Returns, volatility and annualization
Simple returns (P₁/P₀ − 1) are intuitive but not additive; log returns ln(P₁/P₀) add across time and behave better in models — the two diverge meaningfully on large moves. Volatility is the standard deviation of periodic returns; the standard annualization is σ_annual = σ_period × √P under independent periods. The same rule sizes short-horizon VaR: σ_h = σ×√(h/252). Caution: sample volatility with small n is itself volatile — practitioners use multi-year windows and rolling estimates.
Fat tails: why normal is not enough
Real returns are skewed and heavy-tailed: positive excess kurtosis means far-from-mean events happen more often than the normal predicts. Consequences: parametric-normal VaR underestimates the tail; Sharpe with plain volatility overrates short-option strategies (low vol, fat tails); and combining fat-tailed assets gives a fatter-tailed portfolio, not an average one. Practical answers: tail-aware metrics (CVaR, max drawdown), non-normal models and historical stress testing.
Worked example: a fat-tailed index
- Daily vol 1%; parametric 99% VaR with z = 2.33 → 2.33%.
- Real data contains −5% to −10% daily moves — multiples of the parametric number.
- Historical 99% VaR on the same sample: say 4.5% — the tail the normal missed.
- Historical CVaR: the mean of the worst 1% — say 6.2%: 'when it breaks, on average how bad?'
نتیجه: Normal models cannot see the tail; historical models see only the past. Pair both with hypothetical stress tests — that combination is defensible.