VaR & Monte Carlo Risk Simulator

Three answers to the same question

Value at Risk answers one question: over a given horizon, what loss is not exceeded with a given probability. At 95 % over twenty-one trading days, a VaR of 8 % means that in ninety-five months out of a hundred the loss stays under 8 %. It says nothing at all about the remaining five, which is the most common way the number gets misread.

There is no single correct way to compute it, and the three standard methods disagree in ways that are informative. This tool runs all three on the same window and puts them side by side, because the spread between them is itself a result: when the three agree, the estimate is robust; when they diverge, the distribution is telling you something the average hides.

How each figure is produced

Historical VaR takes the actual overlapping returns over the chosen horizon, sorts them, and reads the 5th percentile by linear interpolation between the two neighbouring observations. It assumes nothing about the shape of the distribution, which is its strength, and it can only produce outcomes that have already happened, which is its limit.

Parametric VaR fits a normal distribution to the same sample and reads the same percentile from it, using the one-tailed z-score of 1.645 at 95 %. It is smooth and stable, and it systematically understates tail risk on assets whose returns are fat-tailed, which most of them are.

Monte Carlo VaR draws 10,000 simulated paths from the estimated distribution and takes the 5th percentile of the simulated outcomes. It fills the gaps between observed scenarios, at the cost of inheriting whatever assumption generated the draws.

The outcome histogram is binned across the 1st to 99th percentile range, so the shape of the distribution stays readable instead of being flattened by one extreme observation. The extremes themselves are not discarded from the calculation, only from the bin range. The mean and the median outcome are reported together, and the distance between them is the asymmetry of the sample.

Choosing the window, and what the number is worth

The lookback is selectable: the full available history, or the last one, two, five or ten years. This is the most consequential choice on the page. A window that excludes 2008 and 2020 will produce a comfortable VaR that has never been tested against a crisis, and a window that includes them will look alarming during a calm decade. Neither is wrong; both are conditional on a period, and the period should be a decision rather than a default.

The horizon runs from one trading day to 252, roughly a calendar year. Overlapping horizon returns are used, which is standard practice and which makes successive observations statistically dependent: the effective sample is smaller than the raw count of observations suggests, and the long-horizon figures carry more uncertainty than the short ones.

VaR is a threshold, not a worst case. It says where the fifth percentile sits, not how far past it the loss can go. The losses that matter most in practice are precisely the ones on the far side of that threshold, and no VaR figure, from any of the three methods, describes them. This is a research tool, not investment advice.