Value-at-Risk
Objectives
- Define Value-at-Risk as a quantile of the loss distribution at a stated confidence level and horizon.
- Compute VaR three ways — historical, parametric (variance-covariance), and Monte Carlo — and state the assumptions each one buys.
- Read a SVQ VaR_alpha figure correctly, including what it does and does not tell you about the tail beyond it.
Every risk number you will ever quote is really an attempt to answer one blunt question: how bad is a bad day?
You cannot answer that with an average. The average day is, almost by definition, unremarkable. What keeps risk managers awake is the left tail — the rare, ugly outcomes that the average smooths over and hides. Value-at-Risk is the first and most widely used attempt to put a single, honest number on that tail. Before you can decide whether a price move is genuine evidence of something or just noise, you need a sense of scale for how much loss is normal. Value-at-Risk gives you that scale.
So let us build the intuition first, then make it precise.
Imagine you line up every daily loss your portfolio could plausibly suffer and sort them from gentle to catastrophic. Most days cluster near the middle — small gains, small losses. The far left end holds the disasters. Value-at-Risk simply asks you to draw a vertical line somewhere in that left tail and read off the loss at that line. Pick a confidence level — say ninety-nine percent — and the line falls at the point where only one percent of outcomes are worse. The loss at that line is your one-day, ninety-nine percent Value-at-Risk. In plain words: on ninety-nine days out of a hundred, you expect to lose less than this; on the worst one day in a hundred, you expect to lose at least this much.
Notice what that sentence does and does not promise. It tells you the threshold the worst one percent of days breach. It says nothing about how far past the threshold those days go. That distinction will matter enormously, and we will return to it.
Now the formalism, which is cleaner than the prose. Let your loss over the horizon be a random variable — losses positive, gains negative, by convention, because risk people like the scary number to be positive. Fix a confidence level, call it one minus alpha, where alpha is the tail probability. The Value-at-Risk at level alpha is the (one minus alpha)-quantile of the loss distribution — equivalently, the one-hundred-times-(one minus alpha)th percentile of losses. Written plainly: VaR-alpha is the smallest loss threshold such that the probability of losing more than it is no greater than alpha. If alpha is one percent, you are reading the ninety-ninth percentile of losses. That is the entire definition. Value-at-Risk is a quantile — nothing more exotic than that. Every method below is just a different way of estimating the same quantile.
The first way is historical simulation. You take your actual realized returns over some lookback window, treat that empirical sample as your distribution, and read off the relevant percentile directly. If you have five hundred days of history and you want the ninety-nine percent VaR, you sort the losses and take the fifth-worst — because five out of five hundred is one percent. This method assumes almost nothing about the shape of the distribution, which is its great virtue. Its great weakness is that it assumes the future looks like your window: if your lookback contains no crisis, your VaR will not believe a crisis is possible.
The second way is parametric, often called the variance-covariance method. Here you assume returns are normally distributed, estimate the mean and the standard deviation, and compute the quantile in closed form. The VaR is then the mean loss plus the standard deviation times the normal quantile factor — which, for ninety-nine percent confidence, is about two-point-three-three, and for ninety-five percent, about one-point-six-five. This is fast and elegant. It is also a liar precisely where you need the truth, because real return distributions have fatter tails than the normal curve, so the parametric method systematically understates the extreme losses it was hired to measure.
The third way is Monte Carlo simulation. You specify a model for how returns behave, generate tens of thousands of synthetic future paths, and read the quantile off the simulated losses. This is the most flexible — you can build in fat tails, volatility that clusters, nonlinear instruments like options — but it is only as honest as the model you feed it, and it costs real computation.
Three methods, one quantile. Choosing among them is a question of which assumption you are willing to defend.
When SVQ reports a tail metric labeled VaR_alpha, it is reporting exactly the quantile this chapter defines: the alpha-level percentile of the asset's loss distribution over the chosen horizon. The tail panel reads as a sentence. A one-day VaR at ninety-nine percent of, say, three-point-one percent means that on the worst one trading day in a hundred, SVQ's model expects this position to lose at least three-point-one percent of its value. At ninety-five percent confidence instead of ninety-nine, the number shrinks — the line has moved in from the deep tail toward the middle, so the quoted loss is naturally smaller. The figure is a threshold, not a worst case. On the day the asset blows clean through it, VaR_alpha had already said such a day would come about one session in a hundred — two or three times a year — at ninety-nine percent confidence; what it never promised was how violent that day would be.