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value of all positions and is therefore directly related to what could potentially be lost today in the event of a default. However, other aspects are important in this regard, such as the ability to net transactions in default and the possibility to adjust positions with collateral amounts. Both of these aspects are subject to legal agreements and their potential interpretation in a court of law.

      Contractual features of transactions, such as close-out netting and termination features, refer to replacement costs. MTM is clearly closely related to replacement cost, which defines the entry point into an equivalent transaction(s) with another counterparty. However, the actual situation is more complicated. To replace a transaction, one must consider costs such as bid–offer spreads, which may be significant especially for particularly illiquid products. Note that even a standard and liquid contract might be non-standard and illiquid at the default time. In such a case, one must then decide whether to replace with an expensive non-standard derivative or with a more standard one that does not match precisely the original one. Large portfolios can be replaced one-for-one or macro-hedged. Broadly speaking, documentation suggests that default costs can effectively be passed on via the replacement cost concept, although this is discussed in more detail later via the definition of close-out amount (Section 5.2.6).

      Contractual agreements generally reference replacement costs (and not MTM) in defining a surviving party’s position in a default scenario. Although this represents the economic reality in a default, it can cause further problems. By their nature, replacement costs will include CVA (and more generally xVA) components that create a recursive problem, since one cannot define xVA today without knowing the future xVA. Chapter 14 addresses this topic in more detail (Section 14.6.5). For now, we note that quantification will assume, for reasons of simplicity, that MTM is a good proxy for the real replacement cost and this is in general not a bad approximation.

4.2.2 Credit exposure

      Credit exposure (hereafter often simply known as exposure) defines the loss in the event of a counterparty defaulting. It is also representative of other costs such as capital and funding that appear in other xVA terms. Exposure is characterised by the fact that a positive value of a portfolio corresponds to a claim on a defaulted counterparty, whereas in the event of negative value, a party is still obliged to honour their contractual payments (at least to the extent that they exceed those of the defaulted counterparty). This means that if a party is owed money and their counterparty defaults then they will incur a loss, while in the reverse situation they cannot gain20 from the default by being somehow released from their liability.

      Exposure is clearly a very time-sensitive measure, since a counterparty can default at any time in the future and one must consider the impact of such an event many years from now. Essentially, characterising exposure involves answering the following two questions:

      • What is the current exposure (the maximum loss if the counterparty defaults today)?

      • What is the exposure in the future (what could be the loss if the counterparty defaults at some point in the future)?

      The second point above is naturally far more complex to answer than the first, except in some simple cases.

      All exposure calculations, by convention, will ignore any recovery value in the event of a default. Hence, the exposure is the loss, as defined by the value or replacement cost that would be incurred, assuming no recovery value. Exposure is relevant only if the counterparty defaults and hence the quantification of exposure would be conditional on counterparty default. Having said this, we will often consider exposure independently of any default event and so assume implicitly no “wrong-way risk”. Such an assumption is reasonable for most products subject to counterparty risk, although the reader should keep the idea of conditional exposure in mind. We will then address wrong-way risk, which defines the relationship between exposure and counterparty default, in more detail in Chapter 17.

      Note that exposure from other points of view (most obviously funding-related) need not be conditional on counterparty default.

4.2.3 Default probability, credit migration and credit spreads

      When assessing counterparty risk, one must consider the credit quality of a counterparty over the entire lifetime of the relevant transactions. Such time horizons can be extremely long. Ultimately, there are two aspects to consider:

      • What is the probability of the counterparty defaulting21 over a certain time horizon?

      • What is the probability of the counterparty suffering a decline in credit quality over a certain time horizon (for example, a ratings downgrade and/or credit spread widening)?

      Credit migrations or discrete changes in credit quality (such as those due to ratings changes) are crucial, since they influence the term structure of default probability. They should also be considered, since they may cause issues even when a counterparty is not yet in default. Suppose the probability of default of a counterparty between the current time and a future date of (say) one year is known. It is also important to consider what the same annual default rate might be in four years – in other words, the probability of default between four and five years in the future. There are three important aspects to consider:

      • Future default probability22 as defined above will have a tendency to decrease due to the chance that the default may occur before the start of the period in question. The probability of a counterparty defaulting between 20 and 21 years in the future may be very small – not because they are very creditworthy (potentially, quite the reverse), but rather because they are unlikely to survive for 20 years!

      • A counterparty with an expectation23 of deterioration in credit quality will have an increasing probability of default over time (although at some point the above phenomenon will reverse this).

      • A counterparty with an expectation of improvement in credit quality will have a decreasing probability of default over time, which will be accelerated by the first point above.

      Spreadsheet 4.1 Counterparty risk for a forward contract-type exposure.

      There is a well-known empirical mean-reversion in credit quality, as evidenced by historical credit ratings changes. This means that good (above-average) credit quality firms tend to deteriorate and vice versa. So a counterparty of good credit quality will tend to have an increasing default probability over time, whereas a poor credit quality counterparty will be more likely to default in the short term and less likely to do so in the longer term. The term structure of default is very important to consider.

      Finally, we note that default probability may be defined as real-world or risk-neutral. In the former case, we ask ourselves what the actual default probability of the counterparty is, which is often estimated via historical data. In the latter case, we calculate the risk-neutral (or market-implied) probability from market credit spreads. The difference between real-world and risk-neutral default probabilities is discussed in detail in Chapter 12, but it is worth emphasising now that risk-neutral default probabilities have become virtually mandatory for CVA calculations in recent years due to a combination of accounting guidelines, regulatory rules and market practice.

4.2.4 Recovery and loss given default

      Recovery rates typically represent the percentage of the outstanding claim recovered when a counterparty defaults. An alternative variable to recovery is loss given default (LGD), which in percentage terms is 100 % minus the recovery rate. Default claims can vary significantly, so LGD is therefore highly uncertain. Credit exposure is traditionally measured independently, but LGD is relevant in the quantification of CVA.

      In the event of a bankruptcy, the holders of OTC derivatives contracts with the counterparty in default would generally be pari passu24 with the senior bondholders. OTC derivatives, bonds and CDSs generally reference senior unsecured credit risk and may appear to relate to the same LGD. However, there are timing issues: when a bond issuer defaults, LGD is realised immediately, since the bond can be sold in the market. CDS contracts are also settled within days of the defined “credit event” via the CDS auction that likewise defines the LGD. However, OTC

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