Book 2. Credit Risk

FRM Part 2

CR 19. Future Value and Exposure

Presented by: Sudhanshu

Module 1. Credit Exposure

Module 2. Security Exposure Profiles

Module 3. Collateral and Credit Exposure

Module 1. Credit Exposure

Topic 1. Credit Exposure Metrics

Topic 2. Comparing Credit Exposure to VaR Methods

Topic 3. Credit Exposure Factors

Topic 1. Credit Exposure Metrics

  • Key Metrics and Their Calculation

    • Expected Mark-to-Market (MtM): The expected value of a transaction at a given future point in time. It can be positive or negative. This metric is the starting point for calculating exposure, but it does not represent the exposure itself since a negative value (the counterparty owes you money) would not result in a loss from their default. It's often used in calculating future value and is a key component of the metrics below.

    • Expected Exposure (EE): The expected value of a counterparty's exposure at a given point in the future. It is the expected value of the MtM, but floored at zero, since exposure is non-negative.

    • It's often calculated as:

    • where V(t) is the mark-to-market value at time t. EE is an average measure and, as such, does not capture the "worst-case" loss.

    • Potential Future Exposure (PFE): A worst-case measure of exposure at a given future time. It is a high quantile of the exposure distribution (e.g., 95% or 99%). This is a key metric for setting credit limits and managing the peak risk over the life of a contract. It is the worst exposure an institution could have at a certain time in the future, measured at a specified confidence level.

E E(t)=E[\max (0, V(t))]

  • Expected Positive Exposure (EPE): The average of the expected exposures over the life of the transaction. It is often used as a single number to represent the average exposure of a contract and is a key input for calculating the Credit Valuation Adjustment (CVA), which is the price of the counterparty credit risk. It provides a single, summary measure of the exposure profile.

  • Effective Expected Positive Exposure (EEE): The largest value of expected exposure to date, including the current point in time. It is a non-decreasing metric that helps to capture the full risk profile over time, ensuring a conservative measure that doesn't decrease even if the expected exposure falls later on. It is a non-decreasing, stair-step function.

  • Maximum Exposure: The maximum value of the PFE over the life of the contract. This is the single highest point on the PFE profile curve and represents the absolute worst-case exposure at any point in the future.

Topic 1. Credit Exposure Metrics

Topic 1. Credit Exposure Metrics

Practice Questions: Q1

Q1. Which metric for credit exposure is represented by the “?” in the following graph?

 

 

 

 

 

 

 

 

A. Expected positive exposure (EPE).
B. Potential future exposure (PFE).
C. Effective expected exposure (EE).
D. Effective expected positive exposure (EPE).

Practice Questions: Q1 Answer

Explanation: A is correct.

EPE is equal to average EE over time. It is a useful single amount to quantify exposure.

Topic 2. Credit Exposure & VaR Methods

  • Credit Exposure (EAD): Focuses on the loss given a counterparty's default, considering the replacement value of a contract.

    • Forward-looking: It's an estimate of potential future loss.

    • Directionality: Represents a one-way risk—a loss can only occur if the counterparty defaults while the contract has a positive value to you.

    • Time Horizon: Based on the contract's life, which can be years or decades.

    • Risk Mitigant: Relies on credit risk management tools like netting and collateral.

    • Modeling: Often requires complex Monte Carlo simulations to model the future value of the derivative under various market scenarios.

  • Value at Risk (VaR): Focuss on market risk, estimating the potential loss over a short time horizon at a given confidence level.

    • Backward-looking: Based on historical data or statistical models.

    • Directionality: Represents a two-way risk—a loss can occur from adverse market movements regardless of counterparty health.

    • Time Horizon: Typically 1 to 10 days.

    • Risk Mitigant: Primarily uses hedging and diversification.

    • Modeling: Can be calculated using historical simulation, parametric models, or Monte Carlo simulations over a short time frame.

  • Key Difference: VaR assumes a non-defaulting counterparty, while Credit Exposure explicitly models the risk of default.

