17
CVA

17.1 OVERVIEW

This chapter will introduce the first members of the xVA family, namely credit or counterparty value adjustment (CVA) and debt or debit value adjustment (DVA). It will be shown that, under fairly standard assumptions, CVA and DVA can be defined in a straightforward way via credit exposure, default probability, and loss given default (LGD). We will then discuss computational aspects and show example calculations.

CVA has become a key topic for banks in recent years due to the volatility of credit spreads and the associated accounting (e.g. IFRS¹ 13, Section 5.3.3) and capital requirements (Basel III). However, whilst CVA calculations are a major concern for banks, they are also relevant for other financial institutions and corporations that have significant numbers of over-the-counter (OTC) derivatives to hedge their economic risks.

A key and common assumption made initially in this chapter will be that credit exposure and default probability are independent.² This involves neglecting wrong-way risk (WWR), which will be discussed in Section 17.6. We will also discuss CVA and DVA in isolation from other xVA terms, which will then be dealt with in more detail in Chapters 18 to 20. This is an important consideration since xVA terms cannot, in reality, be dealt with separately and possible overlaps should be considered.

CVA was originally introduced as an adjustment to the risk-free value of a derivative to account for potential default. Rather than using the term ‘risk-free value’, it is possible to define CVA as being the difference between the value with and without counterparty default. Historically, CVA was seen as a ‘credit charge’ for pricing and a ‘reserve’ or ‘provision’ for financial reporting purposes. Such calculations were often made with historical parameters (e.g. volatilities and default probabilities). More recently, accounting requirements (Section 5.3.3) have meant that CVA has become defined via an ‘exit price’ concept and computed with market-implied (risk-neutral) parameters.

In general, CVA is computed with market-implied parameters where practical. Such an approach is relevant for pricing since it defines the price with respect to hedging instruments and supports the exit price concept required by accounting standards. Risk-neutral exposure simulation (Section 15.3.3) is relatively natural due to the parallel with option pricing (e.g. Section 15.1.3) and the fact that many parameters (e.g. forward rates and volatilities) can be derived from market prices. Of course, certain parameters cannot be market-implied since they are not observed in the market (e.g. correlations), or may require interpolation or extrapolation assumptions (e.g. long-dated volatilities). Risk-neutral parameters such as volatilities may generally – but not always – be higher than their real-world equivalents (e.g. historical estimates).

A more controversial issue is the default probability component of CVA. Obtaining market-implied default probabilities in most cases requires proxy credit spread curves to be defined (Section 12.3). As for exposure, the use of market-implied parameters is relevant for pricing purposes. However, the use of market-implied default probabilities may be questioned for a number of reasons:

• market-implied default probabilities are significantly higher than their real-world equivalents (Section 12.1.1);
• a default cannot, in general, be hedged since most counterparties do not have liquid single-name credit default swaps (CDSs) referencing them; and
• the business model of banks is generally to ‘warehouse’ credit risk, and therefore they are only exposed to real-world default risk.

The above arguments are somewhat academic, as most banks (and many other institutions) are generally required by accounting standards to use credit spreads when reporting CVA. There are, however, cases where historical default probabilities may still be used in CVA calculations today, such as in the case of smaller regional banks with less significant derivatives businesses, who may argue that their exit price would be with a local competitor who would apply a similar approach.

In situations such as the above, which are increasingly rare, banks may see CVA as an actuarial reserve and not a market-implied exit price.

17.2 CREDIT VALUE ADJUSTMENT

Standard early reference papers on the subject of CVA calculation include Sorensen and Bollier (1994), Jarrow and Turnbull (1992, 1995, and 1997), Duffie and Huang (1996), and Brigo and Masetti (2005a).

17.2.1 CVA Compared to Traditional Credit Pricing

Pricing the credit risk for an instrument with one-way payments, such as a bond, is relatively straightforward – one simply needs to account for default when discounting the cash flows, and add the value of any payments made in the event of a default. However, many derivatives instruments have fixed, floating, or contingent cash flows or payments that are made in both directions. This bilateral nature characterises credit exposure and makes the quantification of counterparty risk significantly more difficult. Whilst this will become clear in the more technical pricing calculations, a simple explanation is provided in Figure 17.1, which compares a bond to a similar swap transaction. In the bond case, a given cash flow is fully at risk (a portion of its value will be lost entirely) in the event of a default, whereas in the swap case only part of the cash flow will be at risk due to partial cancellation with opposing cash flows. The risk on the swap is clearly smaller due to this effect.³ However, the fraction of the swap cash flows that is indeed at risk is hard to determine, as this depends on many factors such as yield curve shape, forward rates, and volatilities.

Figure 17.1 Illustration of the complexity when calculating the CVA on a derivative instrument such as a swap, compared with pricing credit risk on a debt instrument such as a bond. In the bond, the cash flow circled is fully at risk (less recovery) in the event of default of the issuer, but in the swap the equivalent cash flow is not fully at risk due to the ability to partially offset it with current and future cash flows in the opposite direction (the three dotted cash flows shown circled).

The above will be illustrated in more detail in Section 17.2.4.

17.2.2 Direct and Path-wise CVA Formulas

CVA can be seen as being driven by three separate terms:

• Does the counterparty default?
• If the counterparty defaults, what is the exposure at that time?
• How much of the exposure is lost?

The standard unilateral CVA (UCVA) formula (which assumes the party making the calculation cannot default) can be written as the following expected value (Appendix 17A):

(17.1)

where represents the current time, the (maximum) maturity of the portfolio, and the default time of the counterparty. The terms inside the expected value are as follows:

• is the indicator function, which takes the value 1 if the default time of the counterparty is within the maturity in question and zero otherwise. The expected value of this indicator function should be the cumulative default probability of the counterparty in the interval .
• is the exposure in the event of default discounted to time . The expected value of this at a given default time is the expected positive exposure (EPE), as defined in Section 11.1.5.
• is the loss given default. This is the percentage amount of the exposure expected to be lost if the counterparty defaults. Note that sometimes the recovery rate (R) is used with .

Figure 17.2 Illustration of a direct CVA calculation.

Note that the above three components may be dependent within the calculation specified by Equation 17.1. The above formula could be implemented with a scheme as follows:

1. Simulate a random time of default, , consistent with the underlying credit spread curve (does the counterparty default?).
2. Simulate a value of the portfolio at the future default date discounted back to time , and from this define the exposure at default as simply the positive part of this term (if the counterparty defaults, what is the exposure at that time?).
3. Simulate an LGD value (how much of the exposure is lost?).
4. Multiply the above terms.
5. Repeat and average.

The above process is greatly simplified by assuming that the default time, the exposure at default, and the LGD are independent (no WWR), which also means that only a single average LGD need be used. This will be referred to as a ‘direct CVA calculation’ (Figure 17.2). Note that it is consistent with the direct approach to exposure simulation described in Section 15.3.1, since it is only necessary to evaluate the value of the portfolio at a single potential default date in the future. It is also efficient only to simulate default times that are in the interval and then multiply the final result by the counterparty default probability in this interval.

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Spreadsheet 17.1 Direct CVA calculation for an interest rate swap.

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Another way to write the CVA expression is via a one-dimensional integral over time:

(17.2)

where is the instantaneous default probability, is a ‘risky discount factor’, and is the EPE.

Figure 17.3 Illustration of CVA formula. The component shown is the CVA contribution for a given interval. The formula simply sums up across all intervals and multiplies by the LGD.

The above formula is typically approximated as a discrete sum:

(17.3)

The UCVA depends on the following components:

• LGD. The loss given default, as described above.
• EPE. The discounted EPE for the relevant dates in the future given by . Although discount factors could be represented separately, it is usually most convenient to apply (risk-free) discounting during the computation of the EPE.⁴
• Probability of default (PD). The default probability for the interval , as previously discussed in Section 12.1.3.

The above formula is an alternative way to compute CVA and can be seen as a weighted average of the EPE profile, as illustrated in Figure 17.3. Since the LGD was not assumed to have any time behaviour, it is a simple multiplier.

The above formula is quite intuitive since it represents CVA as a product of the EPE (market risk) and PD (credit risk). Note that default enters the expression via default probability only following the assumption of independence (no WWR). In this approach, whilst one may require a simulation framework in order to compute EPE, it is not necessary to simulate default events. Spreadsheets 17.1 and 17.2 can be used to show that the CVA calculated via Equations 17.1 (with no WWR) and 17.3 is the same.

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Spreadsheet 17.2 Path-wise CVA calculation for an interest rate swap.

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At first glance, it probably seems that the best method for computing CVA is the path-wise formula, rather than the direct approach. One reason for this is that the path-wise formula in Equation 17.3 is intuitive in representing CVA as a summation over the EPE profile, which is the natural way to look at the exposure component, together with the associated PD and LGD terms.

However, direct calculations of xVA terms can be more efficient in terms of computational time. Each simulation in the path-wise approach will require multiple valuations of the underlying portfolio according to the choice of the number of time intervals (m in Equation 17.3). Increasing m will improve the approximation of the integral, but likely at an increased computational cost due to the larger number of portfolio valuations required. It turns out that this can be quite inefficient.

In order to see this, we compare the convergence of the direct and path-wise implementations as a function of the number of valuations of the underlying portfolio that need to be made (since this is often the bottleneck of the CVA calculation). The CVA is calculated for an uncollateralised 10-year receiver interest rate swap with direct and path-wise approaches according to Spreadsheets 17.1 and 17.2. In the path-wise approach, 40 time steps are used, which corresponds to a quarterly grid. This means that for the case of 40,000 valuations, the path-wise method will use 1,000 simulations and 40 time steps, whereas the direct approach will use 40,000 different default times. Figure 17.4 shows the results, which illustrate that the direct approach converges much more quickly.

In order to be more precise about the improvement when using the direct method, the standard deviation of the CVA estimate with each approach (using 20 different calculations, each with a total of 40,000 valuations) is calculated (Figure 17.5).

The reason for the better computational performance of the direct method is that the path-wise simulation is slow to converge due to the autocorrelation between successive points in the simulation. For example, Figure 17.6 shows the strong relationship between successive valuation points in the path-wise simulation. This high positive correlation means that convergence, as a function of the total number of valuations, is slow.

