereby a party loses if the value is positive and does not gain if it is negative – is a defining characteristic of counterparty risk and CVA. The definition of positive exposure is:²

(11.1)

Exposure is therefore directly related to the current and future value of a counterparty portfolio, and calculating this future value is, therefore, key to determining the exposure. Note that the quantification of future value must take into account the relevant netting across different transactions and also any margin that may be held against the exposure.

A key feature of counterparty risk is that it is bilateral, as both parties to a transaction can default and therefore both can experience losses. For completeness, we may need to consider losses arising from both defaults. From a party's point of view, its own default will cause a loss to any counterparty to whom it owes money. This can be defined in terms of negative exposure, which by symmetry is:

(11.2)

A negative exposure will lead to a gain, which is relevant since the counterparty is making a loss.³ This is defined as ‘own counterparty risk’, which is represented by debt value adjustment (DVA).

11.1.2 Definition of Value

The amount represented by the term value in the above discussion is dependent on the value of the individual transactions in question at each potential default time of the counterparty (or the party itself). It must also account for the impact of risk mitigants, such as netting and margin. However, there is a question of whether this is the ‘base value’ or ‘actual value’, as discussed in Section 5.2.1. Whilst the base value would be a more appealing definition, this is not the one typically defined by documentation (Section 6.3.5) and therefore is unlikely to be the value agreed with the counterparty (the administrators of their default).

From one perspective, a party would wish for the relevant documentation and legal practices to align the actual value – agreed bilaterally after the default – to its own unilateral view prior to any default. Without this, there would be a jump in the valuation at the point at which the counterparty defaults (Figure 6.11). However, this actual value does not conform to a definitive, objective representation due to the complexity of valuation adjustments and close-out effects. Furthermore, the actual value contains xVA components, which therefore creates a recursive problem, discussed in Section 6.3.5.

Quantification of exposure and xVA will, therefore, rely on relatively clean measures or base values which can readily drive quantitative calculations. However, it should be remembered that documentation will tend to operate slightly differently. For example, recall that International Swaps and Derivatives Association (ISDA) documentation (Section 6.3.5) specifically references that the ‘close-out amount’ may include information related to the creditworthiness of the surviving party. This implies that an institution can potentially reduce the amount owed to a defaulting counterparty, or increase its claims in accordance with CVA charges it experiences in replacing the transaction(s) at the point where its counterparty defaults. Such charges may themselves arise from the xVA components that depend on exposure in their calculation. The result of this is a recursive problem where the very definition of current exposure depends on potential xVA components in the future. However, note that it is general market practice to link exposure quantification to the concept of base value, which is relatively easy to define and model, and that any errors in doing this are usually relatively small.

When quantifying exposure and other xVA terms, Equations 11.1 and 11.2 are fundamental starting points and will typically rely on a definition of value which may come from a standard valuation model. Whilst this theoretical definition of value cannot practically include aspects such as the type of documentation used, jurisdiction, or market behaviour at the time of default, it will be hoped that issues such as these will only constitute small uncertainties.

A final point to note about the above problems in determining close-out amounts is the time delay. Until an agreement is reached, an institution cannot be sure of the precise amount owed or the value of its claim as an unsecured creditor. This will create particular problems for managing counterparty risk. In a default involving many contracts (such as the number of OTC derivatives in the Lehman bankruptcy), the sheer operational volumes can make the time taken to agree on such valuations considerable. This risk may be taken by the xVA desk in a bank (Section 21.1).

11.1.3 Current and Potential Future Exposure

A current valuation of all relevant transactions and associated margin will lead to a calculation of current value and exposure (admittedly with some uncertainty regarding the actual close-out amount, as noted in Section 11.1.2). However, it is even more important to characterise what the exposure might be at some point in the future. This concept is illustrated in Figure 11.2, which can be considered to represent any situation from a single transaction to a large portfolio with associated netting and margin terms. Whilst the current (and past) exposure is known with certainty, the future exposure is defined probabilistically by what may happen in the future in terms of market movements, contractual features of transactions, netting, and margining, all of which have elements of uncertainty. Hence, in understanding future exposure, one must define the level of the exposure and also its underlying uncertainty.

Schematic illustration of potential future exposure. — **Figure 11.2** Illustration of potential future exposure. The grey area represents positive values and the white area negative values.

Quantifying exposure is complex due to the long periods involved, the many different market variables that may influence the future value, and risk mitigants such as netting and margin; this will be the subject of Chapter 15. This chapter focuses on defining exposure and intuitively discussing the impact of aspects such as netting and margin, and the related concept of funding.

11.1.4 Nature of Exposure

Counterparty risk creates an asymmetric risk profile, as shown by Equation 11.1. When a counterparty defaults, a surviving party loses if the value (from its point of view) is positive, but does not gain if it is negative. The profile can be likened to a ‘short’ option position,⁴ since from the surviving party's perspective, the defaulting counterparty has the option not to pay them. Familiarity with basic options-pricing theory would lead to two obvious conclusions about the quantification of exposure:

since exposure is similar to an option payoff, a key aspect will be volatility (of the value of the relevant contracts and margin); and
options are relatively complex to price (compared with the underlying instruments at least), so quantifying exposure, even for a simple instrument, may be quite complex.

By symmetry, an institution has long optionality from its own default (this is related to the term ‘DVA’).

We can extend the option analogy further – for example, by saying that a portfolio of transactions with the same counterparty will have an exposure analogous to a basket option, and that a margin agreement will change the strike of the underlying options. However, thinking of xVA as one giant exotic options-pricing problem is correct but potentially misleading. One reason for this is that, as already noted in Section 11.1.3, we cannot even write the payoff of the option, namely the exposure, down correctly (since we cannot precisely define the value term in Equation 11.1). Furthermore, xVA contains many other subjective components, such as credit and funding curves (Sections 12.2 and 14.3) and wrong-way risk (Section 17.6). At the core of exposure calculation there is an options-pricing-type problem, but this cannot be treated with the accuracy and sophistication normally afforded to this topic, due to the sheer complexity of the underlying options and the other components that drive xVA. Treating xVA quantification as a purely theoretical options-pricing problem tends to underemphasise other important but more qualitative aspects.

