Understanding Skewness and Kurtosis in Returns Distributions | CFA Level I Quantitative Methods

Welcome! Today, we’re diving into skewness and kurtosis, which are essential concepts for understanding returns distributions. Let’s get started by briefly discussing normal distributions and then moving on to the exciting world of skewed and kurtotic distributions.

Normal Distribution Characteristics

A normal distribution has these key features:

• Equal mean and median
• Described by its mean and variance
• 68% of observations within ±1 standard deviation, 95% within ±2 standard deviations, and 99% within ±3 standard deviations from the mean
• Symmetrical shape

Skewness: Measuring Distribution Asymmetry

Not all distributions are symmetrical. Skewness measures the extent of a distribution’s asymmetry:

• Positively skewed: Long right tail with many outliers
• Negatively skewed: Long left tail with many outliers

For a positively skewed distribution, remember:

• Mode: Highest frequency
• Median: Right of the mode
• Mean: Right of the median (larger than the median and mode)

For a negatively skewed distribution, it’s the opposite:

• Mean: Smaller than the median and mode
• Median: Right of the mean
• Mode: Right of the median

Skewness values over 0.5 indicate significant levels of skewness.

Kurtosis: Peakedness of Distribution

Kurtosis measures the degree to which a distribution is more or less peaked than a normal distribution:

• Leptokurtic: More peaked, fatter tails
• Platykurtic: Flatter, thinner tails
• Mesokurtic: Same kurtosis as a normal distribution

A normal distribution has a sample kurtosis of 3. Higher values indicate leptokurtic distributions, while lower values indicate platykurtic distributions. Excess kurtosis adjusts the scale: positive values mean leptokurtic, and negative values mean platykurtic.

Skewness, Kurtosis, and Risk Management

Understanding skewness and kurtosis is vital for risk management. Most securities returns tend to exhibit skewness and kurtosis. In general, greater positive kurtosis and more negative skew indicate increased risk due to a fatter left tail, suggesting a higher probability of extremely large, negative outcomes.

Key Takeaways

Make sure you can:

• Identify different types of skewed distributions and kurtosis effects
• Understand their risk characteristics
• Apply skewness and kurtosis formulas for large samples

That’s it for this lesson on skewness and kurtosis in returns distributions. With a solid understanding of these concepts, you’ll be better equipped to analyze risk and make informed decisions.