FRM Part 2 - Market Risk Measurement and Management

Chapter: Estimating Market Risk Measures

1952: Harry Markowitz introduced standard deviation as a measure of total risk.
William Sharpe extended this with the Capital Asset Pricing Model (CAPM), focusing on covariance and correlation.
Early models assumed normal distribution of returns.
Post-1987 stock market crash, financial risk managers began exploring other risk measures, leading to the popularity of VaR.

VaR estimates the risk of loss on an investment.
Focus primarily on the left tail of the distribution (worst-case scenarios).
Confidence Interval: If VaR at 95% confidence level indicates a 5% chance of losing more than a specified amount.

Historical Simulation:
- Construct a distribution of losses based on historical data.
- Identify key factors relevant to the asset class (e.g., yields, duration for bonds).
- Order losses to determine the critical value at the desired confidence level.

For 1000 observations at 95% confidence:
- Identify the 51st observation among ordered losses to find VaR.

Assumes returns follow a normal distribution:
- Calculate VaR using mean, standard deviation, and critical value from Z-table.
Example: If mean = 12 million, standard deviation = 24 million,
- VaR at 95% = - (Mean) + (1.645 * Standard Deviation) = -27.5 million.

VaR does not provide insight into the tail distribution (potential for larger losses).
Actual losses could exceed VaR significantly, leading to underestimation of risk.

Averages the losses beyond the VaR threshold, giving a clearer picture of tail risk.
Calculation:
- Divide the tail into slices and compute the average value at risk for those slices.
Provides a better estimate of risk in extreme scenarios compared to VaR.

For a risk measure to be coherent, it must meet four conditions:
1. Sub-additivity: Risk of combined portfolios should not exceed the sum of individual risks.
2. Homogeneity: Doubling the portfolio should double the risk.
3. Monotonicity: If one portfolio always outperforms another, it should have lower risk.
4. Translation Invariance: Adding cash should decrease risk by the same amount.
VaR fails the sub-additivity property; Expected Shortfall meets all criteria and is therefore coherent.

Coherent risk measures can be estimated by manipulating average VaR with different weighting methods.
Weighting by risk aversion allows for a more tailored risk assessment.
Example: Divide the distribution into equal probability slices, applying different weights based on investor risk preferences.

Importance of measuring the precision of estimates is emphasized.
Use of standard error to construct confidence intervals to understand potential estimation error in risk measures.

Tool for assessing if a dataset plausibly comes from a theoretical distribution.
A QQ (Quantile-Quantile) plot compares two datasets visually to determine if they originate from the same distribution.
Discrepancies in tails can highlight extreme events and outliers.

VaR is a valuable tool in risk management but has limitations.
Expected Shortfall provides a more comprehensive view of tail risk.
Coherent risk measures are essential for accurate risk assessments, and quantiles can enhance understanding of risk distributions.