Book 2. Credit Risk

FRM Part 2

CR 20. CVA

Presented by: Sudhanshu

Module 1. Credit Valuation Adjustment

Module 2. Incremental and Marginal CVA, and CVA for a Bilateral Contract

Module 3. Wrong-Way Risk

Module 1. Credit Valuation Adjustment

Topic 1. Credit Value Adjustment (CVA)

Topic 2. CVA Spread

Topic 3. Impact of Changes in Credit Spread and Recovery Rates

Topic 4. Incorporating Netting and Collateralization

Topic 1. Credit Value Adjustment (CVA)

  • Motivation and Purpose:

    • The primary motivation for pricing counterparty risk is to set aside reserves to absorb potential losses due to a counterparty's default.

    • Accurate pricing accounts for risk mitigants like netting and collateralization, which are crucial for managing this risk.

    • Challenges of Pricing Counterparty Risk: The pricing of counterparty risk is complex because it is a function of both the credit exposure and the default probability of the counterparty. It requires sophisticated models to forecast future exposure, which can fluctuate significantly over time, and to estimate default probabilities, which are often non-linear and subject to market-wide factors.

    • ​Formula:
      • ​
      • LGD: Loss Given Default. It represents the percentage of the exposure that will be lost if the counterparty defaults. It is calculated as (1−RR), where RR is the recovery rate.
C V A=-L G D \times \sum_{i-1}^T E P E\left(t_i\right) \times P D\left(t_{i-1}, t_i\right)

    • EPE: Discounted Expected Positive Exposure. At each future time point, it is the exposure value multiplied by the probability that the counterparty has not yet defaulted. It is a forward-looking measure of potential future loss.

    • PD: Marginal Default Probability. This is the probability that the counterparty defaults in a specific time interval, given they have not defaulted before that interval.

  • Definition:

    • CVA is the market value or price of counterparty credit risk. It is an adjustment to the risk-free value of a transaction to account for potential losses from a counterparty's default.

Topic 1. Credit Value Adjustment (CVA)

Practice Questions: Q1

Q1. Which of the following statements is not a motivation for pricing counterparty risk?
A. Accurate pricing should only account for the cost of the trade.
B. Counterparty risk pricing should account for risk mitigants.
C. Best practices organize pricing responsibilities in the organization.
D. Pricing bilateral derivatives contracts.

Practice Questions: Q1 Answer

Explanation: A is correct.

Accurate pricing should account for not only the cost of the trade, but also the cost of counterparty risk.

Topic 2. CVA Spread

  • Objective:

    • The primary objective of the CVA Spread is to convert the upfront CVA, which is a lump-sum, one-time charge, into a continuous running spread that can be charged to a counterparty. This is especially useful for long-term derivative products.

    • Formula:

      •  

    • Components and Mechanics:
    •                  : This represents the total CVA for a transaction from time t to T. It's the full, upfront cost of the counterparty risk.
    •                            : This is the unit premium of a credit default swap (CDS) that the counterparty pays. In this context, it is used as a proxy for the total value of the counterparty's credit risk over time.
    •          : This is the counterparty's credit spread, typically measured in basis points. It reflects the market's assessment of the counterparty's creditworthiness. A wider spread indicates a higher perceived risk of default.
    • Average EPE: The average expected positive exposure. This is a measure of the average potential future exposure to the counterparty over the life of the transaction. It is a key input because it represents the portion of the transaction's value that is at risk.
C D S_{\text {premium }}(t, T)
C V A(t, T)
\frac{C V A(t, T)}{C D S_{\text {premium }}(t, T)}=-X^{C D S} \times \text { Average } E P E
X^{C D S}
  • Purpose: The formula effectively links the CVA to the counterparty's credit spread and the average exposure. By solving for the spread (           ), we can determine the basis point charge that needs to be added to the transaction's pricing to fully compensate for the CVA over its lifespan.
X^{C D S}

Topic 2. CVA Spread

Topic 3. Impact of Changes in Credit Spread and Recovery Rates

  • Credit Spread:

    • Relationship to CVA: An increase in the counterparty's credit spread (as measured by CDS spreads) generally increases CVA. This is because a wider spread indicates a higher perceived probability of default for the counterparty.