There is another way to look at exposure quantification. In financial risk management, value-at-risk (VAR) methods (Section 2.6.1) have, for almost two decades, been a popular methodology for characterising market risk and have more recently been applied to methodologies for initial margin (Chapter 9). Such approaches aim to characterise market risk over relatively short horizons, such as 10 days.⁵ As illustrated in Figure 11.2, the characterisation of exposure is a similar problem since it involves quantifying the market risk of a portfolio. This is indeed true, although we note that in quantifying exposure there are additional complexities, most notably:

Time horizon. Exposure needs to be defined over multiple time horizons (often far in the future) so as to understand the impact of time and the specifics of the underlying contracts. There are two important implications of this:
- Firstly, the ‘ageing’ of transactions must be considered. This refers to understanding a portfolio in terms of all future contractual payments and changes such as cash flows, termination events, exercise decisions, and margin postings. Such effects may also create path dependency, where the exposure at one date depends on an event defined at a previous date. In VAR models, due to the short horizon used (e.g. 10 days), such aspects can often be neglected.
- Secondly, when looking at longer time horizons, the trend (also known as the ‘drift’) of market variables (in addition to their underlying volatility and dependence structure) is relevant (as depicted in Figure 11.2). In VAR-type approaches the drift can be ignored, again since the relevant time horizon is short.
Risk mitigants. Exposure is typically reduced by risk mitigants such as netting and margin, and the impact of these mitigants must be considered in order to estimate future exposure appropriately. In some cases – such as applying the correct netting rules – this requires knowledge of the relevant contractual agreements and their legal interpretation in the jurisdiction in question. In the case of future margin amounts – especially dynamic initial margin (Section 9.3) – another degree of subjectivity is created, since there is no certainty regarding the type of margin and the precise time that it will be received. Other contractual features of transactions – such as termination agreements (Section 7.1.1) – may also create subjectivity, and all such elements must be modelled, introducing another layer of complexity and uncertainty.
Application. Traditional approaches for market risk and initial margin quantification are risk-management approaches. However, exposure must be defined for both risk management and pricing (i.e. xVA). This creates additional complexity in quantifying exposure and may lead to two completely different sets of calculations: one to define exposure for risk-management purposes (‘real world’) and one for pricing purposes (‘risk neutral’). This is discussed in more detail in Section 15.3.3.

11.1.5 Metrics

This section defines the measures commonly used to quantify exposure. The different metrics introduced will be appropriate for different applications and xVA terms. The most common definitions will be used; sometimes the nomenclature used for the terms defined below can differ – in particular, in regulatory definitions (BCBS 2005b) and in previous editions of this book. Such differences will be highlighted below.

Note that the definitions given below are general and can apply to any underlying transaction or groups of transactions. In practical terms, there will also be the question of the aggregation level, which is a group of transactions that must be considered together. For example, for counterparty risk, the obvious aggregation level is all transactions with the counterparty in question, or more specifically all that are covered by a single legal agreement such as a ‘netting set’. Aggregation is discussed in more detail in Section 11.3 and need not be a consideration for the definitions below, where we will simply use the term ‘portfolio’ to refer to the relevant transactions and associated risk mitigants.

Considering first the definition of exposure metrics for a given time horizon, the first definition is expected future value (EFV).⁶ This component represents the forward or expected value of the portfolio at some point in the future. As mentioned above, due to the relatively long time horizons involved in measuring counterparty risk, the expected value can be an important component, whereas for traditional market risk quantification (involving only short time horizons) it is not. EFV at time zero is, by definition, the current value. At other times, EFV represents the expected (average) of the future value calculated with some probability distribution in mind (to be discussed in Chapter 15). EFV may vary significantly from current value for a number of reasons:

Cash flow differential. Cash flows in some transactions may be significantly asymmetric. For example, early in the lifetime of an interest rate swap, the fixed cash flows will typically exceed the floating ones (assuming the underlying yield curve is upward sloping, as is most common). Another example is a cross-currency swap, where the payments may differ by several percent annually due to a differential between the associated interest rates. The result of asymmetric cash flows is that a party may expect a transaction in the future to have a value significantly above (below) the current one due to paying out (receiving) cash flows. Note that – in a portfolio context – this can also apply to transactions maturing due to final payments (e.g. cross-currency swaps), which has an impact on the future value of the remaining portfolio.
Forward rates. Forward rates can differ significantly from current spot variables. This difference introduces an implied drift (trend) in the future evolution of the underlying variables in question (assuming one believes that this is the correct drift to use, as discussed in more detail in Section 15.3.3). Drifts in market variables will lead to a higher or lower future value for a given portfolio even before the impact of volatility. Note that this point is related to the point above on cash flow differential, since this is a result of forward rates being different from spot rates.
Asymmetric margin agreements. If margin agreements are asymmetric (such as a one-way margin posting), then the future value may be expected to be higher or lower, reflecting unfavourable or favourable margin terms, respectively.

In risk management, it is natural to ask what is the worst exposure at a certain time in the future. Potential future exposure (PFE) (Figure 11.2) will answer this question with reference to a certain confidence level. For example, the PFE at a confidence level of 99% will define an exposure that would be exceeded with a probability of no more than 1% (100 minus the confidence level). PFE is an equivalent metric to VAR.

The pricing of some xVA terms requires both expected positive exposure (EPE) and expected negative exposure (ENE), which are the averages of the positive and negative exposures defined previously in Equations 11.1 and 11.2. Note that these definitions are not the average of only the positive (EPE) or negative (ENE) values, but rather the average across all values, but where the negative or positive ones are set to zero. So, for example, when calculating the EPE, only positive values contribute to the total, but zero and negative values contribute in terms of their probability (this may be best understood in the example below).

The above metrics are summarised below and in Figure 11.3.⁷

Expected future value (EFV). The average across all exposure values. Note that this can differ significantly from zero, which represents a more significant positive or negative expected value.
Potential future exposure (PFE). The highest (or lowest) exposure calculated to some underlying confidence level.⁸
Expected positive exposure (EPE). The average of all of the positive exposures (noting that some will be zero). Note that this term, like the PFE, is sensitive to the variability of the distribution (e.g. the volatility). This term is sometimes known as expected exposure (EE).
Expected negative exposure (ENE). The average of all of the negative exposures (noting that some will be zero). This represents the positive exposure from the counterparty's point of view. This term is sometimes known as negative expected exposure (NEE).

