Lecture 25: Tail Risk Measurement: VaR and CVaR (or ES) models

IIT KANPUR-NPTEL

6 chapters7 takeaways14 key terms5 questions

Overview

This video introduces and explains Value at Risk (VaR) and Conditional Value at Risk (CVaR), also known as Expected Shortfall (ES), as measures for assessing tail risk in financial portfolios. It details the theoretical underpinnings, mathematical formulations, and practical computation methods for both VaR and CVaR, including how to handle empirical data and probability distributions. The discussion highlights the limitations of VaR, particularly its inability to capture extreme tail losses, and positions CVaR as a more robust measure for understanding potential losses when extreme events occur.

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Chapters

VaR estimates the maximum potential loss over a specific time horizon (T) at a given confidence level (X%).
It answers the question: 'What is the maximum loss we can expect with X% certainty?'
VaR can be interpreted based on either a gain distribution (looking at the left tail for losses) or a loss distribution (looking at the right tail for losses).
The confidence level (X%) determines the percentile used for calculation (e.g., 95% confidence means looking at the 5th percentile for losses on a gain distribution).

Understanding VaR is crucial for financial institutions to quantify potential downside risk and set risk management limits.

Being 95% certain that losses will not exceed $10 million over a 5-day period.

VaR is formally defined as a percentile or quantile of the loss distribution.
It can be calculated using standardized distributions (like the normal distribution) or directly from empirical data.
When using empirical data, VaR is found by ordering historical returns and selecting the return at the specified percentile.
The choice of time horizon (T) and confidence level (X) are key inputs for VaR calculation.

This section provides the technical basis for calculating VaR, enabling its practical application using either theoretical models or historical data.

Given 1001 daily return observations, the 99th percentile VaR can be found by identifying the 990th ordered observation (assuming data is ranked from highest to lowest return).

When assuming a probability distribution (e.g., normal distribution), returns are first standardized into Z-scores.
The Z-score corresponding to the desired confidence level's tail (e.g., 1% for 99% confidence) is identified.
This Z-score is then converted back to an actual return value, which represents the VaR in percentage terms.
The VaR in monetary terms is calculated by multiplying the percentage VaR by the portfolio's value.

Using probability distributions allows for a more systematic and model-driven approach to VaR estimation, especially when dealing with continuous data.

For a portfolio with a daily mean return of 0.05% and a daily standard deviation of 3.16%, the 1-day 99% VaR is -7.3% (or a loss of $73 on a $1000 portfolio), using a Z-score of -2.326.

VaR does not differentiate between portfolios that have the same maximum loss at a given confidence level but vastly different potential losses beyond that level.
It fails to capture the severity of losses in the extreme tail of the distribution (beyond the confidence level).
Fund managers might be incentivized to take on extreme tail risks if these risks are not captured by VaR, especially if there are potential upside gains.
Conditional Value at Risk (CVaR), or Expected Shortfall (ES), is introduced to address these limitations by focusing on the expected loss *given* that the VaR level is breached.

Recognizing VaR's shortcomings is essential for understanding why more advanced risk measures like CVaR are necessary for comprehensive risk management.

Two portfolios might have the same 95% VaR, but one could have a much larger potential loss if a rare event occurs, a risk not fully captured by VaR.

CVaR is the expected loss given that the loss exceeds the VaR level.
Mathematically, for a continuous distribution, it involves integrating the loss variable from the VaR level to infinity, weighted by the probability density function.
For discrete data, CVaR is the average of all losses that are greater than the calculated VaR.
CVaR calculation requires first determining the VaR level.

This section provides the mathematical definition of CVaR, allowing for its calculation and comparison with VaR.

If the 99% VaR for a 10-day period is $10 million, the CVaR is the average of all losses that are expected to exceed $10 million over that period.

When using empirical data, identify the VaR threshold (e.g., the 990th observation for 99% VaR).
CVaR is then calculated as the average of all observations that fall beyond this threshold (i.e., represent greater losses).
If probabilities are given (probability mass function), CVaR is the sum of probabilities of tail events multiplied by their respective losses, after standardizing the tail probabilities so they sum to 1.
This process ensures that CVaR accounts for the magnitude of losses in the extreme tail.

This explains how to practically compute CVaR from real-world data, making it a tangible risk management tool.

Given a 95% VaR of $1 million, if there are subsequent losses of $2 million (3% probability), $5 million (1% probability), $10 million (0.75% probability), and $20 million (0.25% probability), the CVaR is the weighted average of these losses after standardizing their probabilities to sum to 5%.

Key takeaways

1Value at Risk (VaR) quantifies the maximum potential loss at a specific confidence level over a given period.
2VaR calculations can be performed using theoretical distributions or empirical historical data.
3The interpretation of VaR depends on whether one is analyzing gain or loss distributions.
4VaR's primary limitation is its inability to measure the severity of losses beyond the specified confidence level (tail risk).
5Conditional Value at Risk (CVaR) or Expected Shortfall (ES) addresses VaR's limitations by measuring the average loss given that the VaR threshold has been breached.
6CVaR provides a more comprehensive view of extreme risk by considering the magnitude of losses in the tail of the distribution.
7Both VaR and CVaR are essential tools for financial risk management, with CVaR offering a more robust assessment of extreme downside potential.

Key terms

Value at Risk (VaR)Conditional Value at Risk (CVaR)Expected Shortfall (ES)Tail RiskConfidence LevelTime HorizonPercentileQuantileProbability DistributionEmpirical DataStandard Normal DistributionZ-scoreLoss DistributionGain Distribution

Test your understanding

1What is the primary difference in the information provided by VaR and CVaR regarding extreme losses?
2How does the confidence level (e.g., 95% vs. 99%) affect the calculation and interpretation of VaR?
3Explain how to calculate VaR from a set of historical daily returns.
4Why might a fund manager be incentivized to take on excessive tail risk if only VaR is used for performance evaluation?
5Describe the process of calculating CVaR using empirical data once the VaR threshold has been identified.