Book 1. Market Risk

FRM Part 2

MR 2. Non-Parametric Approaches

Presented by: Sudhanshu

Module 1. NonParametric Approaches

Module 1. NonParametric Approaches

Topic 1.  Bootstrap Historical Simulation Approach

Topic 2. Applying Nonparametric Estimation

Topic 3. Weighted Historical Simulation Approaches

Topic 4. Age-Weighted Historical Simulation

Topic 5. Volatility-Weighted Historical Simulation

Topic 6. Correlation-Weighted Historical Simulation

Topic 7. Filtered Historical Simulation

Topic 8. Advantages and Disadvantages of Nonparametric Methods

Topic 1. Bootstrap Historical Simulation Approach

  • Concept: A simple and intuitive estimation procedure.
  • Process:
    • Draw a sample from the original dataset.
    • Record the VaR from that sample.
    • "Return" the data to the dataset (sampling with replacement).
    • Repeat this procedure multiple times to record various sample VaRs.
  • Best Estimate: The average of all sample VaRs is considered the best VaR estimate from the full dataset.
  • Expected Shortfall (ES): The same procedure can be applied to estimate ES, with the best ES estimate being the average of all sample ES values.
  • Advantage: Bootstrapping consistently provides more precise estimates of coherent risk measures compared to historical simulation on raw data.

Practice Questions: Q1

Q1. Johanna Roberto has collected a data set of 1,000 daily observations on equity returns. She is concerned about the appropriateness of using parametric techniques as the data appears skewed. Ultimately, she decides to use historical simulation and bootstrapping to estimate the 5% VaR. Which of the following steps is most likely to be part of the estimation procedure?

A. Filter the data to remove the obvious outliers.

B. Repeated sampling with replacement.

C. Identify the tail region from reordering the original data.

D. Apply a weighting procedure to reduce the impact of older data.

Practice Questions: Q1 Answer

Explanation:  B is correct.

Bootstrapping from historical simulation involves repeated sampling with replacement. The 5% VaR is recorded from each sample draw. The average of the VaRs from all the draws is the VaR estimate. The bootstrapping procedure does not involve filtering the data or weighting observations. Note that the VaR from the original data set is not used in the analysis.

Topic 2. Applying Non-Parametric Estimation

  • Drawback of Traditional Historical Simulation: The discrete nature of data limits VaR estimation to specific confidence levels (e.g., with 100 observations, only 100 different confidence levels are possible).
  • Non-Parametric Density Estimation Advantage:
    • Avoids restrictive assumptions about the underlying distribution.
    • Uses existing data points to "smooth" the data, enabling VaR calculation at all confidence levels.
  • Simplest Adjustment: Connect the midpoints between successive histogram bars to create a "surrogate density function."
    • This process displaces area between bars, maintaining a valid probability distribution function with a modified shape.
    • Allows for confidence intervals to be utilized regardless of data set size.

Topic 3. Weighted Historical Simulation Approaches

  • Limitation of Traditional HS: Assumes equal weight for all observations within a cutoff period and zero weight otherwise, leading to "ghost effects" and ignoring seasonality.
  • Improvements: This reading identifies four improved methods to the traditional historical simulation.

Practice Questions: Q2

Q2.  All of the following approaches improve the traditional historical simulation approach for estimating VaR except the:

A. volatility-weighted historical simulation.

B. age-weighted historical simulation.

C. market-weighted historical simulation.

D. correlation-weighted historical simulation.

Practice Questions: Q2 Answer

Explanation:  C is correct.

Market-weighted historical simulation is not discussed in this reading. Age- weighted historical simulation weights observations higher when they appear closer to the event date. Volatility-weighted historical simulation adjusts for changing volatility levels in the data. Correlation-weighted historical simulation incorporates anticipated changes in correlation between assets in the portfolio.

Topic 4. Age-Weighted Historical Simulation

  • Concept: Assigns more weight to recent observations and less to older ones.
  • Weighting Scheme (Boudoukh, Richardson, and Whitelaw):
    • Weight for an observation i days old: ​ 
    • λ (decay parameter): 0≤λ≤1. Values close to 1 indicate slow decay.
  • Implication: Reduces the impact of "ghost effects" and older, potentially irrelevant events.
  • Relationship to Traditional HS: Traditional historical simulation is a special case where λ=1 (no decay) over the estimation window.
  • Also known as: The hybrid approach.
w(i)=\frac{\lambda^{i-1}(1-\lambda)}{1-\lambda^n}

Practice Questions: Q3

Q3. Which of the following statements about age-weighting is most accurate?

A. The age-weighting procedure incorporates estimates from GARCH models.

B. If the decay factor in the model is close to 1 , there is persistence within the data set.

C. When using this approach, the weight assigned on day i is equal to

D. The number of observations should at least exceed 250 .

w(i)=\lambda^{i-1} \times(1-\lambda) /\left(1-\lambda^i\right).

Practice Questions: Q3 Answer

Explanation: B is correct.

If the intensity parameter (i.e., decay factor) is close to 1, there will be persistence (i.e., slow decay) in the estimate. The expression for the weight on day i has i in the exponent when it should be n. While a large sample size is generally preferred, some of the data may no longer be representative in a large sample. 