The improvement is quite dramatic, with the standard deviation being six times smaller in the direct approach. Since Monte Carlo error is approximately proportional to the square root of the number of simulations,⁵ this actually represents a speed improvement of times. In other words, we can do 36 times fewer valuations in the direct approach to achieve the same accuracy. Whilst this may sound appealing, it presumes that the valuation step will be a major bottleneck in the calculation. Amdahl's law (Amdahl 1967) gives a simple formula for the overall speed-up from improving one component of a calculation. This formula is , where P is the percentage of the calculation that can be improved and S is the relative speed improvement. For example, if 90% (P = 0.9) of the time is spent on pricing function calls, and these can be sped up by 25 times, then the overall improvement is 7.3 times. There may also be additional overheads in the direct simulation, such as generating the entire path to the default time.⁶

Figure 17.4 The estimate of CVA for a 10-year receiver swap with path-wise and direct simulation approaches as a function of the total number of valuation calls.

Figure 17.5 The estimate of CVA for a 10-year interest rate swap with path-wise and direct simulation approaches and a total of 100,000 valuations. The error bars showing one standard deviation are shown.

Figure 17.6 Correlation between valuations at successive points (two years, and two years and three months) in the path-wise CVA calculation.

A direct simulation approach for CVA may, therefore, be faster, but this will depend on the precise time spent on different components in the Monte Carlo model. This approach is also less obviously applied to portfolios with path dependencies, such as collateralised transactions and some exotics.

Another advantage of the direct approach may be in the modelling of WWR, discussed later in Section 17.6.

17.2.3 CVA as a Spread

For pricing purposes, it is often useful to calculate CVA – and indeed any xVA adjustment – as a running spread (per annum charge) so as to, for example, adjust a rate paid or received. One simple way to do this is to divide by a duration or annuity value for the maturity in question. However, this potentially ignores the ‘CVA of the CVA’. Charging CVA will increase the value of the transaction which will, in turn, increase the total CVA. The correct result could be achieved by solving recursively for the spread that makes the value of the transaction including CVA equal zero. Vrins and Gregory (2011) show that the effect can be bounded by using both risky and non-risky annuity values (see Appendix 17B).

Another approximation to the above (Appendix 17C) assumes that the EPE is constant over time, which yields the following approximation based on the average EPE:

(17.4)

where the UCVA is expressed in the same units as the credit spread (which should be for the maturity of the instrument in question) and the average EPE is as defined in Figure 11.4.⁷ This approximation generally works reasonably well, especially where the EPE profile (or default probability profile) is relatively constant over time. Whilst not used for actual calculations, the approximate formula in Equation 17.4 is useful for intuitive understanding of the drivers of CVA, as it separates the credit component (the credit spread of the counterparty) and the market risk component (the exposure, or average EPE).

Table 17.1 illustrates running CVA calculations for the interest rate swap considered in Section 17.2.2.

For relatively small CVA charges, such as in this example, the recursive effect is small, although Vrins and Gregory (2011) show that it is significant in certain cases (typically more risky counterparties and/or long-dated transactions). Note also that this effect is systematic: an xVA desk charging based on the first-order CVA will always lose money when the trade is booked – all other things being equal – due to the additional ‘CVA on the CVA’.

Table 17.1 Running CVA calculations.

	CVA (bps per annum)
Divide by risky annuity	−1.92
Exact (recursive)	−1.96
EPE approximation	−2.01

17.2.4 Special Cases

There are special cases where CVA can be calculated easily. For example, Figure 17.7 shows CVA for payer and receiver swaps as a function of the swap rate. As noted in Section 15.1.3, in the case where the transaction or portfolio is very in-the-money (ITM) (has a large positive value), the EPE can be approximated by the expected future value (EFV), which leads to a simplification in the calculation.

In such situations, it is possible to either use the CVA formula with the EFV, or alternatively to use a discounting approach, as discussed in Section 16.2.1.⁸ This effect is showing the complex swap-like CVA calculations converging to a simpler case, as previously illustrated in Figure 17.1.

17.2.5 Credit Spread Effects

The shape of the credit curve can have a material impact on CVA. Section 12.1.3 considered upwards-sloping, flat, and inverted credit curves, all of which had the same five-year credit spread of 150 basis points (bps). It was discussed how, whilst these curves gave cumulative default probabilities that were the same at the five-year point, the marginal default probabilities differed substantially. For a flat curve, default probability is approximately equally spaced, whilst for an upwards- (downwards-) sloping curve, defaults are back- (front-) loaded. We show the impact of curve shape on CVA in Table 17.2. The upwards-sloping curve gives the largest value, mainly due to having the largest extrapolated 10-year credit spread. The inverted curve gives the smallest CVA for the opposite reason.

Figure 17.7 CVA for payer and receiver interest rate swaps as a function of the swap rate. Also shown is the analytical CVA calculated with the EFV. The CVA of the payer swap is shown as a positive value for display purposes.

Table 17.2 CVA for 10-year receive fixed interest rate swap using the three credit curves in Figure 12.3. The five-year credit spread is assumed to be 150 bps and the LGD 60% in all cases.

	10 years
Upwards sloping	−24.6
Flat	−20.0
Inverted	−15.7

Another important feature to understand is the change in CVA as credit spread widens, as shown in Figure 17.8. CVA generally increases with increasing credit spread. However, in default, CVA will converge to the exposure of the transaction (or portfolio), which may be zero. Whilst this is not apparent for relatively small credit spread changes, it leads to a large gamma for larger credit spread changes. This behaviour is also important for understanding jump-to-default risk, discussed in Section 21.2.5.

Figure 17.8 CVA for 10-year receiver swap as the credit spread curve is increased. A lognormal scale is used for clarity. The CVA for a very large credit spread change converges to zero since it is a par transaction.

17.2.6 Loss Given Default

Section 12.1.4 discussed the fact that LGD (or recovery rate) could be defined as either the settled or actual value. Substituting the default probability formula (Equation 12.1) into the CVA formula (Equation 17.3) gives:

(17.5)

where the ‘actual’ and ‘market’ LGDs, which should be expected values (although this is not always stated explicitly), are referenced explicitly. The former () should reference the actual expected LGD that would be received in the event that the counterparty defaults. The latter () refers to the value to assume for calibrating market-implied default probabilities, which should, therefore, relate to the seniority of the underlying instrument used to determine the credit spread (usually senior unsecured for most CDS contracts). For example, BCBS (2011b) states that, for regulatory capital purposes:⁹

It should be noted that this , which inputs into the calculation of the CVA risk capital charge, is different from the LGD that is determined for the IRB [internal ratings-based approach] and CCR [counterparty credit risk] default risk charge, as this is a market assessment rather than an internal estimate.

Whilst the above LGDs are different conceptually, if a derivatives claim is of the same seniority as that referenced in the CDS (as is typically the case), then one should typically assume that =. Indeed, this is a requirement in the aforementioned regulatory capital framework (BCBS 2011b), since only is referenced in the formula. In this case, the LGD terms in Equation 17.5 will cancel to first order, and there will be only moderate sensitivity to changing the LGD value.¹⁰ The simple approximation in Equation 17.4 has no LGD input reflecting this cancellation.

Figure 17.9 illustrates the minimal impact of changing the LGD in this case. As expected, changing both LGDs has a reasonably small impact on CVA since there is a cancellation effect: increasing LGD reduces the market-implied default probability but increases the loss in the event of default. The net impact is only a second-order effect: for example, changing the LGD from 60% to 50% changes CVA by less than 2%.

Despite this, in certain cases institutions may consider it appropriate to use a different value for . These cases may include (in order of ease of justification):

• seniority in the exposure waterfall (e.g. trades with securitisation special purpose vehicles) or a significantly different LGD due to the structural nature of the counterparty (e.g. project finance-related transactions);

  

  Figure 17.9 CVA as a function of the LGD used when

• credit enhancements such as margin or other forms of credit support; and
• the consideration that, due to their work-out process experience, they may achieve a higher recovery rate.

The first case – that of different seniority – is the most common and easy-to-justify reason for using a different . An example of this is:

… estimated recovery rates implied by CDS, adjusted to consider the differences in recovery rates as a derivatives creditor relative to those reflected in CDS spreads, which generally reflect senior unsecured credit risk.¹¹

This adjustment for seniority is also envisaged by regulatory capital rules – for example, BCBS (2015a) states:

The market-implied ELGD value used for regulatory CVA calculation must be the same as the one used to calculate the market-implied PD from credit spreads unless it can be demonstrated that the seniority of the derivative exposure differs from the seniority of senior unsecured bonds.

One common situation where lower LGDs are used is project finance. Project finance refers to the funding of infrastructure and industrial projects off-balance sheet, and based upon the projected returns from the project rather than from an underlying balance sheet. The majority of funding is usually via senior debt, with junior debt, equity, and grants also forming part of the financing. Senior debtors will aim for the project to produce sufficient cash flows to repay them in full, and will ensure that the legal structuring of the project will be such that they have priority over all material assets in a default scenario. Project finance is typically associated with strong credit underwriting principles and structural features that enhance the position of senior debtors. This is generally seen via higher recovery rates (approximately 75% on average) in project finance compared to other types of unsecured lending (e.g. approximate 40% average for corporates). For example, Standard & Poor's (2016) states that ‘project finance exhibits a strong average recovery rate of 77%, which is stronger compared to average recoveries observed in the corporate world’.

The impact of using different LGDs is shown in Figure 17.10.

One caveat to the above LGD ‘override’ process is that the LGD must not be a consideration in the credit curve construction process. In cases where different LGDs are used to reflect a more favourable recovery outcome, it is important to ensure that any credit spread proxy that has been determined on the basis of a rating has been done so only via reference to default probability. If this is not the case, then there is the potential for double-counting if the expectation of a relatively high recovery rate leads to a better rating. In such a case, this better rating would, in turn, lead to a lower credit spread, and it would therefore be inappropriate to reflect this a second time using a lower LGD. In order to avoid the above double-counting, it is important that any rating used for the basis of credit spread mapping should reflect only default probability and not expected loss. Standard & Poor's rating definitions are default probability based – for example, AA is defined as ‘The obligor's capacity to meet its financial commitments on the obligation is very strong.’¹² On the other hand, Moody's definitions suggest that recovery rates may be part of the rating process, with, for example, Aa being defined as ‘Obligations rated Aa are judged to be of high quality and are subject to very low credit risk.’¹³

Note that the xVA desk in a bank commonly assumes LGD risk in relation to actual defaults.¹⁴ This risk is due to the potential difference between the value of and the LGD experienced after the work-out process.

Figure 17.10 CVA as a function of the LGD used for equal () and unequal ( varied and ) LGD assumptions.

