Value at Risk (VaR): Definition
Value at Risk (VaR) answers one question: over a set horizon and at a chosen confidence level, how much could a position lose under normal market conditions? Formally, VaR is the quantile of the loss distribution: the loss that will not be exceeded with probability . A one-day 99% VaR of US$5M means that on 99 days out of 100 the loss should stay below US$5M, and on roughly 1 day in 100 it is expected to be worse. VaR always carries two numbers, the confidence level and the horizon, and it is a one-tailed statement about the left (loss) tail only.
Try it yourself
How much could this book lose on a bad day? VaR draws a line in the loss tail at confidence 95%. The parametric line assumes a normal curve, VaR = (z·σ·√h − μ·h)·V with z = 1.645 (one-tailed). The historical and Monte-Carlo methods read VaR off a simulated loss distribution instead. Under normality the three roughly agree; fat tails pull the historical figure deeper.
Why it matters
VaR draws a line in the loss tail and says "we do not expect to cross this on an ordinary day." It compresses a whole distribution of outcomes into a single dollar figure a manager can act on. The catch is that it is a threshold, not a worst case: it tells you the edge of the bad region, never how deep the losses run once you fall past it. Quoting a VaR without its confidence level and horizon is meaningless, like a speed with no units.
Formulas
Worked examples
A desk reports a one-day 99% VaR of US$5M on a US$100M portfolio. Interpret it, and state what it does and does not promise.
On about 99 of every 100 trading days the one-day loss should be smaller than US$5M; on roughly 1 day in 100 the loss is expected to equal or exceed US$5M. That breach is 5 percent of capital. VaR does not say how large the loss is on that bad day, only that it lands beyond US$5M. To gauge the depth of the tail you need expected shortfall, not VaR.
The same desk also quotes a one-day 95% VaR. Why is the 95% figure smaller than the 99% figure for the same portfolio?
A higher confidence level pushes further into the loss tail, so it captures a larger loss. The 95% VaR is the 5th-percentile loss, the 99% VaR is the more extreme 1st-percentile loss. For any fixed horizon, raising from 0.95 to 0.99 always weakly increases VaR because you are asking about a rarer, deeper loss.
Common mistakes
- ✗VaR is the maximum possible loss. It is only a threshold for a given confidence level; losses beyond VaR are not just possible but expected on roughly of days.
- ✗A VaR number stands on its own. VaR is meaningless without both a confidence level and a horizon; "VaR of US$5M" says nothing until you add "one-day, 99%".
- ✗A higher confidence level makes the portfolio safer. The confidence level is a reporting choice; raising it only raises the quoted VaR number, it does not change the underlying risk.
- ✗VaR is two-tailed like a confidence interval. VaR looks only at the loss (left) tail, so it uses a one-tailed quantile, not a symmetric interval.
Revision bullets
- •VaR = the loss not exceeded with probability over a set horizon
- •Always quote both the confidence level and the horizon
- •One-tailed: a 99% VaR is the 1st-percentile loss
- •It is a threshold, not a worst case or maximum loss
- •Says nothing about loss size beyond the threshold
Quick check
A one-day 99% VaR of US$5M means that, under normal conditions,
Why must a VaR figure always state a confidence level and a horizon?
Connected topics
Sources
- Jorion (2007), Ch. 5Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007. ISBN 978-0-07-146495-6.Defines VaR as a quantile of the loss distribution at a stated confidence level and horizon.
- Hull (2018), Ch. 12Hull, J. C. Risk Management and Financial Institutions. 5th ed. Wiley, 2018. ISBN 978-1-119-44811-2.Introduces VaR, the confidence-level / time-horizon convention, and its one-tailed interpretation.