Value at Risk (VaR) Explained: Parametric vs Historical

The Question VaR Actually Answers

Every position carries a distribution of possible outcomes, but most risk conversations collapse into a single fear: how bad could it get? Value at Risk (VaR) is the industry’s attempt to put one number on that fear. It answers a precise question: “At this confidence level, over this time horizon, how much could I lose?”

A 95% one-day VaR of $3,000 on a $100,000 portfolio means there is a 95% chance the portfolio will not lose more than $3,000 in a single day — and, equivalently, a 5% chance that it will. That is the entire idea, and its precision is also its danger: VaR is a threshold, not a ceiling. This guide walks through the two standard ways to compute it, a worked example, and — just as importantly — what VaR quietly refuses to tell you. If you want to run the numbers, the Value at Risk calculator does both methods.

Parametric VaR: Assume a Bell Curve

The parametric (variance-covariance) method assumes returns follow a normal distribution. Once you accept that assumption, VaR is just a point on the bell curve. You need the portfolio’s expected return and volatility per period, and a z-score for your confidence level — the number of standard deviations out into the tail that leaves the chosen probability beyond it:

• 90% confidence → z = 1.2816
• 95% confidence → z = 1.6449
• 99% confidence → z = 2.3263

The formula for the loss fraction over a horizon of T periods is:

• VaR% = z × volatility × √T − mean × T
• VaR ($) = VaR% × portfolio value

Two scaling rules are doing the work. Volatility grows with the square root of time (√T), while the mean return grows linearly with time (×T) — a consequence of assuming returns are independent from one period to the next. The mean term is subtracted because expected drift offsets some of the downside; with a large enough positive mean, VaR can even come out negative, which is an honest result meaning the expected gain outweighs the volatility over that horizon.

Historical VaR: Let the Data Speak

The historical method makes no distribution assumption at all. Instead of a formula, it reads the answer straight from your actual past returns. For a 95% VaR, it finds the 5th percentile of your return series — the loss level that only the worst 5% of periods exceeded — and reports it as a positive VaR figure:

• Historical VaR = the (100 − confidence)th percentile of your real returns, sign-flipped so a loss reads positive

Because it uses whatever shape your returns actually had, historical VaR captures fat tails and skew that a normal curve would smooth away — the sudden 15% crash days that crypto produces far more often than a bell curve predicts. The price is that it needs a real, representative return series, and it can only “see” as much bad behavior as your history contains. If your data window never included a crash, neither will your historical VaR.

Worked Example

Take a $100,000 portfolio with daily volatility of 1.82% and a mean daily return of 0%. The table shows parametric VaR at different confidence levels and horizons, plus a historical VaR read from a sample return series that includes a couple of ugly days.

Method	Confidence	Horizon	VaR %	VaR $
Parametric	95%	1 day	2.99%	$2,994
Parametric	99%	1 day	4.23%	$4,234
Parametric	95%	10 days	9.47%	$9,467
Historical	95%	per period	6.60%	$6,600

Two things stand out. First, the 10-day VaR (9.47%) is not ten times the one-day VaR — it is roughly √10 ≈ 3.16 times larger, because volatility scales with the square root of time. Second, the historical 95% VaR of $6,600 is more than double the parametric 95% figure of $2,994. That gap is the whole argument for the historical method: the real return series contained sharp losing days that a smooth normal curve, calibrated to the same average volatility, simply does not anticipate.

How to Use the Value at Risk Calculator

The tool has two modes. Switch with the Parametric / Historical toggle.

In Parametric mode:

1. Portfolio Value — the total value at risk, in your account currency.
2. Confidence Level — pick 90%, 95%, or 99%; the tool maps this to the matching z-score.
3. Mean Return (per period) — expected return per period in percent. Enter 0 if you do not want to bank on drift.
4. Volatility (per period, std dev) — the standard deviation of returns per period, in percent.
5. Time Horizon (periods) — how many periods ahead, e.g. 1 for one day or 10 for two trading weeks. Volatility is scaled by √horizon.

In Historical mode you provide Portfolio Value, the Confidence Level, and a series of Historical Returns (%, comma or newline separated) — paste real per-period returns and the tool reads the empirical percentile directly. Either way, the results show Value at Risk ($) and Value at Risk (%).

What VaR Does Not Tell You

This is where honesty matters. Both methods express the expected loss magnitude at a threshold — neither caps your loss. By definition, the complement of your confidence level (5% for a 95% VaR) is exactly how often the true loss exceeds the VaR figure. VaR tells you where the tail begins; it says nothing about how deep the tail goes. On the 5% of days worse than a 95% VaR, the loss could be modestly or catastrophically larger.

Parametric VaR carries an extra warning: its normal-distribution assumption tends to understate risk for fat-tailed, skewed assets like crypto, where extreme moves happen far more often than a bell curve implies — the worked example’s $2,994 versus $6,600 gap is exactly this effect. Historical VaR avoids that assumption but is only as honest as its data window. Treat VaR as one input, not a safety guarantee. Pair it with your position sizing and risk management rules so no single day can do outsized damage, watch the depth of realized losses with a drawdown and recovery analysis, and use risk-adjusted return measures like the Sharpe and Sortino ratios to judge whether the return justified the risk in the first place.