What are the key properties of the normal distribution I need to know for CFA Level I?

The normal distribution is the single most important distribution in CFA Level I. It appears in portfolio risk, hypothesis testing, confidence intervals, and VaR. Master these properties.

Key Properties:

1. Bell-shaped and symmetric around the mean
2. Mean = Median = Mode (all at the center)
3. Fully described by two parameters: mean (μ) and standard deviation (σ)
4. Tails extend to ±infinity (never touches zero probability)
5. Skewness = 0 (symmetric)
6. Kurtosis = 3 (mesokurtic, or excess kurtosis = 0)

The 68-95-99.7 Rule (Empirical Rule):

Finance Application:

A portfolio has an expected annual return of 10% with standard deviation of 15%.

• 68% chance the return falls between -5% and 25% (10% ± 15%)
• 95% chance the return falls between -19.4% and 39.4% (10% ± 1.96 x 15%)
• 99% chance the return falls between -28.6% and 48.6% (10% ± 2.576 x 15%)

Standard Normal Distribution (Z):

To compare different normal distributions, standardize:

Z = (X - μ) / σ

The standard normal has μ = 0 and σ = 1.

Example:

What's the probability this portfolio returns less than -20%?

Z = (-20% - 10%) / 15% = -30% / 15% = -2.0

Looking up Z = -2.0: P(Z < -2.0) = 0.0228 or 2.28%

Limitations in finance:

• Real asset returns have fat tails (more extreme events than the normal predicts)
• Returns often exhibit negative skewness (larger downside moves)
• The normal distribution underestimates tail risk — this contributed to the 2008 crisis

Exam tip: Memorize the critical z-values: 1.645 (90%), 1.96 (95%), 2.576 (99%). These appear in hypothesis testing and confidence interval questions constantly.

Practice distribution problems in our CFA Level I question bank.