12
Credit Spreads, Default Probabilities, and LGDs

This chapter discusses default probabilities and recovery rates, or equivalently loss given default (LGD), which are key inputs to define credit value adjustment (CVA) and debt value adjustment (DVA) in counterparty risk quantification. Default probability (Section 3.3.4) defines the likelihood of counterparty default, whilst recovery rates (Section 3.3.5) define the amount lost in the default scenario. The difference between real-world and risk-neutral default probabilities will be discussed, together with the use of the latter in CVA/DVA quantification. This, in turn, leads to the need to define credit spreads, which often need to be estimated from the relatively scarce underlying data.

12.1 DEFAULT PROBABILITY

12.1.1 Real World and Risk Neutral

There is an important difference between so-called real-world and risk-neutral default probabilities that has been at the heart of the increased importance of CVA in recent years. Real-world default probabilities are often estimated from historical default experience via some associated credit rating. Alternatively, they may be based on quantitative models using balance sheet information. A risk-neutral (also known as ‘market-implied’) default probability is derived from market data, using instruments such as bonds or credit default swaps (CDSs).

It would be expected that risk-neutral default probabilities would be higher than their real-world equivalents since investors are risk averse and demand a premium for accepting default risk. This is indeed observed empirically – for example, Altman (1989) tracks the performance of portfolios of corporate bonds for a given rating and finds that the returns outperform a risk-free benchmark (which is a portfolio of treasury bonds). This outperformance shows that the return on the corporate bonds is more than adequate to cover the default losses experienced, and that bond investors are being compensated for material components above expected default rates and recoveries.

We depict the difference between a real-world and a risk-neutral default probability in Figure 12.1. The risk-neutral default probability is typically larger due to an embedded premium that investors require when taking credit risk. There has been research based on understanding the nature and behaviour of the risk premium depicted in Figure 12.1 (see Collin-Dufresne et al. 2001, Downing et al. 2005, and Longstaff et al. 2005).

The difference between real-world and risk-neutral default probabilities has been characterised in a number of empirical studies. For example, Giesecke et al. (2010) used a data set of bond yields that spanned a period of almost 150 years from 1866 to 2008, and found that average credit spreads (across all available bond data) have been about twice as large as realised losses due to default. This would suggest that the two components in Figure 12.1 are, on average, approximately equal. Studies that are more specific include Fons (1987), the aforementioned work by Altman (1989), and Hull et al. (2005). For example, Fons found that one-year risk-neutral default probabilities exceed actual realised default rates (in absolute terms) by approximately 5%. The difference between real and risk-neutral default probabilities from Hull et al. (2005) is shown in Table 12.1 as a function of credit rating. Note that the relative difference can be large, especially for better-quality credits.

Schematic illustration of the difference between real-world and risk-neutral default probabilities. — **Figure 12.1** Illustration of the difference between real-world and risk-neutral default probabilities.

Table 12.1 Comparison between real-world and risk-neutral default probabilities in basis points.

Source: Hull et al. (2005).

	Real world	Risk neutral	Ratio
Aaa	4	67	16.8
Aa	6	78	13.0
A	13	128	9.8
Baa	47	238	5.1
Ba	240	507	2.1
B	749	902	1.2
Caa	1690	2130	1.3

12.1.2 CVA and Risk-neutral Default Probabilities

In the early days of counterparty risk assessment, it was common for firms to use real-world default probabilities (usually based on historical estimates and ratings) in order to quantify CVA, which was not universally considered a component of the fair value of a derivative. This treatment is broadly consistent with the quantification of credit risk across a bank (e.g. for its loan book). Some firms (typically small banks and non-banking firms) may still do this, but it is increasingly less common in the industry. This is highlighted in the following text, which references the implementation of CVA capital charge under Basel III:

In contrast, accounting practices were much more diverse when the Basel Committee, in reaction to CVA losses observed during the crisis, initiated discussions on the implementation of a prudential framework for CVA. Accounting for CVA was not universally fair valued through P&L [profit and loss] and, in many instances, relied almost exclusively on the use of historical default probabilities.¹

Many small banks still use – either partially or completely – historical default probabilities.

The use of historical default probabilities has probably been the largest driver in the disparity in accounting for CVA across market participants. This was one of the reasons that the Basel III capital requirements for CVA, first introduced in 2010 (Section 4.2.5), were outlined with a risk-neutral-based CVA concept, irrespective of whether or not the bank in question actually accounted for its CVA in this fashion.

As discussed in Section 3.1.7, it has been increasingly common in recent years for risk-neutral default probabilities to be used. For example, in an Ernst & Young survey in 2012,² 13 out of 19 participating banks used risk-neutral information (‘market data’) for default probability estimation. The move to risk-neutral has been catalysed by accounting requirements and Basel III capital rules. International Financial Reporting Standard 13 (IFRS 2011; see also Section 5.3.3) requires entities to make use of observable market inputs wherever possible, and Basel III makes explicit reference to the credit spread in the underlying CVA formula (Equation 13.9). Some small regional banks still use real-world default probabilities, but this is becoming increasingly rare and harder to defend to auditors and regulators. For example, Ernst & Young (2014) states ‘The use of historical default rates would seem to be inconsistent with the exit price notion in IRFS 13.’

The use of risk-neutral default probabilities changes the interpretation of CVA to be a market price of counterparty risk, rather than an actuarial assessment of expected future losses due to counterparty defaults. In some sense, this is not surprising given the development of the CDS market and the fact that CVA hedging has become more commonplace. On the other hand, it is important to emphasise that many counterparties are ‘illiquid credits’ in the sense that there is no direct market observable from which to define a risk-neutral default probability. This is particularly true for banks, which may have thousands of counterparties, many of whom are relatively small and do not have bonds or CDSs referencing their own credit risk. The problem of illiquidity is also more significant in regions outside Europe and the US, where CDS and secondary bond markets are more illiquid and sometimes non-existent. Furthermore, many liquid credits are large financial institutions which typically post margin and are therefore less important from a CVA perspective. Non-margin-posting counterparties, such as corporations and small and medium-sized enterprises (SMEs), are almost always illiquid credits but are very important from a CVA perspective.