Schematic illustration of exposure metrics for a single time horizon. — **Figure 11.3** Illustration of exposure metrics for a single time horizon. PFE is assumed to be calculated at a high confidence level.

Example.

Suppose the future exposure is defined by only five different scenarios, each with an equal probability of occurrence. The metrics defined above would then be as in the table below. The calculation of the PFE assumes that the confidence level is more than 80% (in which case it is the highest exposure number).

	Future exposure	Calculation
Scenario 1	70
Scenario 2	50
Scenario 3	30
Scenario 4	−10
Scenario 5	−30
EFV	22
PFE	70	The highest value (in this case)
EPE	30
ENE	−8

Appendix 11A gives formulas for the exposure metrics for a normal distribution.

Example. Suppose future value is defined by a normal distribution with mean 2.0 and standard deviation 2.0. As given by the formulas in Appendix 11A, the EFV, PFE, EPE, and ENE (at the 99% confidence level) are:

EFV	= 2.0 (by definition)
PFE (99%)	= 6.65
EPE	= 2.17
ENE	= 0.17

For a larger standard deviation of 4.0:

EFV	= 2.0
PFE (99%)	= 11.31
EPE	= 2.79
ENE	= 0.79

Note the aforementioned sensitivity of the EPE, ENE, and PFE to the standard deviation. The ENE increases by almost five times when the standard deviation is doubled. Using the options pricing analogy, this is because it represents an out-of-the-money position.

A final metric will approximate exposure over time. Average EPE is defined as the average exposure across all time horizons. It can, therefore, be represented as the weighted average of the EPE across time, as illustrated in Figure 11.4. If the EPE points are equally spaced (as in this example), then it is simply the average. Note that the average EPE is sometimes known simply as EPE, which can cause confusion with the definition of EPE used here. As noted above, the most common definitions are being used, and the Basel definitions (Section 13.4.5) do differ.

This single average EPE number is often called a ‘loan equivalent’, as the average amount effectively lent to the counterparty in question. It is probably obvious that expressing a highly-uncertain exposure by a single EPE or loan-equivalent amount can represent a fairly crude approximation, as it averages out both the randomness of market variables and the impact of time. However, it will be shown later that EPE has a strong theoretical basis for assessing regulatory capital (Section 13.4.5) and quantifying xVA (Section 17.2.3).

Figure 11.4 Illustration of average EPE, which is the weighted average (the weights being the time intervals) of the EPE profile.

11.2 DRIVERS OF EXPOSURE

We now give some examples of the significant factors that drive exposure, illustrating some important effects, such as maturity, payment frequencies, option exercise, roll-off, and default. The aim here is to describe some key features that must be captured; Chapter 15 will give actual examples from real transactions. In the examples below, exposure is shown as a percentage (i.e. assuming a unit notional of the transaction in question). In most cases, the initial value will be assumed to be zero, but the impact of a positive or negative current value will also be shown. Unless stated, the profile shown is the EPE; in most cases, the PFE will have the same behaviour but simply be a larger value. The current time will be represented by images and the maturity of the transaction by images .

Although not generally characterised as counterparty risk, the exposures of debt instruments such as loans and bonds can usually be considered almost deterministic and approximately equal to the notional value. Bonds typically pay a fixed rate and therefore will have some additional uncertainty, since, if interest rates decline, the exposure may increase and vice versa. In the case of loans, they are typically floating-rate instruments, but the exposure may decline over time due to the possibility of pre-payments.

In contrast to the above, products such as derivatives can have complex exposures due to their inherent complexities and the complex impact of risk mitigants such as margin.

11.2.1 Future Uncertainty

The first and most obvious driving factor in exposure is future uncertainty. Some derivatives – such as forward contracts – are usually characterised by having just the exchange of two cash flows or underlyings (potentially netted into a single payment) at a single date in the future (the maturity date of the contract). This means that the exposure is a rather simple, monotonically increasing function, reflecting the fact that, as time passes, there is growing uncertainty about the value of the final cash flow(s). Based on fairly common assumptions,⁹ the exposure of such a profile will follow a ‘square-root-of-time’ rule, meaning that the exposure will be proportional to the square root of the time (t):

(11.3)

This is described in more mathematical detail in Appendix 11B, and such a profile, which is roughly illustrative of a foreign exchange (FX) forward, is illustrated in Figure 11.5. We can see from the above formula that the maturity of the contract does not influence the exposure (except for the obvious reason that there is zero exposure after this date). For similar reasons, much the same shape is seen for vanilla options with an upfront premium, although more exotic options may have more complex profiles (see Section 11.2.6).

11.2.2 Cash Flow Frequency

Many OTC derivatives include the periodic payment of cash flows, which has the impact of reversing the effect of future uncertainty. The most obvious and common example here is an interest rate swap, which is characterised by a peaked shape, as shown in Figure 11.6. The shape arises from the balance between future uncertainties regarding payments, combined with the roll-off of fixed against floating payments over time. This can be represented approximately as:

(11.4)

Figure 11.5 Illustration of a square-root-of-time exposure profile. This example assumes volatility of 15% and the calculation of EPE under normal distribution assumptions.

Graph depicts the exposure of swaps of different maturities. This example assumes volatility of one percent and the calculation of EPE under normal distribution assumptions. — **Figure 11.6** Illustration of the exposure of swaps of different maturities. This example assumes volatility of 1% and the calculation of EPE under normal distribution assumptions, which is roughly illustrative of interest rate swaps.¹⁰

where T represents the maturity of the transaction in question. This is described in more mathematical detail in Appendix 11B. The function shown in Equation 11.4 initially increases due to the images term, but then decreases to zero as a result of the images component, which is an approximate representation of the remaining maturity of the transaction at a future time t. It can be shown that the maximum of the above function occurs at images (see Appendix 11B) – i.e. the maximum exposure occurs at a date equal to one-third of the maturity.

