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Dollar VaR from Log Returns: Quantile Transformation and Approximation

Article Quant Q&A · Author: user16469
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Summary

The document examines how to convert a value-at-risk estimate expressed as a log return into a dollar amount. It presents two candidate conversions: multiplying the position value by the log-return VaR, or multiplying by the exponential of that VaR minus one. The questioner argues that the nonlinear transformation of quantiles suggests these expressions may have been labeled in reverse.

A brief reply agrees with that concern and says the exponential conversion gives the correct amount, while direct multiplication by log-return VaR is an approximation. The exchange does not show a derivation, specify the VaR sign convention, or discuss whether the position is long or short. It therefore highlights the distinction between simple and log returns, but provides little detail on implementation. Readers should confirm conventions and the relevant tail quantile when applying the relationship to a particular risk calculation.

Key ideas

  • Converting a log-return VaR into dollars requires distinguishing log returns from simple returns.
  • The reply identifies the exponential conversion as the correct amount and direct multiplication as an approximation.
  • The discussion does not provide a derivation or define its VaR sign convention.
  • Position direction and tail conventions should be made explicit when applying the conversion.

Tags

Risk managementStatistics
Full text
# Value at risk in dollars vs. log returns


# Value at risk in dollars vs. log returns












I have a quick question about this remark in Tsay's book "Analysis of Financial Time Series" (3rd edition).

He says that $$ \text{dollar VaR} = \text{Value} \times \text{log return VaR} $$ and that $$ \text{Value} \times [\exp(\text{log return VaR}) - 1] $$ is an approximation to that.

Based on how quantiles transform, it seems to me that it should be the other way around!

Eq (7.1) for completeness

Thanks for your help!

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/22374

I think you are right. What he calls the approximation is the correct amount, the other is an approximation.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.