When Adding VaR Estimates Can Be a Conservative Bound

Summary

The document considers whether two value at risk estimates may be added directly instead of combined using a variance and correlation formula. It explains that, without information about dependence between the risk drivers, summing the individual VaRs is commonly treated as a conservative estimate of combined risk. This approach avoids choosing an unsupported correlation assumption, though it may overstate the portfolio risk.

The answer also identifies perfect correlation as an alternative implicit assumption behind direct addition. More generally, a coherent risk measure is expected to satisfy subadditivity: the risk of a combined position should not exceed the sum of its separate risks. VaR does not always satisfy this property, so the inequality is not guaranteed in every setting. The document offers a brief conceptual explanation rather than a worked calculation; it does not specify confidence levels, distributions, or the exact exchange-rate exposures needed to determine the appropriate aggregation method for a particular portfolio.

Key ideas

• Adding individual VaR estimates can be a conservative way to aggregate risk when dependence is unknown.
• Direct addition can also reflect an assumption that risk drivers move perfectly together.
• Subadditivity says combined risk should not exceed the sum of standalone risks.
• VaR may fail subadditivity, so direct addition is not a universal aggregation rule.

Tags

# Summing up two VaRs


# Summing up two VaRs












I know that normally you can't just add two VaRs straight ahead and you need to use the formula with the sum of squares and the square root. However, in the marking scheme for the task in the image two values at risk are added without any squares and root. Can anybody explain why it is valid in this situation?

P.S. All the calculations in the task are performed for exchange rates of 1.23 and 1.4, not 0.4 and 1.28

## Answer by Kermittfrog (score 1, accepted)

https://quant.stackexchange.com/a/51791

Without any additional information, and disregarding potential problems of VaR subadditivity, adding VaR figures is usually deemed conservative. For a meaningful risk measure, we usually require

$$ Risk(X+Y) \leq Risk(X) + Risk(Y) $$

In your example: Simply adding the VaR figures per currency is sufficiently conservative, if no correlation is assumed. Another (implicit) assumption could be that the risk drivers (i.e. the exchange rates) are (again: implicitly) assumed to be perfectly correlated.

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.