Every desk you've studied so far — the one hedging vega, theta and rho included — manages risk position by position, Greek by Greek. But a bank's chief risk officer doesn't sit down with the CEO and recite two hundred deltas. She needs to answer one blunt question, about the whole firm, in one number: how much could we plausibly lose tomorrow? Value at Risk (VaR) is the industry's answer — a single dollar figure that rolls every position, every hedge, and every correlation in the book into one statement a board member with no maths background can act on.

The definition, precisely

The 1-day, 99% VaR of a portfolio is the loss level that a loss will exceed on fewer than 1% of days — equivalently, there is at least 99% confidence that tomorrow's loss will be no worse than this number. In symbols, if L is tomorrow's random loss (a positive L means money lost, a negative L means money made), the VaR at confidence level c is the smallest number V satisfying

Every VaR figure needs two numbers pinned down alongside the dollar amount: a confidence level (typically 95% or 99%) and a time horizon (typically one day, ten days, or a month). "VaR of $4 million" on its own is meaningless — "1-day 99% VaR of $4 million" is a complete, checkable claim: on a normal day, there's a 99% chance tomorrow's loss stays under $4 million, and — just as informatively — you should expect a worse day roughly once every hundred trading days, about two or three times a year.

Picture it: a cutoff on the loss distribution

Think of tomorrow's profit-and-loss as a random draw from some distribution centred near zero. VaR is nothing more than a cutoff point on the far left tail of that distribution — move the confidence slider and watch the cutoff slide with it. A higher confidence level pushes the cutoff further left (a bigger claimed loss), because you're demanding a stronger guarantee.

Notice what the picture makes obvious: the shaded sliver is outside the VaR promise — VaR only bounds the boundary of that region, it says nothing about how thick or how far the sliver stretches. Hold that thought; it's the whole subject of a later lesson.

Worked example: reading VaR off a described distribution

A risk model has produced the following description of tomorrow's possible losses on a portfolio — for several loss thresholds, the probability that the actual loss is at least that large:

What is the 1-day 95% VaR? We need the smallest threshold V with \mathbb{P}(L > V) \le 5\%. Reading down the table, the loss exceeds $1,000,000 with probability 5% — exactly the cutoff we need — so the 95% VaR is $1,000,000. For 99% confidence we need \mathbb{P}(L>V)\le 1\%, which first happens at V=\$1{,}500{,}000: the 99% VaR is $1,500,000. Notice the pattern that always holds — demanding more confidence never makes VaR smaller.

Now the crucial question: what does the 99% VaR of $1,500,000 tell you about the table's very last row — the 0.1% chance of losing $2,500,000 or more? Nothing at all. The VaR figure is silent about everything past its own cutoff. A bank that only reports "99% VaR = $1.5M" has told you a threshold is rarely crossed, but has said nothing about how catastrophic it is when it is — a gap that Expected Shortfall exists specifically to close.