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# Default Probability, Credit Spreads and Funding Costs

For credit valuation adjustments (CVA) and debt valuation adjustments (DVA) in the qualification of counterparty risk to be defined comprehensively, default probability and recovery rates associated with those are required. Relevant funding costs that are required when a position is funded or initial margin is posted should be assessed as required by terms such as funding value adjustments (FVA) and market valuation adjustments (MVA). As a requirement by capital valuation adjustments (KVA), the required return on capital should also be assessed.

# Default Probability

## Real-World and Risk-Neutral

Through some associated credit rating, the approximation of real-world probabilities of default is possible by using historical default data. On the other hand, applying market data, we can get risk-neutral default probabilities using instruments like bonds and credit default swaps (CDS).

Due to the fact that investors are risk-averse and demand a premium for accepting the risk of default, compared to their real-world equivalent, risk-neutral default possibility is likely to be higher.

## The Move to Risk-Neutral

In the recent past, risk-neutral default probabilities have gained significance in the computation of CVA, in comparison to the early days when banks applied real-world probabilities of default instead. Accounting requirements and Basel III capital rules are responsible for catalyzing the move to risk-neutral.

The CVA interpretation is changed to be a market price of counterparty risk as opposed to being some actuarial reserve, through the application of risk-neutral likelihoods of default. Since there lack direct observable markets for a risk-neutral probability of default to be defined, many counterparties become illiquid credits.

When the risk-neutral probability of default is used for illiquid credits, then the challenge of hedging arises as the existence of a hedge is needed by risk-neutral probabilities. However, the existence of such a hedge is impossible as liquid CDSs on the counterparty in question is usually not available.

Nevertheless, deviation from risk-neutral probabilities of default is generally discouraged by regulators and auditors, even in the cases of credits that are illiquid.

## Defining Risk-Neutral Default Probabilities

When credit spreads are observed in the market, then risk-neutral default probabilities are derived from those observations. Market observables, namely single-name CDSs, spreads of asset swaps, prices of loans or bonds, and some proxy or mapping methods are the basis points that can be used to define a credit spread, although it cannot be uniquely defined. However, the most obvious and directly available quote that is clean, when defining a credit spread, is the CDS.

To compute a term such as CVA, the likelihood of default between two sequential dates is required, with the following being the most popular estimation approach:

$$PD\left( { t }_{ i-1 },t_{ i } \right) \approx exp\left( -\frac { { S }_{ { t }_{ i-1 } }{ t }_{ i-1 } }{ LGD } \right) -exp\left( -\frac { { S }_{ { t }_{ i } }{ t }_{ i } }{ LGD } \right)$$

With $$PD\left( { t }_{ i-1 },t_{ i } \right)$$ being the unconditional likelihood of default between $${ t }_{ i-1 }$$ and $$t_{ i }$$, the credit spread at time $$t$$ being $${ S }_{ t }$$, and $$LGD$$ being the loss given default assumed.

## Loss Given Default (LGD)

In case of default by a counterparty, some percentage amount will be lost. This amount is called the loss given default and is very paramount in the establishment of risk-neutral default probabilities. As seen previously, $$LGD=1-Recovery\quad rate$$.

Seniority of claim, sector, and conditions of the economy are the factors causing significant variations in the recovery rates as revealed by historical analysis. Timing also affects recovery in the sense that a default leads to quick settlement of CDS. Bondholders can, in the same way, settle their bonds or offer them in the market for sale.

The settlement of bilateral OTC derivatives cannot be in a timely manner because of their bespoke nature as netting hence the aggregation of most transactions into a single claim, and the impossibility of trading individually.

For a portfolio of trades, the definition of the net claim is quite difficult, hence the creation of two different recovery values, namely:

1. Settled recovery: the achieved recovery due to the credit event by trading out of a claim; and
2. Actual recovery: the received recovery on a derivative due to a bankruptcy or anything similar.

## Credit Curve Mapping

In the computation of the CVA, an approximation of the credit curve is a crucial but subjective input. As opposed to banks who have lots of derivatives counterparties, with most lacking liquid CDS quotes, bond prices or even external ratings associated to them, the number of counterparties generally traded with end users is relatively small with readily available required information.

Due to the challenge of subjectivity, for any given counterparty, the definition of the credit curve lacks a standard method. The following are some of the general issues faced when mapping credit curve: reference instrument, tenor, seniority, liquidity, region, hedging, and capital relief.

## The CDS Market

Liquid CDS quotes are contained in many hundreds of names despite the liquidity of this market not being recently improved, with some credit indices being generally more liquid. Liquid credits trading in the single-name CDS market or secondary bond market is generally referenced by indices.

Some additional technical challenges, associated with applying CDS for the derivation of credit spreads for CVA computation, exists. Failure to pay, restructuring or bankruptcy is the events of credit under an ISDA standard CDS.

