VaR and Expected Shortfall: Quantiles, Scenarios, and Approximations
Summary
The discussion points to sources of analytical Value at Risk and Conditional Value at Risk formulas across continuous distributions. It explains that VaR is obtained from a distribution's cumulative distribution function by taking the relevant quantile, and notes that CVaR is commonly called Expected Shortfall. For analytical formulas, it cites a reference collecting results across many distributions and a research paper providing formulas with proofs for several common distributions.
It also describes practical calculation choices. Scenario analysis estimates portfolio tail risk by simulating outcomes and computing VaR or Expected Shortfall from those scenarios; the answer favors this for forward-looking portfolio analysis. When simulation is unavailable, it suggests a Cornish–Fisher approximation that expresses tail measures using distribution moments, while warning that portfolio implementations can be cumbersome. The post is a collection of pointers and recommendations, not a derivation or comparison of estimator accuracy. Scenario results depend on the chosen distribution and simulation assumptions, and approximations may not capture tail behavior well in every setting.
Key ideas
- VaR is a quantile derived from the cumulative distribution of returns or losses.
- CVaR is widely referred to as Expected Shortfall.
- Simulation provides a scenario-based route to estimate portfolio VaR and Expected Shortfall.
- Cornish–Fisher approximations can express risk measures through distribution moments when scenarios are not used.
- Analytical formulas and approximations depend on distribution assumptions and may be cumbersome for portfolios.
Tags
Full text
# Where can I find a list of VaR and CVaR formulas for continuous distributions? # Where can I find a list of VaR and CVaR formulas for continuous distributions? Where can I find more VaR and CVaR formulas for continuous distributions? I collected a list here: ## Answer by Yulia V (score 9) https://quant.stackexchange.com/a/15127 Values of `VaR` are just the inverses of the cumulative distributions. `CVaR` is not a very commonly used term, its more frequently used synonym is `Expected Shortfall`. See http://www.maths.manchester.ac.uk/~saralees/chap17.pdf for the list of Expected Shortfall values for more than 20 distributions. ## Answer by John (score 3) https://quant.stackexchange.com/a/15129 More often than not, I prefer to work with a scenario representation. That is, I will simulate from the distribution and calculate the VaR and CVaR as appropriate. This is especially the case for forward-looking analysis of portfolios' CVaR, rather than in evaluating the historical returns of some portfolio. If for some reason I can't do the scenario approach, then I will use the Cornish-Fisher approximation. There is a paper by Boudt, Peterson, and Croux that I think provides the formula for both VaR and CVaR, as well as a few others (perhaps refer to other references, and there's already a question on this site about it wrt VaR). If you're using Cornish-Fisher, then you can write the VaR and CVaR in terms of the moments of whatever distribution you're looking at. BPC provides the formula for portfolios as well, but in my experience this is a big pain. ## Answer by Malick (score 2) https://quant.stackexchange.com/a/51309 The paper "Calculating CVaR and bPOE for Common Probability Distributions With Application to Portfolio Optimization and Density Estimation" by Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2018) gives a large number of CVAR analytical formula with full proof. Most of them can also be found on the Expected shortfall (aka CVAR) Wikipedia page.
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.