Book 2. Credit Risk

FRM Part 2

CR 12. Credit Risk

Presented by: Sudhanshu

Module 1. Default Probabilities

Module 2. Credit Risk in Derivatives

Module 3. Default Correlation and Credit VaR

Module 1. Default Probabilities

Topic 1. Credit Risk and Default Probability: Overview

Topic 2. Default Probabilities Using Credit Spreads

Topic 3. Default Probability Estimate Comparisons

Topic 4. Equity Prices for Default Probability Estimates

Topic 1. Credit Risk and Default Probability: Overview

  • Introduction to Credit Risk & Default Probabilities

    • Credit Risk Defined: Credit risk is the exposure an entity faces from the potential default of a counterparty in a derivatives contract or a borrower.

    • Key Factors: Credit ratings, default probabilities, and recovery rates are all significant in assessing credit risk.

    • Credit Rating Agencies: Major agencies like Moody's, S&P, and Fitch provide corporate bond ratings.

    • Investment Grade vs. Lower Rated Bonds: Bonds rated Baa and above (by Moody's) are considered investment grade and have significantly lower default probabilities than lower-rated bonds.

  • Understanding Default Rate Dynamics

    • Cumulative Default Rates: The provided data (Figure 30.1) shows average cumulative default rates for different Moody's ratings from 1970 to 2015.

  • Unconditional Default Probability: This is the probability of default in a specific year, as seen today. For a B-rated bond, the unconditional default probability during Year 4 is calculated as the difference between the 4-year and 3-year cumulative default rates: 17.999%−13.515%=4.484%.

  • Conditional Default Probability (Hazard Rate): The hazard rate is the probability of default in a given year, assuming no earlier default. For a B-rated bond, the conditional default probability during Year 4 (given survival through Year 3) is calculated by dividing the unconditional probability by the survival probability: 0.04484/(1−0.13515)=5.185%.​

  • Recovery Rates: A bond's recovery rate is its market value after a default divided by its face value.

    • Impact of Economic Conditions: Recovery rates have a negative correlation with default rates.

      • Strong Economy: Lower default numbers and higher recovery rates.

      • Weak Economy: Higher default numbers and lower recovery rates.

    • Bond Type: First and second lien bonds naturally have higher recovery rates than subordinated bonds.

Topic 1. Credit Risk and Default Probability: Overview

Topic 1. Credit Risk and Default Probability: Overview

Practice Questions: Q1

Q1.If the unconditional default probability of a Ba-rated bond during Year 3 is 1.914% and the probability of survival through Year 2 is 97.413%, the probability of a default during Year 3, condional on no earlier default, is closest to:

A. 0.673%.

B. 1.914%.

C. 1.965%.

D. 2.587%.

Practice Questions: Q1 Answer

Explanation: C is correct.

The probability of a default during Year 3, conditional on no earlier default, is
equal to: 0.01914 / 0.97413 = 1.965%.

Topic 2. Default Probabilities Using Credit Spreads

  • Estimating Default Probabilities

    • Yield Spread: The yield spread of a bond is the difference between its promised yield and the risk-free rate.

    • Hazard Rate Approximation: The average hazard rate,        ,  can be approximated using the bond yield spread, s(T), and the recovery rate, RR.

      • Formula:​ 

      •  

    • Example Calculation: A one-year bond with a 125 basis point spread over the risk-free rate and a 55% recovery rate has an average hazard rate of: 0.0125/(1−0.55)=0.0278, or 2.78% per year.

    • Risk-Free Rate: This is a crucial factor, often proxied by a Treasury rate. However, credit default swap (CDS) spreads may be a better measure as they do not depend on the risk-free rate.

\bar{\lambda}(T)
\bar{\lambda}(T)=\frac{s(T)}{1-RR}

Topic 3. Default Probability Estimate Comparisons

  • Bond Yield Spreads vs. Historical Data: Higher Estimates: Bond yield spreads generally produce higher estimated default probabilities than historical data. This difference becomes even more significant during economic crises.

    • Expected Excess Return: The table in Figure 30.2 shows the expected excess return on bonds (in basis points) for spreads over Treasuries at various rating categories.

