Value at Risk (VaR) Explained

What VaR actually promises

Value at Risk answers one question: at a given confidence level, over a given horizon, how much could I lose on an ordinary bad day? A 95% one-day VaR of $650 on a $10,000 position means you'd expect to lose less than $650 on 95 of every 100 days — and more than $650 on the other 5. It is a threshold on ordinary losses, not an estimate of the worst day you'll ever see.

The part that rarely gets stated out loud is how that threshold gets computed. There are three common methods, and they are not interchangeable:

• Parametric (variance-covariance): assume returns follow a normal distribution, then VaR = position × volatility × z-score for the chosen confidence. This is what the VaR calculator on this site uses — fast, needs only volatility and confidence as inputs.
• Historical simulation: take the actual past returns, sort them, and read off the percentile that matches your confidence level directly — no distribution shape assumed at all.
• Monte Carlo: simulate thousands of possible future paths from a model of the return process (which can itself include fat tails, jumps, or volatility clustering), then read the percentile off the simulated outcomes.

Same portfolio, same confidence level, same day — three different numbers, because each method encodes a different belief about the shape of the distribution's tails.

Worked example: parametric vs. historical on the same 100 days

Take a synthetic 100-day daily-return sample built to resemble a volatile asset with occasional sharp crashes — mean −0.31%, standard deviation 3.94% (close to "4% daily vol," a realistic order of magnitude for a large-cap crypto asset in normal conditions). The sample includes three crash days (−9%, −12%, −18%) among the 100, which is exactly the kind of negative skew real crypto return series show.

Notice the direction flips. At 95% confidence the parametric model is actually the more conservative (higher) number in this sample — the normal curve isn't automatically wrong near the center of the distribution. But push to 99% confidence, deeper into the tail where the real crash days live, and the normal curve badly understates the loss: $917 parametric vs. $1,206 historical, a 31% gap, on identical underlying data.

The backtest that exposes the problem: breach counting

Risk desks check a VaR model the same way you'd check a weather forecast — count how often reality actually breached the predicted threshold, and compare it to what the model promised.

A 95% VaR model promises breaches on about 5% of days; a 99% VaR model promises breaches on about 1% of days. Running that check on the same 100-day sample:

At 95% the normal model actually over-predicted how often you'd get hurt — 3 real breaches against 5 expected. At 99% it under-predicted by 2x — exactly the tail region where a single missed breach matters most, because 99%-VaR breaches are, by construction, the biggest loss days in the sample.

This is not a quirk of this particular sample. It is the structural signature of negative skew and excess kurtosis — the sample above has a skew of −1.11 and excess kurtosis of 2.86 (a normal distribution has skew 0 and excess kurtosis 0). Fatter, longer left tails mean the deepest percentiles contain losses the normal curve didn't budget for.

The other hidden setting: time-horizon scaling

Most VaR tools quote a 1-day number, then scale it to a longer horizon with the "square-root-of-time" rule: N-day VaR ≈ 1-day VaR × √N. A 10-day VaR under this rule is 1-day VaR × √10 ≈ 3.16×, not 10× — scaling by the square root rather than linearly because risk (standard deviation) grows with the square root of elapsed time under the assumption of independent daily returns.

That assumption — independent, identically distributed daily returns — is the part that quietly breaks in real markets. Crypto volatility clusters: a turbulent day is more likely to be followed by another turbulent day than the i.i.d. assumption predicts, and calm stretches similarly bunch together. When volatility is clustered rather than independent, the true multi-day loss distribution can be wider (if turbulence is persisting) or narrower (if a shock is fading) than the √N shortcut implies. The square-root rule is a convenient approximation, not a law of the market.

Practical guidance

• Before comparing two VaR numbers, find out which method produced each one. "Parametric," "historical," and "Monte Carlo" are three different procedures that happen to report the same unit (a dollar loss threshold).
• A parametric VaR that looks fine at 95% can still understate the 99% tail badly — check the deeper confidence level separately rather than assuming the model's accuracy transfers.
• Backtest with breach counting if you can: over any reasonably long history, a well-calibrated 95% VaR should be breached on roughly 5% of days and a 99% VaR on roughly 1%. Persistent over- or under-breaching is the model telling you its distributional assumption doesn't fit.
• Treat multi-day VaR scaled by √N as a rough estimate, not a guarantee — especially right after a volatility spike, when clustering makes the next few days more likely to stay turbulent than the i.i.d. assumption assumes.
• VaR tells you nothing about the size of a tail breach once it happens. Pair it with maximum drawdown and a hard risk of ruin check before sizing a leveraged position.

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Frequently Asked Questions

What does "95% VaR" actually mean, in plain terms?

A 95% one-day VaR of $650 on a $10,000 position means: on a normal day, you would not expect to lose more than $650, and a loss beyond that threshold should happen on roughly 1 day in 20 (5%). It is a floor on ordinary bad days, not a worst case — it says nothing about how large the losses on the other 5% of days actually get.

Does the VaR method I choose — parametric or historical — change the number for the same portfolio?

Yes, often significantly, because each method makes a different assumption about the shape of the return distribution. On a worked 100-day synthetic return sample with mean -0.3% and standard deviation 3.9%, the parametric (normal-curve) method gives a 99% VaR of 9.17% ($917 on $10,000), while the historical method — simply reading off the actual 1st-percentile loss in the same sample — gives 12.06% ($1,206), a 31% higher figure. Same data, same confidence level, different method, different answer.

Why can a VaR model that looks accurate at 95% confidence badly underestimate the 99% tail?

Because real return distributions (especially crypto) are negatively skewed with fat left tails — crashes are rarer but far more severe than a symmetric normal curve predicts. In the worked 100-day sample, the parametric model's 95% VaR was actually close to the empirical figure, but at 99% confidence the gap widened to 31%, and the number of days that actually breached the parametric 99% VaR was double what the normal model predicted (2 breaches observed vs. about 1 expected). The deeper into the tail you go, the more a normal-distribution assumption understates real risk.

Does VaR scale with the square root of time, and when does that break down?

The common shortcut multiplies 1-day VaR by the square root of the number of days to get a multi-day VaR (e.g. 10-day VaR ≈ 1-day VaR × √10 ≈ 3.16×). That scaling rule assumes returns are independent and identically distributed from one day to the next. It breaks down under volatility clustering — calm periods and turbulent periods tend to bunch together in crypto — which means realized multi-day losses can be larger or smaller than the square-root-of-time estimate depending on whether the following days stay turbulent or calm.