Why Expected Shortfall Has Several Equivalent Names
Summary
The document explains why a family of tail-risk measures appears under names such as expected shortfall, conditional value at risk, average value at risk, and expected tail loss. It connects the names to different mathematical descriptions: an expected loss beyond a quantile, a conditional expectation above value at risk, or an average of value-at-risk levels across the tail. Tail value at risk and several actuarial and statistical variants are also mentioned.
The answers suggest that naming reflects different research traditions and reference books more than a single agreed historical origin. One response notes that formulations can differ in some settings, citing work that distinguishes conditional value at risk from expected shortfall, while saying they are often treated as equivalent in practice. The discussion offers context rather than a definitive history, and it does not establish who first introduced the measure or provide a systematic comparison of the formulations.
Key ideas
- Expected shortfall, conditional value at risk, average value at risk, and expected tail loss are commonly used names for related tail-risk measures.
- The names correspond to different ways of expressing tail losses, including conditional expectation and averaging value-at-risk quantiles.
- Terminology varies across fields and reference traditions, with no definitive origin identified in the discussion.
- Some formulations may differ mathematically, so equivalence should not be assumed in every setting.
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Full text
# Reasoning behind multiple names for the equivalent risk measures AVaR/ETL/ES/CVaR
# Reasoning behind multiple names for the equivalent risk measures AVaR/ETL/ES/CVaR
Doe's any one know the history behind, or background of the multiple naming conventions for the equivalent risk functions. Different quant authors prefer using different names, does any one know why? which came first? why do some quants prefer one over the other. What is the history behind all of this.
```
ES - expected shortfall
CVaR - conditional value at risk
AVaR - average value at risk
ETL - expected tail loss
```
## Answer by Alexey Kalmykov (score 4)
https://quant.stackexchange.com/a/7536
No specific history. I'm not aware who introduced this measure initially. Most probably it came up as an example in the research papers on coherent risk measure. All names make sense to some extent:
Expected shortfall - as it's an expectation of losses Conditional Value at Risk - as it can be written as $E[X |X >VaR_α(X)]$, i.e. conditional expectation Average value at risk - as it can be written as $\frac{1}{1-\alpha}\int_{\alpha}^1VaR_{\beta}(X)d\beta$, which is essentially an average Expected Tail Loss - as it's expectation calculated in the tail of loss distribution
It's also sometimes called Tail VaR.
Personal preferences of people are usually based on their favorite risk management reference book.
## Answer by Quartz (score 3)
https://quant.stackexchange.com/a/7553
I have also seen (in rough decreasing importance order): Mean excess loss, Tail conditional expectation and the variant C.T.E., Tail mean, Mean shortfall...
AVaR doesn't seem as common as the other three you mentioned.
Acerbi and Tasche 2002 discuss the difference between CVaR and ES. In practice there's little mention on reasons for each choice and rarely any differences have a significant role so that all variants become equivalent. The different naming flavours mostly originate historically from various areas: very roughly (from my concentrated and relatively limited sample) CVaR seems more en vogue among economists&traders, ES among risk managers, ETL among "statisticians", TVaR in econometrics, CTE among actuaries etc...
For other formulations of ES check also spectral and distortion risk measures.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.