5
What is xVA?

5.1 OVERVIEW

Prior to 2007 and the global financial crisis (GFC), a number of fundamental assumptions were prevalent when pricing, valuing, and managing derivatives and similar products:

no credit risk;
the existence of a unique, risk-free rate;
the ability to borrow and lend at the risk-free rate (i.e. no funding costs);
no impact of collateral/margin (i.e. collateral was purely a risk mitigant and had no value in itself); and
no regulatory capital costs (although banks would typically impose hurdle rates with respect to economic capital usage).

Traditional derivatives pricing and valuation generally solely examined the impact of cash flows. For simple transactions, this problem was considered relatively straightforward and was often simply a question of applying the correct discount factor. The valuation was only treated as difficult where cash flows were themselves more complex, such as being non-linear, contingent, or multidimensional. These more complex payoffs or ‘exotics’ were difficult to value, but their vanilla equivalents were assumed to be relatively trivial.

In the years since the GFC began, these assumptions have been systematically challenged as a result of market practice (e.g. banks having material credit spreads), accounting and regulatory change. In general, this has led to a series of valuation adjustments that transform a simple valuation into the correct one by taking into account the above components.

Prior to the GFC, credit value adjustments (CVAs) and other valuation adjustments were largely unheard of. Banks may have priced the costs of counterparty risk, funding, and capital when dealing with some clients, but these costs were often not considered prohibitive to trading activity and profitability, even for the riskiest over-the-counter (OTC) derivatives. However, the events from 2007 onwards have resulted in increased costs with respect to these aspects. Additionally, accounting and regulatory standards have evolved to require certain practices around financial reporting and capitalisation, which have had significant cost implications.

Regulation has been a key driver of valuation adjustments. For example, Basel requirements around liquidity and regulatory capital have amplified the importance of funding and capital costs. In addition, the clearing mandate and bilateral margin requirements have created the need for the consideration of initial margin costs, historically rare in OTC derivatives.

5.2 ANALYSIS OF XVA

5.2.1 Definition

Before making further analysis, it will be useful to characterise valuation adjustments (xVAs) and discuss some general points of importance. A general and simple representation is:

(5.1)

where images refers to the different valuation adjustments for all relevant components. This is simply a statement that the correct value (or price) can be thought of as some base or ideal value – probably relatively simple to calculate – together with separate xVA terms.

A first implicit assumption is that of linearity, which means that the xVA components are completely separate from the actual value. In practice, this is not completely true, especially in relation to the consideration of default. This will be discussed later. In practice, the above separation will almost always be considered to be a reasonable, and necessary, approximation.

It is also useful to characterise xVA as allowing an institution to define the economic value of a given transaction. Such valuation is incremental since it represents the value after the transaction with respect to the value before. Off-balance-sheet transactions – such as derivatives – have a complex impact on valuation, which explains the significant complexity of valuation adjustments in contrast to simpler on-balance-sheet transactions such as loans. For example, if a bank lends a customer a certain amount, then they increase their credit risk by a well-defined amount and timescale, and the funding and capital implications are probably also relatively easy to understand. However, when the bank executes a derivatives transaction, the effects on credit risk, collateral, funding and capital are more subtle and complex. An obvious example of this is funding, since derivatives can be both assets and liabilities.

When we refer to value in the above context, it is important to consider what we mean by it. The value of a company can be defined as the sum of shareholder value and the value of its debt. Agency costs arise when shareholders are interested in a company following a risky strategy, since they will benefit from the entire upside whilst suffering limited downside. There is also a clear distinction in default where shareholders receive nothing, but bondholders are still paid some recovery value. These problems will be seen in valuation adjustments related to aspects such as funding where the shareholder and bondholder points of view will lead to different xVAs.

5.2.2 Components

Before making further analysis, it will be useful to characterise the different xVAs. As discussed in Section 3.2.3, the anatomy of valuation adjustments is normally defined as:

(5.2)

where the different terms refer to counterparty risk (CVA and DVA), collateral (ColVA), funding ( images ), capital ( images ), and initial margin ( images ). These are not the only definitions used by authors, although they are now relatively standard and will be used consistently throughout this book. By convention, the valuation adjustments will be represented as above, with a negative amount representing a cost (or loss) and a positive component, therefore, being a benefit (or gain). Some valuation adjustments ( images and images ) can represent a benefit from a valuation point of view. From a pricing point of view, all valuation adjustments can represent an incremental benefit due to portfolio effects.

Schematic illustration of an overview of valuation adjustments. — **Figure 5.1** Overview of valuation adjustments.

The valuation adjustments are also shown in Figure 5.1. Whilst not all the components have been fully explained at this point (we will refer to this figure in later chapters), note that there are three general considerations in defining a valuation adjustment:

Economic. This refers to the need to value components because of due diligence and market practice. Credit provisioning – the need to provision for credit losses – is one such example. Economic capital or funding are also examples.
Accounting. Distinct from economic considerations are accounting rules that may require a valuation adjustment to be included in a specific way. The obvious example here is CVA and DVA (see discussion in Section 5.3.3).
Regulatory. Regulatory considerations may shape the way in which valuation adjustments are treated. The obvious example here is with capital, where regulatory capital is probably considered more relevant than economic capital, as discussed in Section 4.2.1. This is also the case with funding and liquidity requirements, such as the liquidity coverage ratio (LCR) and net stable funding ratio (NSFR) (Section 4.3).

5.2.3 Why Valuation Adjustments?

The role of xVA (Equation 5.1) is to adjust some base value (e.g. one based on simple discounting) to take into account more complex components such as credit and funding and create the actual valuation. Whilst this approach is to some extent historical (i.e. starting from a traditional valuation), there are several reasons why the separation into valuation adjustments makes sense:

Complexity. The base value may be a relatively simple calculation, perhaps only reflecting the valuation of the cash flows as in the traditional approach. It may also conform to a certain type of transaction (for example, one that is centrally cleared).¹ On the other hand, the xVA components are much more complicated and require knowledge of contractual terms together with the economic impact of credit, funding, collateral, and capital. xVA requires a number of subjective parameters (e.g. counterparty credit spread curves, volatilities, and correlations).
Portfolio effects. Whilst the base valuation is typically additive across transactions, xVAs are generally portfolio-level calculations. Hence, whilst base valuations can probably be done at the transaction level, xVA terms must be calculated in a more portfolio-driven framework. Related to this, there is a question of directionality. In general, portfolios which are more unidirectional will be costlier in xVA terms, whilst more balanced positions will have lower costs.
Organisation. Whilst the idealistic, cash flow-driven valuation is suited to separate trading desks specialising in, for example, asset class, market, type of client or region, the xVA component necessitates management by a single dedicated desk. Such a desk is often known as the ‘xVA desk’ or ‘central desk’ (Section 21.1).
Accounting and regulatory. xVA is very much driven by specific accounting and regulatory requirements that impact the way in which various terms are treated. Knowledge of these requirements, both current and future, is a critical aspect of xVA quantification.