As seen in Figure 11.6, a swap with a longer maturity has much more risk, due to both the increased lifetime and the greater number of payments due to be exchanged. An illustration of the swap cash flows (assuming equal semiannual payment frequencies) is shown in Figure 11.7.

An exposure profile can be substantially altered due to the more specific nature of the cash flows in a transaction. Transactions such as basis swaps, where the payments are made more frequently than they are received (or vice versa), will then have more (less) risk than the equivalent equal payment swap. This effect is illustrated in Figure 11.8 and Figure 11.9.

11.2.3 Curve Shape

Another impact the cash flows have on exposure is to create an asymmetry between opposite transactions. In the case of an interest rate swap, this occurs because of the different cash flows being exchanged. In a ‘payer swap’, fixed cash flows are paid periodically at a deterministic amount (the ‘swap rate’), whilst floating cash flows are received. The value of future floating cash flows is not known until the fixing date, although, at inception, their (risk-neutral) discounted expected value will typically be equal to that of the fixed cash flows (‘par swap’). The value of the projected floating cash flows depends on the shape of the underlying yield curve.¹¹ In the case of a typical upwards-sloping yield curve, the initial floating cash flows will be expected to be smaller than the fixed rate paid, whilst later in the swap the trend is expected to reverse. This is illustrated schematically in Figure 11.10.

Schematic illustration of the cash flows of swap transactions of different maturities. Solid lines represent fixed payments and dotted lines floating payments. — **Figure 11.7** Illustration of the cash flows of swap transactions of different maturities (semiannual payment frequencies are assumed). Solid lines represent fixed payments and dotted lines floating payments.

Graph depicts the exposure for swaps with equal and unequal payment frequencies. The latter corresponds to a swap where cash flows are received quarterly but paid only semiannually. — **Figure 11.8** Illustration of the exposure for swaps with equal and unequal payment frequencies. The latter corresponds to a swap where cash flows are received quarterly but paid only semiannually.

Schematic illustration of the cash flows in a swap transaction with different payment frequencies. Solid lines represent fixed payments and dotted lines floating payments. — **Figure 11.9** Illustration of the cash flows in a swap transaction with different payment frequencies. Solid lines represent fixed payments and dotted lines floating payments.

Schematic illustration of the floating cash flows against fixed cash flows in a swap where the yield curve is upward sloping. Whilst the expected value of the floating and fixed cash flows may be equal, the projected floating cash flows are expected to be smaller at the beginning and larger at the end of the swap. — **Figure 11.10** Illustration of the floating cash flows (dotted lines) against fixed cash flows in a swap where the yield curve is upward sloping. Whilst the (risk-neutral) expected value of the floating and fixed cash flows may be equal, the projected floating cash flows are expected to be smaller at the beginning and larger at the end of the swap.

The net result of this effect is that the EPE of the receiver swap is lower due to the expectation of receiving positive net cash flows (the fixed rate against the lower floating rate) in the first periods of the swap, and of paying net cash flows later in the lifetime (Figure 11.11). The ENE is correspondingly more negative. Another way to state this is that the EFV of the swap is negative (by an amount defined by the expected net cash flows). For the opposite ‘payer swap’, this effect would be reversed, with the EPE being higher, the ENE less negative, and the sign of the EFV reversed.

Graph depicts the EFV, EPE, and ENE for a receiver interest rate swap. — **Figure 11.11** Illustration of the EFV, EPE, and ENE for a receiver interest rate swap.

Graph depicts the EFV, EPE, and ENE for a cross-currency swap where the payment currency has a higher interest rate. — **Figure 11.12** Illustration of the EFV, EPE, and ENE for a cross-currency swap where the payment currency has a higher interest rate.

The above effect can be explained in three different ways, all of which are related:

the cash flow differential, as discussed above;
the fact that the yield curve is (usually) upwards sloping, suggesting a market-implied view that interest rates will rise; and
the fact that the forward swap rates will be higher than the current swap rate (and therefore a forward starting swap receiving the swap rate will have a negative value).

This will be discussed in more detail in Section 15.6.1.

The above effect can be even more dramatic in cross-currency swaps, where a high-interest-rate currency is paid against one with lower interest rates, as illustrated in Figure 11.12. The overall high interest rates paid are expected to be offset by the gain on the notional exchange at the maturity of the contract,¹² and this expected gain on the exchange of notional leads to significant exposure for the payer of the high interest rate. In the reverse swap, when paying the currency with the lower interest rates, it is increasingly likely that the future value of the swap will be negative. This creates a negative drift, making the exposure much lower.

The impact of cash flow differential or – equivalently – drift is particularly important in xVA calculations, as will be seen later.

11.2.4 Moneyness

The previous examples have all shown par transactions where the initial value is zero. However, another consideration is ‘moneyness’, where a transaction could be in-the-money (ITM) (positive value) or out-of-the-money (OTM) (negative value). Figure 11.13 shows the exposure profiles for an ITM interest rate swap. Not surprisingly, the ENE is small due to the smaller likelihood of the future value being negative. The EPE is large and quite close to the EFV and therefore is more predictable in its form. This is analogous to an ITM option being more deterministic and having lower sensitivity to volatility.

Graph depicts the EFV, EPE, and ENE for an ITM receiver interest rate swap. — **Figure 11.13** Illustration of the EFV, EPE, and ENE for an ITM receiver interest rate swap.

11.2.5 Combination of Profiles

Some products have an exposure that is driven by a combination of two or more underlying risk factors. An obvious example is a cross-currency swap, which is essentially a combination of an interest rate swap and a FX forward transaction.¹³ This would, therefore, be represented by a combination of the profiles shown in Figure 11.5 and Figure 11.6, and is described in more mathematical detail in Appendix 11C. Figure 11.14 illustrates the combination of two such profiles. Cross-currency swap exposures can be considerable due to the high FX volatility driving the risk, coupled with the long maturities and final exchanges of notional. The contribution of the interest rate swap is typically smaller, as shown. We also note that the correlation between the two interest rates and the FX rate is an important driver of the exposure (in Figure 11.14 a relatively low correlation is assumed, as often seen in practice, which increases the cross-currency exposure).¹⁴

Figure 11.14 Illustration of the EPE of a cross-currency swap (CCS) profile as a combination of an interest rate swap (IRS) and FX forward.