The global financial crisis has not been improved significantly by liquidity in the CDS market. However, the best market-implied price for credit risk is still believed to be provided by the CDS market.

## General Approach

For a particular counterparty, credit spread information has three distinct sources, namely:

1. Direct observables: The observation of the actual counterparty’s credit spread is directly from the market.
2. Single-name proxies: For a particular counterparty, the market trading by another single reference entity usually is considered as a good proxy.
3. Generic proxies: In this case, there lacks a defined credit spread readily mapped directly and some sort of generic mapping through rating. There is also the requirement of region and sector.

Since there may be an incorrect reflection of any idiosyncratic behavior of single-name proxies in the credit spreads mapped, the volatility of the proxy may increase.

# Generic Curve Construction

## General Approach

To achieve a general curve based on market data, some classified points that are relevant are used since this is the crucial aim of generic credit curve mapping. In the CDS market, only a few maturity points may be available as compared to secondary bond markets which have more maturity points.

Granularity should be carefully considered by mapping, with sufficient data points being the factor making granular mapping preferable, for each categorization. Classification approaches may differ between banks.

## Third Party Curves

Some third-party provided generic curves are independent and offer potentially cheaper solutions. However, they may be rigid in their classification thus producing features beyond user control.

## Mapping Approach

The production of bespoke curves by some rating, regional, and sector classification with respect to a chosen universe of liquid single-name CDSs is a typical approach to creating a generic curve. This can be achieved in the following way:

1. Through minimum liquidity threshold, define the available CDS’s universe.
2. The universe is bucketed, depending on the given information, by the classification agreed upon.
3. Outliners in each bucket should be excluded according to some metric provided.
4. The data points that are missing should be filled through numerous procedures of interpolation and extrapolation.
5. Through a weighted average or an average of the relevant points, the resulting data can be defined.

## Hedging

The most obvious hedging instrument for liquid counterparties is the single name CDS. However, capital relief may not be allowed by proxy single-name hedges which may be less liquid for illiquid counterparties. Hence, the most efficient macro-hedges are credit indices.

A single set of indices needs to be constructed for a credit risk to be macro-hedged under a generic curve approach. A regression of the generic curve and a recalibration of the frequency has to be performed.

# Funding Curves and Capital Costs

Due to the following reasons, funding costs in valuing derivatives were never considered by banks and other financial institutions:

1. Funding by banks was rather simple as money could be deposited or raised in the wholesale markets.
2. Funding by banks could be at LIBOR or better as a result of the following reasons.
3. Derivatives were treated by banks as short-term assets hence, where necessary, only short-term funding costs were considered.

Some regulatory response to the global financial crises will make the funding of derivatives positions increasingly expensive, for example: the clearing mandate, bilateral collateral rules, the liquidity coverage ratio, increased capital requirements, and the leverage ratio.

## Funding Costs

Funding costs are asset specific and may arise from a variety of different sources and be different in each case, with variation and initial margin being the obvious difference. The posting of initial margin is not against MTM losses and therefore a direct funding cost. Therefore, the funding costs will be split into FVA and MVA terms.

## Defining a Funding Curve

The following are sources of funding for banks via the debt markets: customer deposits, wholesale money markets, private and public unsecured borrowing, and private or public secured borrowing.

When funding is incorporated into pricing, the following issues should be addressed:

1. The instruments funding costs should be calibrated;
2. Whether a party should use their own cost of funding, the counterparty’s funding cost, or a blended curve based on market participants’ funding costs; and
3. Using a transaction’s final contractual maturity, whether term funding should be used or rather a shorter tenor assumed with respect to term funding not being required or/and early termination of the transaction.

Funding costs should also be characterized as the sum of the following two distinct components:

1. Credit Funding Cost: The cost charged as part of unsecured funding cost, for the party’s credit risk.
2. Funding Liquidity Risk Premium: On top of the pure credit risk, this cost should not be entity-specific and may be relevant in secured funding.

However, in pricing FVA, funding liquidity risk premium is the only relevant inclusion.

# Practice Questions

1) Generally, there are three distinct sources of data pertaining to credit spread for a given counterparty. How is a single-name proxy a source of credit data?

1. There is no defined credit spread that can be readily mapped directly and some sort of generic mapping through rating, region, and sector is required
2. The credit spread of the actual counterparty in question is directly observable in the market
3. Another single reference entity, viewed as a good proxy for the counterparty in question, trades in the market and it may be a parent company or the sovereign of the region in question
4. The proxy may use CDS indices which do not provide default protection but allow spread hedging and will provide partial capital relief

For a particular counterparty, there is market trading by another single reference entity usually considered as a good proxy. It may be a parent company or the sovereign of the region in question and may be applied directly or have additional component added to the credit spread to reflect greater riskiness.

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