Definition
Value at Risk is a loss threshold for a portfolio over a set horizon, estimated so that losses are expected to exceed it only with a small stated probability, such as 5%. A 95% one-day VaR of HK$20,000, for example, means a loss larger than HK$20,000 is expected on about 1 day in 20 under the model's assumptions.
How it works
Three common methods are historical simulation (using the empirical return distribution), parametric (assuming a distribution such as the normal) and Monte Carlo simulation. Under the normal approach with zero mean, VaR = z × σ × portfolio value, where z is 1.645 for 95% and 2.326 for 99%. A longer horizon is often approximated by multiplying by √(days), which assumes independent daily returns.
VaR is a quantile, not a maximum: it marks where the worst 5% of outcomes begin and says nothing about how far losses can go beyond that point. Expected shortfall averages the tail losses and is often used alongside it.
Formula / convention
Parametric VaR (mean ≈ 0) = z × σ × Portfolio value, with z = 1.645 (95%) or 2.326 (99%); h-day VaR ≈ 1-day VaR × √h
Illustrative example
Illustrative: a fictional HK$1,000,000 portfolio has daily volatility of 1.2%. Its 95% one-day VaR is 1.645 × 1.2% × HK$1,000,000 = HK$19,740, so a loss above that is expected on roughly 1 day in 20, and VaR does not say whether such a day costs HK$25,000 or HK$250,000.
What to watch for
VaR is not the worst possible loss: it ignores the size of losses beyond the threshold and depends heavily on the chosen confidence level, horizon, data window and method. Models that assume normal returns tend to understate tail risk, so VaR should be checked by counting how often actual losses exceeded it.