Whilst xVA is generally now accepted as an intrinsic part of derivatives valuation, some banks and other market participants have historically been slow to follow market practice. One of the primary reasons for this is that xVA components are generally costs and therefore have a detrimental impact on profitability over the short to medium term. Furthermore, xVA mispricing cannot generally be taken advantage of (arbitraged), and so banks pricing xVA inappropriately suffer only from an inability to correctly incentivise or disincentivise transactions in terms of their overall impact of the bank through time.

There might also be a question about where it is desirable and practical to incorporate charges such as xVA into transactions and where it is not. In general, xVA is extremely heterogeneous depending on the type of transaction, counterparty, and other aspects. It is, therefore, key to represent the costs at the transaction level for pricing and valuation purposes. This does not mean that xVA will capture all costs. However, other costs (e.g. operational expenses or capital charges that are not transaction specific, such as operational risk) are more homogeneous and therefore do not require specific attention for pricing and valuation.

5.2.4 Mark-to-market and xVA as a Cost (and Benefit)

Transactions such as derivatives are typically valued on a ‘mark-to-market’ (MTM) basis. MTM represents an accounting practice that involves recording the value of an asset or liability to reflect current market levels. MTM, therefore, means that the value is the current market price and not an alternative representation such as the historical cost. A MTM treatment is normally associated with the trading book of a bank that refers to transactions that are regularly traded or have values that can be derived from market observables that are regularly traded. By contrast, the banking book typically refers to assets that are not actively traded and are expected to be held to maturity – an obvious example being loans.

Since xVA terms are used in combination with base valuation via Equation 5.1, they are generally treated as MTM valuations even though they are not traded or based on market observables that are traded. Whilst this creates consistency in the overall valuation of derivatives, it creates inconsistency elsewhere. For example, the treatment of credit (counterparty) risk on a derivative may be different from the treatment of credit risk on a loan. Such inconsistency may be made even starker since banks may transact both products with clients (e.g. in making a floating-rate loan and providing an interest rate swap to convert this to a fixed-rate loan). Such inconsistencies are unavoidable but will provide important context for the way in which banks treat components such as CVAs.

Another important facet of MTM valuation is the treatment of revenue. Suppose a bank executes a long-maturity swap where they receive a rate slightly higher than the market rate – this difference representing their revenue. Contractually, this revenue is accrued over time, but from a valuation perspective it is all realised immediately through a higher MTM. This problem creates a fundamentally incorrect incentive in banks since the front office is potentially rewarded for looking at transactions with large MTMs, even if these have relatively high costs due to being long-dated and having high counterparty risk, collateral, funding, and capital costs.

To a degree, the incorporation of xVA in pricing and valuation can be seen to partially resolve the above dilemma since it ensures that future costs (and benefits) are correctly priced into transactions and then included in the ongoing valuation. Indeed, recent years have seen gradual incorporation of xVA components where banks have evolved from a traditional approach in which they are ignored to one in which they are treated more rigorously. In general, rather than ignoring costs altogether or accruing them over time, these costs are priced into transactions, form a part of the valuation process and are owned and managed centrally.

Table 5.1 shows a general evolution of xVA standards which has occurred at different rates. In general, CVA was the only component considered prior to the GFC, with major banks starting to adopt a more progressive CVA pricing and management approach from the late 1990s onwards. Funding and collateral were moves that started around 2010, heavily catalysed by changes in market conditions and regulation during this period. Finally, capital charges and KVA are not (yet) treated in a similar fashion to the other terms. Whilst capital is priced more rigorously than before, and hurdles are less ‘soft’ (in other words, it is harder to transact when the capital hurdle is not met), KVA does not form part of the valuation. Some argue that this is justified since capital is not a cost in the way the other terms are, but others argue that KVA is analogous to the other xVAs and so should be treated similarly eventually. We will discuss this further in Section 19.3.4.

Table 5.1 Evolution of xVA treatment.

	Traditional non-xVA approach	Market-standard xVA approach
Counterparty risk (CVA/DVA)	Default losses suffered as and when they occur.	CVA inception pricing, valuation, and management.
Funding (FVA/MVA)	Funding costs accrued on a daily basis with unexpected changes.	FVA inception pricing, valuation, and management.
Collateral (ColVA)	Management of operational aspects of collateral. Unexpected costs and benefits from collateral terms.	ColVA inception pricing, valuation, and central management.
Capital (KVA)	Businesses set ‘soft’ return on capital metrics.	KVA is priced more directly into transactions but with no valuation impact.

5.2.5 xVAs by Transaction Type

Very broadly speaking, there are four ways to transact derivatives from a counterparty risk point of view (Table 5.2):

Uncollateralised. An uncollateralised transaction has no agreement for either party to post margin/collateral against their obligations. As discussed previously, many end users, especially non-financial institutions, transact in this way. This situation clearly leads to significant counterparty risk, funding, and capital considerations, and so CVA, DVA, FVA, and KVA terms are important. There is no need to consider collateral or initial margin (ColVA and MVA).
Strongly collateralised. This usually means the traditional form of collateralisation in the interbank market where – generally – both parties post (receive) collateral (variation margin) based on their negative (positive) MTM value. Such collateral exchange clearly mitigates counterparty risk and funding and leads to lower capital charges, but potentially creates the need to consider ColVA.²

Overcollateralised. In this case, which is becoming more common, there is extra collateral (initial margin) which reduces counterparty risk and capital even further but leads to costs in terms of posting initial margin. In such situations, the collateral terms are often simplified (e.g. cash in a single currency)³ and so ColVA may be less important.

Table 5.2 Qualitative illustration of the relative importance of xVA terms in different relationships. Note that DVA adjustments are benefits and FVA adjustments can be costs or benefits.