Topic 5. Volatility-Weighted Historical Simulation

  • Concept: Weights individual observations by volatility rather than proximity to the current date.
  • Purpose: Incorporates changing volatility into risk estimation.
    • Prevents underestimation of current risk if recent volatility has increased.
    • Prevents overstatement of current risk if current volatility is markedly reduced compared to older data.
  • Adjustment Formula: Daily returns (rt,i​) are adjusted based on current volatility forecast (σT,i​) relative to historical volatility forecast (σt,i​).

 

  • VaR Calculation: VaR, ES, and other coherent risk measures are calculated as usual after substituting historical returns with volatility-adjusted returns.
  • Advantages:
    • Explicitly incorporates volatility into estimation.
    • Near-term VaR estimates are more sensible in light of current market conditions.
    • Allows for VaR estimates higher than those from raw historical data.
r^{*}_{t,i}=\left(\frac{\sigma_{T,i}}{\sigma_{t,i}}\right)r_{t,i}

Practice Questions: Q4

Q4. Which of the following statements about volatility-weighting is true?

A. Historic returns are adjusted, and the VaR calculation is more complicated.

B. Historic returns are adjusted, and the VaR calculation procedure is the same.

C. Current period returns are adjusted, and the VaR calculation is more complicated.

D. Current period returns are adjusted, and the VaR calculation is the same.

Practice Questions: Q4 Answer

Explanation: B is correct.

The volatility-weighting method adjusts historic returns for current volatility.
Specifically, return at time t is multiplied by (current volatility estimate /
volatility estimate at time t). However, the actual procedure for calculating VaR using a historical simulation method is unchanged; it is only the inputted data that changes.

Topic 6. Correlation-Weighted Historical Simulation

  • Concept: Incorporates updated correlations between asset pairs.
  • Process: Adjusts the historical correlation (or variance-covariance) matrix to the new information environment.
    • This involves "multiplying" historic returns by the revised correlation matrix to get correlation-adjusted returns.
  • Variance-Covariance Matrix (Σ):
    • Diagonal elements: Updated variances (covariance of asset return with itself).
    • Off-diagonal elements: Current covariance between asset pairs.
  • Advantage: This method is a richer analytical tool than volatility-weighted simulation as it accounts for both updated variances (volatilities) and covariances (correlations).

Topic 7. Filtered Historical Simulation

Complexity: Most comprehensive and complicated of the nonparametric estimators.

  • Combination: Combines the historical simulation model with conditional volatility models (e.g., GARCH, asymmetric GARCH).
  • Features: Captures conditional volatility, volatility clustering, and potential asymmetric effects on volatility (surprise factors).
  • Process:
    • Forecasts volatility for each day in the sample period.
    • Standardizes volatility by dividing by realized returns.
    • Uses bootstrapping to simulate returns that incorporate the current volatility level.
    • Identifies VaR from the simulated distribution.
  • Benefits:
    • Results are sensitive to changing market conditions.
    • Can predict losses outside the historical range.
    • Computationally reasonable even for large portfolios.
    • Empirical evidence supports its predictive ability.

Topic 8. Advantages and Disadvantages of Nonparametric Methods

Advantages of Nonparametric Estimation Methods

  • Intuitive and Computationally Simple: Often easy to implement, even on a spreadsheet.
  • No Restrictive Parametric Assumptions: Not hindered by violations like skewness or fat tails.
  • Avoids Complex Matrices: Does not require complex variance-covariance matrices or deal with dimension problems.
  • Readily Available Data: Data is often available and typically doesn't require adjustments.
  • Flexibility for Complex Analysis: Can accommodate more complex analyses (e.g., combining age-weighting with volatility-weighting).
  • Ability to accommodate skewed data.​

  • Disadvantages of Nonparametric Estimation Methods

    • Critical Dependence on Historical Data: Analysis relies heavily on past data.
    • Sensitivity to Data Periods:
      • Volatile data periods can lead to VaR and ES estimates that are too high.
      • Quiet data periods can lead to VaR and ES estimates that are too low.
    • Difficulty Detecting Structural Shifts: Challenging to identify regime changes in the data.
    • Inability to Accommodate Unseen Events: Cannot account for plausible large-impact events if they haven't occurred in the sample period.
    • Limited Loss Estimation: Difficult to estimate losses significantly larger than the maximum loss observed in the data set (though volatility-weighting offers some improvement).
    • Data Requirements: Needs sufficient data, which might not be available for new instruments or markets.

Practice Questions: Q5

Q5. All of the following items are generally considered advantages of nonparametric estimation methods except:

A. ability to accommodate skewed data.

B. availability of data.

C. use of historical data.

D. little or no reliance on covariance matrices.

Practice Questions: Q5 Answer

Explanation: C is correct.

The use of historical data in nonparametric analysis is a disadvantage, not an
advantage. If the estimation period was quiet (volatile) then the estimated risk measures may understate (overstate) the current risk level. Generally, the largest VaR cannot exceed the largest loss in the historical period. On the other hand, the remaining choices are all considered advantages of nonparametric methods. For instance, the nonparametric nature of the analysis can accommodate skewed data, data points are readily available, and there is no requirement for estimates of covariance matrices.