Topic 3. Credit Exposure Factors

  • Future Uncertainty: Credit exposure is not a fixed amount but a stochastic variable. Its value depends on the future evolution of underlying market factors such as interest rates, foreign exchange rates, commodity prices, and equity prices. These factors are uncertain and must be modeled using probability distributions to project future exposure.

  • Periodic Cash Flows: Periodic cash flows, such as those in a swap, can reset the exposure to or near zero at specific points in time. This is because the mark-to-market value is settled, and the net exposure for that period is paid or received. This is a key reason why swap exposure profiles tend to be "hump-shaped" rather than constantly increasing.

  • Combination of Profiles: The total credit exposure of a portfolio of trades is generally less than the sum of the exposures of the individual trades. This is due to the diversification effect. When trades within a portfolio have negative or low correlation, a loss on one trade may be offset by a gain on another. The netting of these exposures reduces the overall risk.

  • Optionality: Trades with optionality, such as swaps with embedded options, can significantly alter the exposure profile. For the party who is long the option, exposure is one-way, meaning they will only have a positive mark-to-market and therefore credit exposure if the option is "in-the-money." The party who is short the option has no credit exposure to the option itself.

Module 2. Security Exposure Profiles

Topic 1. Typical Exposure Profiles

Topic 2. The Impact of Payment Frequencies & Exercise Dates

Topic 3. Modeling Netting & Collateral

Topic 1. Typical Exposure Profiles

  • Bonds: Exposure is close to the par value and increases as maturity approaches. The exposure is generally tied to the value of the bond, which can fluctuate with interest rates and credit spreads.

  • Interest Rate Swaps: These contracts have a "hump-shaped" exposure profile. Exposure is typically low at the start, grows as the contract gains or loses value, and then decreases as the remaining payments are amortized toward zero at maturity. This profile is due to the opposing "diffusion" (increasing uncertainty over time) and "amortization" (decreasing remaining cash flows) effects.

  • Foreign Exchange Forwards: The exposure for an FX forward tends to increase steadily over the life of the contract, peaking at maturity. This is because the full notional amount is exchanged at the final settlement, making the exposure proportional to the potential change in the exchange rate over the entire life of the contract.

  • Options: For a purchased option, the exposure is positive and can increase with the underlying asset's price, as the value of your option grows. For a written (sold) option, the exposure is zero, as the premium has been received upfront and the counterparty cannot lose money if the option is exercised against them.

  • Credit Default Swaps (CDS): The protection buyer's exposure is zero until a credit event (default) occurs. At that point, the exposure is the notional amount of the swap. The protection seller has exposure for the entire life of the contract.

Topic 1. Typical Exposure Profiles

Topic 2. The Impact of Payment Frequencies & Exercise Dates

  • Interest Rate Swaps:

    • Swaps with equal payment frequencies (e.g., quarterly for both legs) have a smooth, hump-shaped exposure profile.

    • Swaps with unequal frequencies (e.g., semi-annual vs. quarterly) have more volatile exposure profiles. The exposure can be "saw-toothed" as it builds up between payment dates and then resets to a lower level after a payment.

  • Options:

    • Swaption: An option to enter a swap at a future date. It's a long-dated option that has a high exposure close to its exercise date. The value of the option (and therefore the exposure) is highest when there is the most uncertainty about whether it will be "in-the-money."

    • Forward Swap: A forward contract on a swap. It has a forward exposure profile from the start, as it doesn't involve the one-time, optional payment of a swaption.

Topic 2. The Impact of Payment Frequencies & Exercise Dates

Practice Questions: Q1

Q1. Miven Corp. has two trades outstanding with one of its counterparties. Which of the following scenarios would result in the greatest netting advantage for Miven?
A. The two trades have strong positive correlation.
B. The two trades have weak positive correlation.
C. The two trades are uncorrelated with each other.

D. The two trades have strong negative correlation.

Practice Questions: Q1 Answer

Explanation: D is correct.

The greatest netting benefit among the scenarios presented occurs when the two trades have a strong negative correlation. In this case, a large portion of the negative exposures will offset positive exposures.