Graph depicts the exposure for cross-currency swaps of different maturities. — **Figure 11.15** Illustration of the exposure for cross-currency swaps of different maturities.

Figure 11.15 illustrates the exposure for cross-currency swaps of different maturities. The longer-maturity swaps have slightly more risk due to the greater number of interest rate payments on the swap.

11.2.3 Optionality

Although some options have relatively straightforward exposures (Section 11.2.1), the impact of exercise decisions can create some complexities in exposure profiles, since after the exercise date(s) the underlying transaction will have a certain probability of being ‘alive’ or not. This is particularly important in the case of physical settlement. As an example, Figure 11.16 shows the exposure for a European-style interest rate swaption that is physically settled – rather than cash settled¹⁵ compared with the equivalent forward starting swap (as can be seen, the underlying swap has different payment frequencies). Before the exercise point, the swaption must always have a greater exposure than the forward starting swap,¹⁶ but thereafter this trend will reverse, since there will be scenarios where the forward starting swap has a positive value but the swaption would not have been exercised. This effect is illustrated in Figure 11.17, which shows a scenario that would give rise to exposure in the forward swap but not the swaption.

Graph depicts the exposure for a swap-settled interest rate swaption and the equivalent forward swap. — **Figure 11.16** Exposure (PFE) for a swap-settled (physically-settled) interest rate swaption and the equivalent forward swap. The option maturity is one year and the swap maturity five years.

The above example is based on the base value of the underlying swap. This means that the valuation paths illustrated in Figure 11.17 will be the same for both cash- and physically-settled swaps and the exposure can be calculated directly from the future values of the swaption. To use the ‘actual value’ in the calculation would be more complicated, since the future value, exposure, and exercise decision would become linked. Another way of stating this is that, in exercising the option, one should naturally incorporate future xVA adjustments. This, therefore, leads to a recursive problem for the calculation of xVA for products with exercise boundaries,¹⁷ which is similar to the recursive problem discussed in Section 11.1.2.

11.2.7 Credit Derivatives

Credit derivatives represent a challenge for exposure assessment due to wrong-way risk, which will be discussed in Section 17.6. Even without this as a consideration, exposure profiles of credit derivatives are hard to characterise due to the discrete payoffs of the instruments. Consider the exposure profile of a single-name credit default swap (CDS), as shown in Figure 11.18 (long CDS protection), for which the EPE and PFE are shown. Whilst the EPE shows a typical swap-like profile, the PFE has a jump due to the default of the reference entity. This is a rather confusing effect (see also Hille et al. 2005), as it means that the PFE may or may not represent the actual credit event occurring and is sensitive to the confidence level used.¹⁸ Using a measure such as expected shortfall partially solves this problem.¹⁹ This effect will also not be apparent for CDS indices due to a large number of reference credits where single defaults have a less significant impact.

Schematic illustration of exercise of a physically-settled European swaption showing two potential scenarios of future value for the underlying swap. Scenario B corresponds to a scenario where the swaption would be exercised, giving rise to exposure at a future date. In scenario A, the swaption would not have been exercised, and hence the exposure would be zero. The exercise boundary is assumed to be the x-axis. — **Figure 11.17** Illustration of exercise of a physically-settled European swaption showing two potential scenarios of future value for the underlying swap. Scenario B corresponds to a scenario where the swaption would be exercised, giving rise to exposure at a future date. In scenario A, the swaption would not have been exercised, and hence the exposure would be zero. The exercise boundary is assumed to be the x-axis (in reality it would not be constant).

Graph depicts the EPE and 99% PFE for a long-protection single-name CDS transaction. A PFE of 60 percent arises from default with an assumed recovery rate of 40 percent when the probability of default of the counterparty is greater than one minus the confidence level. — **Figure 11.18** EPE and 99% PFE for a long-protection single-name CDS transaction. A PFE of 60% arises from default with an assumed recovery rate of 40% when the probability of default of the counterparty is greater than one minus the confidence level (i.e. 1%).

11.3 AGGREGATION, PORTFOLIO EFFECTS, AND THE IMPACT OF COLLATERALISATION

The aggregation of transactions across a particular portfolio has an important impact on exposure. One example of aggregation is close-out netting (Section 6.3.2), which effectively allows the future values of different transactions to offset one another thanks to a contractual agreement. This requires that all transactions under a given netting agreement with a counterparty be added together for the purposes of determining exposure.²⁰ Netting set aggregation is important for counterparty-specific quantities such as CVA.

There are also other levels at which aggregation may need to be done which relate to aspects such as funding and capital and the corresponding metrics (e.g. FVA and KVA). The discussion below will be generic and consider only the general impact of aggregation on exposure. There are several different aspects to contemplate before understanding the full netting impact on overall exposure.

11.3.1 The Impact of Aggregation on Exposure

Figure 11.19 illustrates the general impact of aggregation. Since the exposure profiles are partly offsetting – meaning that one is positive whilst the other is negative – when aggregated, they produce a reducing effect. This means that the exposure (positive or negative) of the aggregated profile is smaller than the sum of the exposures of each transaction (indeed, the negative exposure is zero since, when aggregated, there is no negative contribution). It is this reduction which is the rationale for risk mitigants such as close-out netting.

Table 11.1 illustrates the impact of aggregation on exposure using a simple example with five scenarios and a single point in time. Note firstly that the EFV is additive across the two transactions,²¹ which should be expected since it is simply the average of the values. The EPE and ENE, on the other hand, are not additive, which is a consequence of their definitions including only positive or negative values (Section 11.1.1): the aggregated EPE or ENE is always less (in absolute terms) than the sum of the individual components.²² This is a general result that the aggregated EPE or ENE cannot be larger (in absolute terms) than the sum of the individual components.²³

Schematic illustration of the impact of aggregation on exposure. — **Figure 11.19** Illustration of the impact of aggregation on exposure.

Table 11.1 Illustration of the impact of aggregation when there is a positive correlation between values. The exposure metrics are shown, assuming each scenario has equal weight.