	ColVA	CVA	DVA	FVA	KVA	MVA
Uncollateralised		✓✓✓	✓✓✓	✓✓✓	✓✓✓
Strongly collateralised	✓✓	✓✓	✓✓		✓✓
Overcollateralised	✓	✓	✓		✓	✓✓✓
Centrally cleared		✓			✓	✓✓✓

Centrally cleared. A centrally-cleared transaction should have minimal counterparty risk and funding requirements and low capital charges. Collateral adjustments will also likely be null due to the nature of collateralisation at a central counterparty (CCP) (cash in the currency of the transaction). However, there will be high costs of posting initial margin to the CCP.

The above analysis is simple and ignores many caveats, such as one-way collateralisation and type of collateral. The notation should not be taken literally (e.g. the MVA cost for an overcollateralised bilateral trade will not be exactly the same as for a centrally-cleared trade), but it is a general starting point for the breakdown of xVA.

An immediate consequence of the xVA differences above is that a risk conversion may occur when the transaction or counterparty terms are changed. In a sense, there is a sort of conservation of xVA where value can merely be pushed around with respect to the different components. Some obvious examples of these conversions are:

Collateralisation. This can reduce counterparty and funding costs but removes beneficial components such as DVA and funding benefits and also requires an analysis of specific optionality in the collateral agreement via ColVA.
Central clearing and bilateral collateral rules. The requirement to post additional collateral in the form of initial margin creates MVA but reduces CVA and KVA.
Hedging. Hedging CVA for accounting purposes may create additional capital requirements and therefore increase KVA. On the other hand, reducing KVA may lead to greater CVA volatility.

It should also be noted that some conversions are not apparent within xVA terms. For example, collateralisation generally reduces counterparty risk and funding costs for both parties. But it also leads to liquidity risk for at least one party due to the need to fund uncertain collateral requirements in the future.⁴ Whilst such components have no specific xVA value, they are certainly important considerations. Indeed, this is the reason why institutions benefiting from one-way margining have sometimes been reluctant to move to two-way collateralisation, even though this would reduce xVA costs (see further discussion in Section 7.6.2).

Finally, note that as some xVA terms are benefits, certain types of transactions may appear less favourable than in other situations. Consider collateral as an example: assets (i.e. lending) benefit from collateralisation, whereas liabilities (i.e. borrowing) do not. Since derivatives can be both assets and liabilities, it stands to reason that one party may see the lack of collateralisation as adding value. In a symmetric set-up this is clearly true, but it can even be true in the more realistic asymmetric set-up that xVA participants use. Put more simply, an end user may (although this is not common) actually see a better price for an uncollateralised derivative than for the same transaction executed on a collateralised basis (see Section 18.2.4).

5.2.6 Overlaps and Portfolio Effects

Some xVA terms will naturally interact. For example, capital requirements (KVA) will exist partly as a buffer against possible counterparty default, but in the event of the actual default these will gradually reduce whilst the accounting CVA increases (see discussion on ‘incurred CVA’ in Section 13.4.1). This is natural as capital held against potential losses reduces as actual accounting losses are taken.

There is also a more complex interaction that is harder to quantify and manage. The definitions for xVA terms (e.g. Figure 5.1) would suggest a valuation approach via the hierarchy of components. However, the real situation is more complex due to non-linearities and overlaps between the various terms. Rather than being a series of mutually exclusive terms, xVA components may share common economic features. Hence, there is a possibility of double counting or overlap which must be considered. Such overlaps are not always obvious and are often not treated rigorously. The common representation where xVAs can be calculated separately and independently is an approximation. In reality, xVA adjustments are interdependent and should, ideally, be computed jointly. For practical reasons, the assumption of independent terms is probably a necessary one, but certain ad hoc adjustments may be made in order to correct for this. These ‘overlaps’ are discussed later in the relevant places. Moreover, xVA can be recursive (i.e. the value today depends on the future strategy), which can be difficult to deal with in a tractable setting.

Another problem with xVA is that it is generally a portfolio-level calculation. Portfolio in this context often applies to a given counterparty, where adjustments such as CVA and ColVA depend on counterparty-specific terms such as netting and collateral agreements. However, the portfolio may also apply to the entire book of derivatives, as is relevant for some regulatory capital calculations and funding under certain assumptions. Indeed, it could even be argued that some xVA calculations (for example, FVA or the leverage ratio) require computation vis-à-vis the entire bank portfolio. This is not surprising since the aim of xVA can be seen to be to allow an institution to fully incorporate the economic value of a given transaction (Equation 5.1). However, from a pragmatic point of view, it is often necessary to make certain simplifying assumptions so as to be able to compute xVA in a practical way for pricing and valuation purposes.

The above explains why it is critical to manage xVA centrally and make consistent decisions regarding pricing, valuation, and risk mitigation so as to optimise aspects such as capital utilisation and achieve the maximum overall economic benefit.

5.2.7 CVA is the Least Real Valuation Adjustment

CVA is the oldest valuation adjustment. It is an explicit component of financial reporting (Section 5.3.3), and there is now a CVA capital charge (Section 4.2.5). Other valuation adjustments are more recent and have yet to become an explicit component of financial reporting or capital rules. However, accounting and regulatory considerations aside, CVA is the least real xVA term. This is due to the fact that CVA relates purely to the possible cost arising from a default event and there are no costs unless this default occurs. Furthermore, for good-quality credits, default is a relatively unlikely event. For similar reasons, DVA is probably so unreal as to be considered inapplicable (Section 17.3).

Hence, if the counterparty never defaults, then the CVA turns out (eventually) to be zero. However, this is not true of funding, collateral, and capital costs. These are typically seen on a daily basis and are necessary ‘production costs’ of a transaction when the counterparty doesn't default (Figure 5.2).

The above comment on CVA should not be taken too literally, but it is an important point to keep in mind. Whilst funding, collateral, and capital are constant needs, the nature of CVA is more binary. Banks may feel that CVA is less real due to having strong relationships with clients across potentially more than one business area, and will feel that the default of these clients is unlikely. Hence, they may feel that the CVA costs – as defined by accounting standards, for example – are not representative of their actual likely losses due to defaults.

Graph depicts the nature of xVA with respect to default. — **Figure 5.2** Nature of xVA with respect to default.