17.3 DEBT VALUE ADJUSTMENT

17.3.1 Accounting Background

A key assumption in the definition of CVA above was that the party making the calculation could not default. This may have seemed like a fairly innocuous and straightforward belief. Indeed, it is consistent with the ‘going concern’ accountancy concept, which requires financial statements to be based on the assumption that a business will remain in existence for an indefinite period. However, as mentioned in Section 5.3.3, international accountancy standards allow (indeed, potentially require) a party to consider its own default in the valuation of its liabilities. It is stated explicitly that an ‘own credit’ adjustment must be made. Furthermore, DVA could be interpreted as being part of the ‘exit price’ due to the fact that a counterparty will charge CVA. DVA is associated, via the expected negative exposure (ENE), with the liability position in the same way that CVA is associated with an asset via the EPE. The use of both CVA and DVA is generally referred to as ‘bilateral CVA’ (BCVA).

The use of BCVA has been largely driven by accounting practices and formally began in 2006 when the FAS 157 (Section 5.3.3) determined that banks should record a DVA entry. FAS 157 states:

Because non-performance risk includes the reporting entity's credit risk, the reporting entity should 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.

This led to a number of large US (and some Canadian) banks reporting DVA in financial statements, with the BCVA taking the form of an exit price rather than a reserve. Most other banks did not report DVA adjustments at this time and still considered CVA to be a reserve (e.g. using historical default probabilities in the calculation).

Following the introduction of IFRS 13 from January 2013 (IFRS 2011), most other large banks have moved to BCVA reporting using market-implied parameters. IFRS 13 requires that transactions such as derivatives be reported at ‘fair value’, the definition of which includes the following comment:

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.

The interpretation of auditors has generally been that IFRS 13 accounting standards require the use of market-implied default probabilities (via credit spreads) and both CVA and DVA components to be reported. This has led to a moderate convergence in recent years, although there are still some exceptions for regions where IFRS 13 has not been implemented, or banks where the CVA and DVA may be considered to be immaterial. However, these exceptions are becoming less common.¹⁵

Despite the general adoption of DVA in accounting, it is important to note that the precise implementation – like CVA – does not follow clear standards across the market (see ESMA 2017). It is also important to note that, whilst banks have generally adopted DVA in financial reporting, this is somewhat undermined by their introduction of funding value adjustment (FVA), as discussed in Chapter 18.

DVA is a double-edged sword. On the one hand, it resolves some theoretical problems with CVA and creates a world where price symmetry can be achieved. On the other hand, the nature of DVA and its implications and potential unintended consequences are troubling. Indeed, as will be discussed in Chapter 18, market practice can be seen to generally disregard DVA in aspects such as pricing and replace it with funding considerations (FVA). However, it is still important to understand DVA from an accounting standpoint and its subsequent relationship to FVA.

17.3.2 DVA, Price, and Value

DVA will be discussed from the point of view of accounting, economic, and regulatory value introduced earlier (Section 5.3.1).

In order to understand the accounting rationale behind DVA, consider two parties with respective valuations of a certain portfolio: and . If these are base valuations (Section 5.2.1), then it is reasonable that the parties agree on a valuation (e.g. they use the same discounting approach) and therefore . If this is not the case, then value has been created or destroyed.

Now, if we bring CVA into the picture, then the valuations between the different parties will not agree since CVA is always a negative adjustment. However, by including DVA then:

(17.6a)

(17.6b)

With and (‘my CVA is your DVA’), then once again it is possible for the actual valuations to agree.

From a pricing point of view, CVA has traditionally been a charge for counterparty risk that is levied on the end user (e.g. a corporate) by their counterparty (e.g. a bank). Historically, banks charged CVAs linked to the credit quality of the end user and the exposure in question. An end user would not have been able credibly to question such a charge, especially since the probability that a bank would default was considered remote (and, indeed, the credit spreads of banks were traditionally very small and their credit ratings strong). This changed during the global financial crisis, and the credit spreads of the ‘strong’ financial institutions became – and have since remained – material. This raises the question of whether banks, too, should be charged a CVA by their counterparties and how two banks could trade together in the interbank market.

One important feature of DVA is that it solves the above issues and creates ‘price symmetry’, where, in theory, parties can agree on prices. However, this raises the question of whether or not a party would be happy to include DVA in its pricing. Whilst this will not give rise to a loss (assuming DVA is also part of valuation), it may be questioned to what extent DVA has an economic value. Whilst there is a clear economic impact of counterparty default losses leading to CVA, the notion of own default leading to DVA gains is more problematic. Furthermore, from a hedging perspective, whilst CVA can be seen as relating to the cost of buying CDS protection on a counterparty, DVA would be linked to selling CDS protection on one's own credit.

A party that does not believe that DVA is part of economic value will be reticent to include it in their pricing. It is important to note that price symmetry is not generally a requirement for markets: for example, banks may determine prices for derivatives, and end users decide whether or not to transact at these quoted prices.

In line with the above view is the fact that regulation requires that DVA gains be derecognised from equity (Section 5.3.3), implying that DVA is not part of the regulatory value.

Rationalising the different accounting, economic, and regulatory DVA viewpoints can be done by considering the view of shareholders and bondholders. In the event of the default of an institution, shareholders receive nothing, whereas bondholders receive a residual – or recovery – value. Suppose a firm enters into an out-of-the-money (OTM) (negative value) transaction, for which it will receive an upfront payment and which will also have a significant DVA component. In the event of default, the other creditors will receive a portion of this amount, leading to a higher recovery rate, and therefore they may see the DVA as real.¹⁶ The shareholders will not consider the DVA to be real since they receive nothing in default. Conflicts between shareholders and bondholders are well known in finance.

Since regulatory capital requirements set the required amount of shareholder equity, it is therefore not surprising that they require the derecognition of DVA. On the other hand, taking the view that financial reporting should reflect the total value of a firm (i.e. not only to shareholders) would support the reporting of DVA as a benefit to bondholders. In terms of pricing, taking a shareholder view would require the exclusion of DVA.

The above shareholder and bondholder view and the link to DVA will be discussed in more detail for FVA in Section 18.2.6.

17.3.3 Bilateral CVA Formula

Under the assumption that both the party making the calculation and its counterparty can default, a BCVA formula is obtained consisting of CVA and DVA components (Appendix 17D):

(17.7a)

(17.7b)

(17.7c)

The suffixes P and C indicate the party making the calculation and its counterparty, respectively. The CVA term is similar to Equation 17.2, but with the factor representing the survival of both parties in the formula. This is sometimes known as the ‘first-to-default’ effect because it ensures that a default event is conditioned on the other party not defaulting first.

The DVA term above is the mirror image of CVA based on the ENE, the party's own default probability and the LGD. DVA is positive due to the sign of the ENE, and it will, therefore, oppose CVA as a benefit. The DVA term corresponds to the fact that in cases where the party itself defaults, it will make a ‘gain’ if it has a liability position (negative exposure). This gain is the opposite of the loss that the surviving counterparty experiences. Since ENE is equal and opposite to the counterparty's EPE, it can be seen that one party's DVA, theoretically, is equal and opposite to the other's DVA, and vice versa.¹⁷

As before, it is possible to discretise the above integrals:

(17.8a)

(17.8b)

Note that the CVA above is different from the previously defined unilateral CVA (UCVA) in Equation 17.3, and a similarly defined unilateral DVA (UDVA), because it contains the survival (no default) probability of the party making the calculation given by . This will be discussed in Section 17.3.4.

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Spreadsheet 17.3 CVA and DVA calculations.

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Table 17.3 shows BCVA calculations for different interest rate swap transactions. In this example, the counterparty's credit quality is worse than the party's own credit quality.¹⁸ The contingent values are smaller, as would be expected, although the impact on BCVA (or UCVA + UDVA) is not clear, since both individual values decrease but have opposite signs. There is a strong asymmetry for BCVA between pay and receive fixed swaps due to the skew in the exposure profiles (see Figure 15.24). This causes the BCVA of the receive fixed swap to be close to zero due to the relatively large DVA, in turn due to the size of ENE. A similar but larger effect is seen for off-market swaps: the OTM swap having a BCVA which is positive overall.

Table 17.3 Upfront CVA and DVA values (in bps) for 10-year interest rate swaps. The party's own credit spread is lower than the counterparty's, and both LGDs are assumed to be 60%. The ITM and OTM transactions are based on a swap rate of 1%, instead of the market rate of 1.49%.

	Pay fixed	Receive fixed	Pay fixed (ITM)	Receive fixed (OTM)
UCVA	−29.9	−17.2	−43.7	−9.9
UDVA	8.1	14.4	4.8	20.7
UCVA + UDVA	−21.8	−2.7	−39.0	10.8
CVA	−28.9	−16.6	−42.4	−9.6
DVA	7.4	13.2	4.3	18.9
BCVA	−21.5	−3.5	−38.1	9.4

Whilst the BCVA price symmetry has some nice theoretical properties, it could be questioned to what extent market participants would actually price these benefits into transactions. For example, would it be realistic to charge so little for the receive fixed swap or to ‘pay through mid’¹⁹ to step into the OTM swap?

To understand BCVA price symmetry more easily, consider the simple formula in Equation 17.4. An obvious extension including DVA is:

(17.9)

where the average ENE is similar to the average EPE defined in Section 11.1.5. Assuming that ,²⁰ we obtain . A party could, therefore, charge its counterparty for the difference in its credit spreads (and if this difference is negative, then this party should pay their counterparty). Weaker counterparties would pay stronger counterparties in order to trade with them based on the differential in credit quality. Theoretically, this leads to a pricing agreement (assuming parties can agree on the calculations and parameters), even when one or both counterparties have poor credit quality. Practically, there are concerns with such an approach, based on the arguments in Section 17.3.2.

17.3.4 Close-out and Default Correlation

The above discussion and formulas for BCVA ignored or simplified three important and interconnected concepts:

• Survival adjustment. UCVA and UDVA are defined without reference to the survival probability of the other party, whilst CVA and DVA include survival probabilities. On one hand, it seems that the first-to-default effect – in that the underlying contracts will cease when the first party defaults and therefore there should be no consideration of the second default – would justify the inclusion of survival probabilities. However, this leads to CVA having sensitivity to a party's own credit quality, which may be seen as unnatural. Regulation may also prohibit doing this (Section 13.3.1).
• Default dependency. Related to the above, the default dependency between the party and its counterparty is not included. If such a dependency is significant, then this would be expected to change the CVA and DVA terms (the formulas above assume that the defaults are independent).
• Close-out. Finally, as discussed in Section 11.1.2, the definitions of EPE and ENE are typically based on standard valuation assumptions and do not reflect the actual close-out assumptions that may be relevant in a default scenario. In other words, in the event of a default, it is assumed that the underlying transactions will be settled at their base values (Section 6.3.4) at the default time, which is inconsistent with the reality of close-out that may reference actual values. For example, as discussed in Section 6.3.5, the 2002 ISDA documentation specifies that the claim ‘may take into account the creditworthiness of the Determining Party,’ which seems to suggest that CVA/DVA may be a part of the close-out value and therefore should change EPE/ENE.