The requirement to use risk-neutral default probabilities for illiquid credits creates a further problem with hedging: risk-neutral probabilities suggest the existence of a hedge, but without a liquid CDS on the counterparty in question, such a hedge does not exist. This is problematic since CVA will generally be much larger and more volatile (compared to using historical default probabilities), but without the availability of the natural hedging instruments to manage this volatility. For these reasons, banks have sometimes attempted to follow an intermediate approach, such as using a blend of historical and risk-neutral default probabilities (including two banks from the aforementioned 2012 Ernst & Young survey). Another approach has been to use risk-neutral default probabilities for ‘liquid credits’ (i.e. those with an active CDS market or equivalent) and historical or blended probabilities for illiquid credits.

However, regulators and auditors generally do not support the deviation from risk-neutral default probabilities, even in cases of illiquid credits. For example, the CVA challenger model imposed by the European Central Bank (ECB) states:³

The CVA challenger model then calculates an estimate of the CVA based on Benchmark PD [probability of default] parameters estimated from current index CDS curves and a market standard LGD parameter. The source of any significant deviations should then be understood.

Basel III capital rules impose similar requirements for capital allocation against CVA, requiring CDS spreads to be used where available, and for non-liquid counterparties stating that a reasonable estimate (‘proxy’) should be used (BCBS 2011b):

Whenever such a CDS spread is not available, the bank must use a proxy spread that is appropriate based on the rating, industry and region of the counterparty.

Using the current credit environment (via CDS spreads) as a reference would seem to be preferable to a backwards-looking and static approach using historical data. However, the large non-default (risk premium) component in credit spreads, the inability to define CDS spreads for most counterparties, and the underlying illiquidity of the CDS market does create problems with such an approach. Furthermore, the unintended consequences of the use of CDS-implied default probabilities has potentially unintended adverse consequences, such as a ‘doom loop’, where CVA hedging can cause an increase in the cost of buying CDS protection (see Section 13.3.6). Some authors (e.g. Gregory 2010) have argued against this requirement and noted that banks do not attempt to mark-to-market much of their illiquid credit risk (e.g. their loan books). However, whilst using subjective mapping methods to determine a credit spread may seem rather non-scientific, it is generally a necessary process for banks to value illiquid assets, such as bonds and loans, held on their trading books. Accordingly, regulators clearly see risk-neutral default probabilities as a fundamental building block for CVA calculations, and they are clearly the basis for defining exit prices for fair value accounting purposes.

In line with the above, market practice (especially in the larger banks) has converged on the use of risk-neutral default probabilities. For example, the European Banking Authority (EBA 2015b) states:

The CVA data collection exercise has highlighted increased convergence in banks' practices in relation to CVA. Banks seem to have progressively converged in reflecting the cost of the credit risk of their counterparties in the fair value of derivatives using market implied data based on CDS spreads and proxy spreads in the vast majority of cases. This convergence is the result of industry practice, as well as a consequence of the implementation in the EU of IFRS 13 and the Basel CVA framework.

12.1.3 Defining Risk-neutral Default Probabilities

Risk-neutral default probabilities are those derived from credit spreads observed in the market. There is no unique definition of a credit spread, and it may be defined in slightly different ways and with respect to different market observables, such as:

single-name CDSs;
asset swaps spreads;⁴
bond or loan prices;⁵ or
using some proxy or mapping method.

All of the above are (broadly speaking) defining the same quantity, but the CDS market is the most obvious clean and directly available quote. In contrast, to calculate a credit spread from a bond price requires various assumptions, such as comparing with some benchmark such as a treasury curve. Where observable, the difference between CDS- and bond-derived credit spreads (the ‘CDS-bond basis’) is significant. The analysis below will discuss the calculation of a risk-neutral default probability, and the definition of credit spreads will be discussed in Section 12.2.

Appendix 12A gives more detail on the mathematics of deriving risk-neutral default probabilities. For quantifying a term such as CVA, we require the default probability between any two sequential dates. A commonly used approximation for this is:

(12.1)

Where images is the default probability between images and images , images is the credit spread at time images and images is the assumed LGD (discussed below). Note that the PD in Equation 12.1 is unconditional (i.e. it is not conditional upon the counterparty surviving to images ). Table 12.2 illustrates this for a simple example using annual default probabilities. To obtain a more granular representation, the most obvious solution would be to interpolate the credit spreads.

Table 12.2 Annual default probabilities for an example credit curve using Equation 12.1. The LGD is assumed to be 60%.

Time	Credit spread	PD
1 year	100 bps	1.65%
2 years	125 bps	2.43%
3 years	150 bps	3.14%
4 years	175 bps	3.79%
5 years	200 bps	4.34%

Graph depicts the three different shapes of credit curve in which all with a five-year spread of 150 bps. — **Figure 12.2** Three different shapes of credit curve, all with a five-year spread of 150 bps.

Equation 12.1 is only an approximation because it does not account for the shape of the credit spread curve prior to the time images (and the more sloped the curve is, the worse the approximation). In Spreadsheet 12.1, it is possible to compare the simple formula with a more accurate calculation.

Suppose we take three different credit curves: flat, upwards sloping, and inverted, as shown in Figure 12.2. The cumulative default probability curves are shown in Figure 12.3. Note that all have a five-year credit spread of 150 bps and assumed LGD of 60%. The only thing that differs is the shape of the curve. Whilst all curves agree on the five-year cumulative default probability of 11.75%, the precise shape of the curve up to and beyond this point gives very different results. This is seen in Figure 12.4, which shows annual default probabilities for each case. For an upwards-sloping curve, default is less likely in the early years and more likely in the later years, whilst the reverse is seen for an inverted curve. In order to calculate risk-neutral default probabilities properly, in addition to defining the level of the credit curve, it is also important to know the precise curve shape. Extrapolation to the 10-year point, if that information is not available, is very sensitive.

