# Conditional Value At Risk

Updated on April 18, 2024
Article byPrakhar Gajendrakar
Edited byRaisa Ali
Reviewed byDheeraj Vaidya, CFA, FRM

## What Is Conditional Value At Risk (CVaR)?

Conditional value at risk (CVaR) defines the expected potential risk associated with an investment in case of a worst-case scenario or extreme market conditions. It is used as a risk-measuring tool and plays a key role in portfolio optimization and risk management.

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The CVaR is typically expressed in percentage, quantifying the scale of expected potential loss in a specific period using the weighted mean of the extreme losses in the distribution tail of possible returns beyond the value at the risk cutoff point. It is a tail risk metric, also referred to as expected shortfall in financial markets.

### Key Takeaways

• Conditional value at risk (CVaR) determines the tail risk of an investment in case of extreme market conditions.
• The tool was developed by American mathematicians Rockafeller and Uryasev in 2000.
•  It is an extension of the value of risk. In other words, it is derived from VaR to give a detailed view by stressing over the average losses in the worst circumstances.
• CVaR helps investors and market analysts in derivatives pricing and hedging, portfolio optimization and asset and risk management.

### Conditional Value At Risk Explained

The conditional value at risk (CVaR ) is a technique to derive the expected loss for a portfolio or individual investment in case the market declines to its worst scenario. CVaR was introduced by Rockafellar and Uryasev in 2000 as a risk management tool. The key components of the conditional value at risk are a probability distribution of losses, which represents the likelihood of different losses linked to an investment, the certainty level in CVaR calculations to describe the worst possible outcomes, along an expected shortfall, which defines the average loss in extreme market circumstances.

The conditional value at risk states the determination of CVaR in three parts. The first is the Monte Carlo simulation applied when dealing with large portfolios and complex distributions simulating multiple potential market scenarios. The second is discrete distribution when the probability distribution is discrete. CvaR is computed as the weighted average of losses. Lastly, the continuous distribution, when the probability distribution is continuous, and CVaR is represented as the integral of losses multiplied by the corresponding probabilities.

The conditional value at risk optimization is based on the interpretation of CVaR, which is done by comparing the CVaR values of different assets and by measuring risk by estimating the expected shortfall in the market’s extreme circumstances. Investors can make or change their investment and imply risk management techniques depending on the interpretation.

Although it has many applications, it is still not accepted universally by all the financial markets. Yet it is used for advanced calculations, especially the conditional value at risk using Python, which helps in risk budgeting, stress testing, monitoring, and reporting. By optimizing the portfolio, an investor reduces the risk exposure and likelihood of facing losses.

### Formula

The basic formula for conditional value at risk (CVaR) is as follows:

### Examples

Below are two examples of conditional value at risk –

#### Example #1

Consider an investor with a portfolio of diverse assets. Portfolio optimization to achieve the best possible return with controlled risk is important to the investor. The investor sets a confidence level to control risk. If the confidence level set is n%, the investor should be managing the portfolio to minimize the expected loss for the remaining (100-n)% of circumstances with the highest potential for unfavorable outcomes. It’s a risk management approach that considers not just the most possible scenarios but also takes into account the tail risks or rare but severe events.

#### Example #2

Consider the application of CVaR in a factory. Delays in raw material delivery can cause unfavorable outcomes. Hence, the operations manager evaluates potential delays and associated costs. If the CVaR of raw material delivery delays at a certain confidence level is 3 days, it means that in the worst cases, the average delay is 3 days. This allows the factory management to plan an effective approach to managing potential extended supply chain delays.

The advantages and disadvantages of conditional value at risk are the following:

• Provides better knowledge for potential loss compared to VaR.
• Instead of a single estimation point, CVaR gauges potential expected loss.
• Conditional value at risk optimization is suitable for portfolios and risk management due to its coherent risk measure property.
• Focuses on tail risk and accounts for average loss in extreme market conditions.

• The tool is not universally accepted, although it has its popularity among risk managers.
• Requires advanced calculation skills for deducing average losses.
• Its accuracy is based on the assumption of the underlying distribution of losses.
• It has complex computational issues, especially when applied to large portfolios.

### Conditional Value At Risk vs Value At Risk

The core distinguishing aspects between conditional value at risk and value at risk are-

• Conditional value at risk reflects the expected loss when the worst-case limit is crossed. Still, value at risk is simply the loss in extreme scenarios with a probability and time horizon.
• CVaR is the extension of value at risk; therefore, it can only be calculated from it, but VaR cannot be derived from the other way round.
• Conditional value at risk is more reliable and detailed than the value at risk as it accounts for the average losses and goes beyond the VaR threshold.

### Frequently Asked Questions (FAQs)

1. What is the importance of CVaR?

Conditional Value at Risk (CVaR) is a useful tool in risk management for quantifying loss, which is a usual scenario in rare scenarios. It provides a wider view than other risk measure tools, considering the tail risk and offering insights into potential losses beyond a specific cutoff point.

2. Why do we use CVaR?

CVaR is a common risk-measuring tool in the investment management domain, particularly in portfolio optimization. It is useful as an extension of VaR by assessing the average loss in extreme conditions, providing a more detailed understanding of potential risks.

3. What are the assumptions of CVaR?

The assumptions of CVaR include a probability distribution of losses, certainty level in calculations, and an expected tail loss representing average losses in extreme market circumstances. It also assumes that CVaR under specific scenarios approximately equals the average of some percentage of the worst-case loss circumstances.

This article has been a guide to what is Conditional Value At Risk (CVaR). Here, we explain its formula, examples, comparison with value at risk, & advantages. You may also find some useful articles here –