Risk Management
Measuring and bounding risk
Risk management converts exposure into a bounded loss budget. Value at Risk states the loss not expected to be exceeded at a given confidence over a horizon; it is not a worst case, is not subadditive, and says nothing about the size of the tail beyond the threshold. Expected Shortfall (CVaR) averages the losses beyond VaR and is coherent. Parametric VaR assumes a distribution, historical VaR replays actual observations, and Monte Carlo simulates paths. Volatility clusters, so GARCH-type updates and stress scenarios complement any single-number measure.
Sizing, drawdown, and leverage
Position sizing is where risk is actually controlled: risk per trade equals position size times stop distance. Drawdown, time under water, and risk of ruin matter more to survival than the Sharpe ratio. Kelly gives the growth-optimal fraction but assumes known edge; practitioners use fractional Kelly because estimation error in edge is large. Correlation tends to rise in crises, so diversification measured in calm markets overstates protection.
Traps and a worked example
Common errors: scaling VaR linearly with time instead of by the square root of time; treating VaR as a maximum loss; assuming normality when returns are fat-tailed; and ignoring liquidity when sizing a stop.
Worked example: a $1,000,000 portfolio with 1.2% daily volatility. One-day 95% parametric VaR = 1.645 x 1.2% = 1.97%, about $19,700. Ten-day VaR, assuming independent returns, scales by the square root of 10: 1.97% x 3.162 = 6.24%, about $62,400. Linear scaling would wrongly give $197,000. Rule of thumb: scale risk by the square root of time, and pair every VaR figure with a stress-loss estimate.