	Transaction 1	Transaction 2	Aggregated
Scenario 1	50	15	65
Scenario 2	30	5	35
Scenario 3	10	−5	5
Scenario 4	−10	−15	−25
Scenario 5	−30	−25	−55
EFV	10	−5	5
EPE	18	4	21
ENE	−8	−9	−16

Aggregation can be seen to produce a diversification effect. When considering the aggregation benefit of two or more transactions, the most obvious consideration is, therefore, the correlation between their future values (and therefore their exposures also). A high positive correlation between two transactions means that future values are likely to be of the same sign. This means that the aggregation benefit will be small or even zero, as is the case in Table 11.1, where the diversification effect is small. Aggregation only produces a reduction in exposure in scenarios where the values of the transactions have opposite signs, which occurs only in scenario 3. The aggregated EPE and ENE – compared to the sum of the individual values – are only reduced by a small amount.²⁴

On the other hand, negative correlations are clearly more beneficial as values are much more likely to have opposite signs, and hence the aggregation benefit will be stronger; this is illustrated in Table 11.2.

Appendix 11D gives a simple formula for the impact of aggregation on the exposure of a portfolio. It shows that the exposure reduction increases with the size of the portfolio and as the correlation between individual transactions reduces.

11.3.2 Off-market Portfolios

Offsetting effects in the aggregation are not always driven by the structural correlation between the future values of different transactions, but also by relative moneyness – for example, the extent to which the current value of a portfolio is significantly positive (ITM) or negative (OTM).

Figure 11.20 illustrates the impact of an OTM portfolio on the exposure of a new transaction. The OTM (negative value) portfolio is unlikely to have positive exposure unless the value of the transactions moves significantly. This, therefore, damps the positive exposure contribution of a new transaction in aggregate.²⁵

An ITM portfolio can also produce a beneficial aggregation effect on positive exposure, as shown in Figure 11.21. The negative value of a new transaction will have an impact on offsetting the positive exposure of the ITM portfolio. Put another way, the ENE of a new transaction is offset by the ENE of the portfolio.

Table 11.2 Illustration of the impact of aggregation when there is a negative correlation between values. The exposure metrics are shown, assuming each scenario has equal weight.

	Transaction 1	Transaction 2	Aggregated
Scenario 1	50	−25	25
Scenario 2	30	−15	15
Scenario 3	10	−5	5
Scenario 4	−10	5	−5
Scenario 5	−30	15	−15
EFV	10	−5	5
EPE	18	4	9
ENE	−8	−9	−4

**Figure 11.20** Schematic illustration of the impact of a negative future value on netting.

**Figure 11.21** Schematic illustration of the impact of a positive future value on netting.

The above effects are important since they show that even directional portfolios can have significant aggregation effects. They will help us to understand the behaviour of xVA at a portfolio level in later examples.

11.3.3 Impact of Margin

In general, margin has the effect of reducing exposure, and it can, therefore, simply be subtracted from the value of the portfolio to determine the positive and negative exposures. Equations 11.1 and 11.2 therefore become:

(11.5)

(11.6)

Table 11.3 Illustration of the impact of margin on exposure. The exposure metrics are shown, assuming each scenario has equal weight.

	Value (no margin)	Margin amount	Value (with margin)
Scenario 1	25	20	5
Scenario 2	15	12	3
Scenario 3	5	3	2
Scenario 4	−5	−3	−2
Scenario 5	−15	−16	1
EPE	9		2.2
ENE	−4		−0.4

Margin that is received is positive in the above equations and will, therefore, reduce positive exposure and increase negative exposure. Posted margin will do the opposite. This is in line with the fact that receiving margin mitigates counterparty risk, but posting margin may create counterparty risk (to the extent that it is not offset by the value). Concepts such as segregation (Section 11.4.3) will complicate the above description.

A simple example is given in Table 11.3, loosely assuming two-way collateralisation without initial margin. In scenarios 1–3, the positive exposure is significantly reduced since margin is held. The exposure is not perfectly collateralised, which may be the case in practice due to factors such as a rapid increase in value, or contractual aspects such as thresholds and minimum transfer amounts (Section 7.3.4). In scenario 4, the value of the portfolio is negative, and margin must, therefore, be posted, but this does not increase the positive exposure (again, in practice, due to aspects such as thresholds and minimum transfer amounts). Finally, in scenario 5, the posting of margin creates positive exposure.²⁶ In comparison with the benefits shown in the other scenarios, this is not a particularly significant effect, but it is important to note that margin can potentially increase as well as reduce exposure. Overall, both the EPE and ENE are reduced due to the two-way collateralisation. These effects will be seen in actual cases in Section 15.6.5.

Margin typically reduces exposure, but there are many (sometimes subtle) points that must be considered in order to assess the true extent of any risk reduction. To correctly account for the real impact of margin, parameters such as thresholds and minimum transfer amounts must be properly understood and represented appropriately. Furthermore, the margin period of risk (MPoR) must be carefully analysed to determine the true period of risk with respect to margin transfer. Quantifying the extent of the risk-mitigation benefit of margin is not trivial and requires many, sometimes subjective, assumptions.

To the extent that collateralisation is not a perfect form of risk mitigation, there are three considerations, which are illustrated in Figure 11.22:

Firstly, there is a granularity effect, because it is not always possible to ask for all of the margin required due to parameters such as thresholds and minimum transfer amounts. This can sometimes lead to a beneficial overcollateralisation (as seen in Figure 11.22), where the margin amount is (for a short time) greater than the exposure. Note that the analysis of exposure should consider the impact of posted, not just received, margin.

Figure 11.22 Illustration of the impact of margin on (positive) exposure, showing the delay in receiving margin and the granularity of receiving and posting margin amounts discontinuously. Also shown is the impact of the volatility of margin itself (for ease of illustration, this is shown in the last period only).
Secondly, there is a delay in receiving margin, which involves many aspects such as the operational components of requesting and receiving margin or the possibility of margin disputes. These aspects are included in the assessment of the MPoR (Section 7.2.3).
Thirdly, we must consider a potential variation in the value of the margin itself (if it is not cash in the currency in which the exposure is assessed).

