Book 1. Market Risk

FRM Part 2

MR 6. Validating BHCs VaR Models

Presented by: Sudhanshu

Module 1.  VaR Model Validation

Module 1.  VaR Model Validation

Topic 1. Introduction to VaR Model Validation

Topic 2. Conceptual Soundness of VaR Models

Topic 3. Sensitivity Analysis for VaR

Topic 4. Benefits and Challenges of Sensitivity Analysis

Topic 5. Confidence Intervals for VaR

Topic 6. Challenges in Estimating Confidence Intervals

Topic 7. Benchmarking VaR Models

Topic 8. Backtesting for Benchmarking

Topic 9. Topic 8. Backtesting for Benchmarking

Topic 1. Introduction to VaR Model Validation

• Value-at-Risk (VaR) estimates the maximum potential loss over a given time horizon at a specific confidence level.
• Used extensively in Basel regulatory capital requirements.
• Three key elements:
  • Loss Size: The monetary value the institution could lose.
  • Confidence Level: Typically 99% under Basel guidelines.
  • Time Frame: Usually one trading day.
• Why Validate VaR?
  • Regulatory compliance.
  • Avoid undercapitalization.
  • Ensure models adapt to market volatility and structural changes in portfolios.

Practice Questions: Q1

Q1. Which of the following items is crucial when evaluating the conceptual soundness of a VaR model?
A. The portfolio return distribution should be normal.
B. Inputs need to be adjusted to determine the change in VaR.
C. The model should use actual historical returns to calculate VaR.
D. The model should be designed to meet specific risk management objectives.

Practice Questions: Q1 Answer

Explanation:  D is correct.

Conceptually sound models are designed to meet the specific risk management objectives of the bank. Understanding the intended use of the VaR model (e.g., regulatory capital calculation, internal risk assessment) is essential for evaluating its appropriateness. Portfolio return distributions tend to be nonnormal, so the VaR model should not assume a normal distribution. Adjusting key inputs is part of sensitivity analysis and is not used to evaluate the conceptual soundness of the model. Historical returns would not be a good input for calculating VaR when the portfolio composition is dynamic.

Topic 2.  Conceptual Soundness of VaR Models

• Model Validation Requirements:
  • Appropriate model design aligned with bank-specific risk objectives.
  • Accurate and complete input data.
  • Sound methodological assumptions (distribution, time horizon, correlation).
• Common Conceptual Pitfalls:
  • Rigid reliance on historical returns that ignore changing portfolio compositions.
  • Ignoring nonlinear instruments like derivatives.
  • Missing data for illiquid assets or rarely traded securities.
• Pseudo-Historical Returns:
  • Recalculated returns reflecting current portfolio composition.
  • Example: Portfolio includes a new CDS overlay—VaR should adjust accordingly.

• Failures in Conceptual Soundness
  • VaR that doesn’t reflect dynamic risk exposure fails conceptual soundness.
  • Real-world scenario
    • A bank implements a hedging strategy → model must reflect reduced risk.
  • Key Test:
    • Does the VaR estimate decrease when portfolio risk actually decreases?
• Model assumptions should be stress-tested and reviewed regularly.

Topic 3. Sensitivity Analysis for VaR

• Sensitivity analysis checks how small changes in inputs (weights, prices, volatility) affect VaR.
• Step-by-Step:
  • Identify Key Inputs – e.g., equity weights, interest rate exposures.
  • Adjust Inputs One at a Time – change 5%, 10%, etc.
  • Recalculate VaR – observe the change.
• Marginal VaR:
  • Measures the impact of a single asset on portfolio VaR.
  • Estimated via regression slope: ΔPortfolio VaR vs. ΔComponent VaR.
• Sensitivity ≠ linear: higher weights can amplify or dampen risk depending on correlation with rest of portfolio.

Practice Questions: Q2

Q2. Which of the following actions is least likely a benefit of sensitivity analysis?
A. Model validation.
B. Regulatory compliance.

C. Improved decision-making.
D. Reducing regulatory capital.

Practice Questions: Q2 Answer

Explanation:  D is correct.

The benefits of sensitivity analysis include model validation, risk assessment, regulatory compliance, and improved decision-making. Regulatory capital depends on the level of VaR, which in turn depends on the risk exposures of the portfolio—and not on sensitivity analysis, which validates the VaR model.