Graph depicts the cumulative default probabilities for flat, upwards-sloping, and inverted credit curves. — **Figure 12.3** Cumulative default probabilities for flat, upwards-sloping, and inverted credit curves. In all cases, the five-year spread is 300 bps and the LGD is assumed to be 60%.

Bar chart depicts the annual default probabilities for flat, upwards-sloping, and inverted curves, as described in the text. — **Figure 12.4** Annual default probabilities for flat, upwards-sloping, and inverted curves, as described in the text. In all cases, the five-year spread is 150 bps and the LGD is assumed to be 60%.

12.1.4 Loss Given Default

In order to estimate risk-neutral default probabilities, an assumption for the LGD is typically required. The LGD refers to the percentage amount that would be lost in the event of a counterparty defaulting (all creditors having a legal right to receive a proportion of what they are owed). Equivalently, this is sometimes defined as one minus the recovery rate. LGD depends on the seniority of the over-the-counter (OTC) derivative claim – normally this ranks pari passu (of the same seniority) with senior unsecured debt, which in turn is referenced by most CDS contracts. However, sometimes derivatives may rank more senior (typically in securitisations) or may be subordinated, in which case further adjustments may be necessary.

Historical analysis on recovery rates shows that they vary significantly depending on the sector, the seniority of the claim, and economic conditions. As an example, Table 12.3 shows some experienced recovery values for financial institutions, which span the whole range from virtually zero to full recovery. For CVA computation, if the seniority is pari passu to that of the observable credit instruments and so the estimate is not of primary importance due to a cancellation effect. In situations where the LGD is expected to be higher or lower than that of other creditors (e.g. due to structural seniority), the estimation is more important. We discuss this more in Section 12.2.3.

A final point on recovery is related to the timing. CDSs are settled quickly following a default and bondholders can settle their bonds in the same process (the CDS auction) or simply sell them in the market. However, more complex contracts such as derivatives cannot be settled in a timely manner. This is partly due to their bespoke nature and partly due to netting (and margin), which means that many transactions are essentially aggregated into a single claim and cannot be traded individually. The net claim (less any margin) is then often quite difficult to define for the portfolio of trades (see Figure 2.8). This potentially creates two different recovery values:

Table 12.3 Recovery rates for CDS auctions for some credit events in 2008. The Fannie Mae and Freddie Mac subordinated debt traded at higher levels than the senior debt due to a ‘delivery squeeze’ caused by a limited number of bonds in the market to deliver against CDS protection.

Reference entity	Seniority	Recovery rate
Fannie Mae	Senior	91.5%
	Subordinated	99.9%
Freddie Mac	Senior	94.0%
	Subordinated	98.0%
Washington Mutual		57.0%
Lehman		8.6%
Kaupthing Bank	Senior	6.6%
	Subordinated	2.4%
Landsbanki	Senior	1.3%
	Subordinated	0.1%
Glitnir	Senior	3.0%
	Subordinated	0.1%
Average		38.5%

Settled recovery. This is the recovery that could be achieved following the credit event by trading out of a claim – for example, by selling a defaulted bond.
Actual recovery. This is the actual recovery received on a more complex portfolio (e.g. including derivatives) following a bankruptcy or similar process.

In theory, settled and actual recoveries should be very similar, but in reality, since bankruptcy processes can take many years, they may differ materially. This is illustrated in Figure 12.5. It should be possible to agree on the claim with the bankruptcy administrators prior to the actual recovery, although this process may take many months. This would allow an institution to sell the claim and monetise the recovery value as early as possible. In the case of the Lehman Brothers bankruptcy, the settled recovery was around 9%, whereas some actual recoveries traded to date (e.g. derivatives portfolios) have been substantially higher (in the region of 30–40%).

Schematic illustration of recovery settlement after a credit event. The settled recovery rate is achieved shortly after the credit event time. The final recovery occurs when the company has been completely wound up. The actual recovery for a derivative claim may be realised sometime between the settled and final recoveries via trading out of the claim. — **Figure 12.5** Schematic illustration of recovery settlement after a credit event. The settled recovery rate is achieved shortly after the credit event time (e.g. by participating in the CDS auction). The final recovery occurs when the company has been completely wound up. The actual recovery for a derivative claim may be realised sometime between the settled and final recoveries via trading out of the claim.

It should also be noted that recoveries on derivatives may be improved due to offsetting against other claims or other assets held (e.g. see the discussion on set-off in Section 6.3.6). These components may not be included in CVA but may give some additional benefit in a default workout process (discussed in Section 21.1.4).

12.2 CREDIT CURVE MAPPING

12.2.1 Overview

A credit curve is a key but often subjective input into a CVA calculation. Banks will have many hundreds or even thousands of counterparties, which will be entities such as corporates, SMEs, financial institutions, and SSAs (sovereigns, supranational entities, and agencies). The vast majority of these counterparties will not have liquid CDS quotes, bond prices, or even external ratings associated with them. End users of derivatives will be in a somewhat different situation, with only a relatively small number of (bank) counterparties. However, even then CDS quotes for some counterparties may be illiquid or unavailable. All firms, therefore, face some sort of credit curve mapping challenge.