One could make an analogy between CVA and a short out-of-the-money option position. Such an option is unlikely to expire in-the-money and therefore may be viewed (in the real world) as having no value. Now, suppose a trading desk has an entire portfolio of such options: perhaps the best strategy is to do nothing and wait for all the options to expire when they will hopefully be worthless. If the trader hedges the options book, then losses will be crystallised through the hedging (although this will avoid large losses in case any of the options move in-the-money). Of course, the ‘do nothing’ strategy is difficult to justify since the price of options will be observable in the market and will also be hedgeable (via the underlying asset), and so controls such as risk limits will force hedging. However, for CVA, such market prices do not exist, and the hedging strategy is far more difficult to execute. Hence, CVA may seem partially unreal – especially for high-quality credits – as viewed from a purely economic perspective, and may be seen more like an accounting or regulatory requirement. This is not true for other valuation adjustments that are seen on a real and continuous basis, irrespective of counterparty defaults.

The above point will be relevant for later discussion on the use of market-implied default probabilities (Section 12.1.2), the management of CVA (Section 21.2), and the overlap between CVA and KVA (Section 19.4.1).

5.3 VALUATION

5.3.1 Price and Value

Price and value are not the same thing. A price is a relatively straightforward and objective quantity as seen by a given transaction in a given market (assuming such a transaction is observed). Value is a more subjective component as it may be considered to be linked to the gain from holding an asset or by disposing of it in the market. Furthermore, accounting and regulation impact the value of a given asset due to certain requirements or restrictions in relation to holding that asset. It may be useful to define the following different versions of value:

Economic value. This is often used to refer to the specific value of a derivative to a particular party. As such, it is a private valuation and can include entity-specific valuation adjustments. Economic value from a bank's point of view may well be derived from (practical) hedging considerations, including effects such as market inefficiencies. The potential to exit a transaction at some point in the future may play a role here but is probably not the primary concern.
Accounting value. Accounting value is the appropriate value that a transaction should be marked at for official balance sheet purposes and as such should be general and not dependent on entity-specific considerations. This is not necessarily the same as the economic view of a given bank, although there should be a clear connection.
Regulatory value. This component is increasingly important as more regulation begins to impact the derivatives market. Such regulation can essentially artificially reduce (or even increase) the value of certain transactions. For example, a bank may view a given transaction as beneficial from an economic and accounting point of view, but it may not be so beneficial in light of the underlying regulatory capital requirements or the leverage ratio (LR).

The above distinctions are important. In particular, some authors tend to focus on economic value, whilst others take more of an accounting-driven viewpoint. Arguments about the form or validity of xVA adjustments may be different when viewed from different perspectives. When the term ‘value’ or ‘valuation’ is used below, it should be assumed to be accounting driven unless otherwise stated.

The ‘law of one price’ states that there should be only a single price for a derivative transaction. Otherwise, arbitrageurs would simply buy and sell in different markets and remove the price difference. With respect to valuation adjustments, there could be two concerns here:

Asymmetry. Valuation adjustments are generally costs and may violate the law of one price since they create a pricing framework that is not symmetric.
Entity-specific charges. Some components, such as the funding and capital cost of a bank, are entity specific, meaning that different market participants will quote different prices.

The law of one price stems from no-arbitrage assumptions. However, the obvious arbitrageurs in the derivatives market with respect to valuation adjustments are end users that transact on a largely uncollateralised basis and are charged xVA components for counterparty risk, funding, and capital. However, such entities are not market makers, and their business model is not to exploit perceived inconsistencies in the pricing of banks. Moreover, it is generally not possible to trade xVA more directly – for example, in contracts such as contingent credit default swaps (Section 3.1.8).

Ultimately, xVA should be considered as a case of significant market incompleteness, and certain asymmetric and entity-specific charges (that are clearly a component of economic value) may still exist.

5.3.2 xVA Markets

It is also relevant to characterise two broad markets in which derivatives are traded:

Consumer market. This refers generally to end users using derivatives to fulfil hedging or investment purposes. An end user may only transact in one direction (e.g. always paying the fixed interest rate or buying oil). End users often do not exit transactions early, and when they do the price penalties may be quite significant. Transactions in these markets are rather one-way and asymmetric and are likely to be inefficient. That said, the end users will not be set up to take advantage of arbitrage opportunities. Directionality and lack of collateralisation can make xVA components very significant in these cases.
Intermediary market. This generally refers to the more sophisticated financial institutions that manage their positions with their clients via each other (e.g. in the interbank market). Some transactions will be centrally cleared. Since the players in this market are generally more sophisticated and manage two-way flows, the transactions tend to be more symmetric and efficient. Valuation adjustments will, in general, be smaller due to the use of risk-reduction mechanisms such as collateral. Arbitrage opportunities will be easier to exploit and the market will be more efficient.

The above characterisations, though general, are important since the xVA adjustments considered here originate mainly in consumer markets. Due to the nature of such markets, arguments based on considerations such as no-arbitrage and market efficiencies may be harder to apply.

5.3.3 Accounting Standards

Changes in accounting standards have been a key driver for the implementation of CVA. This has been driven by components of Financial Accounting Standards (FAS) 157, which are part of US GAAP (generally accepted accounting principles), and International Financial Reporting Standards (IFRS) 13, which are relevant for most other regions. Although there are some differences, there are many common elements between IFRS and US GAAP with respect to the measurement of fair value. These accounting standards provide principles on how to measure fair value, but this does not remove a certain level of subjectivity. In general, accounting standards are less prescriptive than regulation, such as Basel capital and liquidity requirements.

More generally, the fair price is defined as being an exit price. The wording in Financial Accounting Standards Board 157 (FASB 2006) reads:

Fair value is the price that would be received to sell an asset or paid to transfer a liability in an orderly transaction in the principal (or most advantageous) market at the measurement date under current market conditions (ie an exit price) regardless of whether that price is directly observable or estimated using another valuation technique.

Exit price would most obviously reference the current traded price of an asset. For example, the exit price of a bond would represent the current price at which it is trading in the secondary market (assuming this is a liquid market).

Exit price creates a problem for many transactions subject to significant xVA adjustments because a market price may not be observable. Even though many OTC derivatives markets are actively traded (especially for the most common products), such prices may not be relevant for similar transactions held by the reporting entity due to the specifics of the trading relationship. For example, the price of an interest rate swap on a swap execution facility (SEF) or in the interbank market (an intermediary market) would not be reflective of the exit price of the same transaction traded on an uncollateralised basis (in the consumer market). A problem with the fair value application to xVA is that a market of some type is assumed. Kenyon and Green (2014b) go so far as to argue that an uncollateralised market price is an oxymoron.