    • Rating and Hazard Rates: For investment-grade companies, the ratio of the hazard rate from bond prices to the hazard rate from historical data is very high, but this ratio decreases as credit ratings decline.​

  • Real-World vs. Risk-Neutral Probabilities: Risk-Neutral Probabilities: These are implied from credit spreads and are used for pricing instruments and valuing credit derivatives.

    • Real-World Probabilities: These are based on historical data and are used for scenario analysis to calculate potential future losses.

  • Difference: Risk-neutral probabilities are typically higher than real-world probabilities. This difference is mainly due to the systematic risk of bond defaults, which cannot be diversified away. Other factors include the relative illiquidity of corporate bonds and the subjective probabilities of bond traders.

Topic 3. Default Probability Estimate Comparisons

Practice Questions: Q2

Q2. A 2-year corporate bond yields 190 basis points above the risk-free rate. With a recovery rate of 35%, the average hazard rate for Years 1 and 2 is closest to:

A. 0.67%.

B. 1.55%.

C. 2.92%.

D. 5.43%.

Practice Questions: Q2 Answer

Explanation: C is correct.

A 2-year bond yielding 190 basis points above the risk-free rate with a recovery rate of 35% would yield an average hazard rate of: 0.0190 / (1 − 0.35) = 0.0292, or 2.92% per year.

Topic 4. Equity Prices for Default Probability Estimates

  • The Merton Model

    • Equity as an Option: The Merton model views a company's equity         as a call option on the company's asset value           , with the strike price being the required debt repayment (D).

      • Formula: 

    • Value of Equity: The value of equity today (E0​) can be calculated using the Black-Scholes-Merton formula.

    • Value of Debt: The value of debt today (D0​) is the difference between the company's asset value and its equity value (D0​=V0​−E0​).

    • Model Accuracy: Historical data suggests that the Merton model produces solid rankings of default probabilities in both risk-neutral and real-world scenarios

(E_T)
(V_T)
E_T=max(V_T-D,0)

Module 2. Credit Risk in Derivatives

Topic 1. Credit Risks of Derivatives

Topic 2. CVA and DVA

Topic 3. Credit Risk Mitigation

Topic 1. Credit Risks of Derivatives

  • Risk for Non-Defaulting Party: A non-defaulting party in a derivatives transaction will likely incur a loss in two primary situations.

    1. The total value of the transactions for the non-defaulting party is positive and exceeds the collateral posted by the defaulting party. In this case, the non-defaulting party becomes an unsecured creditor for the amount of the transactions minus the collateral.

    2. The total value of the transactions for the defaulting party is positive and is less than the collateral posted by the non-defaulting party. Here, the non-defaulting party becomes an unsecured creditor for the return of the excess collateral they posted.

  • ISDA Master Agreement: The International Swaps and Derivatives Association (ISDA) Master Agreement is typically used to govern bilaterally cleared derivatives transactions between two companies. This agreement outlines initial and variation margin requirements for financial institutions and defines what constitutes an "event of default".

Practice Questions: Q1

Q1. In a bilaterally cleared derivatives transaction between two companies (Company A and Company B), Company B defaults. The value for Company A is a positive $50,000 and the collateral posted by Company B is $30,000. In this situation, Company A is a(n):

A. secured creditor in the amount of $20,000.

B. secured creditor in the amount of $80,000.

C. unsecured creditor in the amount of $20,000.

D. unsecured creditor in the amount of $80,000.

Practice Questions: Q1 Answer

Explanation: C is correct.

The total value for Company A (as the nondefault party) is positive, and at $50,000, it exceeds the $30,000 collateral posted by Company B (as the default party). Company A will be an unsecured creditor for an amount equal to $50,000 −$30,000 = $20,000.

Topic 2. CVA and DVA

  • Defining CVA and DVA

    • CVA (Credit Valuation Adjustment): From a bank's perspective, CVA is the present value of the expected cost to the bank if its counterparty defaults.

  • DVA (Debt Valuation Adjustment): DVA is the present value of the expected cost to the counterparty if the bank itself defaults.

    • Valuing the Portfolio: The total value of a bank's derivatives transactions, accounting for the possibility of default, is calculated as the no-default value minus CVA plus DVA:

      • Formula: 

      • DVA as a Benefit: A bank's default means it won't have to make payments on its derivatives positions, so DVA is considered a benefit to the bank, increasing the value of its portfolio.

