VaR Non-Subadditivity and Choosing Quantile Bounds

Summary

The document concerns an example used to show that value at risk can fail to be subadditive, a property often desired of portfolio risk measures. The question asks why particular loss or outcome bounds are used when computing quantiles for two portfolios. The response offers only a brief hint: the bounds in the example were selected so that the relevant event probabilities come close to the target VaR confidence levels.

This points to the role of quantile construction in examples of risk aggregation: chosen outcomes and their probabilities determine the portfolio quantiles, which can then be compared with the quantile of the combined portfolio. However, the document does not include the cited example, its probability distributions, or a derivation of the bounds. It therefore gives little detail for reproducing the result or evaluating the specific choices, and should be read as a pointer rather than a complete explanation of VaR non-subadditivity.

Key ideas

• Value at risk can fail the subadditivity property when portfolio risks are aggregated.
• Quantile bounds depend on the outcome values and their probabilities.
• The cited example chooses bounds to bring event probabilities near the desired confidence levels.
• The response is only a hint and does not provide enough detail to reconstruct the example.

Tags

# Showing that VaR is not sub additive


# Showing that VaR is not sub additive












I found on pages 2 and 3 of Martin Haugh's "Risk Measures, Risk Aggregation and Capital Allocation" from 2010 an example showing non sub-additivity of VaR (excerpts given at the end).

I understood the example on the whole, but I would like to clear out this doubt: when we compute the quantile, why for portfolio A we use -5 and l<-5, whereas for portfolio B we use +5 and +4?

Page 2

Page 3

## Answer by Bob Jansen (score 1)

https://quant.stackexchange.com/a/71903

What Alper says but from a quick look, I'll give you hint:

They figured out that (by trail and error for B) these bounds result in the various $P(\cdot)$ to have value close to the desired VaR levels (0.98 and 0.95).

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