No standard method exists for defining a credit curve for a given counterparty. This is not surprising given the subjectivity of the problem, and although some basic principles apply, there are many different approaches to credit curve mapping. Much of the regulatory guidance is quite broad and only makes reference to general aspects such as the rating, region, and sector being considered when determining the appropriate credit spread. One more detailed example is a publication by the EBA (2013) that discusses the determination of credit spreads for CVA purposes. Some of the general issues to be faced with credit curve mapping are:

Reference instrument. As noted above (Section 12.1.3), there are a number of potential sources of credit spread information, such as CDSs or bonds.
Tenor. It is also important to define fully the term structure of credit spreads up to the maturity of the counterparty portfolio in question. Available market data may mean that some tenors may be easier to map than others. Defining long-dated tenors (e.g. above 10 years) is particularly challenging.
Seniority. It may be that the instrument used to define the credit spread has different seniority to that of the potential derivative claim with the counterparty.
Liquidity. Some instruments, such as CDS indices, will be more liquid but less appropriate from a fundamental point of view for mapping a given credit. Other more relevant sources may be rather illiquid. A certain threshold for market liquidity (e.g. the number of entities quoting a price and the frequency of trading) needs to be established for a given quote to be used.
Region. Whilst European and US debt markets may provide a reasonable amount of liquidity, other regions (e.g. Asia) are generally much more limited. Regional banks may, therefore, face an even greater challenge for determining credit spreads.
Hedging. Related to the above liquidity comments, some reference instruments may provide reasonable mapping information but may not facilitate the hedging of counterparty risk. This could be due to a lack of liquidity or for practical reasons (e.g. not being able to short a corporate bond).
Capital relief. Related to hedging will be the potential capital relief that will be available from various hedges, which can sometimes create problems. For example, proxy single-name CDSs may be considered good hedges,⁶ but under current capital rules they achieve no capital relief (Section 13.3.6).⁷ Index hedges do allow capital relief, but the magnitude of this may not align with the view of the bank and/or the accounting CVA.

The above will lead to some rather difficult and subjective decisions over the choice of mapping methodology – for example, whether it is appropriate to map to an illiquid bond price observed in the secondary market for the counterparty in question, or to use a CDS index that is much more liquid and can provide a hedge. Another difficult decision might be whether a firm should use a single-name CDS on a similar credit, which is believed to represent an excellent reference point but under regulatory rules does not attract any capital relief as a hedge.

12.2.2 The CDS Market

There are a few hundred reference entities with liquid single-name CDS quotes, mainly large financial institutions and sovereigns, although the liquidity of this market has not been improving in recent years. There are also credit indices, which are generally more liquid. Figure 12.6 gives an overview of the main CDS instruments available for mapping purposes. The indices are managed by IHS Markit.⁸ Reading from the bottom, the first choice would obviously be to map to a single-name CDS or a relevant proxy such as a parent company. If such information were not available, then the counterparty would be mapped to the relevant index depending on whether it is a corporation, financial, or sovereign entity. Corporations may be further sub-divided according to credit quality. For example, the European Crossover index and CDX High Yield indices contain non-investment-grade names.

Schematic illustration of classification of counterparties according to global credit indices. — **Figure 12.6** Illustration of classification of counterparties according to global credit indices.

Table 12.4 lists some characteristics of credit indices globally. Generally, indices reference liquid credits that trade in the single-name CDS market or secondary bond market. Note that more detailed classifications exist that are not shown. For example, iTraxx SovX is sub-divided into various regions (Western Europe, CEEMEA – Central and Eastern Europe, Middle East and Africa – Asia Pacific, Global Liquid Investment Grade, G7, BRIC – Brazil, Russian, India, and China – and Latin America). The main non-financials index is sub-divided into sectorial indices (TMT – technology, media, and telecommunications – industrials, energy, consumers, and autos). Whilst these sub-divisions potentially give a more granular representation, they have to be balanced against the available liquidity in the CDS market, which is poor beyond the iTraxx and CDX main indices. The liquid indices trade at maturities of 3, 5, 7, and 10 years, whilst for the less liquid ones the 5- and 10-year tenors are the most traded. In general, the 5-year CDS quote for any reference entity or index is the most liquid.

There are some additional technical issues with using CDSs to derive credit spreads for calculating CVA. First, the credit events under an International Swaps and Derivatives Association (ISDA) standard CDS are failure to pay, bankruptcy, or restructuring. Restructuring is specific to CDS agreements in Europe and emerging markets and is not considered a credit event in CDSs referencing North American credits.

Table 12.4 The universe of key credit indices globally.

	Index	Size	Comment
iTraxx Europe	Main	125	Most actively traded investment-grade names
	Non-financials	100	Non-financial credits
	Financials senior	25	Senior subordination financial names
	Financials sub	25	Junior subordination financial names
	Crossover	40	Sub-investment-grade credits
	High volatility	30	Widest spread credits from the main index
	LevX	30	European first lien loan CDSs
CDX	Main	125	Most actively traded investment-grade credits
	High vol	30	High-volatility investment-grade CDSs
	High yield	100	High-yield credits
	Crossover	35	CDSs that are at the crossover point between investment grade and junk
	Emerging markets	14	Emerging markets CDSs
	Emerging markets diversified	40	Emerging markets CDSs
	LCDX	100	First lien leverage loans CDSs
iTraxx Asia	Asia	50	Investment-grade Asian (ex-Japan) credits
	Asia HY	20	High-yield Asian (ex-Japan) credits
	Japan	50	Investment-grade-rated Japanese entities
	Australia	25	Liquid investment-grade Australian entities

From the perspective of hedging CVA, it is important that a counterparty failing to make a payment in a derivative contract would trigger a payment on the underlying CDS used for hedging. Ideally, there would therefore be a ‘cross-default’ of these obligations in the documentation so that a derivatives default will trigger the CDS contract. Indeed, sometimes CDSs do include such a trigger explicitly, although they are inevitably more expensive. A recent development in the CDS market has also been to add a new credit event triggered by a government-initiated bail-in.

