Part I ยท Chapter 3
Probability & Statistics Review
Two assets with the same mean return can have wildly different risk. Variance, skewness, and kurtosis reveal what the mean hides.
\( \mu = E[X], \quad \sigma^2 = E[(X-\mu)^2] \)
Learning objectives
Understand mean, variance, skewness, kurtosis.
Compare normal vs fat-tailed distributions.
Read histograms and density curves.
Connect distribution shape to financial risk.
Distribution & Risk
Seed:
Sample statistics
Sample mean: โ
Sample variance: โ
Skewness: โ
Excess kurtosis: โ
Student-t has kurtosis โซ 0; normal has โ 0. Negative skew = downside fatter.
๐ Moments
\[ \text{Skewness} = E\!\left[\!\left(\frac{X-\mu}{\sigma}\right)^{\!3}\right],\quad \text{Excess kurtosis} = E\!\left[\!\left(\frac{X-\mu}{\sigma}\right)^{\!4}\right]-3 \]
ฮผ mean (location)
ฯ spread (volatility)
skew asymmetry: + = right tail; โ = left tail
kurt tail thickness: 0 = normal, > 0 = fat tails
๐ Reading the plots
- The histogram shows the empirical distribution; switching to Student-t produces visible extreme tails.
- For the same ฮผ, two distributions can have very different probability of large losses.
- Sample skewness/kurtosis bounce around their true value โ bigger n = more stable.
โ ๏ธ Pro Tip: What to Avoid
Student says
"Mean return is positive, so the strategy is safe."
Why this is wrong
A positive mean can coexist with huge volatility, fat tails, and devastating drawdowns. Mean is one number โ risk needs the whole distribution.
Correct interpretation
Report mean + ฯ + kurtosis + worst-case quantile (VaR / drawdown). Mean alone is uninformative for risk.
๐ Mini-quiz
๐ Key Takeaways
| Statistic | Meaning | Exam tip |
|---|---|---|
| Mean ฮผ | Average outcome | One-number summary โ never sufficient for risk |
| Variance ฯยฒ | Spread around the mean | Equal to second central moment |
| Skewness | Asymmetry | Stock returns typically slightly negative |
| Excess kurtosis | Tail thickness vs normal | Equity returns: 3โ10 (very fat tails) |