Topic 3. Modeling Netting & Collateral

  • Netting: A legal arrangement that allows the reduction of gross exposure to a single net amount, reducing potential losses. Instead of being exposed to the full value of each contract, a firm is only exposed to the net value of all contracts in a netting set.

  • Key Considerations:

    • Gross Exposure vs. Net Exposure: Netting can significantly reduce a firm's total credit exposure, as positive and negative MtMs across different trades with the same counterparty can offset each other.

    • Correlation: The effectiveness of netting depends on the correlation of the underlying assets. Low or negative correlation provides the greatest netting benefit. For instance, if two trades are perfectly negatively correlated, their MtMs will offset each other, resulting in zero net exposure.

  • ​Netting Factor: A metric to quantify the benefit of netting. It is calculated as the ratio of net exposure to gross exposure.
    • ​Formula:
      • ​n: number of trades in the netting set
      •   : average correlation among trades
1 / \sqrt{1+(n-1) * \rho}
\rho
  • Example: If two trades are uncorrelated (ρ=0), the netting factor is                     meaning the net exposure is reduced to 71% of the gross exposure. For four uncorrelated trades, the factor is                       a 50% reduction.
  • The maximum negative correlation must be bounded by [−1/(n−1)] to prevent the expression under the square root from becoming negative.
1 / \sqrt{2} \approx 0.71,
1 / \sqrt{4}=0.50,

Topic 3. Modeling Netting & Collateral

Practice Questions: Q2

Q2. Which of the following security types will most likely result in a peaked shape for the exposure profile represented by potential future exposure (PFE)?
A. Long option position.
B. Foreign exchange product.
C. 10-year loan with a floating rate payment.
D. Swap.

Practice Questions: Q2 Answer

Explanation: D is correct.

Exposure profiles of swaps are typically characterized by the peaked shape that results from balancing future uncertainties over payments and roll-off risk of swap payments over time.

Practice Questions: Q3

Q3. Which of the following statements best describes the benefit of netting risk exposures? The benefits of netting are realized when:
A. marked-to-market (MtM) values have high structural correlations for two trades.
B. marked-to-market (MtM) values have opposite signs for two trades.
C. expected exposure (EE) values are minimal.
D. expected future exposure (EFE) values have zero correlation.

Practice Questions: Q3 Answer

Explanation: B is correct.

The benefits of netting are realized when MtM values have opposite signs for two trades.

Module 3. Collateral and Credit Exposure

Topic 1. Margin Period of Risk (MPOR)

Topic 2. Modeling Collateral

Topic 3. Differences Between Funding and Credit Exposure

Topic 4. Impact of Collateral on Counterparty Risk and Funding

Topic 1. Margin Period of Risk (MPOR)

  • Definition: The time period from the last exchange of collateral until a defaulting counterparty's positions are closed out and re-hedged in the market. This is the period during which a firm's exposure is uncollateralized and a loss could occur.

  • Calculation: The MPOR is not a single number but is influenced by several factors, including:

    • The time between margin calls.

    • The time for collateral to be transferred.

    • The time from a margin call to the default declaration.

    • The time needed to close out and re-hedge the defaulting counterparty's positions.

  • Regulatory Standard: The length of the MPOR varies by asset class.

    • OTC Derivatives: A minimum of 5 business days for cleared contracts.

    • Non-cleared Derivatives: Typically a 10-day horizon is used, reflecting a higher perceived risk.

Topic 2. Modeling Collateral

  • Impact: Collateral reduces exposure by requiring counterparties to post assets to cover their obligations. This creates a buffer that can be used to cover any losses from the defaulting party.

  • Key Risks Introduced:

    • Liquidity Risk: The collateral itself may be illiquid. If the collateral is hard to sell quickly, it may not cover the exposure in a timely manner.

    • Operational Risk: The logistical challenges of managing collateral. This includes the risk of manual errors, disputes, and the failure to make or receive timely margin calls.

