VaR (Value at Risk)

Value at Risk (VaR) is the most cited risk metric in retail portals because it compresses a distribution into one number: the maximum loss not expected to be exceeded at a given confidence level. At α = 0.95 over a daily horizon, the 95% VaR is the loss that should be exceeded only 5% of the time. The downside is what it hides: VaR is silent about how bad the loss gets when it does breach the threshold.

Formula (historical)

Sort the historical return series ascending. The α-quantile is VaR_α.

VaR_α = − q_(1-α) ( returns )

On 1,000 daily returns with α = 0.95 the VaR is the 50th worst observation (5% × 1000). For Gaussian-distributed returns the closed form is VaR_α = μ + σ · z_(1-α), but real returns are rarely Gaussian — historical or Cornish-Fisher VaR is preferable.

Where VaR breaks

• ·Not sub-additive: a diversified portfolio can have a VaR larger than the sum of its components — the opposite of what diversification should deliver. This violates "coherent risk measure" axioms.
• ·Tail blindness: VaR is the same whether the worst 5% of days lose −2% on average or −12%.
• ·Optimization instability: minimising VaR can push the optimizer towards concentrated portfolios with cliff-edge risk.

When VaR is still useful

For regulatory disclosure with a fixed standard (the EU PRIIPs KID still uses VaR-based stress tests), for quick communication to a non-technical audience, and as a sanity check alongside CVaR. The honest use of VaR is to report it next to CVaR — the spread is the tail premium.

How MEDGE Capital uses VaR

VaR is reported at both 95% and 99% on every portfolio analysis, alongside CVaR. The risk-decomposition table separates contribution to VaR and contribution to CVaR per holding so you can see which positions drive the tail vs the body of the loss distribution.