CDF Ordering and Comparing Value at Risk

Summary

This note asks whether pointwise ordering of two cumulative distribution functions is enough to determine which distribution has higher value at risk. It observes that if one CDF lies below another, the corresponding distribution places relatively more probability in its lower tail, but questions whether that alone establishes a VaR ordering.

The key tool for answering is the quantile definition: at a fixed probability level, ordered CDFs imply oppositely ordered quantiles, subject to the usual care around atoms and quantile conventions. For a loss variable, VaR is typically a loss quantile; for a return variable, the relevant lower-tail quantile is often expressed with a sign change. The document itself provides no derivation, example, or resolution, and does not specify whether its distributions describe returns or losses. That convention and the confidence level must be fixed before translating tail ordering into a statement about higher VaR.

Key ideas

• Pointwise CDF ordering can be translated into quantile ordering at a fixed probability level.
• A lower-tail return quantile and a loss VaR use different sign conventions.
• The document poses the comparison question but supplies no worked answer.
• Quantile conventions matter when distributions contain point masses.

Tags

# Which distribution has higher VaR?


# Which distribution has higher VaR?












Let say I have 2 distributions with cdf $F_1$ and $F_2$. And I know that $F_1 \leq F_2$. With this information I know that $F_1$ has bigger lower tail than $F_2$ but I don't think this right away doesn't mean that $F_1$ has bigger VaR. How can I conclude which distribution have higher VaR?

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