The definition of restructuring as a credit event is quite complex but intends to cover a circumstance where a reference entity (due to a deterioration of its credit) negotiates changes in the terms with its debtors as an alternative to formal insolvency proceedings. However, in 2012 the restructuring of Greek debt avoided triggering CDSs via a restructuring credit event (potentially so as to ensure the stability of major European banks selling protection on Greece). A similar problem has occurred more recently with Banco Popular Español SA.⁹

Another important aspect of CDSs is a ‘succession event’, which could refer to proceedings such as a merger, demerger, transfer of assets or liabilities, or another similar event in which one entity assumes the obligations of another entity. For banks, ring-fencing and bad-bank-type restructuring are important considerations. A potential problem is ‘orphaning’, where a restructuring process leaves CDS contracts referencing an old entity and not referencing any outstanding debt, meaning that they become worthless.¹⁰

Note also that the deliverable in a CDS contract is typically a bond or loan, and not a derivative receivable. This leaves a potential basis risk between the LGD on the derivative and the payout on the CDS, as discussed in Section 12.1.4. Contingent CDSs, which do reference derivatives transactions or portfolios directly, are very illiquid.

Liquidity in the single-name CDS market has fallen since the global financial crisis, although some indices have become more liquid. There is a general issue with the depth and liquidity of the single-name CDS market and the calculation and management of CVA. Despite this, there is still a requirement to reasonably estimate credit spreads in order to calculate CVA for any counterparty.

12.2.3 Loss Given Default

Ideally, LGDs would be derived from market prices, but this is not generally possible since the relevant market information does not exist. Generally, a credit spread curve can be used to estimate risk-neutral default probabilities, with an assumed LGD value being an important part of the process (Equation 12.1). It is not possible to estimate market-implied LGDs unless different seniorities of credit spread are traded by the same reference entity (which is rarely the case). A ‘recovery lock’ or ‘recovery swap’ is an agreement between two parties to swap a realised recovery rate (when/if the relevant recovery event occurs) with a fixed recovery rate (fixed at the start of the contract and generally the same as the standard recovery rates outlined above). Recovery locks do not generally trade, except occasionally for distressed credits. Likewise, a ‘fixed recovery CDS’ (when compared to a standard CDS) would allow information on implied recovery rates to be observed, but these typically do not trade either.¹¹

There are defined LGDs in the trading of CDS contracts (e.g. in order to value transactions) which depend on the underlying reference entity (e.g. 60% for iTraxx Europe and CDX NA). However, these LGD values only represent a convention and are not, therefore, truly market-implied or risk-neutral values. Generally, these standard LGDs, where known, are used to derive risk-neutral default probabilities. The choice of LGD is therefore driven by market convention but is not implied directly from market prices.

Sometimes, more favourable (lower) LGDs may be used to reflect aspects such as (in order of ease of justification):

Structural seniority of the portfolio – for example, derivative transactions with securitisation special purpose vehicles.
Credit enhancements or other forms of credit support, such as collateral available either directly (although margin specific to the contracts in question should generally be modelled within the exposure simulation, as discussed later in Section 15.5) or as a result of a broad relationship with a client. In the latter case, legal opinion as to whether the collateral can be used will be important.
The assumption of a favourable workout process, based on past experience. Often banks may have actual recovery rates on their derivative portfolios which are generally higher than the standard recovery rates, even though in terms of seniority they should be the same. One reason for this is the Lehman case mentioned in Section 12.1.4. Auditors and regulators may not accept such assumptions without stronger evidence.

An example of a bank commenting on the use of the first approach above 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’.¹² The above arguments may sometimes be hard to justify, especially for regulatory capital purposes (e.g. the incoming FRTB-CVA capital requirements only seems to allow, with evidence, the first case; see Section 13.3.5).

12.2.4 General Approach

The most obvious observable market price to determine a credit spread is generally accepted to be that of a CDS, as this instrument cleanly references credit risk without other effects (such as a significant interest rate component in a bond). Not surprisingly, CDSs are viewed as being the primary source of credit spread information. Note, however, that regulation does tend to allow other liquid trading credit risk instruments (such as bonds) where CDSs are not available, as long as such spreads correspond to the appropriate rating, region, and industry combination.¹³

In general, there are three different sources of credit spread information for a given counterparty:

Direct observables. In this situation, the credit spread of the actual counterparty in question is directly observable in the market. Note that even when this data exists, there may only be one liquid tenor (e.g. typically five years for a single-name CDS), which is a clear problem, especially for long-dated portfolios. If it is possible to short the credit (e.g. buy CDS protection), then it may be possible to hedge and gain capital relief.
Single-name proxies. This is a situation where another single reference entity trades in the market, which is viewed as a good proxy for the counterparty in question. This may be a parent company or the sovereign of the region in question, and may be used directly or have an additional component added to the credit spread to reflect greater riskiness. Hedging with such a name may provide a spread hedge but not a default hedge, and capital relief will not be achieved under current regulatory rules.¹⁴
Generic proxies. This is the case where there is no defined credit spread that can be readily mapped directly, and some sort of generic mapping via rating, region, and sector is required. CDS indices do not provide default protection¹⁵ but may be used as macro credit spread hedges, which will provide partial capital relief.

More discussion of the capital relief of various credit hedges will be given in Section 13.3.6. In particular, future regulation will recognise single-name credit hedges that either are linked through the legal entity or share the same sector.

A summary of the above approaches is given in Table 12.5.

If there is no relevant credit spread information available, then it may be acceptable for credit spreads to be constructed more indirectly using historical default probabilities and risk premium components (Figure 12.1), although it is clear that credit spreads should be in line with market levels.¹⁶ For example, the EBA (2017) states:

When no time series of credit spreads is observed in the markets of any of the counterparty's peers due to its very nature (eg project finance, funds), a bank is allowed to use a more fundamental analysis of credit risk to proxy the spread of an illiquid counterparty. However, where historical PDs (‘probabilities of default’) are used as part of this assessment, the resulting spread cannot be based on historical PD only – it must relate to credit markets.

Table 12.5 Comparison of different credit spread approaches.

