What does Value at Risk actually measure?
Every risk department in the world produces one number every morning, and it is almost always the same number: Value at Risk. It is the industry's answer to a question that sounds simple and isn't — how much could we lose?
The honest answer to that question is "everything," which is useless. So risk managers narrowed it: how much could we lose on a normal bad day, where "normal bad" is defined precisely.
The sentence that defines it
A VaR figure is only meaningful as a full sentence:
The 1-day 95% VaR of this portfolio is $200,000.
Translated: on 95 days out of 100, the one-day loss should not exceed $200,000. Equivalently, about one trading day in twenty — roughly one day a month — the loss should be larger than that.
Notice what the sentence does and does not claim. It marks a threshold, a line drawn at a percentile of the loss distribution. It says how often you cross the line. It says nothing whatsoever about how far past the line you go when you do — a silence that becomes this course's central theme.
Where the number comes from
Take every daily return the portfolio experienced over some past window — say the last 500 trading days, about two years. Sort them from worst to best. Five percent of 500 is 25, so count in to the 25th worst day. Whatever that day's loss was, that's your historical 95% VaR.
That's it. No calculus. A percentile of a sorted list.
A worked example
A $10 million portfolio. Over the last 500 trading days, the sorted daily returns look like this at the bad end (illustrative, rounded):
- Worst day: −5.8%
- 10th worst: −2.9%
- 25th worst: −2.0%
- 50th worst: −1.4%
The 25th worst day is the 5% cutoff. So the 1-day 95% VaR ≈ 2.0% × $10,000,000 = $200,000.
Push the confidence level to 99% and you count in to the 5th worst day instead (1% of 500). If that day was −3.6%, the 1-day 99% VaR is $360,000. Same portfolio, same data — a bigger number, because you asked about a rarer day.
Why the industry adopted it
VaR became standard for one reason above all: it compresses a whole portfolio — equities, bonds, currencies, derivatives, dozens of desks — into a single currency figure a non-specialist can read. A board member who cannot follow a covariance matrix can follow "$200,000 on a bad day, and here is last month's number for comparison."
That compression is genuinely useful and genuinely dangerous. Every compression discards information. The rest of this course is about which information VaR discards, and what professionals bolt on to recover it.
Try it now
- A year of daily bars is below — roughly 250 sessions, which is the whole population you are about to take a percentile of. A chart is a poor spreadsheet, but it is an excellent way to see what the percentile is a percentile of.
- 5% of 250 sessions is about 12. So find the level that roughly a dozen sessions fell further than: Measure the tall red bars, work down from the worst, and stop when you have counted twelve. That threshold is your 1-day 95% VaR, read off a chart rather than sorted in a column. Multiply it by an imaginary $100,000 position to see it in currency.
- Now do it for 1% — about two or three sessions out of 250. That is the 99% VaR. Note how much bigger it is, and write one sentence describing the gap as an observation about this window of data, not a forecast of the next one.
Next: the three dials that change the answer.