    • Non-linear Relationship: The relationship is not linear. CVA is sensitive to changes in credit spreads, and the sensitivity is not uniform across all spread levels.

    • Spread Curve Shape: The shape of the credit spread curve also impacts CVA. A downward-sloping (inverted) credit spread curve leads to a higher CVA compared to a flat or upward-sloping curve. This is because a downward-sloping curve implies that the market is expecting a higher short-term risk of default, which is when the majority of CVA losses would be expected to occur.

    • Recovery Rate (RR):

      • Relationship to CVA: A higher recovery rate reduces the Loss Given Default (LGD), which in turn reduces CVA. The relationship is inverse and linear. Since LGD is calculated as 1−RR, an increase in the recovery rate directly translates to a decrease in the potential loss and therefore a lower CVA.

  • Settled vs. Actual: A higher actual recovery rate than a settled recovery rate will produce a lower CVA. This highlights the importance of using accurate and up-to-date recovery rate assumptions in the CVA calculation.

  • Impact of Market: The recovery rate is not a fixed variable; it can be influenced by market conditions at the time of default. For example, during a systemic crisis, the market value of assets available for recovery may be lower, resulting in a lower recovery rate.

Topic 3. Impact of Changes in Credit Spread and Recovery Rates

Topic 4. Incorporating Netting and Collateralization

  • Netting:

    • Definition: Netting is the process of offsetting the value of multiple positions or payments due from one counterparty against those due to the same counterparty.

    • Impact on CVA: Netting is a highly effective risk mitigant because it reduces the overall exposure across multiple transactions with the same counterparty. Instead of having separate exposures for each trade, the total exposure is based on the net value of the portfolio. This significantly lowers the Expected Positive Exposure (EPE) and, therefore, reduces the CVA.

    • Calculation: The incremental CVA for a new trade is the difference in CVA before and after adding the trade to the netting set.

    • Formula:

    • Collateralization:

      • Definition: Collateralization involves exchanging cash or other assets as security to cover the exposure on a transaction. It directly mitigates counterparty risk by changing the Expected Exposure (EE).

\Delta C V A_{N S, i}=C V A(N S, i)-C V A(N S)

  • Initial Margins: Act as a buffer against potential future exposure. They are typically held by a third party and reduce CVA by providing a cushion against mark-to-market losses.

  • Thresholds & Minimum Transfer Amounts: These are contractual amounts that allow for a certain level of uncollateralized exposure. A threshold is the amount of exposure that can build up before a party is required to post collateral. A minimum transfer amount is the smallest amount of collateral that can be exchanged. Both of these increase CVA as they allow for greater uncollateralized risk.

  • Margin Period of Risk (MPoR): This is the time interval between the last successful exchange of collateral and the closeout of a defaulting counterparty's positions. A longer MPoR increases the CVA because it increases the period during which exposure can build up, and the counterparty may be unable to post collateral.

Topic 4. Incorporating Netting and Collateralization

Practice Questions: Q2

Q2. A trader wants to know the approximate CVA for a counterparty in a swap transaction. The expected potential exposure (EPE) is 7%, and its credit spread is 475 basis points. What is the CVA as a running spread?
A. ‒0.33%.
B. ‒1.48%.
C. ‒2.25%.
D. ‒9.75%.

Practice Questions: Q2 Answer

Explanation: A is correct.

Calculation of the CVA as a running spread entails multiplying the counterparty’s EPE by its credit spread:

-7 \% \times 4.75 \%=-33 \mathrm{bps}

Practice Questions: Q3

Q3. Regarding the impact of changes in the credit spread and recovery rate assumptions on the CVA, which of the following statements is true?
A. A decrease in the credit spread will most often increase the CVA.

B. For an upward-sloping curve, the CVA will be higher compared to a downward-sloping curve.
C. Increasing the recovery rate will reduce the CVA.
D. If the actual recovery rate is higher than the settled recovery rate, the CVA will most likely be higher compared to a situation where both recovery assumptions are the same for both rates.

Practice Questions: Q3 Answer

Explanation: C is correct.

Increasing the recovery rate will increase the implied probability of default but reduce the resulting CVA. The CVA will most often increase given an increase in the credit spread.