We also emphasise that the treatment of margin is path dependent, since the amount of margin required at a given time depends on the amount of margin called (or posted) in the past. This is especially important in the case of two-way margin agreements.

Figure 11.23 shows the qualitative impact of collateral on exposure for three broadly defined cases:

Partially collateralised. Here, the presence of contractual aspects such as thresholds means that the reduction of exposure is imperfect. A threshold can be seen as approximately capping the exposure.

Figure 11.23 Illustration of the EPE of an interest rate swap with different levels of collateralisation.
Strongly collateralised. In this case, it is assumed that parameters such as thresholds are zero, and therefore the exposure is reduced significantly. We assume there is no initial margin and the MPoR leads to a reasonably material exposure.
Overcollateralised. In this case, we assume there is initial margin, and therefore the exposure is reduced further compared to the strongly collateralised case (and potentially to zero if the initial margin is large enough).

The impact of margin on exposure is discussed further in Section 15.5.

11.4 FUNDING, REHYPOTHECATION, AND SEGREGATION

11.4.1 Funding Costs and Benefits

Over recent years, a market consensus has emerged that uncollateralised exposures that give rise to counterparty risk and CVA also need to be funded and therefore give rise to additional costs. Such funding costs are generally recognised via funding value adjustment (FVA). It is, therefore, appropriate to discuss exposure from the point of view of funding.

The more detailed explanation and arguments around funding costs will be discussed in Chapter 18, but a basic explanation is as follows. A positive value represents an asset that needs to be funded, whilst a negative value represents a funding benefit (Figure 11.24). To some extent, funding costs and benefits cancel out since it is only necessary to consider them at a portfolio level (e.g. it is not necessary to fund transactions individually). The precise discussion of what ‘portfolio’ means in this context is discussed later in Section 18.3.2.

Note that not all assets give rise to funding costs. For example, it is possible to repo many bonds with good liquidity and low credit risk (e.g. treasuries). The receipt of cash from the repo transaction to a large extent offsets the funding cost of buying the bond. On the other hand, assets – such as derivatives – that cannot be ‘repoed’ in this way may be considered to have a funding cost.

Similar to the repo example above, if margin is received against an asset, then it may mitigate the funding cost, as long as it is reusable. Accordingly, margin that needs to be posted against a liability will likely negate any funding benefit. Hence, it is generally the portfolio value less the margin posted or received that determines the funding position. Loosely, the funding position is defined as:

(11.7)

Schematic illustration of the impact of a positive or negative value and margin on funding. — **Figure 11.24** Illustration of the impact of a positive or negative value and margin on funding.

There is a funding cost when the above term is positive, and a funding benefit when the term is negative. Posted margin will be negative and so create a positive funding cost, whilst received margin will be positive and represent a benefit.

11.4.2 Differences Between Funding and Credit Exposure

The concept of funding costs and benefits can, therefore, be seen to have clear parallels with the definitions of positive and negative exposure given earlier (compare Figure 11.24 with Figure 11.1). A positive exposure is at risk when a counterparty defaults, but is also the amount that has to be funded when the counterparty does not. A negative exposure is associated with own default and is also a funding benefit (we will see later that these two components are generally thought to overlap). Accordingly, any margin held against a positive exposure reduces both counterparty risk and the associated funding cost. Margin posted against a negative exposure reduces own counterparty risk and the funding benefit.

However, whilst positive and negative exposures defined for counterparty risk (CVA and DVA) purposes have clear parallels with funding positions, there are some distinct differences that must be considered:

Definition of value. The definition of value in the equations defining positive and negative exposure depends on close-out assumptions, since they only arise in default scenarios. Such valuations are subjective and need to be agreed between both the defaulting and surviving parties, based on the documentation and possibly subject to legal challenge (Section 6.3.5). With respect to funding, this can be considered to be an objective measure made by the party in question, since funding arises in non-default scenarios. However, the potential recursive problem (Section 6.3.4) with respect to the definition of value and the calculation of xVA (FVA in this case) does still exist.
MPoR. The MPoR is a concept that is defined assuming the default of the counterparty and is relevant for credit exposure. In assessing the equivalent funding delay in receiving margin against a derivatives portfolio, a more normal (i.e. not contingent on counterparty default) margin posting frequency (which is likely much shorter) should be assumed. As discussed in Section 18.2.5, this is one reason why the FVA of a collateralised derivative may be considered to be zero, even though the equivalent CVA is not.
Aggregation. Close-out netting is a concept that applies in a default scenario and hence credit exposure is defined at the netting set level (which may correspond to or be a subset of the counterparty level). On the other hand, funding may apply at the overall portfolio level, since values for different transactions are additive, and margin received from counterparties may be reused.
Wrong-way risk (WWR). Wrong-way risk is generally (although not always) a concept applied in relation to credit exposure as it relates to a linkage between the event of default and the underlying value of a portfolio. WWR, therefore, is less important to consider in the case of funding (although it will be discussed in Section 18.3.7).
Segregation. As discussed in Section 11.4.3, segregation has different impacts on credit exposure and funding because it prevents the reuse of margin.

Despite the above differences, credit, debt, and funding value adjustments (CVA, DVA, and FVA) have many similarities and are usually quantified using shared methodologies.

11.4.3 Impact of Segregation and Rehypothecation

Margin in derivative transactions can be seen to serve two purposes: it has a traditional role in mitigating counterparty risk, but it can also be seen as neutralisation funding. This is why margin can be considered to impact both credit exposure and funding costs.²⁷ Historically, the primary role of margin in OTC derivatives was to reduce counterparty risk. One way to observe this is in the prominence of one-way margin agreements (Section 7.3.2) for counterparties with excellent credit quality, meaning that the counterparty risk is low. In a more funding-sensitive regime, such margin terms are more costly because the associated funding costs may not be seen to be driven by the credit risk of the counterparty, but rather by the cost of raising funds by the party itself.

Whilst the traditional use of margin is to reduce counterparty risk, its role in defining funding costs and benefits has become increasingly important in recent years as funding has received closer consideration. Margin may, therefore, be complementary in mitigating both counterparty risk and funding costs. For example, receiving margin from a counterparty against a positive value has a two-fold benefit.