The above points have been studied by various authors. Gregory (2009a) shows the impact of survival probabilities and default correlation on BCVA, but in isolation of any close-out considerations. Brigo and Morini (2010) consider the impact of close-out assumptions in the unilateral (i.e. one-sided exposure) case.

Not only are close-out assumptions difficult to define, but quantification is challenging because it involves including the future BCVA at each possible default event in order to eventually determine the current BCVA. This creates a difficult recursive problem. Brigo and Morini (2010) show that, for a loan, the expectation that DVA can be included in the close-out assumptions (‘risky close-out’) leads to cancellation, with the survival probability of the party making the calculation. This means that the formula in Equation 17.8a without the survival probability term is correct in a one-sided situation with risky close-out. Gregory and German (2013) consider the two-sided case and find that a simple result does not apply, but that the bilateral formulas used in Equations 17.8a and 17.8b without survival probabilities are probably the best approximation in the absence of a much more sophisticated approach.

Generally, most market participants do not follow a more advanced approach and simply include survival probabilities (or not) directly. For example, in an Ernst & Young Survey in 2012,²¹ six out of 19 respondents report making CVA and DVA ‘contingent’ (survival probability adjusted) and seven non-contingent, with the remainder not reporting DVA at the time. Anecdotal evidence (e.g. Totem, Section 5.3.5) seems to suggest that there is still a mix between participants that use contingent or non-contingent calculations. As discussed in Section 6.3.4, this problem is not specific to CVA and DVA, but is a general consideration for any xVA term.

17.3.5 The Use of DVA

The issue of DVA in counterparty risk is part of a broader issue: the general incorporation of credit risk in liability measurement. Accountancy standards have generally evolved to a point where ‘own credit risk’ can (and should) be incorporated into the valuation of liabilities. For example (relevant for the US), the Financial Accounting Standards Board (FASB) issued Statement of Financial Accounting Standards (SFAS) 157 in 2006 (which became effective in 2007), relating to fair value measurements. This permits a party's own credit quality to be included in the valuation of its liabilities, stating ‘The most relevant measure of a liability always reflects the credit standing of the entity obliged to pay.’ Amendments to IAS 39 by the International Accounting Standards Board (IASB) in 2005 (relevant for the European Union) also concluded that the fair value of a liability should include the credit risk associated with that liability. This position was reinforced with the introduction of IFRS 13 from the beginning of 2013.

DVA was a very significant question for banks in the years following the global financial crisis, since their own credit risk (via credit spreads) experienced unprecedented volatility. Banks reported massive swings in accounting results as their credit spreads widened and tightened. Articles reporting such swings did not seem to take them seriously, making statements such as:

The profits of British banks could be inflated by as much as £4bn due to a bizarre accounting rule that allows them to book a gain on the fall in the value of their debt.²²

[DVA is] a counter-intuitive but powerful accounting effect that means banks book a paper profit when their own credit quality declines.²³

There is logic to the use of DVA on own debt, since the fair value of a party's own bonds is considered to be the price that other entities are willing to pay for them. However, it is questionable whether a party would be able to buy back its own bonds without incurring significant funding costs. It therefore became typical for equity analysts to remove DVA from their assessment of a company's ongoing performance, with the view that DVA is no more than an accounting effect and of no interest to shareholders.

DVA in derivatives has probably received more scrutiny than that in own debt since derivatives valuation has received much attention and is based on rigorous hedging arguments. The criticism of DVA stems mainly from the fact that it is not easily realisable (Gregory 2009a). Other criticisms include the idea that the gains coming from DVA are distorted because other components are ignored. For example, Kenyon (2010) makes the point that if DVA is used, then the value of goodwill (which is zero at default) should also depend on a party's own credit quality. Losses in goodwill would oppose gains on DVA when a party's credit spread widens.

This debate really hinges on to what extent a party can ever realise a DVA benefit. Some of the arguments made in support of DVA have proposed that it can be monetised in the following ways:

• Defaulting. A party can obviously realise DVA by going bankrupt but, like an individual trying to monetise their own life insurance, this is a clearly not a very good strategy.
• Unwinds and novations. A party unwinding, novating, or restructuring a transaction might claim to recognise some of the DVA benefits since they are paid out via a CVA gain for the counterparty. For example, monolines derived substantial benefits from unwinding transactions with banks,²⁴ representing large CVA-related losses for the banks and associated DVA gains for the monolines. The monoline MBIA monetised a multibillion-dollar derivatives DVA in an unwind of transactions with Morgan Stanley.²⁵ However, these examples occurred since the monolines were so close to default that the banks preferred to exit transactions prior to an actual credit event. The banks would have been much less inclined to unwind in a situation where the monolines' credit quality had not been so dramatically impaired. Furthermore, if a transaction is unwound, then it would generally need to be replaced. All things being equal, the CVA charged on the replacement transactions should wipe out the DVA benefit from the unwind.
• Close-out process. As discussed above (Section 17.3.4), another way to realise DVA might be in the close-out process in the event of the default of the counterparty. In the bankruptcy of Lehman Brothers, these practices have been common, although courts have not always favoured some of the large DVA (and other) claims.²⁶
• Hedging. An obvious way to attempt to monetise DVA is via hedging. The obvious hedging for CVA is to short the counterparty's credit risk. This can be accomplished by shorting bonds or buying CDS protection. It is possible that neither of these may be achievable in practice, but they are theoretically reasonable ways to hedge CVA. The CVA (loss) is monetised via paying the carry in the repo transaction (to finance the bond purchase) or the premiums in the CDS protection position. However, in order to hedge DVA, a party would need to go long its own credit risk. This could be achieved by buying back one's own bonds, which requires funding (see discussion in Section 18.2.5), or by selling CDS protection referencing one's own credit risk (which is not possible).²⁷ An alternative is, therefore, to sell protection on similar correlated credits. This clearly creates a major problem as banks would attempt to sell CDS protection on each other, and it is also inefficient to the extent that the correlation is less than 100%. An extreme example of the latter problem was that some banks had sold protection on Lehman Brothers prior to their default as an attempted ‘hedge’ against their DVA.

Most of the above arguments for monetising DVA are fairly weak. It is therefore not surprising that (although not mentioned in the original text), the Basel Committee (BCBS 2011d) determined that DVA should be derecognised from the CVA capital charge (Section 13.3.1). This would prevent a more risky bank having a lower capital charge by virtue of DVA benefits opposing CVA losses. This is part of a more general point with respect to the Basel III capital charges focusing on a regulatory definition of CVA and not the CVA (and DVA) defined from an accounting standpoint. Even accounting standards have recognised problems with DVA with the FASB – for example, determining that DVA gains and losses be represented in a separate form of earnings known as ‘other comprehensive income’.

Market practice has always been somewhat divided over the inclusion of DVA in pricing, as shown in Figure 17.11, with many banks giving some, but not all, of the DVA benefit in pricing new transactions. Even those quoting that they ‘fully’ include DVA would not do this on all transactions (an obvious exception being where the DVA is bigger than the CVA and they would not pay through mid). Anecdotally, most banks' submissions to the Totem consensus pricing can be clearly seen to contain no DVA component (although they do contain FVA, as discussed in Section 18.2.5).

Figure 17.11 Market practice around including DVA in pricing.

Source: Deloitte/Solum CVA Survey (2013).

Market practice has generally resolved the debate over DVA by considering it a funding benefit. Indeed, in the hedging argument above, buying back one's own debt could be seen as a practical alternative to the obviously flawed idea of selling CDS protection on one's own credit. However, buying back debt clearly requires that it is first issued, creating a link to funding which must, therefore, be considered. More broadly, it can be shown that different ‘strategies’ give rise to different valuation adjustments. Whilst there are strategies that support the use of DVA, these are hard to justify as being economically realistic.

When FVA is considered, it is possible to see DVA as a funding benefit. This will be discussed in more detail in Chapter 18.

17.4 CVA ALLOCATION

Risk mitigants, such as netting and margin, reduce CVA, but this can only be quantified by a calculation at the netting set level. It is, therefore, important to consider the allocation of CVA to the transaction level for pricing and valuation purposes. This, in turn, leads to the consideration of the numerical issues involving the running of large-scale calculations rapidly.

17.4.1 Incremental CVA

When there is a netting agreement, the impact will reduce CVA and cannot increase it (this arises from the properties of netting described in Section 11.3.1). It follows that, for a netting set (a group of transactions with a given counterparty under the same netting agreement):

(17.10)

where is the total CVA of all transactions under the netting agreement and is the standalone CVA for transaction i (i.e. computed in isolation). The above effect (CVA becoming less negative) can be significant, and the question then becomes how to allocate the netting benefits to each individual transaction. Note that these netting benefits constitute a portfolio effect and will be strong for balanced and offsetting portfolios and weaker for directional portfolios (Section 15.2.3).

The most obvious way to allocate the portfolio netting effect is to use the concept of ‘incremental CVA’, analogous to the incremental EPE discussed in Section 15.6.3.²⁸ Here the CVA as a result of any trading behaviour (most obviously a new transaction) is calculated based on the incremental effect this transaction has on the netting set:

(17.11)

where and represent the (portfolio) CVA before and after the change, respectively, and represents the incremental effect of that change. The above formula ensures that the CVA of a new transaction is represented by its contribution to the overall CVA at the time it is executed. Hence, it makes sense when CVA needs to be charged to individual salespeople, traders, businesses, and, ultimately, clients. CVA depends on the order in which transactions are executed, but does not change due to subsequent transactions. An xVA desk (Chapter 21) charging this amount will directly offset the instantaneous impact on its total CVA from the change in CVA as a result of the new transaction. Note that Equation 17.11 may not only correspond to new transactions but also to:

• unwinding;
• restructuring transactions; and
• changes to contractual terms (e.g. margin agreements).

