10. Kurtosis in Return Distributions

j. explain skewness and the meaning of a positively or negatively skewed return distribution;

k. describe the relative locations of the mean, median, and mode for a unimodal, nonsymmetrical distribution;

l. explain measures of sample skewness and kurtosis;

What is Kurtosis? Kurtosis is a measure that tells us how more-or-less “peaked” a distribution is compared to a normal distribution, based on the size of the distribution’s tails.

What is “platykurtic”? What does it mean? It is a distribution with smaller tails that is less peaked than a normal distribution. It means a return distribution has more returns with large deviations from the mean.

What is “leptokurtic”? What does it mean? It is a distribution with relatively larger tails and means the distribution is more peaked than a normal distribution. It represents return distributions with deviations clustered around the mean.

What is “mesokurtic”? A distribution with the same kurtosis as the normal distribution is called “mesokurtic.”

Why is kurtosis critical in risk management settings? It is critical because most securities returns exhibit both skewness and kurtosis. Most risk managers focus more on kurtosis than standard deviation because it focuses on the distribution of returns in the tails of the distribution, since this is where the risk is.

What is the mathematical formula to calculate skewness? What does the value mean? \(\frac{\sum (X_i - \mu)^3}{N \sigma^3}\) - \mu = population mean - \sigma = standard deviation - The normal distribution has a skew of 0 because it is symmetric - If skewness is positive, it means that the average magnitude of positive deviations is larger than the average magnitude of negative deviations