Kurtosis: Unlocking the Secrets of Data Distribution
Kurtosis is a statistical measure used to describe the characteristics of a dataset. When normally distributed data is plotted on a graph, it typically forms a shape known as the bell curve. Kurtosis gives insight into how much of the data lies in the ’tails’— the extremities furthest from the mean.
Distributions with high kurtosis have more data points in the tails, bringing the tails in closer to the mean. Conversely, distributions with low kurtosis exhibit fewer data points in the tails, pushing the tails further from the mean.
In the investing world, high kurtosis indicates significant price fluctuations, both positive and negative, far from the average returns. Investors thus may experience substantial price swings, a phenomenon known as kurtosis risk.
Key Takeaways
- Kurtosis assesses the ‘fatness’ of the tails in probability distributions.
- There are three kurtosis categories: mesokurtic (normal distribution), platykurtic (less than normal), and leptokurtic (more than normal).
- Kurtosis risk measures how often an investment’s price changes dramatically.
- Knowing a curve’s kurtosis characteristic reveals the risk levels of the investment being evaluated.
Understanding Kurtosis
Kurtosis measures the combined weight of a distribution’s tails relative to the center of the curve. For normally distributed data displayed via a histogram, most data points reside within three standard deviations of the mean. High kurtosis extends these tails beyond three standard deviations.
Common Misunderstanding
Often confused with the peakedness of a distribution, kurtosis instead describes the shape of a distribution’s tails. For instance, a distribution can have a high peak with low kurtosis or a low peak with high kurtosis. Thus, kurtosis measures ’tailedness,’ not ‘peakedness.’
Formula and Calculation of Kurtosis
Automated Calculation with Spreadsheets
Using Excel or Google Sheets makes kurtosis calculation straightforward. Consider a dataset of sample values: 4, 5, 6, 3, 4, 5, 6, 7, 5, and 8. To compute kurtosis:
=KURT(A1:A10)
When entered in Google Sheets or Excel, this formula yields a kurtosis of -0.1518, indicating a platykurtic curve with lighter tails.
Manual Calculation
Manual calculation is intricate but effective for understanding the essentials of kurtosis. To simplify, consider data points: 27, 13, 17, 57, 113, and 25. Calculations involve obtaining mean, deviations, and moments:
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Compute the Mean: $$ mean = \frac{sum}{count} = \frac{42}{6} = 42 $$
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Find Deviations and Moments: $$ S2 = \sum(y_i - \bar {y})^2 $$ $$ S4 = \sum(y_i - \bar {y})^4 $$
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Calculate: $$ S2 = 7,246 \quad S4 = 26,694,358 $$ $$ M2 = \frac {S2}{n} = 1,207.67 \quad M4 = \frac {S4}{n} = 4,449,059.67 $$
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Final Kurtosis: $$ K = \frac{M4}{M2^2} - 3 = .05 $$
Types of Kurtosis
Mesokurtic (Kurtosis = 3.0)
This type acts like a normal distribution, indicating a moderate level of risk and a balanced presence of extreme values. Stocks falling under this category suggest moderate fluctuation, making them stable investments.
Leptokurtic (Kurtosis > 3.0)
Leptokurtic distributions have longer tails and more outliers, resembling a ‘skinny’ center with ‘fat’ tails. They indicate high risks but the potential for substantial returns.
Platykurtic (Kurtosis < 3.0)
This type exhibits shorter tails and fewer outliers, signaling more stability with past trends showing minimal extreme price movements. Investments here portray lower-than-moderate risk.
Using Kurtosis in Investment Strategy
Kurtosis is critical in understanding an investment’s price volatility. High kurtosis suggests significant price variability, offering both large gains and losses. Low kurtosis indicates more stable returns, preferred for conservative investment strategies.
For instance, a stable portfolio setup could benefit from assets with low kurtosis to mitigate risk. Conversely, momentum-driven portfolios might leverage assets with high kurtosis for occasional larger returns.
Kurtosis vs. Other Financial Metrics
Kurtosis serves various analytical purposes distinct from Alpha, Beta, R-Squared, and Sharpe Ratio. While each evaluates specific attributes of investment performance—return estimates, volatility, correlation with benchmarks, or risk-return tradeoffs—kurtosis offers insights into tail risk and distribution shape.
Why Is Kurtosis Important?
Understanding kurtosis provides insight into the probability of extreme events, known as ’tail risk,’ affecting investments. High kurtosis means rare events happen more frequently, whether positive or negative.
Excess Kurtosis Explored
Excess kurtosis compares against a standard normal distribution. With normal kurtosis assumed to be three, any deviation from this value denotes excess kurtosis.
Kurtosis vs. Skewness
Kurtosis measures data spread in tails relative to mean-centered data, whereas skewness measures relative symmetry around the mean.
The Bottom Line
Kurtosis is a pivotal measure of data distribution, offering key insights into investment risks. For investors, understanding kurtosis helps manage tail risk by predicting the frequency of ‘infrequent’ occurrences in price returns.
Related Terms: Alpha, Beta, Sharpe Ratio, Tail Risk.
References
- Microsoft. “KURT Function”.
- Google Docs Editors Help. “KURT”.
- St. Olaf College. “Sample Size”.
- Statistics Canada. “4.5.3 Calculating the Variance and Standard Deviation”.
- University of California Los Angeles. “FAQ: What’s With the Different Formulas for Kurtosis?”
