Limitations of Value at Risk
Value at Risk reports a threshold loss: the most you expect to lose over a horizon at a chosen confidence level. Its great weakness is that it says nothing about how bad the tail beyond that threshold can be. A 99% one-day VaR of $10m is consistent with a worst-case loss of $11m or of $500m. VaR is also not subadditive in general, so the VaR of a combined book can exceed the sum of its parts, which contradicts the idea that diversification reduces risk. Estimates are highly model-dependent (normal, historical, Monte Carlo each give different numbers), and a single VaR figure can lull a desk into ignoring the rare but ruinous event.
Why it matters
VaR draws a line in the loss distribution and tells you the size of a "bad but not unusual" day. It is silent about the catastrophe lurking past that line. Think of it as a fence height that the flood will clear 1 day in 100; it never tells you how deep the water gets once it does. That blind spot is precisely where institutions fail, which is why VaR needs a tail-aware companion measure and a backtest.
What does a one-day 99% VaR of $8m fail to tell you?
Formulas
Worked examples
Two trading books each have a one-day 99% VaR of $5m. The head of desk claims the combined VaR must be $10m or less because "diversification always helps." Is the claim safe?
No. VaR is not generally subadditive, so the combined 99% VaR can exceed $10m when the books share concentrated tail exposures (for example both short deep out-of-the-money options on the same underlying). The intuition that risk measures fall when you pool positions holds for a coherent measure such as expected shortfall, but it is not guaranteed for VaR.
Common mistakes
- ✗VaR is the worst possible loss. VaR is only the threshold the loss exceeds with a fixed small probability; the loss beyond it can be far larger and is exactly what VaR ignores.
- ✗A lower portfolio VaR always means lower risk. Because VaR can violate subadditivity, merging positions can raise it, and two books with identical VaR can have very different tail severity.
- ✗VaR is an objective number. The figure depends heavily on the method (parametric, historical, Monte Carlo), the window, and the assumed distribution, so reported VaR is a modelling choice as much as a fact.
Revision bullets
- •VaR is a threshold loss, not the maximum loss
- •It is silent about the severity of the tail beyond the threshold
- •VaR is not subadditive in general, so it can penalise diversification
- •Numbers are model-dependent (normal vs historical vs Monte Carlo)
- •A single VaR figure can mask catastrophic, low-probability risk
Quick check
What does a one-day 99% VaR of $8m fail to tell you?
Why can combining two portfolios produce a VaR larger than the sum of the individual VaRs?
Connected topics
More in VaR: Validation & Alternatives
Sources
- Jorion (2007), Ch. 5Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007.Discusses what VaR measures and its principal shortcomings as a single-number risk summary.
- Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. "Coherent Measures of Risk." Mathematical Finance, 9(3), 203-228, 1999.Shows VaR can fail subadditivity, the formal basis for the diversification critique.