Coherent risk measures

Coherent risk measures represent a class of mathematical tools used within risk management to quantify financial risk in a consistent and theoretically sound manner. These measures adhere to a set of desirable properties, or axioms, that ensure they behave logically in financial contexts, particularly for assessing potential losses in investments and portfolios. They are a fundamental concept in portfolio theory, guiding how financial institutions measure and allocate capital requirements against various forms of financial exposure.

History and Origin

The concept of coherent risk measures was formally introduced in a seminal 1999 paper titled "Coherent Measures of Risk" by Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath³. Their work provided an axiomatic framework for defining what constitutes a "good" risk measure, addressing shortcomings found in earlier methods. Prior to this, various ad-hoc measures were in use, but the Artzner et al. paper laid down a rigorous mathematical foundation that revolutionized the field of risk measurement in quantitative finance. This framework allowed for a systematic evaluation and comparison of different risk quantification techniques.

Key Takeaways

Coherent risk measures satisfy four fundamental axioms: monotonicity, translation invariance, positive homogeneity, and subadditivity.
These axioms ensure that the measure behaves logically and promotes risk diversification.
Expected Shortfall (ES) is a prominent example of a coherent risk measure, widely adopted in finance.
Coherent risk measures are crucial for setting appropriate regulatory capital and for internal risk management within financial institutions.
While theoretically robust, practical challenges and critiques, particularly regarding the positive homogeneity axiom, exist.

Formula and Calculation

A risk measure (\rho) is considered coherent if it satisfies the following four axioms for any financial positions (random variables representing losses) (X) and (Y), and for any constant (c > 0):

Monotonicity: If (X \le Y), then (\rho(X) \le \rho(Y)).
- Interpretation: If one position always results in losses no greater than another, its risk measure should be no greater.
Translation Invariance: For a risk-free amount (c), (\rho(X + c) = \rho(X) + c).
- Interpretation: Adding a certain amount of cash to a position reduces the risk by that same amount.
Positive Homogeneity: For (c \ge 0), (\rho(cX) = c\rho(X)).
- Interpretation: Scaling up a position by a factor (c) scales its risk by the same factor. This implies that the risk of a portfolio is proportional to its size.
Subadditivity: (\rho(X + Y) \le \rho(X) + \rho(Y)).
- Interpretation: The risk of a combined portfolio is less than or equal to the sum of the risks of its individual components. This axiom mathematically captures the benefit of diversification.

The exact calculation of a coherent risk measure depends on the specific measure being used. For instance, Expected Shortfall (ES), a common coherent risk measure, is typically calculated as the expected loss given that the loss exceeds a certain percentile (like the 95th or 99th percentile).

Interpreting the Coherent Risk Measures

Coherent risk measures provide a unified framework for understanding and comparing risk across different assets or portfolios. Their interpretation stems directly from the axioms they satisfy. For example, the subadditivity axiom means that if two portfolios are combined, their total risk will not be greater than the sum of their individual risks. This property is vital for promoting sound portfolio construction and encourages diversification. When a financial institution calculates its capital requirements using a coherent measure, it inherently accounts for the benefits of combining different exposures. Such measures aim to give a "worst-case" yet realistic assessment of potential losses that an entity might face.

Hypothetical Example

Consider a hypothetical investment firm with two independent trading desks, Desk A and Desk B.

Desk A's Losses (X): Could be $10 million with 1% probability, otherwise $0.
Desk B's Losses (Y): Could be $15 million with 1% probability, otherwise $0.

A non-coherent measure like Value at Risk (VaR) at 99% could incorrectly assess combined risk. If the rare loss events for Desk A and Desk B happen independently, their combined VaR might appear higher than a coherent measure would suggest.

However, a coherent risk measure, such as Expected Shortfall (ES), would apply its subadditivity property. Let's assume the combined portfolio (X+Y) could result in a total loss of $25 million if both unlikely events occur simultaneously (e.g., if their losses are correlated) but might be lower if they are not. An ES calculation would evaluate the expected loss beyond a very high percentile (e.g., 99.5%) of the combined loss distribution. If the ES of X is $1 million and the ES of Y is $1.5 million, the ES of the combined portfolio (X+Y) would be less than or equal to $2.5 million, reflecting the benefits of potential diversification, even if losses are extreme. This demonstrates how coherent risk measures encourage a unified approach to market risk assessment.