The above situations are all cases where it may be necessary to charge (or refund) a counterparty for the change in CVA, and where the incremental calculation is relevant.

As shown in Appendix 17E, it is possible to derive the following fairly obvious and intuitive formula for incremental CVA:

(17.12)

which is the same as Equation 17.5 but with the incremental EPE replacing the standalone EPE. This assumes that the behaviour in question does not change the PD or LGD of the counterparty, which may be debated in some situations (e.g. a counterparty undertaking to post more margin may be more likely to default, although this would be captured in the credit spread). The quantification of incremental EPE was covered in detail in Section 15.6.3 and will require aggregations (Equation 15.2) at the netting set level. For new transactions this will just require comparing the netting set properties with and without the new transaction, whereas for more complex restructurings it will require a full calculation of the netting set EPE:

(17.13)

Table 17.4 Incremental BCVA calculations (bps upfront) for a seven-year USD swap paying fixed with respect to different existing transactions and compared to the standalone value. The counterparty credit curve is assumed to be flat at 150 bps, the own credit curve is 100 bps and both LGDs are equal to 60%.

	Standalone	Directional	Neutral/Offsetting
CVA	−26.3	−25.1	−13.7
DVA	6.0	5.3	−2.5
BCVA	−20.3	−19.9	−16.3

Incremental EPE can be negative, due to beneficial netting effects, which will lead to CVA being positive; in such a case, it would be a benefit and not a cost (e.g. unwinding a transaction would be expected to lead to this effect).

It is worth emphasising that, due to the properties of EPE and netting, incremental CVA in the presence of netting will never be lower (more negative) than standalone CVA without netting. The practical result of this is that an institution with existing transactions under a netting agreement will be likely to offer conditions that are more favourable to a counterparty with respect to a new transaction. Cooper and Mello (1991) quantified such an impact many years ago, showing specifically that a bank that already has a transaction with a counterparty can offer a more competitive rate on a forward contract. Anecdotally, some clients in jurisdictions with debatable netting enforceability (Section 6.3.2) may note that they only trade with banks who deem netting to be enforceable, since the other banks are unable to price in any incremental benefits.

The treatment of netting makes the treatment of CVA a complex and often multidimensional problem. Whilst some attempts have been made at handling netting analytically (e.g. Brigo and Masetti 2005b, as noted in Section 15.1.3), CVA calculations incorporating netting accurately typically require a general Monte Carlo simulation for exposure (EPE) quantification. For pricing new transactions, such a calculation must be run more or less in real time.

An example of incremental CVA following the previous results for incremental exposure in Section 15.6.3 is shown in Table 17.4. The example is a seven-year EUR payer interest rate swap in directional and neutral/offsetting cases. The counterparty and party's own credit spreads are assumed to be 150 bps and 100 bps respectively, and an LGD of 60% is assumed for both.

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Spreadsheet 17.4 Incremental CVA and DVA calculations.

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Considering CVA or DVA in isolation, the above results are expected since the directional case leads to only a small reduction (in absolute terms), whilst the reduction in the neutral/offsetting case is much larger. Indeed, incremental DVA changes sign in this case, which might be described as offsetting, whereas CVA reduces but does not change sign, which might be more accurately defined as neutral.

Note that there is another important feature in this example, which is that BCVA does not change substantially due to a cancellation. Indeed, for equal credit spread, BCVA is always the same and has no incremental effect. This is analogous to a similar result for FVA, discussed in Section 18.3.3.

Figure 17.12 Incremental CVA (as a spread in bps per annum) for the balanced/risk-reducing case in Table 17.4.

Another point to emphasise is that the benefit of netting seen in incremental CVA of a new transaction also depends on the relative size of the new transaction. As the transaction size increases, the netting benefit is lost, and CVA (or DVA) will approach the standalone value. This is illustrated in Figure 17.12, which shows incremental CVA for the seven-year swap in the last example, a function of the relative size of this new transaction. For a smaller transaction, CVA decreases to a lower limit of -6.2 bps, whereas for a large transaction size, it approaches the standalone value (-17.9 bps).

17.4.2 Marginal CVA

Following the discussion in Section 15.2.2, it is possible to define marginal CVA in a similar way by simply including marginal EPE in the relevant formula. Marginal CVA may be useful to break down CVA for any number of netted transactions into transaction-level contributions that sum to total CVA. Whilst it might not be used for pricing new transactions (due to the problem that marginal CVA changes when new transactions are executed, implying adjustment to the value of existing trades), it may be required for pricing transactions executed at the same time (perhaps due to being part of the same deal) with a given counterparty.²⁹ Alternatively, marginal CVA is the appropriate way to allocate CVA to transaction-level contributions at a given time – for example, for accounting purposes. This may be useful to give an idea of transactions that could be usefully restructured, novated, or unwound.

Table 17.5 shows incremental and marginal CVA corresponding to the same interest rate swap shown in Table 17.4 together with a cross-currency swap. DVA results are not shown but can be reproduced in Spreadsheet 17.5.

Table 17.5 Illustration of the breakdown of CVA of an interest rate swap (IRS) and cross-currency swap (XCCY) via incremental and marginal allocation. The counterparty credit curve is assumed to be flat at 150 bps, and the LGD is 60%.

	Incremental (XCCY first)	Incremental (IRS first)	Marginal
IRS	−13.7	−26.3	−17.7
XCCY	−23.2	−10.7	−19.2
Total	−36.9	−36.9	−36.9

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Spreadsheet 17.5 Marginal CVA and DVA calculations.

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Incremental CVA clearly depends very much on the ordering of the transactions, with the contribution from the second transaction being much smaller. The marginal allocation is more ‘fair’. Clearly, the amount of CVA charged can be very dependent on the timing of the transaction. This may be problematic and could possibly lead to ‘gaming’ behaviour. However, this is not generally problematic for two reasons:

• a given client will typically be ‘owned’ by a single trader or salesperson and will, therefore, be exposed only to the total charge on a portfolio of transactions (although certain transactions may appear beneficial at a given time); and
• most clients will execute relatively directional transactions (due, for example, to their hedging requirements) and netting effects will therefore not be large.

17.5 IMPACT OF MARGIN

17.5.1 Overview

The impact of margin on CVA follows directly from the assessment of the impact of margin on exposure in Sections 15.6.5 and 15.6.6. As with netting, the influence of margin on the standard CVA formula given in Equation 17.3 is straightforward: margin only changes EPE, and hence the same formula may be used with EPE based on assumptions of collateralisation.

Whilst strong collateralisation would imply that CVA would be small and potentially negligible, it has become increasingly common for market participants to model the impact of variation margin, even when thresholds are zero. Although the minimum margin period of risk (MPoR) of 10 days applies only for regulatory capital purposes, it has become standard across other CVA calculations, such as for accounting purposes. Whilst margining may reduce CVA significantly, this reduction is not as strong as might be thought, as shown in Section 15.6.5. Furthermore, whilst CVA to ‘collateralised counterparties’ may be materially reduced, an institution may have more of such counterparties and/or bigger portfolios with them.

As an illustration of the above, consider the results in Figure 3.4 in Chapter 3. The bank in question has a material CVA to banks and ‘other financial institutions’, even though most of these counterparties would be expected to be strongly collateralised. It is therefore important to model the impact of margin on CVA.

It should be noted that the ability to charge collateralised counterparties for CVA may be limited. Whilst financial end users posting margin may still be charged some CVA, it is clearly not possible to charge interbank counterparties for the purposes of hedging. There is less incentive, therefore, to price collateralised CVA for pricing purposes.

17.5.2 Example

Figure 17.13 shows CVA for the same pay and receive interest rate swaps considered previously, using the aforementioned Spreadsheet 17.4, with a classical model for the MPoR (Section 15.5.3). The exposure profiles are shown in Figure 17.13 for the pay fixed swap and compared to the uncollateralised exposure. Since the MPoR modelling assumptions are symmetric, the receive fixed swap will be the opposite of this case. The MPoR is assumed to be 10 business days.

Table 17.6 shows CVA and DVA values for uncollateralised and collateralised interest rate swaps. Note that the collateralised values are more symmetric for pay and receive swaps (this may not be the case when more advanced modelling of cash flows is done, as discussed in Section 15.5.3). As shown in Appendix 15C, the simple formula of the reduction in CVA (via EPE) due to collateralisation would be about 8.4 times.³⁰ CVA reduction of the pay fixed swap (or DVA of the receive fixed swap) is better than this due to the positive skew of EPE (see Figure 15.24). CVA reduction of the receiver swap is lower due to the negative skew.

The fact that BCVA is small for collateralised transactions may provide some comfort as it may be difficult to charge such costs in these situations (e.g. interbank transactions). However, note that the DVA calculation for collateralised counterparties is potentially even more controversial than for uncollateralised cases. This is because it stems from the implicit assumption that, 10 days prior to a party's own default, they would cease to make margin payments and this would, in turn, lead to a benefit. Even if present, this benefit would only be experienced by bondholders, for whom the margin not returned would enhance their recovery rate.

Figure 17.13 Incremental CVA (as a spread in bps per annum) for the balanced/risk-reducing case in Table 17.4.

Table 17.6 Upfront CVA and DVA values (in bps) for 10-year uncollateralised and collateralised interest rate swaps.

	Pay fixed		Receive fixed
	Uncollateralised	Collateralised	Uncollateralised	Collateralised
CVA	−28.9	−3.0	−16.6	−3.2
DVA	7.4	1.5	13.2	1.3
BCVA	−21.5	−1.5	−3.5	−1.9

17.5.3 Initial Margin

As discussed in Section 15.5.4 and demonstrated in Section 15.6.6, the initial margin will reduce exposure – and therefore CVA/DVA – towards zero. Figure 17.14 illustrates this for CVA by showing the impact of a fixed initial margin or threshold.³¹ Note that a threshold can be seen to be a negative initial margin and vice versa. For high thresholds, CVA tends to the uncollateralised value, whilst for high initial margin it tends to zero. As previously discussed in Section 15.5.4, the modelling of dynamic initial margins is a complex undertaking. Given that CVA in the presence of initial margin should be small, the degree of complexity that is warranted in order to calculate this is not clear. In terms of accounting CVA, it may be that parties argue that where a high coverage of initial margin is held, CVA can be assumed to be zero. Alternatively, a party may ignore the benefit of initial margin, which will lead to a material CVA but avoid the complexity of the calculation.

Figure 17.14 Impact of the initial margin and threshold (negative values) on the CVA of a 10-year receiver interest rate swap with an MPoR of 10 business days.