Topic 4. Benefits and Challenges of Sensitivity Analysis

• Benefits:
  • Identifies which positions are most influential in risk exposure.
  • Useful for capital optimization and risk budgeting.
  • Builds confidence with internal stakeholders and regulators.
• Challenges:
  • Data Gaps: Missing data → unreliable sensitivity output.
  • Use of proxies (e.g., index returns instead of security-level returns) may distort output.
  • Complex portfolios require simulation-based approaches.

Practice Questions: Q3

Q3. Which of the following findings is incorrect regarding the empirical analysis of VaR confidence intervals?
A. Confidence intervals are not symmetric.
B. Larger datasets lead to tighter confidence intervals.
C. Order statistics produces tighter confidence intervals for VaR compared to bootstrap techniques.
D. GARCH VaR tends to produce much tighter confidence intervals compared to historical simulation VaR.

Practice Questions: Q3 Answer

Explanation: C is correct.

Confidence intervals are not symmetric. Using more data leads to tighter
confidence intervals. GARCH VaR also tends to produce tighter confidence
intervals compared to historical simulation VaR. Regarding order statistics and bootstap techniques, neither approach produces a tighter confidence interval.

Topic 5.  Confidence Intervals for VaR

• Used to quantify uncertainty in VaR estimation.
• Why rarely used?
  • Complex statistical modeling.
  • Often ignored due to computational limitations.
• Key Inputs for Confidence Intervals:
  • Return distribution (often wrongly assumed as normal).
  • Variance of pseudo returns.
  • Time-series dynamics (e.g., GARCH).
• Example: 1-day 99% VaR = $5M ± $1M → true VaR lies between $4M and $6M with 99% confidence.

Practice Questions: Q4

Q4. Which of the following statements regarding benchmarking VaR models is most accurate?
A. Benchmarking VaR models is not used for validating their performance.
B. Banks routinely benchmark their VaR model against several competing models.
C. In the statistical backtesting of VaR models, the errors are independently, but not identically, distributed.
D. Benchmarking is usually conducted for only a short time period during a bank’s transition to a new model.

Practice Questions: Q4 Answer

Explanation: D is correct.

Benchmarking is usually done for a short time period when the bank is planning on transitioning to a new model. Benchmarking VaR models is crucial for validating their performance and ensuring that they provide accurate risk assessments. Because trading portfolios change frequently, the errors in formal statistical backtesting used to conduct benchmark tests are not independently and identically distributed, especially for regression-based results. In practice, banks rarely conduct benchmarking on an ongoing basis because of the time and resources needed to develop another VaR model to benchmark against.

Topic 6. Challenges in Estimating Confidence Intervals

• Data Issues:
  • Incomplete or biased datasets lead to skewed results.
• Modeling Assumptions:
  • Financial returns are often non-normal (fat tails, skewness).
  • Assuming normality underestimates tail risk.
• Bootstrap and Order Statistics:
  • Bootstrap: Resample historical returns with replacement.
  • Order Stats: Use ranked data to find quantiles.

Topic 7. Benchmarking VaR Models

• Benchmarking = Comparing a bank’s VaR with an alternative (e.g., GARCH-based VaR).
• Most useful when transitioning between models.
• Why difficult?
  • Banks rarely maintain multiple models.
  • High cost and effort to develop and maintain a benchmark.
• Simplistic Methods (e.g., line plot comparison) show limited insight.
• Robust Method: Statistical backtesting (e.g., coverage tests, loss function comparisons).

Topic 9. Backtesting for Benchmarking

• Backtesting Positional VaR vs. Realized P&L (GARCH VaR):
  • Positional VaR often more conservative (due to regulatory capital incentives).
  • GARCH P&L VaR tends to be more responsive and accurate.
• Empirical Findings:
  • Positional VaR leads to higher capital requirements.
  • GARCH-based backtests show better alignment with actual losses.
• Use backtesting results to:
  • Improve VaR estimates.
  • Identify model weaknesses.
  • Communicate with regulators.

Topic 10.  Comparison of Validation Tools

Comparison of Validation Tools

Use all tools in combination for robust VaR model validation.