    • Wrong-Way & Right-Way Risk:

      • Wrong-way risk occurs when the probability of default is positively correlated with exposure.

      • Right-way risk occurs when the probability of default is negatively correlated with exposure.

f_{nd}-CVA+DVA
  • CVA and DVA Calculations

    • Formulas:

       

      •  

      •  

    • Key Terms:

      •    : The risk-neutral probability of the counterparty defaulting during the ith interval.

      •     : The present value of the expected loss to the bank if the counterparty defaults at the midpoint of the ith interval.

      •      : The risk-neutral probability of the bank defaulting during the  ith interval, calculated from the bank's credit spreads.

      •      : The present value of the expected loss to the counterparty (which is a gain to the bank) if the bank defaults at the midpoint of the ith interval.

v_i^{*}
q_i^{*}
v_i
q_i
CVA: CVA=\sum_{i=1}^N q_i \times v_i
DVA: DVA=\sum_{i=1}^N q_i^{*} \times v_i^{*}

Topic 2. CVA and DVA

Practice Questions: Q2

Q2. A bank is assessing the impact of a new transaction on CVA and DVA. If the new transaction is negatively correlated to existing transactions, the impact will likely be a(n):

A. increase to both CVA and DVA.

B. decrease to both CVA and DVA.

C. increase to CVA and decrease to DVA.

D. decrease to CVA and increase to DVA.

Practice Questions: Q2 Answer

Explanation: B is correct.

If a new transaction is negatively correlated to existing transactions for the bank and the counterparty, the new transaction will likely decrease both the CVA and the DVA.

Topic 3. Credit Risk Mitigation

  • Key Mitigation Techniques

    • Netting: This reduces exposure by offsetting positive and negative transaction values with a counterparty. For example, if a bank has four transactions with values of +$5M, -$7M, +$10M, and -$2M, the total exposure is not the sum of positive values ($15M) but the net value, which is +$6M.

    • Collateral Agreements: In the event of a default, the non-defaulting party can keep the cash or marketable securities posted as collateral by the defaulting party.

    • Downgrade Triggers: These are clauses in agreements that allow the non-defaulting party (e.g., a bank) to either close out outstanding transactions at market value or demand collateral if the counterparty's credit rating falls below a specified level. The value of these triggers can be reduced by significant rating downgrades or if the counterparty has similar triggers with multiple dealers

Practice Questions: Q3

Q3. Bank TGF has three uncollateralized transactions with JL Co. The transactions have values to the bank of: +$12 million, −$4 million, and −$3 million. If the bank mitigates its credit risk through netting, the impact of this technique will result in an exposure reduction of:

A. $5 million.

B. $7 million.

C. $8 million.

D. $12 million.

Practice Questions: Q3 Answer

Explanation: B is correct.

If the bank treats all three transactions as individual transactions, the exposure is $12 million. If the transactions are netted, the exposure is $5 million. This equates to an exposure reduction of $7 million.

Module 3. Default Correlation and Credit VaR

Topic 1. Significance of Estimating Default Correlation for Credit Portfolios

Topic 2. Gaussian Copula Model

Topic 3. Credit VaR

Topic 1. Significance of Estimating Default Correlation for Credit Portfolios

  • Why Default Correlation Matters

    • Definition: Default correlation is the likelihood that two companies will default around the same time.

    • Causes: Default correlation can be caused by external events impacting companies in the same region or industry, overall economic conditions, or contagion where one company's default causes another's.

    • Diversification Limits: Because of default correlation, credit risk cannot be fully diversified away.

    • Models for Evaluation: Two main models are used to evaluate default correlation:

      • Reduced-form models: Assume hazard rates are correlated with macroeconomic variables and follow random processes. They are mathematically straightforward and reflect economic cycles, but their achievable default correlations are low.

      • Structural models: Based on models like the Merton model. Default correlations are introduced by assuming correlated stochastic processes for the companies' assets. These models can be set to have a high correlation, but they are computationally intensive.

Practice Questions: Q1

Q1. Which of the following statements is most accurate regarding reduced-form versus structural models used to estimate default correlation?

A. Structural models tend to take a long time to process.

B. Reduced-form models are more computationally intensive.

C. Reduced-form models allow for very high default correlations.

D. Economic cycle impacts on default correlation trends are best reflected in structural models.

Practice Questions: Q1 Answer

Explanation: A is correct.

