Value-at-Risk & Risk Metrics Calculator
Compute Value-at-Risk (VaR) and Expected Shortfall three ways — parametric, historical and Monte-Carlo — for a single position or a two-asset portfolio. The signature calculation of the FRM, PRM and CAIA risk readings, with a tail-shaded return distribution and a plain-English reading.
| Method | VaR | Expected Shortfall |
|---|---|---|
| Parametric (normal) | — | — |
| Historical | — | — |
| Monte-Carlo (10k) | — | — |
How it works
Value-at-Risk answers one question: “over the next h days, how much could I lose, with c confidence, under normal conditions?” This tool computes it three independent ways and shows them side by side, because the methods make different assumptions and often disagree.
1. Parametric (variance–covariance)
where σh = σannual·√(h/252), μh = μannual·(h/252), z = N−1(c) and φ is the standard-normal density.
Assuming returns are normally distributed, the loss quantile is just a z-score number of standard deviations from the mean. The familiar z-values are 1.282 @ 90%, 1.645 @ 95%, 1.960 @ 97.5% and 2.326 @ 99%. Expected Shortfall uses the normal density at z, divided by the tail probability — the mean of the normal tail. Fast and clean, but it understates risk when returns have fat tails.
2. Historical simulation
No distribution is assumed. Your scaled returns are sorted ascending; the VaR is the loss at the (1−c) percentile, and ES is the average of every return worse than that cutoff. With a pasted series this uses your real data directly; otherwise it ranks a normally-generated sample. The honesty of historical VaR is also its weakness: it can only ever show losses that already happened in the window.
3. Monte-Carlo simulation
We draw 10,000 random period returns from r ~ N(μ·h/252, σ²·h/252), sort them, and read the empirical quantile for VaR with the tail mean for ES — the same percentile logic as historical, but on simulated rather than realised paths. It converges to the parametric answer for a linear position and is the go-to method once a portfolio contains options or other non-linear payoffs.
Square-root-of-time scaling
Volatility scales with the square root of time, so a 10-day VaR is about 3.16× the 1-day figure, not 10×. We annualise on a 252 trading-day year. The rule assumes returns are independent and identically distributed; it breaks down when volatility clusters or returns trend, so treat long horizons with care.
Two-asset portfolios
Diversification lowers risk whenever assets aren’t perfectly correlated. The blended volatility is σp = √(w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂). With ρ < 1 the portfolio vol is below the weighted average of the two — that gap is the diversification benefit, and it feeds straight into a lower VaR.
A threshold
The loss you should not exceed at confidence c — not the worst case. Breaches still happen (1−c) of the time.
The tail's depth
If you do breach VaR, ES is the average loss beyond it. Coherent and Basel-preferred.