### Credit Exposure and Funding

The main objective of this chapter is going to be a detailed definition of exposure and its key characteristics. The various forms of exposure will be discussed and the crucial metrics will be used for their quantification.

# Credit Exposure

The asymmetry of potential losses based on the value of the underlying transaction is a key characteristic of counterparty risk. The main feature of credit exposure is whether the effective contract values are positive or negative. For positive values, default by a counterparty results in an inability to undertake future commitments. Therefore, as an unsecured creditor, the surviving party will have a claim on the positive value at the time of the default. Exposure is simply defined as:

$$Exposure=max\left( Value,0 \right) \quad \quad \quad \quad \quad \left( a \right)$$

## Bilateral Exposure

Due to the bilateral property of counterparty risk, both parties can experience losses since they can both default. The default of a party causes a loss to any counterparty they owe. To define it in terms of negative exposure, the symmetry is:

$$Negative\quad Exposure=min\left( Value,0 \right) \quad \quad \quad \quad \quad \left( b \right)$$

Since the counterparty is making a loss, a negative exposure will lead to a gain.

## The Close-Out Amount

The effective value of the relevant contract at the counterparty’s default time is the value as explained discussion above. This value may not conform to any definite representation defined and modeled. In the appropriate jurisdiction, the close-out amount is defined by the relevant documentation and its legal interpretation. Equations ($$a$$) and ($$b$$) are crucial for the quantification of exposure and other xVA terms even though they don’t include aspects like documentations used and jurisdiction.

## Exposure as a Short Option Position

As shown by equation ($$a$$),an asymmetric risk profile is created by counterparty risk. For a positive value, default by counterparty causes the party to lose but does not gain if the value is negative. The conclusions drawn about the quantification exposure is that a key aspect will be volatility due to the similarity of exposure to an option payoff, and it may be complex to quantify exposure as options are relatively complex to price. By symmetry, a party has a long option from their own default.

## Future Exposure

The definition of future exposure is probabilistic as a probable future event is based on uncertain market movements and the contractual features of the transactions. Defining the level of exposure and its underlying uncertainty is important in understanding future exposure. Due to long periods involved and lots of different market values influencing the exposure and risk mitigants, it becomes extremely complex to quantify exposure.

## Comparison to Value-at-Risk

There are shared similarities in the characterization of $$VaR$$ and exposure. However, in the quantification of exposure, additional complexities are faced such as:

Time horizon: The definition of exposure has to be over multiple time horizons, as opposed to $$VaR$$, in order to comprehend the effect of time and specifics of the underlying contracts. The main implications is that we must consider the aging of the transactions, and the trend of market variables in addition to their underlying volatility and market structures is relevant when considering longer time horizons.

Risk mitigants: To estimate and reduce future exposure correctly, we consider the impact of risk mitigants. Another degree of subjectivity is created in case of future collateral amounts as it is impossible to know with certainty the timing of the cash flows and the amount of collateral to be received.

Application: The definition of exposure must be for both risk management and pricing as this creates additional complexity in the quantification of exposure leading to two different sets of calculations.

This implies that, compared to $$VaR$$, exposure is more complex yet it is the main component of counterparty risk.

# Metrics for Exposure

Metrics are the measures that are commonly used to quantify exposure. The total number of relevant transactions appropriately netted and comprising of any collateral amounts that are relevant is the exposure we are referring to as the netting set. Different introduced metrics are useful to distinct applications.

## Expected Future Value (EFV)

The forward or expected value of the netting set at some future point is represented by the expected future value. This component is the representation of the average of the future value calculated keeping in mind some probability measures. The variation of the EFV from the current value may be due to:

1. Cash flow differentials: cash flows in derivative transactions may be asymmetric;
2. Forward rates: they significantly differ from current spot variables; and
3. Asymmetric collateral agreements: the future value will be unfavorable if collateral agreements are asymmetric thus reflecting favorable or unfavorable collateral terms, respectively.

## Potential Future Exposure (PFE)

The PFE mainly addresses the question of the worse future exposure that could be encountered, based on a given confidence level. It is important to know that PFE is a similar metric to $$VaR$$ and can also significantly differ from zero, therefore, representing expected values that are significant positive or negative in transactions.

## Expected Exposure (EE)

EE can be described as the average of all exposure values which arises due to positive values as other values’ contribution is only in terms of their probabilities. The implication is that the EE will be above the EFV. This concept is similar to the fact that, compared to an underlying forward contract, an option is more valuable.

## Maximum PFE

By representing the worst case exposure over an entire interval, the highest PFE value is simply represented by peak or maximum PFE. It is used as a metric in credit limits management in other times.

## Expected Positive Exposure (EPE)

Represented as the weighted average of the EE across time, EPE can be described as the average exposure across all time horizons. The EPE is said to have an average if the EE points are equally spaced. Because it is an average of both the market random variables and the time impact, expressing an uncertain exposure by a single amount of EPE can obviously represent an approximation that is fairly crude.