When considering the shape of the credit spread curve, the CVA will be lower for an upward-sloping curve compared to a downward-sloping curve.

Finally, a higher actual recovery rate will most likely lead to a lower CVA compared to a situation where the recovery assumptions are the same for both actual and settled rates.

Practice Questions: Q4

Q4. When incorporating netting and collateralization into the CVA calculation, which of the following statements is incorrect?
I. Netting increases the CVA price because it reduces exposure when trades are settled.
II. Collateralization does not change the CVA because it only changes the counterparty’s expected exposure.
A. I only.
B. II only.
C. Both I and II.
D. Neither I nor II.

Practice Questions: Q4 Answer

Explanation: C is correct.

Both statements are incorrect. Netting reduces the CVA price as it reduces exposure when trades are settled. Collateralization also reduces the CVA, changing only the counterparty’s expected exposure (EE), but not its default probability.

Module 2. Incremental and Marginal CVA, and CVA for a Bilateral Contract

Topic 1. Incremental and Marginal CVA

Topic 2. Converting CVA Into a Running Spread

Topic 3. Applying CVA to Exotic Products and Path Dependency

Topic 4. CVA for a Bilateral Contract (BCVA)

Topic 5. BCVA Spread

Topic 1. Incremental and Marginal CVA

  • Incremental CVA: Incremental CVA is the change in the total CVA of a portfolio that results from adding a new trade to an existing netting set.

    • Purpose: It is a key metric for pricing a new trade, as it measures the exact increase in counterparty risk that the new trade contributes. It ensures that the trade is profitable enough to cover this additional risk. Incremental CVA is a forward-looking, pre-deal measure.

    • Calculation:                                                                    where CVA(NS,i) is the CVA of the portfolio with the new trade, and CVA(NS) is the CVA of the existing portfolio.

  • Marginal CVA: Marginal CVA is a method for decomposing a netted portfolio's total CVA into the contribution of each individual trade.

    • Purpose: This provides an ex-post (after the fact) view of the portfolio. It is used to determine which specific trades are the primary drivers of the CVA and can be used for performance attribution, risk reporting, and to understand the historical impact of each trade on the portfolio's risk profile. It answers the question, "How much CVA is this particular trade responsible for within the existing portfolio?"

\Delta CVA_{NS,i}=CVA(NS,i)-CVA(NS),

Topic 2. Converting CVA Into a Running Spread

  • Objective and Rationale: The primary goal is to transform the CVA, which is a lump-sum, upfront capital charge, into a continuous running spread that can be factored into the pricing of a transaction over its entire life. This makes the cost of counterparty risk transparent and integrates it directly into the profitability analysis of the deal.

    • This conversion is particularly useful for products like interest rate swaps, where a basis point spread can be added to the fixed leg of the swap to compensate for the CVA.

  • Calculation: The upfront CVA is divided by the risky duration and notional amount to arrive at a spread. This spread is often expressed in basis points (bps).

    • The calculation can be iterative or recursive. This is because the new spread, once added to the transaction, can change its mark-to-market (MtM) value and, consequently, alter the CVA itself. This feedback loop requires a recursive solution to arrive at a stable spread.

  • ​Alternative Perspective: The CVA spread can be thought of as a continuous premium on a credit default swap (CDS) that the counterparty would need to pay to hedge the counterparty risk. The formula provided in the document highlights this relationship.

Practice Questions: Q1

Q1. With respect to the CVA calculation, which of the following statements is correct when a risk manager wishes to understand which trades have the greatest impact on a counterparty’s CVA? The manager would use:
A. incremental CVA because it accounts for the change in CVA once the new trade is priced, accounting for netting.
B. marginal CVA because he could break down netted trades into trade level contributions.
C. incremental CVA because he could break down netted trades into trade level contributions.
D. marginal CVA because it accounts for the change in CVA once the new trade is priced, accounting for netting.

Practice Questions: Q1 Answer

Explanation: B is correct.

Understanding which trades have the greatest impact on a counterparty’s credit value adjustment requires use of the marginal CVA. Incremental CVA, by contrast, is useful for pricing a new trade with respect to an existing one.