A fair value measurement should, therefore, take into account the elements that market participants would consider when setting the price for a transaction. Counterparty risk (CVA) is clearly defined as being a component of fair value. For example, IFRS 13 (IFRS 2011) states:

The entity shall include the effect of the entity's net exposure to the credit risk of that counterparty or the counterparty's net exposure to the credit risk of the entity in the fair value measurement when market participants would take into account any existing arrangements that mitigate credit risk exposure in the event of default.

The CVA for an uncollateralised derivative will clearly be materially greater than one that is collateralised (e.g. an interbank transaction), and this must clearly be accounted for. Although not explicitly mentioned, the concept of exit price would also indicate that other components that market participants would consider when setting a price, such as funding, should also be included in the valuation.

Related to the above point is that IFRS 13 and FASB 157 clearly define the need to include a risk premium. For example, IFRS 13 (IFRS 2011) states:

Market participants generally seek compensation (ie a risk premium) for bearing the uncertainty inherent in the cash flows of an asset or a liability. A fair value measurement should include a risk premium reflecting the amount that market participants would demand as compensation for the uncertainty inherent in the cash flows. Otherwise, the measurement would not faithfully represent fair value. In some cases determining the appropriate risk premium might be difficult. However, the degree of difficulty alone is not a sufficient reason to exclude a risk premium.

This implies the use of market-implied (or risk-neutral) parameters (e.g. implied volatilities and credit spreads) regardless of whether such parameters can be observed directly from market prices. This clearly has a fundamental impact on the calculation of CVA and other adjustments. The most significant aspect of this is the use of credit spreads, and not historical default probabilities, in the quantification of counterparty risk (CVA), as discussed in Section 12.1.2. This is important because banks, in general, may consider historical default probabilities to be more relevant for other assets such as loans.⁵

In accounting, whilst there is the concept of credit risk applicable to the fair value of assets, there is correspondingly the concept of own credit applicable to liabilities. This creates symmetry, where the borrower considers their own credit risk in the same way as their counterparty does. Since derivatives can be both assets and liabilities, the concept of exit price leads to the need for an own credit component to be included, which is usually known as debt or debit value adjustment (DVA). The relevant wording is:

IFRS 13: The fair value of a liability reflects the effect of non-performance risk. Non-performance risk includes, but may not be limited to, an entity's own credit risk.
FASB 157: The reporting entity shall consider the effect of its credit risk (credit standing) on the fair value of the liability in all periods in which the liability is measured at fair value.

The use of DVA can be questioned since, for example, a party with a worse credit quality may achieve better terms when exiting a transaction due to their counterparty experiencing a larger cost via CVA (which is removed after exiting the transaction). Hence, a counterparty with a large CVA may pay this in order to exit, which is seen by the other party as a DVA. Collective CVA and DVA adjustments conform to the law of one price (Section 5.3.1).

However, DVA is problematic since it appears on a bank's balance sheet as a profit which increases as the credit quality of the bank deteriorates. Moreover, the benefit is in relation to the default of the bank and so – unless the bank does indeed default – this depreciates due to time decay (Section 21.1.3). Furthermore, Basel III capital requirements disallow DVA accounting benefits from a capital point of view, BCBS 2011d stating:

Derecognise in the calculation of Common Equity Tier 1, all unrealised gains and losses that have resulted from changes in the fair value of liabilities that are due to changes in the bank's own credit risk.

This rule is to prevent a bank seeing an increase in its equity as a result of an increase in its credit risk and a reduction in the value of its liabilities.

DVA is therefore a good example of the difference between economic, accounting, and regulatory value (Section 5.3.1). Whilst it is an accounting requirement, it may not be viewed as economically meaningful and is certainly not part of the value from a regulatory point of view.

The market the exit price corresponds to should also be considered. Both IFRS 13 and FASB 157 define this as the ‘principal (or most advantageous) market for the asset or liability’. This can create problems, as the following comment illustrates:

In the case of Danske Bank, accountants are telling it that the exit price should reflect what one of its Nordic peers would pay for a trade; but Danske trades far more frequently with global dealers.⁶

From a CVA/DVA perspective, this can cause problems. For example, an institution may argue that it would exit a transaction with a counterparty who used historical default probabilities. Here, the explicit reference to components such as risk premiums would tend to suggest that this would be inappropriate. Given a smaller bank's relative inactivity in some derivatives markets, it would seem natural to view the exit price as relating more to a global dealer than a smaller peer bank.

5.3.4 Accounting Trends

As discussed above, accounting standards reference only CVA and DVA components directly. However, they implicitly suggest that other components should be included if they are a component of the exit price of a transaction. Indeed, it has become relatively common practice to also include funding value adjustment (FVA) in financial reporting. For example:

… a fair-value adjustment was applied to account for the impact of incorporating the cost of funding into the valuation of uncollateralised derivatives.⁷

Regarding pricing and valuation of xVA components, banks generally want the economic value to align with official (e.g. accounting-based) numbers. The reason for this is that banks measure performance annually and base the performance of their derivatives business primarily on the basis of MTM valuation (as opposed to other measures such as return on capital). Hence, anything that a bank considers to be a real economic cost should be included in the official valuation so as to avoid reporting misleading valuations and distributing profits gross of such a cost. This may explain the motivation for banks to account for other components such as FVA. Another motivation is the overlap between DVA and FVA, which will be discussed in Section 18.2.5.

However, adjustments such as FVA lead to further complexities. Firstly, they create a framework where the law of one price – seemingly a key feature of the requirement to include DVA – no longer holds. They also create a recursive problem since one party's exit price is another's entry price, and so suggests that the funding costs should be those of the party with whom the transaction would be exited (e.g. a novation to another bank),⁸ rather than the entity making the calculation. Note also that both IFRS 13 and FASB 157 state that a ‘fair value is a market-based measurement, not an entity-specific measurement’. Hence, a bank seems unable to value with its own cost of funding, but rather with something closer to the ‘market cost of funding’. It can, of course, price with its own cost of funding, but it may then see a deviation between price and valuation.

