Financial returns are not independent draws from a fixed bell curve. Across asset classes and decades, a small set of statistical regularities — the stylised facts — appear again and again. Any risk model that ignores them will be confident at exactly the wrong moments.
§01The stylised facts
Cont (2001) surveyed the empirical properties shared by a wide range of financial return series. Among the most important for risk management:
- Heavy tails: extreme returns occur far more often than a normal distribution predicts.
- Volatility clustering: large moves tend to be followed by large moves, of either sign; calm periods persist too.
- Little linear autocorrelation in returns, but strong, slowly decaying autocorrelation in absolute or squared returns.
- Leverage effect: in many equity markets, volatility tends to rise more after price falls than after rises.
§02Modelling clustering: ARCH and GARCH
Engle (1982) introduced the ARCH model, in which today's variance depends on recent squared shocks. Bollerslev (1986) generalised it to GARCH, adding dependence on past variance. The GARCH(1,1) specification remains a workhorse:
The persistence α + β governs how long a volatility shock lasts. With typical daily estimates such as α = 0.08 and β = 0.90, persistence is 0.98 and the half-life of a shock is about 34 trading days — volatility regimes last weeks to months, not days. RiskMetrics popularised a simpler special case, the exponentially weighted moving average with decay λ = 0.94 for daily data.
§03Implications for risk and sizing
- Use conditional, not unconditional, volatility when setting limits; yesterday's calm is a poor guide to tomorrow after a shock.
- Size positions to target risk rather than notional. Moreira and Muir (2017) found that scaling exposure inversely to recent variance improved risk-adjusted performance for several factor portfolios.
- Stress-test beyond the model: heavy tails mean historical and hypothetical scenarios belong alongside parametric measures such as value-at-risk.
- Monitor correlations as well as volatilities; in stress, correlations between assets tend to rise, and diversification can shrink when it is needed most.