	Liquidity	Hedging	Capital relief
Direct observables	Poor	Spread and default hedge	High
Single-name proxies	Medium	Partial spread hedge only	None^a
Generic proxies	Good	Partial spread hedge only	Partial

a Future regulatory rules will provide partial relief; see Section 13.3.6.

The future FRTB-CVA capital rules (BCBS 2017) make a similar statement:

When no credit spreads of any of the counterparty's peers is available due to the counterparty's specific type (eg project finance, funds), a bank is allowed to use a more fundamental analysis of credit risk to proxy the spread of an illiquid counterparty. However, where historical PDs are used as part of this assessment, the resulting spread cannot be based on historical PD only – it must relate to credit markets.

Any credit spread mapping needs to be implemented via some sort of decision tree, as illustrated in Figure 12.7. The typical benchmark choice is a CDS where available, and other instruments such as bonds will normally only be considered where the single-name CDS is not liquid. Other quotes, such as bond spreads, will have to be derived using some methodology and then potentially basis-adjusted to attempt to estimate the equivalent CDS value. Single-name proxies may attract a small spread adjustment to account for a perceived higher (or lower) credit risk. This spread adjustment could be because a parent company is being used as a proxy (but does not offer an explicit guarantee and the child company is viewed as riskier). Sovereign CDSs are also quite common proxies, especially in markets where single-name CDSs are limited, and a spread will be added to reflect the additional idiosyncratic risk with respect to the sovereign credit quality. Clearly, if a significant spread adjustment needs to be made, then this suggests that the proxy is not a particularly good choice. Is it also important to note that single-name proxies may increase volatility, since any idiosyncratic behaviour of the proxy will be incorrectly reflected in the mapped credit spread. Using historical data may only be justifiable when no relevant credit spread data on similar credits is available.

Schematic illustration of the decision tree in order to map a given counterparty credit spread. — **Figure 12.7** Example decision tree in order to map a given counterparty credit spread.

An important consideration in the decision tree above is what constitutes a liquid CDS quote. Banks may define a certain threshold for this by considering, for example, the average number of CDS quotes for the reference entity over a certain period of time. Alternatively, a third-party source could be used – for example, IHS Markit defines CDS liquidity scores of 1–5, with 1 being the most liquid and 5 being the least liquid, based on aspects such as bid-ask spreads, number of active market makers, and number of quotes.¹⁷ Fitch Ratings also offers a similar service, where each entity is given a score across the global CDS universe.¹⁸

With respect to the decision tree in Figure 12.7, it is important to note that a typical bank will end up with many generic proxies. This is due to the likelihood of having many clients who have relatively small balance sheets (e.g. corporates, SMEs) and therefore not having liquid instruments traded in the credit markets. Although the individual exposure to each of these clients may be relatively small, collectively the total exposure is likely to be very significant, and therefore the construction of general curves as proxies is important.

12.3 GENERIC CURVE CONSTRUCTION

12.3.1 General Approach

The aim of generic credit curve mapping is to use some relevant liquid credit spread quotes to estimate a general curve based on observable market data, as illustrated in Figure 12.8. Some methodology will be required to combine points at a given tenor (perhaps with some underlying weighting scheme also used to bias towards the more liquid quotes). Some other and possibly separate approach will be required to interpolate between tenors.

Graph depicts the generic curve construction procedure. The crosses represent observable credit spreads as a function of maturity. — **Figure 12.8** Illustration of a generic curve construction procedure. The crosses represent observable credit spreads as a function of maturity.

Bar chart depicts the market practice for marking non-tradable credit curves. — **Figure 12.9** Market practice for marking non-tradable credit curves.

Source: Deloitte/Solum CVA Survey (2013).

Granularity is an important consideration in generic curve construction. A very granular definition will be problematic as relevant data may not be available to define various components. On the other hand, a very coarse approach may not define specific credit spreads well. Clearly, in regions such as the US and Europe the most granular definition may be possible, whereas in smaller regions industry classification will almost certainly need to be excluded. Counterparties such as municipalities may represent a particular challenge, with limited credit spread information, apart from that of the sovereign, being available.

Figure 12.9 illustrates market practice for the construction of generic curves for non-tradable credits. In addition to the obvious use of indices, it can be seen that bespoke curves are generated as a function of rating, region, and industry. Not surprisingly, whilst classification via rating is common, the use of region and industry grouping is less common. In other words, banks will classify by all three if possible and drop the industry and possibly also the regional categorisation if necessary. Internal spread corresponds to using some internal spread estimation, potentially from the pricing of loans to the same or similar counterparties. This is clearly less in line with the concept of defining spreads with respect to external pricing and market observables.

Regulators generally propose a mapping based upon rating, region, and industry; they necessarily accept more sparse representation when data does not clearly allow this. Even within these categories, the classification is generally kept fairly narrow – for example, EBA (2013) states that, ‘where a CDS for a counterparty is not available, institutions shall use a proxy spread that is appropriate having regard to the rating, industry and region of the counterparty’ and suggest the following classification as a minimum:¹⁹

A pre-defined hierarchy of internal and/or external credit ratings. Note that for non-externally rated names, banks will typically use their internal rating mapped to external ratings.
Industry, based on:
- public sector;
- financials; and
- others.
Region, based on:
- Europe;
- North America;
- Asia; and
- rest of the world.

Despite the relatively general definitions above, it is recognised that firms may have to depart from the minimum granularity if there is insufficient data.

The approach to classification may differ between firms. For example, a large global bank may believe it has an exposure that is not concentrated from a regional or sectorial point of view. On the other hand, a local bank will necessarily have a more geographically-concentrated exposure and may be more exposed to certain industry sectors that are more active in its own region.

In general, there are two general approaches that have become popular for defining generic credit curves:

Intersection (or ‘bucketing’) approach. This is a basic approach that takes a measure such as the average or median across all liquid names in the relevant rating, region, and sector sub-categories to imply the generic credit spread. Whilst simple, this suffers from the sparseness of the underlying data, with some sub-categories having no data whatsoever. As such, it would be expected not to be robust and to be potentially unstable.
Cross-section methodology. This is a more sophisticated approach, based on a multidimensional regression across rating, region, and industry sector. The use of regression avoids problems that arise from a lack of data.

