You quote a one-day VaR using a normal distribution and the trading floor laughs. Equity returns have fat tails, payoff diagrams kink, and your historical sample lacks the 2008 days that matter. Two tools rescue your VaR. The right return convention and a simulation engine that does not assume normality.
A simple return over one period is . A continuously compounded (log) return over one period is .
The numerical difference is small for daily moves: a 1% simple return is a 0.995% log return. The conceptual difference is large.
Time aggregation: Continuously compounded returns ARE time-additive. A two-period log return equals the sum of two one-period log returns. Simple returns are not, they multiply.
Cross-sectional aggregation: Simple returns aggregate cleanly across portfolios, portfolio simple return is the weighted average of constituent simple returns.
Common mistakes
- Adding simple returns across time. Simple returns aggregate by multiplication, not addition. A two-day total simple return given daily returns of 1% and 2% is (1.01)(1.02) − 1 = 3.02%, not 3%. Trap: choice B is 3%. Catches candidates who add. The exam tests this conversion regularly.
- Using log returns to compute portfolio returns. Portfolio simple return is the weighted average of constituent simple returns. Log returns do not aggregate cleanly across assets. Trap: a 2-asset portfolio with log returns 5% and 10% at 50/50 weights is NOT 7.5% portfolio log return. Convert to simple, weight, then convert back.
- Treating a JB-test failure as proof of fat tails specifically. JB rejects the joint null of zero skew and zero excess kurtosis. Rejection could be from skew alone, kurtosis alone, or both. Inspect the components before concluding fat tails.
Bottom line
- Continuously compounded (log) returns are time-additive across periods; simple returns are not. Convert with R-cc = ln(1 + R-simple). Simple returns aggregate across assets, log returns do not.
- Volatility, variance rate, and implied volatility are distinct objects. Implied is forward-looking from option prices; the others are realized. Square-root-of-time scaling assumes iid returns.
- Equity returns typically show negative skew and excess kurtosis above 2; two moments are insufficient because fat tails dominate risk.
- Jarque-Bera tests joint zero skew and zero excess kurtosis under chi-square with 2 dof; realistic financial samples reject normality almost every time.
Exam shortcut
When a question asks for a multi-period return, check the convention: log returns add across time, simple returns multiply. When it asks for portfolio aggregation, the opposite. Simple returns weight cleanly. When it asks about VaR and gives a Jarque-Bera failure, the normal-based VaR understates tail risk; the answer flags fat tails. For Monte Carlo, antithetic variates work on monotonic outputs only.
The full lesson (about 2,725 words, 18 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
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