Topic 3. Applying CVA to Exotic Products and Path Dependency

  • Exotic Products:

    • Valuing CVA for exotic products, such as swaptions or convertible bonds, is complex. Their payoffs are highly non-linear, making a simple, closed-form solution for CVA impossible.

    • Monte Carlo Simulation: This is the primary method for valuing CVA for exotic products. It involves simulating thousands of possible future paths for the underlying asset. For each path, the exposure at different time steps is calculated, which is then used to determine the expected positive exposure (EPE) and the final CVA.

    • Approximations: In some cases, to simplify the problem, a more exotic product may be approximated by a simpler one. For example, a swaption might be treated as a forward-starting swap. However, this method can introduce significant error and should be used with caution.

  • Path Dependency:

    • For products where the payoff depends on the entire history of the underlying asset's price, such as an Asian option or a lookback option, the entire path from the present to the future date is required to accurately assess future exposure.

  • Path-Dependent Payoffs: The final value of a path-dependent option is determined not just by the final price of the underlying, but by the average price (in the case of an Asian option) or the maximum/minimum price (in the case of a lookback option) over a specified period.

  • Impact on CVA: This path dependency means that the exposure at any given time depends on the path taken to reach that time, and cannot be determined by the value of the underlying at that time alone. This significantly complicates the CVA calculation, reinforcing the need for Monte Carlo simulation to generate and evaluate a wide range of potential pa

Topic 3. Applying CVA to Exotic Products and Path Dependency

Topic 4. CVA for a Bilateral Contract (BCVA)

  • Definition: Acknowledges that both counterparties in a contract can default. BCVA is the sum of the CVA and the Debt Value Adjustment (DVA). It represents the cost of potential losses from both your counterparty defaulting on you and you defaulting on your counterparty.

  • Formula:

  • Components:

    • CVA (Credit Value Adjustment): The adjustment for the risk that your counterparty will default on you. This is calculated using your Expected Positive Exposure (EPE).

    • DVA (Debt Value Adjustment): The adjustment for the risk that you will default on your counterparty. This is calculated using your Expected Negative Exposure (ENE).

  • Relationship to DVA and BCVA:
    •           is the Loss Given Default of your institution, and         is the probability of default for your institution.
    • BCVA, therefore, considers the value of all future expected losses and gains due to default by both parties. It is a more comprehensive measure of counterparty risk than CVA alone.
BCVA=CVA+DVA
D V A=-L G D_I \times \sum_{i-1}^m \operatorname{ENE}\left(t_i\right) \times P D_I\left(t_{i-1}, t_i\right)
P D_I
LGD_I

Topic 5. BCVA Spread

  • Purpose: The BCVA Spread expresses the bilateral CVA as a running spread or basis point charge. This is a practical way to account for bilateral counterparty risk and incorporate it directly into the pricing of a transaction. Instead of a one-time, lump-sum charge, the spread allows the risk to be amortized over the life of the contract.

  • Formula:

    •  
  • Derivation and Significance: The formula is an extension of the CVA spread calculation, incorporating the DVA component.

    • The resulting spread represents the fee that must be added to a transaction's rate (e.g., a swap rate) to compensate for the combined risk of both your counterparty and your own institution defaulting.

    • It provides a more accurate and comprehensive measure of the true cost of entering into a bilateral derivative contract.

  • Components:
    •             The counterparty's CDS spread.
    •             The institution's own CDS spread.

    • average EPE: Average expected positive exposure.

    • average ENE: Average expected negative exposure.

\frac{B C V A(t, T)}{C D S_{\text {premium }}(t, T)}=-X_C^{C D S} \times \text { average EPE }-X_I^{C D S} \times \text { average ENE }
X_C^{C D S}:
X_I^{C D S}:

Module 3. Wrong-Way Risk

Topic 1. Wrong-Way Risk (WWR) and Right-Way Risk (RWR)

Topic 2. Examples of Wrong-Way and Right-Way Risk

Topic 3. WWR Modeling

Topic 4. Impact of Collateral and CCPs on WWR

Topic 1. Wrong-Way Risk (WWR) and Right-Way Risk (RWR)

  • Defining the Relationship: WWR and RWR describe a crucial dependence between a counterparty's creditworthiness and the credit exposure on a transaction. This link is often driven by the same macro or global factors that impact both the value of the transaction and the counterparty's ability to pay.