These approaches will be outlined in more detail below.

12.3.2 Intersection (Bucketing) Approach

In the intersection methodology, the proxy spread for a given obligor shall be determined by aggregating data across the relevant rating, region, and sector sub-categories forming a bucket. If this classification is rather broad (e.g. a single-A European financial services counterparty), then there will be a large number of data points but less distinction between different counterparties. In contrast, a more granular classification (e.g. a single-A Asian utility company) distinguishes better between different counterparties but provides less data for each curve calibration. Mapping has to consider granularity carefully; more granular mapping is preferable only if there are sufficient data points for each categorisation. With few data points, there is a danger of the idiosyncratic risk of a particular credit creating unrealistic and undesirable volatility.

Chourdakis et al. (2013) illustrate the problems with an intersectional approach by considering the following classification:

Regions (five). North America, Europe, Japan, Asia ex-Japan, and rest of the world.
Rating (six). AAA/AA, A, BBB, BB, B, and CCC.
Sectors (five). Financial services, non-financial services, industrial production, raw material, and other sectors.

Note that even the above broad classification gives a total of images possible combinations. Not surprisingly, there are data issues with this classification. Whilst these authors observe 1,551 CDS quotes available across the defined universe, making an average of 10.3 quotes per sub-category, the data is concentrated in certain regions, and there are 42 sub-categories without a single CDS quote. For example, Figure 12.10 shows a breakdown of the quotes for financial services firms by rating and region, illustrating that the majority of the data is North American and European credits and intermediate (A and BBB) credit ratings.

A potential process for generic curve construction based on an intersectional methodology could be as follows:

Define the universe of available CDSs via a minimum liquidity threshold.
Bucket this universe by the agreed-upon classification (rating, region, and sector).

Figure 12.10 Granularity of intersectional CDS data for the financial services industry by credit rating and geographic region.

Source: Chourdakis et al. (2013).
Depending on the data available, potentially exclude outliers in each bucket according to some given metric, such as more than a certain number of standard deviations from the median (this clearly requires a reasonable number of observations).
Define the resulting generic curve(s) via an average of the relevant points or a weighted average depending on the relative liquidity of different points.
Fill in missing data points via various rules, interpolation, and extrapolation methods.

The main advantage of the above approach is its simplicity and ease of understanding and implementation. Clearly, without a more liquid underlying CDS market, it is not a robust approach and will require override, interpolation, and extrapolation assumptions in order to be able to define generic credit spreads for each sub-category.

12.3.3 Cross-section Methodology

The intersectional approach has drawbacks driven by the limited liquid credit spread data available in the market. A very broad definition of generic curves is less descriptive, whereas a detailed categorisation is limited by the illiquidity of the CDS market, meaning that buckets will have limited or no data points. As a result, there will be potentially large jumps in credit spreads due to the idiosyncratic behaviour of names in a given bucket. This behaviour will be particularly adverse for buckets with fewer CDS quotes to which to calibrate.

An alternative approach to credit spread mapping is proposed by Chourdakis et al. (2013) and is based on a cross-section methodology using a multidimensional linear regression.²⁰ This approach still uses a categorisation based across rating, region, and sector (and potentially other categories), but it generates the spread via a factor approach rather than a direct mapping to the names in a given bucket. A given spread is generated as the product of several factors, such as:

global ();
industry sector ();
region ();
rating (); and
seniority ().

Each factor will contain several sub-categories (e.g. different credit ratings), for which the coefficients need to be estimated. The spread is defined by:

(12.2)

or equivalently by:

(12.3)

which is a linear regression problem from which the coefficients can be estimated. A key assumption behind such a methodology is that CDS spreads can be defined by a set of factors that are independent of the other components (e.g. there is a single multiplicative credit spread factor for all European credits, which is independent of the sector, rating, and seniority of those obligors). This means that the estimate of a BBB Asian credit spread, for example, will use all BBB data and all Asian data, rather than just the data intersecting the two-dimensional classification. This credit spread can be estimated even if there are no BBB Asian reference points, as long as there are separate points for BBB and Asia.

The calibration of the factors above to liquid credit spread data is straightforward, as it involves a linear regression which is the result of minimising the squared differences in log spreads. The advantage of a cross-sectional approach is that there will be much more data available to calibrate each of the factors. This would be expected to give rise to smoother and more reasonable behaviour (such as credit spreads changing monotonically across the credit rating spectrum), as is shown by Chourdakis et al. (2013).

A simple example of cross-sectional regression coefficients and resulting credit spreads is shown in Table 12.6. This example ignores seniority and has a limited granularity over industry and region.

There are a number of ways to check the accuracy of this type of approach. One is to examine the residuals (the deviation of observed credit spreads compared to those implied by the regression formula), which is an ‘in-sample’ test. Alternatively, an ‘out-of-sample’ test could be used to examine the prediction of data (e.g. CDS spreads) not used in the regression. Weak and distressed credits can be a particular problem in this type of approach, since their behaviour is quite idiosyncratic and may not be well explained by the model in Equation 12.2. For example, CCC credits can have particularly high or low spreads as a result of industry and region coefficients that are mainly driven by other ratings.

images — **Table 12.6** Example coefficients and implied credit spreads for a simple example. Note that the credit spread is in basis points per annum.