Counterparty risk reduction. In the event of the counterparty defaulting, it is possible to hold on to (or take ownership of) the margin to cover close-out losses.
Funding benefit. The margin can be used for other purposes, such as being posted against a negative value in another transaction.²⁸ Indeed, it could be posted against the hedge of the transaction.

However, as Table 11.4 illustrates, the type of margin must have certain characteristics to provide benefits against both counterparty risk and funding costs. Firstly, in order to maximise the benefits of counterparty risk mitigation, there must be no adverse correlation between the margin and the credit quality of the counterparty (WWR). Note that wrong-way margin does still provide some benefit as a mitigant as long as it retains some value when the counterparty defaults. A second important consideration is that, for margin to be used for funding purposes, it must be reusable. This means that margin must not be segregated and must be reusable (transferred by title transfer or allowed to be rehypothecated). In the case of cash margin, this is trivially the case, but for non-cash margin, rehypothecation must be allowed so that the margin can be reused or pledged via repo.

Table 11.4 Impact of margin type on counterparty risk and funding. In this context, WWR refers to an adverse relationship between counterparty default and the value of the margin (e.g. a counterparty posting its own bonds).

	Margin can be used	Segregated or rehypothecation not allowed
No WWR	Counterparty risk reduction Funding benefit	Counterparty risk reduction No funding benefit
WWR present	Limited counterparty risk reduction Funding benefit	Limited counterparty risk reduction No funding benefit

Consider the counterparty risk mitigation and funding benefit from various types of margin under certain situations:

Cash that does not need to be segregated. As discussed above, this provides both counterparty risk and funding benefits.
Securities that can be rehypothecated. As above, as long as the haircuts are sufficient to mitigate against any adverse price moves and also the corresponding haircuts associated with reusing the securities (e.g. the repo market or the margin terms for another transaction).
Cash or securities that must be segregated or cannot be rehypothecated. These provide a counterparty mitigation benefit since they may be monetised in a default scenario, but they do not provide a funding benefit since they cannot be reused in a non-default scenario. Note that this is one of the scenarios used in the Totem xVA consensus pricing service (Section 18.2.5), where it can be seen as a means to separate counterparty risk and funding-related costs.
Counterparty posting own bonds (that can be rehypothecated). These provide a questionable counterparty risk-mitigation benefit since they will obviously be in default when needed.²⁹ However, as long as they can be rehypothecated (and the haircuts are sufficient for this purpose), then they provide a funding benefit.

One example of the above balance can be seen in the recent behaviour of sovereigns, supranationals, and agencies (SSAs) counterparties, who have traditionally enjoyed one-way credit support annexes (CSAs) with banks and not posted margin due to their high credit quality (typically triple-A). SSAs have begun to move towards two-way margining, sometimes in the form of their own bonds.³⁰ This is because the traditional one-way agreement creates a very significant funding obligation for the banks, which is, in turn, reflected in the cost of the swaps SSAs use to hedge their borrowing and lending transactions. As banks have become more sensitive to funding costs, which in turn have become higher, the move towards a two-way margining means that a counterparty can achieve a significant pricing advantage (see later example in Section 21.3.4). Posting own bonds may be seen as optimal for a high-credit-quality counterparty because it minimises the liquidity risk it faces from posting other margin. Furthermore, thanks to its strong credit quality, the counterparty risk it imposes is less significant than the funding costs, and hence it most obviously needs to reduce the latter (put another way, it is focusing on minimising costs via the bottom left of the four scenarios in Table 11.4).³¹

11.4.4 Impact of Margin on Exposure and Funding

Given aspects such as rehypothecation and segregation, when considering the benefit of margin on exposure, it is important to carefully define the exposure and funding components with reference to the nature of the underlying margin. In general, margin should be subtracted (added) to the exposure when received from (posted to) the counterparty. However, segregation and rehypothecation create distinct differences. From the more general Equation 11.5, the positive exposure from the point of view of counterparty risk is:

(11.8)

where images is the value of the total margin received from the counterparty,³² and images is the margin posted to the counterparty that is not segregated. Any margin received, irrespective of segregation and rehypothecation aspects, can be utilised in a default situation. However, if margin posted is not segregated, then it will create additional counterparty risk, since it cannot be retrieved in the event that the counterparty defaults.³³

On the other hand, the funding position (generalised from Equation 11.7) is:

(11.9)

where images represents the margin received that can be rehypothecated (or, more generally, reused), and images represents all margin posted, irrespective of segregation and rehypothecation aspects.

Finally, to make these definitions more precise, we can distinguish between the two general types of margin (Section 7.2.3):

Initial margin. As seen from Equation 11.8, to minimise counterparty risk this needs to be segregated, otherwise posting initial margin will increase positive exposure. Furthermore, if initial margin is both posted and received, then the amounts will offset one another (in practice, initial margin received could be simply returned to the counterparty!). Two-way initial margin must, therefore, be segregated (as required by bilateral margin rules; Section 7.4),³⁴ and unilateral initial margin should ideally be segregated.³⁵
Variation margin. From Equation 11.9, we can see that variation margin should be rehypothecable (or reusable) so as to reduce funding costs. Whilst this potentially creates more counterparty risk when posting, since variation margin is typically already owed against a negative value, this should not be a major concern. Variation margin is never segregated and can typically be rehypothecated, since it is posted against a negative valuation and does not represent overcollateralisation.

Making the above assumptions regarding initial and variation margins, we can write the above formulas more specifically as:³⁶

(11.10)

(11.11)

The exposure for counterparty risk purposes can be offset by variation margin (which may be positive or negative), and initial margin received images . The funding position is fully adjusted by variation margin and increased by initial margin posted images . Note finally that in the absence of initial margin, the formulas above become similar, with caveats from the first four points in Section 11.4.2. Note that there are also some other points that may need to be considered above. For example, initial margin received may be considered to have a funding cost due to the need to segregate it with a third-party custodian.

Whilst both variation and initial margin impact funding, it is convenient to separate their effects into FVA and MVA (margin value adjustment) terms. This will become more obvious in the example below.