Portfolio Risk Simulator — Monte Carlo VaR

Monte Carlo Portfolio Risk

Horizon 63 d

Expected return μ (annual) 8%

Volatility σ (annual) 20%

Confidence level

Trials run

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Mean return

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VaR (loss)

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Expected Shortfall

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Std dev

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P(loss)

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How it works

Each simulated trial draws a full geometric Brownian motion price path for the portfolio, one trading day at a time. Daily log-returns are normally distributed, so compounding many small random steps produces the familiar lognormal spread of possible outcomes by the end of the horizon.

r(t) = (μ − ½σ²)·dt + σ·√dt·Z,  Z ~ N(0,1)
V(t) = V₀ · exp(Σ r)
VaR_α  = −Quantile(R, 1−α)
ES_α   = −mean(R | R ≤ −VaR_α)

Value-at-Risk (VaR) — the loss that will not be exceeded with probability α, read directly off the simulated return distribution (empirical quantile, no normal-distribution shortcut).
Expected Shortfall (ES) — the average loss given that the loss already exceeds VaR — the tail risk VaR itself doesn't show.
μ / σ — the assumed annual drift and volatility feeding every simulated path; higher σ fattens the spread of outcomes and pushes VaR/ES further from zero.
Trials accumulate live — the histogram and VaR/ES lines settle down as the Monte Carlo estimate converges toward the true distribution.

Real-world relevance: this empirical Monte Carlo VaR/ES workflow is the same one used by bank trading desks and asset managers to size risk limits and capital buffers — it captures fat tails and non-normal effects that closed-form parametric VaR misses.

Equity paths Loss tail (≤ −VaR) VaR / ES markers

Portfolio Risk Simulator — Monte Carlo VaR & Expected Shortfall