Note that any initial margin posted would not show up in any of the above calculations as long as it is segregated (Section 7.2.5). However, it will represent a cost from the point of view of margin value adjustment (MVA) (Chapter 20).

17.5.4 CVA to CCPs

Episodes such as the recent Nasdaq default (Section 10.1.3) provide a reminder that central counterparties (CCPs) are risky. Of particular note is the fact that clearing members can make losses as a result of their default fund contributions, even if the CCP itself is not insolvent. Such default fund exposure would suggest that clearing members should attempt to quantify their CVA to a CCP. This is also in line with the fact that default fund-related capital requirements are relatively high (Section 13.6.3), reflecting this risk.

Nowadays, CVA is generally calculated for all counterparties, even those that are strongly collateralised (for example, see Figure 3.4), but the calculation of CVA to CCPs is not standard. However, it would seem that this is likely to become more routine, given the increasing use of CCPs and awareness of the risk. Accountants may also require such CVA calculations given the likely materiality of the numbers.

However, the calculation of CVA to a CCP is a very challenging problem since the underlying parameters for CVA – namely PD, EPE, and LGD – are hard to determine. In order to determine these inputs, the follow points need to be quantified.

• Default probability. This is the probability of one or more defaults amongst the other clearing members of the CCP. This would require the assessment of many different credit spread curves together with – in theory – assumptions regarding the dependency between the defaults. CVA will also be sensitive to the LGD used to imply default probability, since it will not cancel as it does in more traditional CVA calculations.
• Total exposure. In the event of one or more defaults, it would be necessary to determine the potential total loss in excess of the defaulter-pays resources. Note that the initial margin and default fund contributions of each individual member are generally not known, although there may be CCP disclosures regarding the aggregate amounts. Since CCP initial margins are taken to a high coverage, this may be expected to be small. However, the possibility of extreme market moves and difficulty in auctioning portfolios can create potentially large losses, as in the aforementioned Nasdaq case. It is not clear how – or if – it would be possible to model the likely efficiency of an auction.
• Allocation of exposure. Even knowing the total loss in a default fund, it would be necessary to estimate the proportion of this loss that would be experienced by a given clearing member. Methods such as rights of assessment (Section 8.3.4) are usually based on a pro rata allocation which is predictable. However, the possibility for heterogenous allocation via the use of auction incentive pools (Section 10.2.3) or other loss-allocation methods, such as variation margin gains haircutting (Section 10.2.4), further complicates this.
• LGD. The assessment of any recovery of losses will also be extremely difficult.³²

Another important consideration is the meaning of the CVA value to a given CCP. CVA, being an expected loss, is relevant in the event that the exposure is part of a diversified portfolio and/or is hedgeable. It could be argued that CVA to a CCP does not meet either of these criteria as there are few CCPs and a clearing member may have a material CVA to a given CCP that can be neither diversified nor hedged.

Regarding the computation of CVA to a CCP, Arnsdorf (2019) has proposed a simple approach based on the posted initial margin of the clearing member (and therefore not requiring detailed information about the structure of the CCP). This assumption holds if default fund requirements are linked to initial margin requirements and also if default fund losses are allocated homogenously, both of which are approximately true. Arnsdorf also proposed to use the credit spreads of the other clearing members to calculate default probabilities, and extreme value theory to calculate EPE based on the distribution of losses above the defaulter-pays resources.

17.6 WRONG-WAY RISK

17.6.1 Overview

So far, this chapter has described the calculation and computation of CVA under the commonly-made simplification of no WWR, which assumes that EPE, PD, and LGD are not related. WWR is the phrase generally used to indicate an unfavourable dependence between exposure (EPE) and counterparty credit quality: the exposure is high when the counterparty is more likely to default, and vice versa. Such an effect would have a clear impact on CVA and DVA. Moreover, certain WWR features can also apply to other situations and impact other xVA terms through dependencies related to collateral, funding, and other factors. WWR is difficult to identify, model, and hedge due to the often subtle macro-economic and structural effects that cause it.

Whilst it may often be a reasonable assumption to ignore WWR, its manifestation can be potentially dramatic. In contrast, ‘right-way’ risk can also exist in cases where the dependence between exposure and credit quality is a favourable one. Right-way situations will reduce CVA.

Losses due to WWR have also been clearly illustrated. For example, many dealers suffered heavy losses because of WWR during the Asian crisis of 1997/1998. This was due to a strong link between the default of sovereigns and of corporates and a significant weakening of their local currencies. A decade later (starting in 2007), the global financial crisis caused heavy WWR losses for banks buying insurance from so-called monoline insurance companies (Section 2.4.4).

WWR is often a natural and unavoidable consequence of financial markets. One of the simplest examples is mortgage providers who, in an economic recession, face both falling property prices and higher default rates by homeowners. In derivatives, classic examples of trades that obviously contain WWR across different asset classes are:

• Put option. Buying a put option on a stock (or stock index) where the underlying reference in question has fortunes that are highly correlated to those of the counterparty is an obvious case of WWR (e.g. buying a put on one bank's stock from another bank). Correspondingly, equity call options should be right-way products.
• FX forward or cross-currency products. Any foreign exchange (FX) contract should be considered in terms of a potential weakening of the currency and simultaneous deterioration in the credit quality of the counterparty. This would obviously be the case in trading with a sovereign and paying its local currency in an FX forward or cross-currency swap (or, more likely in practice, hedging this trade with a bank in that same region). Another way to look at a cross-currency swap is that it represents a loan collateralised by the opposite currency in the swap. If this currency weakens dramatically, the value of the ‘margin’ is strongly diminished. This linkage could be either way: first, a weakening of the currency could indicate a slow economy and hence a less profitable time for the counterparty. Alternatively, the default of a sovereign, financial institution, or large corporate counterparty may itself precipitate a currency weakening.
• Interest rate products. Here it is important to consider a relationship between the relevant interest rates and the credit spread of the counterparty. A corporate paying the fixed rate in a swap when the economy is strong may represent WWR for a bank receiving fixed, since interest rates would be likely to be cut in a recession. On the other hand, during an economic recovery, interest rates may rise and so paying fixed may represent a right-way situation.
• Commodity swaps. A commodity producer (e.g. a mining company) may hedge the price fluctuation to which it is exposed with derivatives. Such a contract should represent right-way risk since the commodity producer will only owe money when the commodity price is high, when the business should be more profitable. The right-way risk arises due to hedging (as opposed to speculation).
• Credit default swaps. When buying protection in a CDS contract, the exposure will be the result of the reference entity's credit spread widening, and the default probability will be linked to the counterparty's credit spread. In the case of a strong relationship between the credit quality of the reference entity and the counterparty, clearly there is extreme WWR. A bank selling protection on its own sovereign would be an obvious problem. On the other hand, with such a strong relationship, selling CDS protection should be a right-way trade.

There is also empirical evidence supporting the presence of WWR. Duffee (1998) describes a clustering of corporate defaults during periods of falling interest rates, which is most obviously interpreted as a recession, leading to both low interest rates (due to central bank intervention) and a high default rate environment. This has also been experienced in the last few years by banks on uncollateralised receiver interest swap positions, which have moved ITM together with a potential decline in the financial health of the counterparty (e.g. a sovereign or corporate). This effect can be seen as WWR creating a ‘cross-gamma’ effect (Section 21.2.3) via the strong linkage of credit spreads and interest rates, even in the absence of actual defaults. Regarding the FX example above, results from Levy and Levin (1999) look at residual currency values upon default of the sovereign and find average values ranging from 17% (triple-A) to 62% (triple-C). This implies the amount by which the FX rate involved could jump at the default time of the counterparty.

Regulators have identified characteristics of both general (driven by macro-economic relationships) and specific (driven by causal linkages between the exposure/margin and default of the counterparty) WWR, which are outlined in Table 17.7.

Table 17.7 Characteristics of general and specific WWR.

General WWR	Specific WWR
Based on macro-economic behaviour	Based on structural relationships that are often not captured via real-world experience
Relationships may be detectable using historical data	Hard to detect except by a knowledge of the relevant market, the counterparty, and the economic rationale behind their transaction
Potentially can be incorporated into pricing models	Difficult to model and dangerous to use naïve correlation assumptions; should be addressed qualitatively via methods such as stress testing
Should be priced and managed correctly	Should, in general, be avoided as it may be extreme

Specific WWR is particularly difficult to quantify since it is not based on any macro-economic relationships and may not be expected to be part of historical or market-implied data. For example, in 2010, the European sovereign debt crisis involved deterioration in the credit quality of many European sovereigns and a weakening of the euro. However, historical data did not bear out this relationship, largely since neither most of the sovereigns concerned nor the currency had ever previously been subject to any strong adverse credit effects.

17.6.2 Quantification of WWR in CVA

Incorporation of WWR in the CVA formula is probably most obviously achieved by simply representing the exposure conditional upon the default of the counterparty. For example, in the path-wise formula, this would simply involve replacing EPE in Equation 17.3 with , which represents EPE at time , conditional on this being the counterparty default time . This approach supports approaching WWR quantification heuristically by qualitatively assessing the likely increase in the conditional – compared to unconditional – exposure. An example of a qualitative approach to WWR is in regulatory capital requirements and the alpha factor (Section 13.4.5). A conservative estimate for alpha, together with the requirement to use stressed data in the estimation of the exposure, represents a regulatory effort partially to capitalise general WWR.

Appendix 17F shows a simple formula for the conditional EPE for a forward contract-type exposure (an extension of the previous unconditional case given in Appendix 11A). The relationship between EPE and counterparty default is expressed using a single correlation parameter. This correlation parameter is rather abstract, with no straightforward economic intuition, but it does facilitate a simple way of quantifying and understanding WWR.

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Spreadsheet 17.6 Simple wrong-way risk example.

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Figure 17.15 shows the impact of wrong-way (and right-way) risk on EPE. With 50% correlation, WWR approximately doubles the expected positive exposure (EPE), whilst with -50% correlation, the impact of right-way risk reduces it by at least half. This is exactly the type of behaviour that is expected: positive correlation between the default probability and exposure increases conditional EPE (default probability is high when exposure is high), which is WWR. Negative correlation leads to smaller EPE and so-called right-way risk.

Figure 17.15 Illustration of wrong-way and right-way risk exposure profiles using a simple model with correlations of 50% and -50%, respectively.