    • Legal Risk: Potential for legal challenges in seizing collateral, especially in a cross-border default scenario. The legal enforceability of collateral agreements is a key concern.

    • Granularity Effect: Collateral can fail to be an effective risk mitigant if the exposure is concentrated in a few, large trades. Collateral may not fully cover a very large, single exposure.

Topic 3. Differences Between Funding and Credit Exposure

  • Credit Exposure: The potential loss due to a counterparty's failure to pay what is owed. It is a one-way risk, as the loss is incurred only if a counterparty with a positive MtM defaults. This risk is typically measured by metrics like EPE and PFE.

  • Funding Exposure: The cost or risk of funding a position over time. It is a two-way risk, as both parties face funding costs regardless of who is "in-the-money." This is particularly relevant when a counterparty needs to post collateral and has to borrow the funds to do so.

  • Key Concept: The Credit Value Adjustment (CVA) captures the cost of credit exposure, while the Funding Value Adjustment (FVA) captures the cost of funding exposure. These are two separate components of the price of a derivative.

    • CVA: The market price of counterparty credit risk. It is the difference between the risk-free value of a derivative and its value, taking into account the possibility of counterparty default. CVA is always a cost to the institution holding the positive exposure.

    • where LGD is Loss Given Default, EE is Expected Exposure, and PD is Probability of Default.

C V A=L G D * \sum_i\left(E E_i * P D_i\right)

  • DVA (Debit Valuation Adjustment): The value of a firm's own credit risk to the counterparty. It is a gain for the firm and is symmetric to CVA.

  • FVA: An adjustment to the value of a derivative to account for the funding costs associated with posting collateral. The value of a derivative is affected by the cost of funds that are required to be posted as collateral.

  • Total Value: The total value of a derivative is typically expressed as:

  •  

\text { Value }=\text { Risk }- \text { FreeValue }+C V A-D V A+F V A

Topic 3. Differences Between Funding and Credit Exposure

Topic 4. Impact of Collateral on Counterparty Risk and Funding

  • Counterparty Risk: Collateral reduces counterparty risk by limiting the potential for loss in a default scenario. When a counterparty defaults, the firm can seize the posted collateral to cover its losses, effectively reducing its credit exposure.

    • Credit Support Annex (CSA): Collateral is governed by a Credit Support Annex, which is a legally binding document that is part of the ISDA Master Agreement. The CSA specifies the rules for posting collateral, including thresholds and minimum transfer amounts.

    • Threshold: A threshold is an uncollateralized exposure amount that the counterparty is allowed to have before they are required to post collateral. A lower threshold means less uncollateralized exposure and a greater reduction in counterparty risk.

    • Minimum Transfer Amount (MTA): The smallest amount of collateral that can be transferred. This reduces the number of small, frequent transfers.

    • "Jump to Default" Risk: Even with collateral, there is always a risk that a counterparty's exposure can jump significantly between the last collateral exchange and the point of default, leaving a gap in protection.

  • Funding: Collateral requirements introduce a funding cost for the collateral itself. The party required to post collateral must either fund this collateral from its own balance sheet or borrow it from the market, incurring a cost. This cost is represented by the FVA.

    • Collateral as a CVA Hedge: Collateral acts as a hedge against CVA. By posting collateral, the counterparty reduces the potential for a large loss, thus reducing the CVA charge.

    • Funding Cost: The cost of borrowing or holding the collateral is a real cost to the business, and this cost must be factored into the pricing of the derivative. The funding cost of collateral is typically linked to the firm's own cost of funding, which can be different from the risk-free rate.

Topic 4. Impact of Collateral on Counterparty Risk and Funding

Practice Questions: Q1

Q1. Time steps that enter into the calculation of the number of days in the margin period of risk include all of the following except:
A. valuation/margin call.
B. posting collateral.
C. settlement.
D. close-out and re-hedge.

Practice Questions: Q1 Answer

Explanation: B is correct.

The time period from which the request for collateral is received to which it is released refers to the receipt of collateral, but it does not involve its actual posting. All of the remaining items are part of the MPoR.