  • Wrong-Way Risk (WWR): WWR is an unfavorable dependence where the exposure to a counterparty is positively correlated with that counterparty's probability of default.

    • In this scenario, as the counterparty's financial health deteriorates and their credit spread widens (indicating a higher chance of default), our exposure to them from the transaction simultaneously increases.

    • This compounding effect is particularly dangerous, as it means the potential loss is greatest precisely when the counterparty is most likely to default.

  • Right-Way Risk (RWR):

    • RWR is a favorable dependence where the exposure to a counterparty is negatively correlated with that counterparty's probability of default.

    • Here, as the counterparty's credit quality worsens, the value of the transaction to us decreases, thereby reducing our exposure.

    • This creates a natural hedge, where the risk of the transaction is mitigated by the counterparty's deteriorating credit quality.

Topic 1. Wrong-Way Risk (WWR) and Right-Way Risk (RWR)

Topic 2. Examples of Wrong-Way and Right-Way Risk

  • Overview: WWR and RWR are often linked to macroeconomic events (e.g., interest rates, inflation) and global factors that impact both the creditworthiness of a counterparty and the value of a transaction.

  • Over-the-Counter Put Option: A classic example of WWR. A put option gives the buyer the right to sell an underlying asset at a strike price.

    • If a macroeconomic event causes the asset's price to fall, the value of the put option increases, which increases your exposure to the counterparty.

    • If this same event also causes the counterparty's creditworthiness to deteriorate, it creates a "wrong-way" situation: as your exposure grows, so does the risk of the counterparty defaulting. The document also notes that out-of-the-money put options have more WWR than in-the-money puts.

  • Commodity Contracts:

    • WWR Example: A counterparty in the oil industry is a client for a long-term forward contract to sell oil. If oil prices fall sharply, the counterparty's credit quality may decline, but your exposure to them (the receivable) increases.

    • RWR Example: You buy a forward contract from a company that produces cocoa. If cocoa prices rise, the company's financial health may improve, while your exposure to them (in terms of a receivable) decreases.

  • Foreign Currency Transactions:

    • RWR Example: An American exporter has a long position in a foreign currency. As the foreign currency depreciates against the USD, the exporter's credit risk increases, but their exposure (as a US company with USD receivables) to you decreases. The document notes that a negative correlation between exposure and default probability lowers the conditional expected exposure, showing RWR.

Topic 2. Examples of Wrong-Way and Right-Way Risk

Topic 3. WWR Modeling

  • Conceptual Framework: Wrong-way risk is modeled based on a conditional expected exposure. This means we are estimating the CVA under the unfavorable condition that the exposure and the counterparty's probability of default are positively correlated. This positive correlation is what constitutes WWR.

  • Relationship to Credit Spread: More specifically, WWR is a phenomenon where the exposure to a counterparty is positively correlated with that counterparty's credit spread. A widening credit spread indicates deteriorating credit quality and a higher default probability.

  • Quantitative Approach: The conditional expected exposure is calculated to account for this correlation. The higher the positive correlation between exposure and default probability, the higher the CVA will be. Conversely, a negative correlation (right-way risk) lowers the conditional expected exposure, which in turn reduces the CVA.

  • Complexities: Accurate WWR modeling often requires sophisticated techniques, such as simulating the underlying risk factors (e.g., interest rates, commodity prices) and the counterparty's credit spread together, accounting for their interdependence.

Topic 4. Impact of Collateral and CCPs on WWR

  • Collateral:

    • Collateralization is a primary risk mitigant that reduces exposure and, therefore, the impact of WWR.

    • It helps to limit the potential loss in a wrong-way scenario by requiring a counterparty to post collateral when exposure increases.

    • This is particularly effective in limiting the "snowball effect" of WWR, as the exposure that is correlated with default is actively reduced.

  • Central Counterparties (CCPs):

    • CCPs act as a central hub, or a "hub-and-spoke" model, for clearing transactions. This model fundamentally changes the risk profile by introducing a single, highly creditworthy counterparty.

    • By centralizing transactions, CCPs eliminate the direct bilateral exposures between individual counterparties. This, in turn, significantly lowers the overall wrong-way risk in the market, as the exposure to an individual counterparty's default is replaced by a single exposure to the CCP, which is managed separately and is considered much lower.