The size of EPE will now depend on the counterparty default probability. Figure 17.16 shows EPE for differing counterparty credit quality, showing that the exposure increases as the credit quality of the counterparty increases. This result might at first seem counterintuitive, but it makes sense when one considers that for a better credit quality counterparty, default is a less probable event and therefore represents a bigger shock when it occurs. We note an important general conclusion, which is that WWR increases as the credit quality of the counterparty increases.

A more sophisticated approach to modelling the relationship between default probability and exposure will be harder to achieve and may introduce computational challenges. At a high level, there are a number of problems in doing this, which are:

• Uninformative historical data. Unfortunately, WWR may be subtle and not revealed via any empirical data, such as a historical time series analysis of correlations.
• Misspecification of relationship. The way in which the dependency is specified may be inappropriate. For example, rather than being the result of a correlation, it may be the result of causality – a cause-and-effect-type relationship between two events. If the correlation between two random variables is measured as zero, then this does not prove that they are independent.³³

  

  Figure 17.16 Illustration of exposure under the assumption of WWR for different credit quality counterparties.

• Direction. The direction of WWR may not be clear. For example, low interest rates may typically be seen in a recession, when credit spreads may be wider and default rates higher. However, an adverse credit environment where interest rates are high is not impossible.

17.6.3 Wrong-way Risk Models

An obvious modelling technique for WWR is to introduce a stochastic process for the credit spread and correlate this with the other underlying processes required for modelling exposure. A default will be generated via the credit spread (‘intensity’) process, and the resulting conditional EPE will be calculated in the usual way in either a path-wise or direct implementation (but only for paths where there has been a default). This approach can be implemented relatively tractably, as credit spread paths can be generated first, and exposure paths need only be simulated in cases where a default is observed. The required correlation parameters can be observed directly via historical time series of credit spreads and other relevant market variables.³⁴ This will be referred to as the intensity approach.

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Spreadsheet 17.7 Direct simulation of wrong-way risk for an interest rate swap.

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Figure 17.17 Future values of the swap with 90% (top) and -90% (bottom) correlation between credit spreads and interest rates. The potential future exposure (5% and 95%) is shown by the dotted black lines, and EPE and ENE are shown by the solid black lines.

This approach is illustrated for the interest rate swap example previously discussed in Section 15.6.1. Using a lognormal model for credit spreads correlated to the interest rate process, the exposure conditional on default in Equation 17.3 is calculated to be compared to the previous Figure 15.23. Figure 17.17 shows the swap values generated conditional on default for high positive and negative correlations.

Figure 17.18 Illustration of the structural approach to modelling general WWR, assuming some underlying bivariate distribution.

Since this is a swap paying the fixed rate, a positive correlation with credit spreads leads to interest rates being higher in default scenarios, leading to a higher positive exposure (WWR). For a negative correlation, the negative exposure increases (right-way risk). The relationship between changes in interest rates and default rates has been empirically shown to be generally negative.³⁵

Note that the above approach can be seen to generate only weak dependence between exposure and default, given that the correlations used are close to their maximum and minimum values. Hence, whilst this approach is the most obvious and easy to implement, it may underestimate the true WWR effect.

An even simpler and more tractable approach to general WWR is to specify a dependence directly between the counterparty default time and the exposure distribution, illustrated in Figure 17.18 (for example, see Garcia-Cespedes et al. 2010). In this ‘structural approach’, the exposure and default distributions are mapped separately onto a bivariate distribution. Positive (negative) dependency will lead to an early default time being coupled with a higher (lower) exposure, as is the case with WWR (right-way risk). Note that there is no need to recalculate the exposures as the original unconditional values are sampled directly. The advantage of this method is that pre-computed exposure distributions are used, and WWR is essentially added on top of the existing methodology. However, this is also a disadvantage since it may not be appropriate to assume that all the relevant information to define WWR is contained within the unconditional exposure distribution.

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Spreadsheet 17.8 Exposure distribution using a Gaussian copula approach.

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Figure 17.19 shows EPE of the payer swap with a structural model and a correlation of 50%, and compared to an intensity model with the same correlation.

Figure 17.19 EPE of a payer interest rate swap calculated with WWR using intensity and structural models, each with a correlation of 50%.

Figure 17.20 shows CVA as a function of correlation for the intensity and structural approaches. Whilst both approaches produce a qualitatively similar result, with higher CVA for negative correlation, the effect is much stronger in the structural model.

The big drawback with the structural model is that the correlation parameter described above is opaque and, therefore, difficult to calibrate. Discussion and correlation estimates are given by Fleck and Schmidt (2005) and Rosen and Saunders (2010). More complex representations of this model are suggested by Iscoe et al. (1999) and De Prisco and Rosen (2005), where the default process is correlated more directly with variables defining the exposure. Estimation of the underlying correlations is then more achievable.

Figure 17.20 CVA as a function of the correlation between counterparty default time and exposure.

One other possible approach to WWR is a more direct, parametric one. For example, Hull and White (2011) have proposed linking the default probability parametrically to the exposure using a simple, functional relationship. They suggest using either an intuitive calibration based on a what-if scenario, or calibrating the relationship via historical data. This latter calibration would involve calculating the portfolio value for dates in the past and examining the relationship between this and the counterparty's credit spread. If the portfolio has historically shown high values together with larger than average credit spreads, then this will indicate WWR. This approach obviously requires that the current portfolio of trades with the counterparty is similar in nature to that used in the historical calibration, in addition to the historical data showing a meaningful relationship.

17.6.4 Jump Approaches

Jump approaches for WWR may be more realistic, especially in cases of specific WWR. For example, this has been clearly shown to be relevant for FX examples where the CDS market gives some clear indication of the nature of FX WWR. Most CDSs are quoted in US dollars, but sometimes simultaneous quotes can be seen in other currencies. For example, Table 17.8 shows the CDS quotes for Italian sovereign protection in both US dollars and euros. These CDS contracts should trigger on the same credit event definitions, and thus the only difference between them is the currency of cash payment on default. There is a large ‘quanto’ effect, with euro-denominated CDSs being cheaper by around 30% for all maturities.

Figure 17.21 shows the historical evolution of the Japanese sovereign quanto basis, which also shows a considerable difference between the price of CDS protection purchased in US dollars and Japanese yen. The CDS market, therefore, allows WWR effect in currencies to be observed and potentially also hedged.

The above data seem to suggest that the relationship involved is a causal one and the FX rate ‘jumps’ around the time of a sovereign default. Indeed, it has been shown (Ehlers and Schönbucher 2006) that an intensity-type modelling approach, such as that described in Section 17.6.3, is not able to match the CDS data. In such an approach, there would be a correlation between the FX rate and the credit spread, but the effect that this produces is generally not strong enough, even with +/-100% correlation.

Table 17.8 CDS quotes (mid-market) for Italian sovereign protection in both US dollars and euros, from April 2011.

Maturity (years)	USD	EUR
1	50	35
2	73	57
3	96	63
4	118	78
5	131	91
7	137	97
10	146	103

Figure 17.21 The Japan sovereign quanto basis for the dollar–yen pair.

Source: Chung and Gregory (2019).

Figure 17.22 Illustration of the currency jump approach to WWR for FX products.

A simple jump approach was first proposed by Levy and Levin (1999) to model FX exposures with WWR. This assumes that the relevant FX rate jumps at the counterparty default time, as illustrated in Figure 17.22.

The nature of implied jumps related to sovereign defaults has been well characterised. Levy and Levin (1999) used historical default data on sovereigns and showed that the implied jump is larger for better-rated sovereigns – for example, 83% for AAA and 27% for BBB sovereigns. This is probably because their default results in or is caused by a more severe financial shock and the conditional FX rate, therefore, should move by a greater amount. For example, a default of a large corporation should be expected to have quite a significant impact on the local currency (albeit smaller than that due to sovereign default).

The RVs can also be implied from available CDS quotes. For example, the implied jumps for the euro against the US dollar for European sovereigns around the time of the sovereign debt crisis in Europe were 9%, 17%, 20%, and 25% for Greece, Italy, Spain, and Germany, respectively.³⁶ This is again consistent with a higher-credit-quality – and potentially more-systemically-important – sovereign, creating a stronger impact.

It is not surprising that the above jump effect is observed for sovereign counterparties. However, it can also be observed for other counterparties as well. For example, Chung and Gregory (2019) use data from the Japanese CDS market to show that there are material implied jumps for financial institutions (average 13.5%) and non-financial corporations (average 8%), as well as for the sovereign (average 38.4%).

Whilst there is rarely market data to illustrate the above effect, in the absence of CDS data, Finger (2000) suggests how non-sovereign counterparties might be modelled if the sovereign quanto effect is observed.

17.6.5 Credit Derivatives

Credit derivatives are a special case, as the WWR is unavoidable (buying credit protection on one party from another party) and may be specific (e.g. buying protection from a bank on its sovereign). A number of approaches have been proposed to tackle counterparty risk in credit derivatives, such as Duffie and Singleton (2003), Jarrow and Yu (2001), and Lipton and Sepp (2009).

Appendix 17G describes the pricing for a CDS with counterparty risk using a simple model, ignoring the impact of any collateralisation. Due to the highly contagious and systemic nature of CDS risks, the impact of margin may be hard to assess and indeed may be quite limited. It is interesting to calculate the fair price for buying or selling CDS protection as a function of the correlation between the reference entity and the counterparty (the counterparty is selling protection). Figure 17.23 shows the fair premium – i.e. reduced to account for CVA – that an institution should pay in order to buy CDS protection.³⁷ Selling protection will require an increased premium. We can observe the very strong impact of correlation: one should be willing only to pay around 200 bps at 60% correlation to buy protection, compared with 250 bps with a ‘risk-free’ counterparty. CVA in this case is 50 bps (per annum), or one-fifth of the risk-free CDS premium. At extremely high correlations, the impact is even more severe and CVA is huge. At a maximum correlation of 100%, the CDS premium is just above 100 bps, which relates entirely to the recovery value.³⁸ When selling protection, the impact of CVA is much smaller and reduces with increasing correlation due to right-way risk.³⁹

Figure 17.23 Fair CDS premium when buying protection subject to counterparty risk, compared with the standard (risk-free) premium. The counterparty CDS spread is assumed to be 500 bps.