	Sub-category	Coefficient
	Global	−5.90
	AAA	0.00
AA	0.63
A	1.11
BBB	1.42
BB	2.37
	Financials	0.00
Non-financials	−0.05
	US	0.00
Europe	0.16
Asia	0.23

	Global	Rating	Industry	Region	Proxy credit spread
European AA financial	−5.90	0.63	0.00	0.00	51.4 bps
US AA financial	−5.90	0.63	0.00	0.07	55.2 bps
US BBB non-financial	−5.90	1.42	−0.05	0.07	115.6 bps
US BB non-financial	−5.90	2.37	−0.05	0.07	299.0 bps

EBA (2015b)²¹ shows some time series of generic spreads for various hypothetical and real credits for banks using the advanced CVA capital charge methodology (Section 13.3.4).²² These results illustrate the variation between the methodologies of different banks and the differing stability of different approaches.

It should be possible to improve cross-sectional approaches by extending them across factors other than liquid CDS spreads to produce more reliable and stable results. For example, Sourabh et al. (2018) show that the additional use of equity returns produces more accurate estimates of CDS spreads. Depending on the amount of data available, it may also be useful to include other credit spread information from loans and bonds, with appropriate representation of aspects such as seniority and basis.

12.3.4 Curve Shape, Interpolation, and Indices

Quite often, generic curve methodologies may only characterise a single liquid point, and indices can be used to fill in missing points. For example, suppose a generic credit spread is defined at only a single point (e.g. five years). It is then necessary to imply the curve shape: an obvious solution is to use a representative index and to scale this shape to fit the defined point, as shown in Figure 12.11. Due to the importance of term structure noted in Section 12.1.3, it is essential not to make crude assumptions. Indices can also be used to fill in missing rating points. For example, one may look at the ratio of single-A to triple-B spreads in iTraxx or CDX and use this ratio to infer one rating curve from another in the more granular generic curve representation.

Graph depicts the representation of defining a curve shape based on the shape of the relevant index. — **Figure 12.11** Illustration of defining a curve shape based on the shape of the relevant index. The cross shows a single (e.g. five-year) point that is assumed to be known for the curve in question.

12.3.5 Third-party Providers

A potential source of generic credit spread curves is a third-party provider. This may offer a potentially cheaper solution compared to a firm implementing its own methodology and is also independent (which may be desirable from an auditor's perspective). On the other hand, there are potential drawbacks, such as the methodology being rigid and producing unexpected behaviour beyond the control of the user (e.g. the volatility of hedged CVA).

A non-exhaustive list of third-party providers providing generic credit spread information is as follows:

IHS Markit CDS Enhanced Sector Curves.²³ This provides generic CDS curves for all tenors across seven regions, eleven sectors, and seven rating classes (Table 12.7), using end-of-day CDS prices and a ‘multivariate factor value model’ (cross-sectional-type approach). CVA computation is a key motivation for the provision of this service.
Standard & Poor's Credit Default Swap Market Derived Signals (CDS MDS).²⁴ This uses CDS spread data in a linear regression model against rating, sector,²⁵ corporate/sovereign classification, CDS documentation type (credit event definitions), and CDS currency denomination. This allows the determination of generic benchmark CDS spreads across these classifications.
Moody's Fair Value CDS (FVS-CDS) spreads.²⁶ These are CDS spreads derived from an expected default frequency (EDF^TM) measure published daily.²⁷ Because the underlying model is based on balance sheet information (e.g. equity prices), it can be used directly rather than to generate generic credit spread curves.
Fitch. Fitch provides a Bank Credit Model, which is a statistical model producing a daily implied CDS spread for over 9,500 banks globally, and CDS Benchmark Curves to support the pricing of illiquid names using average CDS values calculated by rating, sector, and region for the full-term structure across each currency, seniority, and restructuring.²⁸
Kamakura Risk Information Services (KRIS). KRIS provides quantitative measures of creditworthiness such as implied CDS spreads for 40,000 global companies.

Table 12.7 Regions, sectors, and ratings for IHS Markit CDS Enhanced Sector Curves.

Source: www.ihsmarkit.com.

Region	Sector	Rating
North America	Basic materials	AAA
Europe	Consumer goods	AA
Eastern Europe	Consumer services	A
Middle East	Energy	BBB
Japan	Financials	BB
Asia ex-Japan	Government	B
Oceania	Healthcare	CCC
	Industrials
	Technology
	Telecommunications services
	Utilities

12.3.6 Hedging

An important consideration in the choice of mapping methodology is the potential hedging of CVA. Here, the appropriate strategy depends on the liquidity of the counterparty. For liquid counterparties, the single-name CDS is the most obvious hedging instrument. For illiquid counterparties, proxy single-name hedges may be less liquid and do not currently allow capital relief (Table 12.5). Credit indices are, therefore, the most efficient macro-hedges. Whilst a more granular mapping methodology may be considered to more accurately reflect the underlying economic behaviour, it may make hedging less effective.

Ultimately, mapping and hedging can be self-fulfilling prophecies since the mapping mechanism ultimately defines the effectiveness of the hedge. For example, consider a European firm that intends to hedge its CVA using the iTraxx Europe index. It is likely that using only the single-name CDSs contained within this index for a generic curve construction methodology will lead to lower volatility (of the hedged CVA). However, this ignores other reasonably liquid European credits that – whilst not liquid and/or large enough to be part of the index – may be expected to provide useful credit spread information. Whether or not such an approach is acceptable would be up to the firm's auditors and regulators.

In order to macro-hedge credit risk under a generic curve approach, it is necessary to construct a ‘beta mapping’ to a given index or set of indices. This involves performing a regression of the generic curve against the index to obtain the optimum hedge ratio. There is no definitive consensus as to over what time period such regression should be performed; longer shows will be less noisy, but shorter periods may be more accurate. Another important consideration is the recalibration frequency; daily recalibration minimises the potential for large discrete changes but may be operationally cumbersome. From a practical perspective, a periodic (e.g. monthly) recalibration may be more appropriate but will lead to potentially significant changes, leading to mark-to-market impacts and required adjustment of hedges. Some banks actually map to the beta-adjusted indices directly, which produces more stability in between recalibration dates but may be harder to defend to regulators and auditors. CVA hedging is discussed in more detail in Section 21.2.