Practical Applications

Coherent risk measures are widely applied in the financial industry for various purposes:

Regulatory Compliance: Many financial regulations, such as those prescribed by the Basel Committee on Banking Supervision (BCBS), guide financial institutions in determining their regulatory capital requirements. While VaR has historically been used, there's a growing shift towards coherent measures like Expected Shortfall due to their more desirable properties for assessing tail risk.
Internal Risk Management: Banks and investment firms use these measures for internal capital allocation, setting risk limits for trading desks, and evaluating the overall risk exposure of their portfolios, encompassing credit risk, market risk, and operational risk.
Portfolio Optimization: In academic research and advanced quantitative finance, coherent risk measures serve as objective functions in portfolio optimization problems, aiming to construct portfolios that minimize risk while achieving target returns.
Stress Testing and Scenario Analysis: They are used to quantify potential losses under extreme but plausible market conditions, helping firms understand their resilience to adverse events.

Limitations and Criticisms

Despite their theoretical appeal, coherent risk measures are not without limitations and criticisms. One significant point of contention often revolves around the positive homogeneity axiom. Critics argue that scaling up a position might, in reality, lead to a disproportionately higher risk due to illiquidity, market impact, or concentrated positions that exceed market capacity. This implies that the risk of doubling a portfolio might be more than twice the original risk, a scenario not captured by positive homogeneity².

Another challenge is the practical estimation of coherent risk measures, especially Expected Shortfall, for complex portfolios or illiquid assets where historical data may be sparse or unrepresentative. While Expected Shortfall is theoretically superior to VaR in terms of coherence, its estimation can be more sensitive to tail events and requires more robust statistical methods. Some academic work suggests that the very coherence, particularly positive homogeneity, can make these measures "ineffective" in curbing certain types of investor behavior, especially in markets with statistical arbitrage opportunities¹.

Coherent Risk Measures vs. Value at Risk

The distinction between coherent risk measures and Value at Risk (VaR) is central to modern risk management. While VaR is widely used, it is generally not a coherent risk measure because it often fails the subadditivity axiom. This means that the VaR of a combined portfolio could be greater than the sum of the VaRs of its individual components, which contradicts the principle of diversification and can disincentivize combining risks.

For example, two risky assets, each with a VaR of $1 million, might, when combined, have a VaR of $2.5 million if their extreme losses occur simultaneously (e.g., they are highly positively correlated and their individual VaRs are calculated assuming independence or a simpler correlation structure). This behavior of VaR can lead to a misrepresentation of true portfolio risk and potentially encourage an entity to break up its risks into smaller, seemingly less risky parts, rather than managing them holistically.

In contrast, coherent risk measures, by definition, satisfy subadditivity. This property ensures that combining risks always yields a total risk that is less than or equal to the sum of the individual risks, thereby correctly reflecting the benefits of diversification. Furthermore, coherent measures like Expected Shortfall provide a more comprehensive view of tail risk by averaging losses beyond a certain percentile, whereas VaR only indicates the maximum loss at a given confidence level.

FAQs

What makes a risk measure "coherent"?

A risk measure is considered "coherent" if it satisfies four specific mathematical properties: monotonicity, translation invariance, positive homogeneity, and subadditivity. These axioms ensure that the measure provides a consistent and logical assessment of risk, aligning with fundamental financial principles like diversification.

Is Value at Risk (VaR) a coherent risk measure?

Generally, no. Value at Risk (VaR) typically fails the subadditivity axiom. This means that the VaR of a portfolio might be greater than the sum of the VaRs of its individual assets, which does not accurately reflect the benefits of diversification.

What is an example of a coherent risk measure?

The most widely recognized example of a coherent risk measure is Expected Shortfall (ES), also known as Conditional Value at Risk (CVaR) or Tail VaR. ES quantifies the expected loss given that the loss exceeds a certain high percentile, providing a more comprehensive view of tail risk than VaR.

Why are coherent risk measures important for financial institutions?

Coherent risk measures are crucial for financial institutions because they provide a sound mathematical framework for assessing and managing risk. They help in setting appropriate regulatory capital requirements, guiding portfolio construction through effective risk management strategies, and promoting prudent risk-taking by accurately reflecting the benefits of diversification.