There have been various arguments about how to resolve the above contradictions. For example, Albanese et al. (2015) consider an exit price based on ‘a competitive auction where entities of all types, including unlevered real-money funds with negligible funding costs, are allowed to participate’. Whilst exit prices permit reference to the most advantageous market – which may imply counterparties with minimal funding costs – it is not practically possible to novate derivatives transactions to such counterparties since they do not play such a role in the market. Hence, it may be argued that whilst, in theory, they should not exist, market inefficiencies may lead to real adjustments such as FVA, which are reasonable components of the actual exit price.

Whilst funding and FVA have become fairly standard adjustments in valuation, as well as pricing, other valuation adjustments such as MVA (initial margin) and KVA (capital) have not yet been treated in a similar way.⁹ However, since these are clear components of entry prices, it could be argued that it is only a matter of time before they are treated in a similar manner to FVA. This will be discussed in more detail in Sections 19.3.4 and 20.3.2.

5.3.5 Totem

IHS Markit's Totem service is a well-established utility for valuation consensus, enabling market participants to price test their own valuations for various products: each firm submits prices for specific trades and IHS Markit normalises the data and returns anonymous data to participants.¹⁰

One part of Totem is an xVA service that covers transactions subject to valuation adjustments involving credit risk, funding, collateral, and initial margin (at the time of writing, capital-related valuation adjustments or KVA are not considered). Only contributors see the results of the exercise, which are not public domain, and so the discussion below describes only the general approach. Products covered are mainly interest rate swaps with a variety of maturities, moneyness, and margin/collateral terms. Whilst xVAs are not seen explicitly, the coverage is comprehensive so as to allow extraction of the assumptions for pricing credit and funding in collateralised and uncollateralised trades across the anonymous list of banks.

The Totem xVA results can be used to answer questions such as:

What general valuation adjustments (e.g. CVA, DVA, FVA) does a bank consider when pricing a transaction?
What specific assumptions does the bank use in the calculation of a valuation adjustment (e.g. loss given default quantification)?
What cost does the bank attach to entity-specific components (e.g. its funding cost)?
How do banks deal with collateral in transactions given that they have a relationship to counterparty risk, funding, and collateral adjustments?

Since results are displayed bank by bank (anonymously), it is possible to see a degree of convergence in certain aspects. The prices submitted are, by convention, supposed to relate to the cost of assignment or novation (i.e. the participant is pricing the cost of stepping into an existing transaction with a given counterparty) and so have a clear notion of an entry price, which in turn implies that the aggregative data says something about the exit price.

Since the consumer market is not observable for xVA prices, Totem is the only place where certain pricing is clearly observable, and it is possible to try to extract a consensus. This may be used to defend the use of a given accounting adjustment (for example, the adoption of FVA was likely justified and driven partly by this initiative). Totem xVA has also created some convergence within xVA pricing and valuation, although there are still significant differences. Some of these differences may be expected to disappear over time as market practice becomes more well defined (e.g. with mathematical formulas and numerical implementations), but some differences might always be expected to persist (e.g. banks having different funding costs).

Contributing to Totem can give a bank verification of its own pricing and valuation approach and may give internal and external credibility. However, there is also the potential drawback that the Totem results may indicate that a bank is an outlier with respect to certain individual assumptions, which may then need to be changed, especially if they are seen as not conservative with respect to market practice. Some example results will be shown in Section 18.2.5.

5.3.6 Contractual Terms and Value

Certain contractual terms require a definition of value in order to facilitate collateralisation or some settlement of a transaction.

Collateral (margin). Collateral exchange will require a definition of the value of a portfolio in order to determine the collateral amount.
Resets. Some transactions, notably cross-currency swaps (sometimes called MTM swaps), have a reset feature (see Section 7.1.2) where a cash payment is made to neutralise the value of the transaction at pre-specified dates.
Termination. Clauses may exist (Section 7.1.1) whereby the transaction may be terminated at given points, either optionally or based on certain events (e.g. a rating downgrade). In such cases, the transaction would be settled at its prevailing market value.
Close-out. In the event of a counterparty default, the value of a portfolio with respect to a party defines their claim (or liability) with respect to the defaulted counterparty.

All of the above require some definition of value. From a high level, the question is, therefore, whether the value term should be defined (Equation 5.1) as being the actual value (with xVA) or the base value (without xVA).

Not surprisingly, it is base values that are generally referenced within contractual definitions. The possible exemption to this is close-out, as discussed in more detail in Section 6.3.4. The use of base value is partly for historical reasons (in the past, xVA was not seen as important) and partly for reasons of simplicity. However, this creates a potential issue: for example, a transaction being terminated will jump from its xVA-inclusive value to a value without xVA. A more theoretically-appealing solution would be, therefore, to base contractual terms on actual valuations, but this would create significant complications via more subjectivity and also a recursive problem whereby contractual terms would impact xVAs, which in turn would define contractual terms.

5.4 PRICING

5.4.1 Reality or Creating the Right Incentive?

Traditional derivatives pricing is based on so-called risk-neutral valuation principles which are directly linked to hedging. There is, therefore, a fairly direct relationship between the costs and benefits reflected in the price and those monetised as a result of the hedging strategy.

However, the costs and benefits reflected in valuation adjustments are generally more indirect, and it could be argued that a bank would not need to adjust its trading activities or capital structure every time it entered into a new transaction. However, the way in which xVA is priced will probably reflect an implicit assumption that this actually is the case. For example:

Credit. The pricing of CVA may implicitly assume that a bank would immediately buy credit default swap (CDS) protection on the counterparty in question. In reality, this may not be possible, and the bank may not buy CDS protection in any form as a result of one transaction, even though it may incur additional credit risk.
Funding. The pricing of FVA and MVA is linked essentially to an assumption that the bank will issue unsecured debt of some type(s) as a result of the transaction. In reality, unless the transaction is very large, the funding position will not need to be altered.
Capital. The pricing of KVA or the requirement to meet a certain capital hurdle will inherently assume that the bank will need to raise more equity capital via a rights issue.

In reality, of course, alternations to a capital structure are ‘sticky’ as they cannot happen continuously and occur relatively discretely, and a bank will have some capacity. For example, capital buffers ensure that new business does not cause a bank to breach its regulatory capital requirements. Even hedging with illiquid transactions such as CDS must be done on a relatively discrete basis. The same is true in reverse, as transactions that ‘realise’ xVA may be priced as if the benefit from cost reduction can immediately be monetised by retiring debt, buying back shares, or selling CDS protection.

