---
title: A note on subadditivity of value at risks (VaRs): A new connection to comonotonicity
url: https://www.ml-quant.com/papers/doi/10-1017-jpr-2025-31/
site: ML-Quant (https://www.ml-quant.com)
updated: 2026-09-26
license: Summaries CC BY 4.0; links go to the original sources
index: https://www.ml-quant.com/llms.txt
identifier: doi:10.1017/jpr.2025.31
source_url: http://dx.doi.org/10.1017/jpr.2025.31
featured: 2025-09-22
citations: 5
topic: Risk, Credit & Banking
---


# A note on subadditivity of value at risks (VaRs): A new connection to comonotonicity

The research reveals a new characteristic of value at risk (VaR), stating that its subadditivity holds for any confidence level only if the loss random variables are comonotonic.

- Source: http://dx.doi.org/10.1017/jpr.2025.31
- Identifier: doi:10.1017/jpr.2025.31
- Released: 2025-09-16
- First featured: Quant Letter No. 113 (2025-09-22): https://www.ml-quant.com/issues/2025-09-22/
- Citations (Semantic Scholar): 5
- Published in: J. Appl. Probab.
- Topic: Risk, Credit & Banking

## Related

- [Quantiles under ambiguity and risk sharing](https://www.ml-quant.com/papers/arxiv/2412.19546/): The study introduces Choquet Expected Shortfall, a new class of risk measures, and provides optimization algorithms and examples using financial data.
- [Time-Series Foundation AI Model for Value-at-Risk Forecasting](https://www.ml-quant.com/papers/arxiv/2410.11773/): The research highlights the superior performance of a time-series model, TimesFM, in forecasting Value-at-Risk, with fine-tuning further enhancing the results.
- [Optimal insurance design with Lambda-Value-at-Risk](https://www.ml-quant.com/papers/arxiv/2408.09799/): The paper studies optimal insurance solutions using the Lambda-Value-at-Risk model, revealing that a truncated stop-loss indemnity is ideal under certain conditions and discusses the effect of model uncertainty.
- [Adaptive Multilevel Stochastic Approximation of the Value-at-Risk](https://www.ml-quant.com/papers/arxiv/2408.06531/): The paper introduces a multilevel stochastic approximation algorithm that adaptively selects the number of inner samples to compute the value-at-risk of a financial loss, improving the previous scheme's complexity.
- [Risk sharing with lambda value-at-risk under heterogeneous beliefs](https://www.ml-quant.com/papers/arxiv/2408.03147/): The research investigates risk distribution among multiple parties using Lambda value at risk, offering formulas for optimal allocations under differing beliefs.
- [CAESar: Conditional Autoregressive Expected Shortfall](https://www.ml-quant.com/papers/arxiv/2407.06619/): Conditional Autoregressive Expected Shortfall: The Conditional Autoregressive Expected Shortfall (CAESar) methodology is introduced for estimating Value at Risk (VaR) and Expected Shortfall (ES), providing a more comprehensive measure of tail risk and outperforming existing regression methods in forecasting performance.