## Negative Exposure

Since exposure has always been represented as positive future values, negative exposure will be from the counterparty’s point of view. Consequently, measures like negative expected exposure (NEE) and expected negative exposure, which precisely are the opposite of EE and EPE, are defined and can be used for the calculation of metrics like debt valuation adjustment (DVA) and funding valuation adjustment (FVA).

## Effective Expected Positive Exposure (EEPE)

EEPE is a metric whose motivation is a version of EPE which is more conservative and deals with the following problems: neglected large exposures by the fact that EPE is a representation of the average of the exposure, and rollover risk that is not properly captured, as exposure for short-dated transactions may be underestimated by EPE. Being the effective EE (EEE) average, EEPE is a non-decreasing version of the EE profile.

# Factors Driving Exposure

These are some of the significant factors that drive exposure and their key features.

## Loans and Bonds

Instruments of debts like loans and bonds are deterministic and, compared to notional values, they are equal despite the fact that they are not characterized as counterparty risk. Bonds have some additional uncertainty since they pay a fixed rate. As a result, a decline in interest rates will result in a corresponding increase in the exposure, and the converse is also true. As for loans, exposure may decline over time as they are floating rate instruments.

## Future Uncertainty

The characteristic of forward contracts is that they have two cash flows exchanges or underlyings at the maturity of the contract. The implication is that uncertainty increases with time concerning the final exchange value since exposure is a rather simple increasing function. Future uncertainty will be proportional to the square root of time, with respect to common assumptions.

$$Exposure\propto \sqrt { t }$$

## Periodic Cash flows

Periodic payment of cash flows is popularly known for reversing future uncertainty effects. The approximate representation of this effect is:

$$Exposure\propto \left( T-t \right) \sqrt { t }$$

$$T$$ is the maturity of the transaction under scrutiny. Because of $$\sqrt { t }$$, the given function will initially increase but the ($$T – t$$) component will make it decrease to zero which is the approximate representation of maturity that remains at a time $$t$$ in the future. At $$\frac { T }{ 3 }$$, the maximum exposure of the function occurs. The specific nature of cash flows in a transaction will result in an alteration of exposure profile. Moreover, an asymmetry between transactions in opposite sides is an effect of the cash flows on the exposure. For an interest rate swap to happen, different cash flows must be exchanged as opposed to the payer swap where at a periodic payment of fixed cash flows happens as floating cash flows are received.

## Combination of Profiles

The combination of profiles represents an exposure driven by two or more underlying risk factors’ combination of some products such as cross-currency swaps. As a result of high FX volatility driving the risk, coupled with long maturities, exposure of the cross-currency swap can be taken into consideration. However, the interest rate swap’s contribution is typically smaller. Noticeably, two interest rates and FX rates are correlated in a way that can crucially increase the exposure.

## Optionality

Certain probabilities, occurring after the exercise dates, is as a result of complexities in exposure created by the impact of exercise decisions. Views on aspects such as counterparty risks should be incorporated in the exercise. This implies that the decision on whether to exercise or not should have a component of future xVA.

## Credit Derivatives

Wrong-way risk is the cause exposure assessment challenge represented by credit derivatives. Because of the instruments’ discrete payoffs, credit derivatives’ exposure payoffs are difficult to characterize. There is a probability that the actual occurring credit rate event may or may not be represented by PFEs. A measure like ES can solve this occurrence partially. For CDS indices, large reference credit numbers ensure that this effect is not possible as single defaults have a less significant impact.

## The Impact of Netting on a Future Exposure

Exposures are taken to be additive due to a lack of legal agreements allowing netting; hence, there is no one another-offsetting by the positions. Before calculating the exposure, values can be added at the netting set level if there is a netting permission. Since the shown profiles at all future points will give a zero exposure, a zero transaction will consequently be given by two transactions that are positive.

## Netting and the Impact of Correlation

Correlation between future values is a common approach in the consideration of two or more transactions’ netting benefits. Between two transactions, future values are likely to be of similar sign provided that a high positive correlation exists. The implication is that the befit of netting will be zero if not very small. For cases of transactions with opposite sign values, netting will be of help. Stronger netting benefit is due to the likely event of opposite signs in future values, therefore, making negative correlations more useful.

Small correlation in transactions of different asset classes is where the majority of the netting may occur, therefore creating a positive benefit. Based on exposure, the following netting factor is derived assuming that future values follow the multivariate normal distribution:

$$Netting\quad factor=\frac { \sqrt { n+n\left( n-1 \right) \bar { \rho } } }{ n }$$

Where $$n$$ represents the number of exposures and $$\bar { \rho }$$ is the average correlation.

In case there is no correlation:

$$Netting\quad factor=\frac { 1 }{ \sqrt { n } }$$

## Netting and Relative MTM

Not only does netting benefit rely on future values’ correlation but also their relative offset. Despite a structural correlation between transactions, netting benefits are a result of negative future values as it is unlikely for out-of-the-money portfolios to have exposure unless there is a significant movement of the MTM transactions. Therefore, Netting enables the useful effect of a positive future value.