Why, therefore, is it important to price credit, funding, or capital into a transaction if a bank has the capacity already available and will, therefore, bear no direct economic costs? The answer is that xVA pricing is creating the right incentive and penalising (motivating) transactions that will lead to additional costs (benefits) in general. However, the lack of direct linkage to costs and benefits may lead to xVA pricing being seen as not necessarily as rigorous and well defined as otherwise might be the case.

5.4.2 Approach for Capital

An extreme case of the comments above relates to capital costs. Whilst banks have always had a notion of achieving the correct return on capital when pricing transactions and making business decisions, this may be done relatively passively. Accordingly, not meeting a capital hurdle may not necessarily mean that a transaction could not be executed, as there are other perceived benefits (such as building client relationships).

Furthermore, there is the question of whether or not capital is a cost (Figure 5.1). Buying CDS protection (CVA) or issuing fixed-rate debt (FVA/MVA) incurs fixed costs, but raising equity capital does not, since dividends paid to common shareholders that form the majority of Common Equity Tier 1 capital (Section 4.2.1) are discretionary and will depend on the performance of the bank in question. This leads to two opposing points of view:

Capital is not a direct cost. Capital is not a direct cost and so need not form an explicit valuation adjustment. Imposing a fixed return on capital for all transactions and business is inappropriate, especially since banks may offer multiple services to some clients. Hence, the phrase ‘return on capital’ is more appropriate than KVA, and some businesses may contribute less return per unit capital but cannot necessarily be closed down as a result, since they may add franchise value. There should be no accounting adjustment in relation to capital.
Capital is a direct cost. Whilst common stock dividends are discretionary, investors do implicitly require a certain return and will sell their shares if they do not achieve this. Hence, capital is another cost and the term KVA can be used to express this. If a business or transaction cannot achieve the correct KVA, then it cannot be justified and the bank should look at opportunities elsewhere. Like FVA, KVA is a component of the exit price, as all market participants require a return on capital, and so there should be an accounting KVA adjustment.

At the time of writing, it is hard to say which of the above views will prevail. Certainly, no bank has yet gone as far as making a full accounting adjustment for KVA. There are reasonable arguments in both points of view, and we will discuss this again when considering the management of KVA in Section 19.3.4.

5.4.3 Approach to Regulatory Ratios

As described in Chapter 4, the leverage ratio (LR) (Section 4.2.7) and liquidity ratios (Section 4.3) require a bank to conform to a metric which is a regulatory assessment of their capital structure with respect to capital or funding. Even if a bank prices these components into a transaction via a traditional approach to xVA, there is the question of the impact of the new transaction on these ratios and whether they will deteriorate. Of course, a bank will naturally build in a buffer and so no single transaction will cause a breach. However, in line with the discussion about pricing to incentivise (Section 5.4.1), it is natural to incentivise (disincentivise) a transaction that worsens (improves) the regulatory metric.

In general, the required ratios for capital and liquidity can be seen as being driven by a resource component, such as capital or available funding, divided by a risk component, such as exposure or required funding. We can denote these components as images and images and assume that a new transaction will cause the risk component to change by an amount images , for which the bank may change the resource component by images . Suppose that the bank's desired ratio is images and that its current ratio is images :

(5.3)

This will imply that:

(5.4)

The second term in Equation 5.4 is an excess amount of the ratio that can be utilised by the new business. This means that the bank is happy for the ratio to reduce from images to images . If the bank is not happy to allow this, or equivalently if images , then the above formula reduces to:

(5.5)

This requires that new business be charged in accordance with keeping the regulatory ratio constant. We will refer to Equation 5.5 as ‘ratio invariance pricing’.

Table 5.3 Simple LR example.

	Tier 1 capital	Exposure	LR	LR-implied capital	Actual capital	Ratio
Business 1	15	200	7.5%	10	15	7.5%
Business 2	10	400	2.5%	20	15	3.75%
Total	25	600	4.2%	30	30	5%

Consider the simple example in Table 5.3. Suppose that the bank's required LR is images and:

Business 1 has a Tier 1 capital requirement (without consideration of the LR) of against an exposure of . This leads to an implied LR of 7.5%.
Business 2 holds less capital against an exposure due to being less risky, and it has an implied LR of only 2.5%.

The bank's overall LR would be 4.2%, which is not above the required amount. For the bank to meet the LR target overall, Business 2 may be charged an extra five units (on top of 10) of Tier 1 capital based on Equation 5.4.¹¹ Note that Business 2 on a standalone basis has an LR of only 3.75% now, but, thanks to the position of Business 1, the overall LR of the bank is at the desired LR.

In order to price based on ratio invariance, the bank will need to charge for an increase in resources in line with the relative increase in the risk being added. For example, a NSFR invariant price would require pricing in additional funding based on the NSFR ratio, which will naturally be different from the amount of funding that would be required otherwise. An LR invariant price would price in raising enough capital to keep the bank's LR constant. Again, this will almost inevitably be different from the amount of capital the bank would assess without reference to the LR.

Whilst many assets may be assumed to be naturally LCR and NSFR compliant, i.e. their natural funding strategy would generate more available stable funding (ASF) than required stable funding (RSF), some others will not be. Some aspects of derivatives clearly fall into the latter category. Indeed, derivatives do not generate any net ASF and so cannot possibly have a standalone NSFR of more than 100% (the more derivatives liability, the worse the NSFR contribution). This is not the case for the LR which may imply capital that is either larger or smaller than the amount defined by traditional capital rules. In general, for relatively complex transactions such as derivatives, metrics such as the NSFR and LR are simple and have specific features that may be seen as being particularly conservative and even non-economic. Hence, it may be expected that maintaining invariant regulatory metrics is more expensive than more economic definitions of capital and funding. This would imply that terms such as KVA and FVA should have such metrics built in on a worst-case basis (i.e. a transaction must cover the increase in basic capital costs and maintain the bank's LR at the same or a better level).

The counterargument to the above is that a bank has many activities that can be collectively beneficial. Suppose a bank has a particular division that generates capacity in a given regulatory metric. One example could be lending to relatively weak credits which will incur accordingly high capital charges but will probably have a favourable effect on the LR since it is credit-quality insensitive (Section 4.2.7). It could be viewed that this area will generate ‘leverage ratio capacity’ that other businesses can utilise. Ultimately, there is no point in charging for the LR if there is not a material possibility that the business in question may contribute to an eventual breach of the regulatory requirement.