Structural models are computationally intensive (relative to reduced-form models), and, therefore, take a long time to process. However, an advantage to structural models is that they allow for higher default correlations. Economic cycle impacts on default correlation trends are best relected in reduced-form models.

Topic 2. Gaussian Copula Model

  • Modeling Time to Default

    • Purpose: The Gaussian copula model is used to quantify the correlation between the probability distributions of times to default for multiple companies. It assumes that all companies will eventually default.

    • Probability Types: Both real-world and risk-neutral default probabilities can be used.

      • Real-world probabilities: The left tail is estimated from rating agency data.

      • Risk-neutral probabilities: The left tail is estimated from bond prices.

    • Transformation: The model transforms the non-normal times to default into normal variables with zero mean and unit standard deviation.

    • The Copula: The assumption that the joint distribution of these variables is bivariate normal is called the Gaussian copula. It allows the correlation structure to be estimated independently of the unconditional distributions. This default correlation is also known as the copula correlation.

  • One-Factor Gaussian Copula Model

    • Simplification: A one-factor model can be used to avoid needing different correlations for each company pair.

    • Formula: The model assumes that a company's default is impacted by a common factor (F) and a company-specific factor​

      •  

      • Here,      ​is the correlation of company i's equity returns with a market index.

    • Conditional Default Probability: The default probability for company i by time T, conditioned on the common factor F, is given by:

       

      •  
(Z_i)
a_i
x_i=a_iF+\sqrt{1-a_i^2}Z_i
Q_i(T|F)=N(\frac{N^{-1}[Q_i(T)]=a_iF}{\sqrt{1-a_i^2}})

Topic 2. Gaussian Copula Model

Practice Questions: Q2

Q2. The Gaussian copula model transforms which of the following factors into normal variables?

A. Recovery rates.

B. Times to default.

C. Credit spread risk.

D. Default probabilities.

Practice Questions: Q2 Answer

Explanation: B is correct.

The Gaussian copula model transforms the times to default into normal variables with means of zero and unit standard deviations. The other choices are all relevant in the credit risk world, but they are not transformed into normal variables via this model.

Topic 3. Credit VaR

  • Understanding Credit VaR

    • Definition: Credit value at risk (VaR) is the credit risk loss that will not be exceeded over a given period of time, at a specific confidence level.

    • Calculation using Gaussian Copula: The percentage of losses on a large portfolio that will be less than V(T,X) is given by a formula using the probability of default by time T (Q(T)) and the copula correlation (ρ)

      •  

         

    • Estimating Credit VaR: The actual credit VaR can be estimated as the loan portfolio size (L) multiplied by the expected loss given default:

      • Formula: L×(1−RR)×V(T,X), where RR is the recovery rate.

V(T, X)=N(\frac{N^{-1}[Q(T)]+\sqrt{\rho}N^{-1}(X)}{\sqrt{1-\rho}})
  • The CreditMetrics Approach​

    • Alternative Model: The CreditMetrics model, developed by JPMorgan, is another approach to estimating credit VaR.

    • Simulation: It determines a probability distribution of credit losses by applying a Monte Carlo simulation to the credit rating changes of all counterparties.

    • Advantages: This approach accounts for both defaults and downgrades and can incorporate credit mitigation strategies.

    • Rating Transitions: The model uses a rating transition matrix as a basis for its simulations. A joint probability distribution for rating changes is constructed using a Gaussian copula model. The copula correlation between rating transitions for two companies is set equal to their equity return correlations

Topic 3. Credit VaR

Practice Questions: Q3

Q3. Assuming a loan portfolio of L, a recovery rate of RR, and the percentage of losses on a portfolio less than V(T, X), which of the following formulas is used to estimate credit VaR?

A. L × (RR) × V(T, X).

B. L × (1 − RR) / V(T, X).

C. V(T, X) / [L × (1 − RR)].

D. L × (1 − RR) × V(T, X).

Practice Questions: Q3 Answer

Explanation: D is correct.

The appropriate formula for estimating credit VaR is: L × (1 − RR) × V(T, X).

CR 12. Credit Risk

By Prateek Yadav

CR 12. Credit Risk

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