## Impact of Collateral on Exposure

After considering many points, it is noticed that collateral has a reducing effect on exposure. Proper understanding and appropriate representation of collateral parameters like thresholds and minimum transfer amounts are crucial for accurate accounting. Moreover, to determine the true risk horizon based on the collateral transfer, a careful analysis of the margin period of risk (MPR) is important. To understand better that the extent that collateral is not a perfect form of risk mitigation, the following are important considerations:

• Due to parameters like threshold and lack of possibility to ask for all the required collaterals, there is a granularity effect leading to beneficial over-collateralization.
• Receipt of collateral involving most operational components’ aspects of requesting and receiving collateral to the possibility of collateral disputes is delayed.
• The potential of variation in the collateral value itself must be considered.

The following are common cases:

• Partially collateralized: Exposure reduction is imperfect due to the presence of contractual aspects.
• Collateralized: The assumption is a significant reduction in exposure resulting from aspects such as thresholds.
• Over-collateralized: The assumption is that a further reduction in the exposure is due to the initial margin.

# Funding, Re-hypothecation and Segregation

## Funding Costs and Benefits

Logically, a positive exposure/MTM is a funding cost whereas a negative exposure/MTM corresponds to a funding benefit. An economically un-monetized derivative is represented by a positive MTM, and therefore, it has to be funded like any other asset having an associated cost. Conversely, a derivative liability is created by a negative MTM representing a loss needing no immediate repayment, providing a funding benefit hence acting like a loan.

## Differences between Funding and Credit Exposure

Recall that in case of counterparty default, then a positive value is considered to be at risk and the funding benefits are related to a positive exposure. Nevertheless, there are some specific differences between funding and credit exposure that are elaborated as follows:

• Close-out: Potential close-out adjustments for credit exposure are not applicable to MTM-based funding aspects.
• Margin period of risk: MPR is relevant in credit exposure. For a delay in receiving collateral against a portfolio assessment, the normal collateral posting frequency should be assumed. Hence, despite the equivalent CVA not being zero, the FVA collateralized derivative is considered to be zero.
• Netting: Since credit exposure is defined at the netting service level, in default scenarios, close-out netting is an applicable concept.
• Segregation: On funding and credit exposure, segregation has different impacts.

## Impact of Segregation and Re-hypothecation

In OTC derivative transactions, the collateral serves the two purposes, namely providing funding positions and mitigating counterparty risk. It is beneficial in the following ways to receive counterparty’s collateral against a positive MTM; one is that it reduces counterparty risk by holding on to the collateral to cover close-out losses, and secondly, it has a funding benefit as it can be used for other purposes. There should be no adverse correlation between collateral and the counterparty’s credit quality for maximum counterparty risk-benefit. Moreover, collaterals should be purposely reusable. Various collateral types under certain situations have the following counterparty risk mitigation and funding benefits: cash that needs no segregation, securities that can be re-hypothecated, cash or securities that must be segregated or cannot be re-hypothecated, and finally, a counterparty posting its own bonds.

## Impact of Correlation on Credit and Funding Exposure

Generally, upon receiving collateral from the counterparty, it should be subtracted from the exposure. However, there is a distinct difference created by segregation and re-hypothecation. Based on counterparty risk, the following is a general equation:

$${ Exposure }_{ Funding }=MTM-{ CR }_{ RH }+CP$$

Instead of Value, $$MTM$$ is applied, $${ CR }_{ RH }$$ is the re-hypothecated collateral received, and $$CP$$ is the representation of all posted collaterals.

The distinction between two general types of collaterals will make this definition more precise in the following ways:

• Initial margin: The credit exposure will be increased given that the initial margin is not segregated.
• Variation margin: In order to provide funding benefits, it should be re-hypothecable.

Specifically, the above formula can be rewritten making the above assumptions regarding the initial and variation margin:

$${ Exposure }_{ CCR }=max\left( value-VM-{ IM }^{ R },0 \right)$$

$${ Exposure }_{ Funding }=MTM-VM+{ IM }^{ P }$$

Where $${ IM }^{ R }$$ is the initial value received which, together with the variation margin, can offset exposure for counterparty risk purposes; and

$${ IM }^{ P }$$, which is the initial margin posted, can increase the funding exposure and is fully adjusted by the variation margin.

# Practice Questions

1) Given that the number of exposure is 19 and the average correlation is 0.87, determine the netting factor under the assumption is that the future values follow a multivariate normal distribution.

1. 0.9841
2. 0.9605
3. 0.8233
4. 0.9364

$$Netting\quad factor=\frac { \sqrt { n+n\left( n-1 \right) \bar { \rho } } }{ n }$$
$$n$$ is given as 19 and $$\bar { \rho }$$ is given as 0.87. Therefore:
$$Netting\quad factor=\frac { \sqrt { 19+19\left( 19-1 \right) \times 0.87 } }{ 19 } =0.9364$$