It is clearly important – but not trivial – for banks to decide to what extent additional LCR, NSFR funding costs, and LR costs are passed on to originating businesses. Examples of NFSR invariance (Section 18.3.4) and LR invariance (Section 19.2.6) will be given later.

5.4.4 Lack of Arbitrage

Traditional pricing of complex components such as exotic derivatives (exotics) is a useful comparison to xVA. With exotics, there is a large amount of expertise required in terms of quantitative analytics, trading, and risk management capabilities. Banks mispricing can be arbitraged by more sophisticated competitors. Traditionally, this meant that smaller banks would not try to compete in pricing certain exotic products because they lacked the economy of scale and the ability to build the required expertise.

Whilst xVA pricing shares similarities with exotics pricing in terms of quantitative techniques, the underlying environment is very different. Firstly, complex xVA adjustments arise from products that have been viewed historically as being simple, such as interest rate swaps. A bank must, therefore, approach xVA by necessity and not by choice.

There is no arbitrage in xVA markets. Banks quote prices to end user clients who mainly transact in only one direction. It is, therefore, not possible for banks to exploit the mispricing of a competitor. Since xVAs are mainly costs, a bank mispricing will therefore only suffer from a ‘winner's curse’ by taking on business too cheaply.¹² Such mispricings may not be immediately obvious but may potentially lead to accounting losses at some point in the future.¹³ This is one reason why the adoption of market-standard xVA approaches has often been slow, especially amongst smaller and regional banks. Clearing prices for xVA-heavy products are often driven by the ‘lowest common denominator’, with banks incentivised to price more aggressively because competitors may ignore or underprice certain components. The herd mentality can sometimes be a strong consideration: a bank may treat something in a particular way because that is what ‘market practice’ represents, even if the bank believes this to be inappropriate.

It probably shouldn't be expected that, like exotics pricing, xVA prices will all converge on a well-defined value, with banks quoting very close to this value. With so much intrinsic complexity within the modelling, unobservable parameters, and potentially entity-specific costs such as funding and capital, some reasonable dispersion is not surprising. Note also that banks who are most aggressive on pricing one transaction may not be so on another. One example of this is with respect to funding: since there are both costs and benefits, some transactions may achieve the best price from a bank with a low funding cost, whilst others may actually be more aggressively priced by a bank with a relatively high cost of funding (see Section 18.2.4).

5.4.5 Entry and Exit Pricing

As xVAs have become more significant in recent years, transacting derivatives has generally become more expensive. Within banks, sales and trading have found xVA charges to be problematic to pass on to clients and have thus suffered from reduced profitability. However, it is also important to bear in mind that exit prices change together with entry prices.

Suppose (Figure 5.3) that a bank charges (implicitly or otherwise) a certain amount at the inception of a transaction (entry), of which some is xVA and the rest is profit. Assume that the xVA component is fully accounted for and so is not realised as profit. Now, if the transaction is exited at some point in the future (exit), then the replacement counterparty will want to charge xVA and also make some additional profit.¹⁴ The original counterparty will be able to pay the xVA charge directly from their xVA accounting reserves, although any extra charge will result in a loss.

This illustrates that high xVA charges at inception can potentially be monetised when exiting the transaction, if the xVA charged at exit is less than the accounting xVA prior to exiting the transaction. This will be the case even if the xVA is higher than at inception because the corresponding accounting losses will have been taken (and may have been hedged), and the xVA desk will have the incentive to pay out so as to achieve xVA optimisation. Hence, whilst client relationship may be less profitable at inception, there is better potential for gains when restructuring, unwinding, and novating transactions later. This is different from the traditional way in which client-driven derivative markets work, where initial transactions are profitable but there is little incentive for salespeople to focus effort on restructuring such transactions later.

Schematic illustration of xVA charges at entry and exit of a transaction. — **Figure 5.3** Illustration of xVA charges at entry and exit of a transaction.

This example may also be seen as an argument for components such as KVA being part of the accounting xVA (Section 5.4.2). If this is not the case, then it will be released as P&L at the entry of the transaction, which would suggest that exiting the transaction can only be done by taking a loss (via paying KVA to the replacement counterparty as an additional charge).

On the other hand, xVA benefits at the entry of transactions may make it harder to exit such transactions due to the loss of these benefits, although counterparties should be able to pay for these benefits (e.g. by paying for the funding generated when entering into an OTC portfolio).

5.4.6 xVA Quantification

Previously, Figure 3.10 gave a generic overview of the computation of an xVA term. A general formula for xVA can be written as:

(5.6)

where there is an integration from the current time to the maximum maturity of the portfolio in question over the following three terms:

A curve, . This defines the unit cost of the underlying cost or benefit. For example, a counterparty credit spread curve defines the cost of the credit risk, whilst funding and capital curves define the cost of funding or capitalising a position. This component is normally assumed to be deterministic (e.g. a funding cost is fixed).
A utilisation component, . This defines the magnitude of the utilisation with respect to cost or benefit above. Generally, this component is not deterministic and has to be modelled. From a quantification point of view, an expected value, , is required.
Discounting assumptions, . Since xVA represents the present value of future costs and benefits, these need to be discounted. However, this is not trivial and may reflect credit risk as well as interest rates.

The three terms above are often assumed to be independent and/or deterministic, meaning that they can be modelled and quantified separately. However, there are some situations when recognising dependencies may be viewed as important. The most well-known of these cases, is known as wrong-way risk (WWR), which usually applies to CVA and relates to a dependency between the counterparty credit spread and the exposure. This and other situations will be discussed when relevant.

5.4.7 Special Cases

The calculation of the profile images in Equation 5.6 is generally a significant quantification challenge, with issues over model choice, calibration, numerical tractability, path dependency, and portfolio effects. In general, it requires the valuation of option-like payoffs and indeed may well represent a giant option on a multi-asset portfolio. However, in certain special cases, the valuation collapses to essentially pricing forward contracts and is therefore largely model independent and separable across transactions.

These special cases arise since the value of images in the future equates to the future MTM of the portfolio, either due to collateralisation or because this represents the amount of the funding position. They will, therefore, be discussed later in Sections 16.2.1 and 18.2.3. These special cases can be dealt with by simply changing discounting assumptions (i.e. Equation 5.6 is not required specifically).

The special cases also relate to the starting point for xVA calculations (Equation 5.1) and may be incorporated directly into the ‘base value’. There is no obvious way in which to decide on this starting point, which may relate to the easiest way to calculate and manage xVA across an organisation.