Practice Questions: Q1

Q1. How many of the following statements regarding wrong-way risk (WWR) and right-way risk (RWR) are correct?

  • Co-movement in risk exposure and default probability producing a decline in overall risk is an example of wrong-way risk.
  • Co-movement in risk exposure and default probability producing an increase in overall counterparty risk is an example of right-way risk.
  • Co-movement in risk exposure and default probability producing neither a decline nor an increase in the overall counterparty risk is an example of wrong-way risk.
  • Co-movement in risk exposure and default probability producing a decline in risk exposure but an increase in counterparty default probability is an example of right-way risk.

A. None.
B. All.
C. Two.
D. Three.

Practice Questions: Q1 Answer

Explanation: A is correct.

A decline in overall counterparty risk is an example of right-way risk. An increase in overall counterparty risk is an example of wrong-way risk. An increase in overall counterparty risk is a condition for the emergence of wrong-way risk. A decline in risk exposure but increase in counterparty default probability may or may not lower overall counterparty risk.

Practice Questions: Q2

Q2. Which of the following events would likely lead to an increase in WWR?
I. The borrower and the guarantor are business partners.
II. A monoline insurer sold protection concentrated in a business or industry.
A. I only.
B. II only.
C. Both I and II.
D. Neither I nor II.

Practice Questions: Q2 Answer

Explanation: C is correct.

WWR will increase if the borrower and guarantor are business partners. The guarantees offered by a monoline insurer may turn out to be worthless if the risk exposure increases and the guarantor is hit by a flood of claims due to a concentrated position in an industry or business.

Practice Questions: Q3

Q3. Which of the following statements regarding WWR and RWR is correct?
A. A long put option is subject to WWR if both risk exposure and counterparty default probability decrease.
B. A long call option experiences RWR if the interaction between risk exposure and counterparty default probability produces an overall decline in counterparty risk.
C. Declining local currency can decrease the position gain in a foreign currency transaction, while increasing risk exposure of the counterparty.
D. The 2007–2009 credit crisis provides an example of WWR from the perspective of a long who had sold credit default swaps (CDSs) as protection against bond issuers’ default.

Practice Questions: Q3Answer

Explanation: B is correct.

A long call option experiences RWR if risk exposure and counterparty default probability results in decreased counterparty risk. A long put option is subject to WWR if both risk exposure and counterparty default probability increase. Declining local currency can increase the position gain in a foreign currency transaction, while increasing counterparty risk exposure. The 2007–2009 credit
crisis provides an example of WWR from the perspective of a long who had bought CDSs as protection against bond issuers’ default.

Practice Questions: Q4

Q4. How many of the following statements regarding counterparty risk are correct?

  • Speculation in normal-functioning derivatives markets automatically produces RWR.
  • RWR has been the center of attention in historical context, whereas WWR has not been paid much relative attention.
  • The counterparty default probability does not enter into the equation for estimating the overall counterparty risk.
  • Unlike exposure to OTC derivatives, which is normally assumed to be a fixed amount for a specified time period, exposure to bank loans fluctuates depending on market conditions.

A. None.
B. All.

C. Two.
D. Three.

Practice Questions: Q4 Answer

Explanation: A is correct.

Hedging, and not speculation, in normal functioning markets automatically produces RWR. Historically, RWR was relatively neglected by institutions for planning purposes. The counterparty default probability is one of the key elements in estimating overall counterparty risk. OTC exposures fluctuate based on market conditions.

Practice Questions: Q5

Q5. Which of the following statements is correct?
I. Depreciation of the yen after the default of Lehman Brothers gave a substantial gain to Japanese bank foreign currency swaps positions to obtain dollar funding in interest rate swaps.
II. Fixed-rate receivers experience a value gain to the extent that the swap rate increases.
A. I only.
B. II only.
C. Both I and II.
D. Neither I nor II.

Practice Questions: Q5 Answer

Explanation: D is correct.

Appreciation, and not depreciation, of the yen generated a substantial gain for Japanese banks with foreign currency swaps positions. A fixed-rate receiver experiences a value gain to the extent that the swap rate declines.