Financial Risk Tower Crumbling

Unlocking Financial Stability: Why Traditional Risk Measures Fall Short

Lena Kashyap in Business & Economy December 2025 • 4 min read.

"A critical look at comonotonic additive risk measures and their limitations in modern financial scenarios"

In the complex world of finance, accurately measuring risk is essential. Risk measures are used every day by institutions and investors alike to understand potential losses and fluctuations. Regulators also use them to determine the necessary capital reserves for banks, investment firms, and insurance companies, acting as a buffer against financial instability. The goal is to translate those intuitive, almost gut feeling to well-defined mathematical axioms.

One approach is comonotonic additive risk measures, which have become popular due to their simplicity and appealing properties. These measures assume that if assets move in the same direction, they don't hedge each other, and the risk of the sum should be the sum of the risks. Sounds intuitive, right? However, the financial landscape is rarely that straightforward, recent studies showing that this is not always the case.

This article explores the limitations of comonotonic additive risk measures, drawing from academic research to show where these methods fall short. We'll examine scenarios where they prove incompatible with key financial principles, potentially leading to inaccurate risk assessments and flawed decision-making. Understanding these limitations is crucial for anyone involved in financial regulation, investment, or risk management.

The False Security of Comonotonic Additivity

Comonotonic additive risk measures operate on the principle that if two assets always move in the same direction (i.e., they are comonotonic), they offer no hedging benefit. This leads to the assumption that the risk of a combined position is simply the sum of individual risks. However, this assumption breaks down in several important contexts.

One major issue arises when dealing with limited liability. In many real-world financial situations, firms are only liable for the amount they have invested. This means that any surplus beyond that point is irrelevant from a regulatory perspective. However, comonotonic additive risk measures often fail to account for this, leading to an overestimation of the capital required to cover potential losses.

Surplus Invariance: Regulatory capital should focus on potential losses, not the size and probability of surpluses.
Excess Invariance: Risk measures should be insensitive to gains, focusing solely on the negative aspects of a position.

Studies have demonstrated that comonotonic additive risk measures satisfying surplus invariance are necessarily quantiles. This significantly restricts the types of risk measures that can be used, potentially excluding more sophisticated measures that better capture tail risk.

Beyond the Limitations: A More Realistic Approach

While comonotonic additive risk measures offer a simplified view of risk, they often fail to capture the complexities of modern financial markets. By understanding these limitations, regulators and financial professionals can move towards more robust and accurate risk assessment methods. This may involve incorporating factors like limited liability, market dynamics, and the potential for diversification, ultimately leading to greater financial stability and more informed decision-making.

About this Article -

This article was crafted using a human-AI hybrid and collaborative approach. AI assisted our team with initial drafting, research insights, identifying key questions, and image generation. Our human editors guided topic selection, defined the angle, structured the content, ensured factual accuracy and relevance, refined the tone, and conducted thorough editing to deliver helpful, high-quality information.See our About page for more information.

This article is based on research published under:

DOI-LINK: https://doi.org/10.48550/arXiv.2212.13864,

Title: The Limitations Of Comonotonic Additive Risk Measures: A Literature Review

Subject: q-fin.rm

Authors: Samuel Solgon Santos, Marcelo Brutti Righi, Eduardo De Oliveira Horta

Published: 28-12-2022

Everything You Need To Know

What are comonotonic additive risk measures and why are they used?

Comonotonic additive risk measures are a type of risk assessment method used in finance, favored for their simplicity. They operate on the principle that if assets move in the same direction, they offer no hedging benefit, thus the risk of a combined position is simply the sum of individual risks. These measures have become popular in finance and regulation because they are intuitive and easy to implement, translating what might be a gut feeling into well-defined mathematical axioms.

What are the limitations of using comonotonic additive risk measures in finance?

Comonotonic additive risk measures have limitations due to their underlying assumptions. A key issue is the failure to account for limited liability, where firms are only liable for their investment. These measures often overestimate the capital needed to cover potential losses by not recognizing that surplus beyond the invested amount is irrelevant from a regulatory perspective. These measures don't reflect market dynamics or potential diversification benefits.

What is 'surplus invariance' and why is it important in risk measurement?

Surplus invariance refers to the idea that regulatory capital should focus on potential losses and not the size and probability of surpluses. Surplus invariance is important because the goal of risk measurement is to determine the capital reserves needed to guard against financial instability. If a risk measure is sensitive to the potential for gains (surplus), it may not accurately reflect the true risk exposure of an entity.

How does the concept of limited liability affect the accuracy of comonotonic additive risk measures?

Limited liability significantly impacts the accuracy of comonotonic additive risk measures. Limited liability means a firm is only responsible for the amount invested, rendering any surplus irrelevant from a regulatory standpoint. Since comonotonic additive risk measures often fail to account for limited liability, they overestimate the required capital reserves, as they treat all gains as relevant even though the firm's liability is capped at the investment amount. This overestimation can lead to inefficient allocation of capital.

If comonotonic additive risk measures have limitations, what are some alternative or complementary approaches to risk assessment that could provide a more realistic view?

Given the limitations of comonotonic additive risk measures, regulators and financial professionals can adopt more robust methods. These include incorporating factors like limited liability, market dynamics, and the potential for diversification, to enhance the accuracy of risk assessment. The incorporation of market dynamics can capture potential hedging benefits, while accounting for diversification can reduce the overestimation of risk that occurs when simply adding individual risks. In addition, measures satisfying surplus invariance are necessarily quantiles, this might exclude more sophisticated measures that better capture tail risk.