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Interpreting Value at Risk for Positive Return Tails

Article Quant Q&A · Author: Nourhaine Nefzi
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Summary

The document asks how to interpret a VaR estimate for extreme positive returns after using extreme value theory to model both negative and positive tails. The response distinguishes downside risk from the upper tail: when losses correspond to negative returns, positive-return VaR is not naturally a measure of loss risk. An upper-tail quantile can instead describe unusually large gains.

It also relates the lower distribution quantile to the negative of a confidence-level VaR and points to the Omega ratio as a way to compare gains and losses in the tails. No model specification, estimated quantiles, or empirical example is supplied. The interpretation depends on the analyst’s objective: upper-tail quantiles can describe upside outcomes, while VaR’s conventional risk meaning concerns potential losses.

Key ideas

  • Conventional VaR measures downside loss exposure when negative returns represent losses.
  • An upper-tail quantile can describe the distribution of unusually positive returns.
  • The lower-tail quantile corresponds to the negative of VaR under the stated convention.
  • The Omega ratio compares gains and losses across distribution tails.
  • The document provides interpretation guidance but no empirical estimates or EVT implementation details.

Tags

Risk managementStatisticsVolatility
Full text
# VaR interpretation for positive returns


# VaR interpretation for positive returns












I used Extreme Value Theory to separate extreme negative returns from extreme positive returns, then, I calculated the VaR for both. I need to know what could be the interpretation of VaR for positive returns? Thanks in advance.

## Answer by Richi Wa (score 0, accepted)

https://quant.stackexchange.com/a/21588

What do you model? If negative returns are losses, then what is your interest in the "risk" of the positive ones. Most naturally you could look at quantiles of your distribution. The 1%-quantile is the negative of $VaR_{99\%}$. The $99\%$-quanile could be of interest if you want to know about the right end of the distribution.

The concept of Oemga ration compares losses and gains in the tails.

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