Value at Risk (VaR)
The maximum loss expected over a given horizon at a given confidence level.
Definition
Value at Risk answers the question: "What is the worst I can lose with X% confidence over T days, assuming normal market conditions?" A 95% one-day VaR of $10,000 means there is a 5% chance of losing more than $10,000 tomorrow. It became the regulatory standard after Basel II. VaR has well-known weaknesses: it ignores the size of losses in the tail and assumes the distribution behaves nicely. That is why it is usually paired with Conditional VaR.
Formula
Parametric VaR(alpha, T) = Z_alpha * sigma * sqrt(T) * Portfolio_Value
Historical VaR = alpha-percentile of historical returns
Monte Carlo VaR = alpha-percentile of simulated returns
Z_alpha at 95% = 1.645, at 99% = 2.326Worked example
A $1M portfolio has daily volatility of 1.5%. Parametric 95% one-day VaR = 1.645 * 0.015 * 1,000,000 = $24,675. With Monte Carlo simulation generating 10,000 paths, the 5th-percentile loss might come out higher (e.g. $31,000) because real markets have fatter tails.
How ARIA Analyst uses it
ARIA computes parametric, historical and Monte Carlo VaR in the Risk module and surfaces them in every portfolio analysis at 95% and 99% confidence.
Related terms
Conditional VaR (Expected Shortfall)
The average loss in the worst (1 - alpha)% of cases — a coherent tail-risk measure.
Monte Carlo Simulation
A probabilistic technique that simulates thousands of random scenarios to estimate the distribution of outcomes.
Volatility (σ)
Standard deviation of returns — the most common dispersion-based risk measure.
Maximum Drawdown
The largest peak-to-trough decline of equity over a given window.
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