17.6.6 Collateralisation and WWR

Margin is typically assessed in terms of its ability to mitigate exposure. Since WWR potentially causes exposure to increase significantly, the impact of margin on WWR is very important to consider. However, this is very hard to characterise because it is very timing dependent. If the exposure increases gradually prior to default, then margin can be received, whereas a jump in exposure deems margin useless. This is illustrated, for example, by Chung and Gregory (2019), who show that margining has very little impact on an FX portfolio under the jump approach described in Section 17.6.4.

Not surprisingly, jump approaches tend to show collateralisation as being near useless in mitigating WWR, whereas more continuous approaches (such as the intensity and structural approaches described in Section 17.6.3) suggest that margin is an effective mitigant against WWR. The truth is somewhere in between, but is likely very dependent on the type of transaction and counterparty. Pykhtin and Sokol (2013) consider that the quantification of the benefit of margin in a WWR situation must account for jumps and a period of higher volatility during the MPoR. They also note that WWR should be higher for the default of more systemic parties, such as banks. Overall, their approach shows that WWR has a negative impact on the benefit of collateralisation. Interestingly, counterparties that actively use margin (e.g. banks) tend to be highly systemic and will be subject to these extreme WWR problems, whilst counterparties that are non-systemic (e.g. corporates) often do not post margin anyway.

WWR may also be present in terms of the relationship between the value of margin and the underlying exposure. Consider a payer interest rate swap collateralised by a high-quality government bond. This would represent a situation of general WWR, since an interest rate rise would cause the value of the swap to increase, whilst the margin value would decline. In the case of a receiver interest rate swap, the situation is reversed and there would be a beneficial right-way margin position. Given the relatively low volatility of interest rates, this is not generally a major problem.

A more significant example of general wrong-way (or right-way) margin could be a cross-currency swap collateralised by cash in one of the two underlying currencies. If margin is held in the currency being paid, then an FX move will simultaneously increase the exposure and reduce the value of the collateral.

There can also be cases of specific WWR where there is a more direct relationship between the type of margin and the counterparty credit quality. An entity posting its own bonds is an example of this: this is obviously a very weak mitigant against credit exposure and CVA, although it may mitigate FVA, as discussed in Section 11.4.3. A bank posting bonds of its own sovereign is another clearly problematic example.

17.6.7 Central Clearing and WWR

Given their reliance on collateralisation as their primary protection via variation and initial margin, CCPs may be particularly prone to WWR, especially those that clear products such as CDSs. A key aim of a CCP is that losses due to the default of a clearing member are contained within resources committed by that clearing member (the so-called ‘defaulter-pays’ approach described in Section 8.3.4). A CCP faces the risk that the defaulter-pays resources of the defaulting member(s) may be insufficient to cover the associated losses. In such a case, the CCP would impose losses on its members and may be in danger of becoming insolvent itself.

CCPs tend to disassociate credit quality and exposure. Parties must have a certain credit quality (typically defined by the CCP and not external credit ratings) to be clearing members, but will then be charged initial margins and default fund contributions driven primarily by the market risk of their portfolio (that drives the exposure faced by the CCP).⁴⁰ In doing this, CCPs are in danger of implicitly ignoring WWR.

For significant WWR transactions such as CDSs, CCPs have the problem of quantifying the WWR component in defining initial margins and default funds. As with the quantification of WWR in general, this is far from an easy task. Furthermore, WWR increases with increasing credit quality, shown both quantitatively and empirically in Section 17.6.4. Similar arguments are made by Pykhtin and Sokol (2012) in that a large dealer represents more WWR than a smaller and/or weaker credit quality counterparty. These aspects perversely suggest that CCPs should require greater initial margin and default fund contributions from better-credit-quality and more-systemically-important members.⁴¹

Related to the above is the concept that a CCP waterfall may behave rather like a collateralised debt obligation (CDO), which has been noted by a number of authors, including Murphy (2013), Pirrong (2013), and Gregory (2014). The comparison, illustrated in Figure 17.24, is that the ‘first loss’ of the CDO is covered by defaulter-pays initial margins and default funds, together with CCP equity. Clearing members, through their default fund contributions and other loss-allocation exposures, have a second loss position on the hypothetical CDO. Of course, the precise terms of the CDO are unknown and ever changing, as they are based on aspects such as the CCP membership, the portfolio of each member, and the initial margins held. However, what is clear is that the second loss exposure should correspond to a relatively unlikely event, since otherwise it would imply that initial margin coverage was too thin.

Figure 17.24 Comparison between a CCP loss waterfall and a CDO structure.

The second loss position that a CCP member is implicitly exposed to is therefore rather senior in CDO terms. Such senior tranches are known to be heavily concentrated in terms of their systemic risk exposure (see, for example, Gibson 2004, Coval et al. 2009, and Brennan et al. 2009) as discussed in Section 10.2.2.

CCPs also face WWR on the margin they receive. They will likely be under pressure to accept a wide range of eligible securities for initial margin purposes. Accepting more risky and illiquid assets creates additional risks and puts more emphasis on the calculation of haircuts that can also increase risk if underestimated. CCPs admitting a wide range of securities can become exposed to greater adverse selection, as clearing members (and clients) will naturally choose to post margin that has the greatest risk (relative to its haircut) and may also present the greatest WWR to a CCP (e.g. a European bank may choose to post European sovereign debt where possible).

NOTES

1. 1 International Financial Reporting Standards.
2. 2 As well as the recovery value.
3. 3 It is also smaller due to the lack of a principal payment, but this is a different point.
4. 4 In case explicit discount factors are required, care must be taken – for example, in an interest rate product where high rates will imply a smaller discount factor and vice versa. To account for this convexity effect technically means quantifying the underlying exposure using the ‘T-forward measure’ (Jamshidian 1989). By doing this, discount factors depend on expected future interest rate values, not on their distribution. Hence, moving the discount factor out of the expectation term (for exposure) can be achieved.
5. 5 This is a consequence of the central limit theorem.
6. 6 This would be required in some models and any path-dependent cases (e.g. collateralisation).
7. 7 This is the simple average of the EPE values in our example, although for non-equal time intervals it would be the weighted average. In the approximate formula, the undiscounted EPE is required, although in a low interest rate environment the discounting differences may be small, especially for short-dated transactions.
8. 8 This would have to account for the LGD.
9. 9 We note that regulatory capital requirements are more prescriptive than accounting requirements, which is why we make reference to them here. However, it does not necessarily follow that accounting CVA must follow Basel standards. The future regulatory framework for computing CVA capital (BCBS 2017) will use a bank's accounting CVA for determining regulatory capital (standardised CVA, SA-CVA), which may completely align accounting standards with regulatory capital ones.
10. 10 This is because for small values of . This is therefore more accurate for smaller spread values.
11. 11 J.P. Morgan (2015). Annual Report.
12. 12 www.standardandpoors.com.
13. 13 www.moodys.com.
14. 14 International Association of Credit Portfolio Managers (IACPM) Survey (2018).
15. 15 Davis, C. (2019). Japan banks face huge CVA hit, dealers say. Risk (21 January). www.risk.net.
16. 16 Assuming the creditors for the transaction in question have no security or seniority over the other creditors.
17. 17 This assumes that the parties will make the calculations in the same way (e.g. models and parameters).
18. 18 The precise credit curves can be seen in Spreadsheet 17.3.
19. 19 Meaning giving a better price than the case with no valuation adjustments, such as an interbank trade.
20. 20 This is sometimes a reasonable approximation in practice, especially for a collateralised relationship, but we will discuss the impact of asymmetry below.
21. 21 Ernst & Young (2012). CVA Survey. www.ey.com.
22. 22 Wilson, H. (2011). Banks' profits boosted by DVA rule. Daily Telegraph (31 October). www.telegraph.co.uk.
23. 23 Wright, W. (2011). Papering Over the Great Wall St Massacre. Financial News (26 October). www.fnlondon.com.
24. 24 Due to there being different accountancy standards for insurance companies, the monolines did not see this gain as a DVA benefit.
25. 25 Rappaport, L. (2011). MBIA and Morgan Stanley settle bond fight. Wall Street Journal (14 December). www.wsj.com. Although MBIA paid Morgan Stanley $1.1bn, the actual amount owed (as defined by Morgan Stanley's exposure) was several billion dollars. The difference can be seen as a DVA benefit gained by MBIA in the unwind, although MBIA may not have seen it like this.
26. 26 PricewaterhouseCoopers (2017). Lehman Brothers International (Europe) – In Administration. Joint Administrators' eighteenth progress report, for the period from 15 March 2017 to 14 September 2017 (9 October). www.pwc.com.
27. 27 Either because it is illegal or because of the extreme WWR it will create (Section 17.6.5), meaning that no party should be willing to enter such a trade except at a very low premium.
28. 28 The reader may wish to refer back to the discussion around incremental exposure in this section as many of the points made there will apply to incremental CVA also.
29. 29 This could also cover a policy where CVA adjustments are only calculated periodically and several trades have occurred with a given counterparty within that period.
30. 30 Using an MPoR of 10 days and a maturity of 10 years, and assuming 250 business days in a year, this comes from .
31. 31 A one-way margin agreement in favour of the party in question is assumed.
32. 32 Karagiannopoulos, L. (2018). Clearing members warn recovering money after Nasdaq deal could take a long time. Reuters (16 November). www.reuters.com.
33. 33 A classic example of this is as follows. Suppose a variable X follows a normal distribution. Now choose Y = X². X and Y have zero correlation but are far from independent.
34. 34 Noting that the credit spread may be determined by some proxy or generic curve, in which case this historical time series should be used.
35. 35 Longstaff and Schwartz (1995), Duffee (1998), and Collin-Dufresne et al. (2001).
36. 36 Moses, A. (2010). Quanto swaps signal 9 percent Euro drop on Greek default. Bloomberg (21 June). www.bloomberg.com.
37. 37 This is assumed to be free of counterparty risk. We assume that the reference entity CDS spread is 250 bps, whereas the counterparty CDS spread is 500 bps. Both LGDs are assumed to be 60%.
38. 38 The premium based only on recovery value (i.e. where there is no chance of receiving any default payment) is .
39. 39 For zero or low-correlation values, the protection seller may possibly suffer losses due to the counterparty defaulting when the CDS has a positive mark-to-market (requiring a somewhat unlikely tightening of the reference entity credit spread). However, for high correlation values, the mark-to-market of the CDS is very likely to be negative at the counterparty default time and (since this amount must still be paid) there is virtually no counterparty risk.
40. 40 Some CCPs do base margins partially on credit quality, but this, if used, is not a large effect.
41. 41 Of course, better-credit-quality members are less likely to default, but the impact if they do is likely to be more severe.