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📊 '''Tail value at risk''' — also known as conditional value at risk (CVaR) or expected shortfall — is a [[Definition:Risk measure|risk measure]] that quantifies the expected loss in the worst-case tail of a loss distribution, beyond a specified confidence level. In the insurance and [[Definition:Reinsurance|reinsurance]] industry, tail value at risk (TVaR) has gained prominence as a complement to — and in many applications a replacement for — [[Definition:Value at risk|value at risk]] (VaR), because it captures not just the threshold at which extreme losses begin but also the average severity of those losses once the threshold is breached. This property makes TVaR especially relevant for insurers, whose portfolios are often exposed to [[Definition:Catastrophe risk|catastrophe events]], [[Definition:Long-tail liability|long-tail liabilities]], and other heavy-tailed loss distributions where the magnitude of extreme outcomes matters enormously.

🔬 TVaR at the 99th percentile, for example, represents the average loss that occurs in the worst 1% of scenarios — providing a richer picture of tail exposure than VaR, which only identifies the loss level at the boundary of that 1%. [[Definition:Actuary|Actuaries]] and [[Definition:Risk management|risk managers]] in insurance use TVaR extensively in [[Definition:Enterprise risk management|enterprise risk management]], [[Definition:Economic capital|economic capital]] modeling, and [[Definition:Reinsurance|reinsurance]] purchasing decisions. Several regulatory frameworks incorporate TVaR directly: the [[Definition:Swiss Solvency Test|Swiss Solvency Test]] (SST) uses expected shortfall as its core capital calibration metric, and various internal model approaches under [[Definition:Solvency II|Solvency II]] and [[Definition:C-ROSS|C-ROSS]] may employ TVaR alongside or instead of VaR. In the United States, the [[Definition:National Association of Insurance Commissioners|NAIC]] and actuarial standards of practice reference TVaR-type measures in the context of [[Definition:Reserves|reserving]] and capital adequacy testing.

⚠️ The appeal of TVaR for insurers rests on a fundamental insight: in an industry defined by its obligation to pay large, infrequent losses, knowing only the threshold of a bad outcome is insufficient — understanding the expected depth of the loss beyond that threshold is critical for ensuring [[Definition:Solvency|solvency]] and setting appropriate [[Definition:Premium|pricing]]. TVaR also possesses the mathematical property of coherence, meaning it satisfies axioms (including subadditivity) that VaR does not, making it more suitable for aggregating risks across business lines or legal entities within an insurance group. As [[Definition:Catastrophe model|catastrophe models]] become more sophisticated and insurers face emerging risks with limited historical data — such as [[Definition:Cyber risk|cyber risk]] and [[Definition:Pandemic risk|pandemic risk]] — TVaR provides a disciplined framework for allocating capital to the tail, ensuring that extreme but plausible scenarios receive the financial attention they demand.

'''Related concepts:'''
{{Div col|colwidth=20em}}
* [[Definition:Value at risk]]
* [[Definition:Economic capital]]
* [[Definition:Catastrophe model]]
* [[Definition:Swiss Solvency Test]]
* [[Definition:Enterprise risk management]]
* [[Definition:Risk measure]]
{{Div col end}}

Definition:Tail value at risk - Revision history

PlumBot: Bot: Creating definition