Option Skew — Part 6: The Skewness and Kurtosis for a Lognormal
In this article first I will define skewness and kurtosis and then I will explain how to calculate the skewness and kurtosis for a lognormal distribution.
Understanding the Forecast Statistics
Most distributions can be defined up to four moments. The first moment describes its location or central tendency (expected returns), the second moment describes its width or spread (risks), the third moment its directional skew (most probable events), and the fourth moment its peakedness or thickness in the tails (catastrophic losses or gains). All four moments should be calculated in practice and interpreted to provide a more comprehensive view of the project under analysis. Risk Simulator provides the results of all four moments in its Statistics view in the forecast charts.
Measuring the Center of the Distribution — the First Moment
The first moment of a distribution measures the expected rate of return on a particular project. It measures the location of the project’s scenarios and possible outcomes on average. The common statistics for the first moment include the mean (average), median (center of a distribution), and mode (most commonly occurring value).
The figure below illustrates the first moment — where, in this case, the first moment of this distribution is measured by the mean (μ) or average value.
Measuring the Spread of the Distribution — the Second Moment
The second moment measures the spread of a distribution, which is a measure of risk. The spread or width of a distribution measures the variability of a variable, that is, the potential that the variable can fall into different regions of the distribution — in other words, the potential scenarios of outcomes.
The second figure below illustrates two distributions with identical first moments (identical means) but very different second moments or risks. The visualization becomes clearer in second figure below.
As an example, suppose there are two stocks and the first stock’s movements (illustrated by the darker line) with the smaller fluctuation is compared against the second stock’s movements (illustrated by the dotted line) with a much higher price fluctuation. Clearly an investor would view the stock with the wilder fluctuation as riskier because the outcomes of the more risky stock are relatively more unknown than the less risky stock.
The vertical axis in the second figure below measures the stock prices, thus, the more risky stock has a wider range of potential outcomes. This range is translated into a distribution’s width (the horizontal axis) in the first figure below, where the wider distribution represents the riskier asset. Hence, width or spread of a distribution measures a variable’s risks.
Notice that in the first figure below, both distributions have identical first moments or central tendencies but clearly the distributions are very different. This difference in the distributional width is measurable. Mathematically and statistically, the width or risk of a variable can be measured through several different statistics, including the range, standard deviation (σ), variance, coefficient of variation, and percentiles.
Measuring the Skew of the Distribution — the Third Moment
The third moment measures a distribution’s skewness, that is, how the distribution is pulled to one side or the other. The first figure below illustrates a negative or left skew (the tail of the distribution points to the left) and the second figure below illustrates a positive or right skew (the tail of the distribution points to the right).
The mean is always skewed toward the tail of the distribution while the median remains constant. Another way of seeing this is that the mean moves but the standard deviation, variance, or width may still remain constant.
If the third moment is not considered, then looking only at the expected returns (e.g., median or mean) and risk (standard deviation), a positively skewed project might be incorrectly chosen! For example, if the horizontal axis represents the net revenues of a project, then clearly a left or negatively skewed distribution might be preferred as there is a higher probability of greater returns (the first figure below) as compared to a higher probability for lower level returns (the second figure below).
Thus, in a skewed distribution, the median is a better measure of returns, as the medians for both the first figure below and the second figure below are identical, risks are identical, and hence, a project with a negatively skewed distribution of net profits is a better choice. Failure to account for a project’s distributional skewness may mean that the incorrect project may be chosen (e.g., two projects may have identical first and second moments, that is, they both have identical returns and risk profiles, but their distributional skews may be very different).
Measuring the Catastrophic Tail Events in a Distribution — the Fourth Moment
The fourth moment or kurtosis, measures the peakedness of a distribution. The figure below illustrates this effect.
The background (denoted by the dotted line) is a normal distribution with a kurtosis of 3.0, or an excess kurtosis of 0.0. Excel’s results show the excess kurtosis value, using 0 as the normal level of kurtosis, which means that a negative excess kurtosis indicates flatter tails (platykurtic distributions like the Uniform distribution), while positive values indicate fatter tails (leptokurtic distributions like the Student’s T or Lognormal distributions).
The distribution depicted by the bold line has a higher excess kurtosis, thus the area under the curve is thicker at the tails with less area in the central body. This condition has major impacts on risk analysis as for the two distributions in the figure below, the first three moments (mean, standard deviation, and skewness) can be identical but the fourth moment (kurtosis) is different.
This condition means that, although the returns and risks are identical, the probabilities of extreme and catastrophic events (potential large losses or large gains) occurring are higher for a high kurtosis distribution (e.g., stock market returns are leptokurtic or have high kurtosis). Ignoring a project’s kurtosis may be detrimental.
Typically, a higher excess kurtosis value indicates that the downside risks are higher (e.g., the Value at Risk of a project might be significant).
Definition of Skewness and Kurtosis
The skewness of a series of price data can be measured in terms of the third moment about the mean. If the distribution is symmetric, the skewness will be zero.
where
is the mean of the observations, σ is the standard deviation, and n is the number of observations. A normal distribution always has zero skewness, being a symmetric distribution.
The kurtosis describes the relative peakedness of a distribution. The kurtosis is measured by the fourth moment about the mean. To make things more confusing, there is more than one definition of kurtosis.
Pearson kurtosis is defined as
Fisher kurtosis, on the other hand, is defined as
Fisher kurtosis is thus simply Pearson kurtosis minus 3.
- The normal distribution has a Pearson kurtosis of 3 (Fischer kurtosis of 0) and is called mesokurtic.
- Distributions with Pearson kurtosis larger than 3 (Fisher higher than 0) are called leptokurtic, indicating higher peaks and fatter tails than the normal distribution.
- Pearson kurtosis smaller than 3 (Fischer lower than 0) is termed platykurtic.
- Pearson kurtosis higher than 3 is also called excess kurtosis, or simply “fat tails.” Before calculating skewness and kurtosis from asset prices.
The Skewness and Kurtosis for a Lognormal Distribution
The skewness and kurtosis of a lognormal distribution will vary across different lognormal distributions depending on the volatility and time horizon. The skewness and kurtosis for different lognormal distribution can be calculated by the following expressions:
where
σ is the annualized volatility, and T is the time horizon for our analysis (typically the expiration of a derivative contract). Notice that all lognormal distributions have a positive skewness. In other words, the lognormal distribution is always skewed to the right.
For example, what is the skewness and kurtosis for the stock price in a BSM economy, where the stock price follows a geometric Brownian motion, the asset returns volatility is 30% and the time horizon is three months?
The skewness and Fischer kurtosis of the asset returns are still zero, since the returns are normally distributed, when the asset price is lognormally distributed.
Summary
Ever wonder why these risk statistics are called moments? In mathematical vernacular, moment means raised to the power of some value.
In other words, the third moment implies that in an equation, three is most probably the highest power.
In fact, the equations below illustrate the mathematical functions and applications of some moments for a sample statistic.
For example, notice that the highest power for the first moment average is one, the second moment standard deviation is two, the third moment skew is three, and the highest power for the fourth moment is four.
About the Author
Actuary Roi Polanitzer is the owner and chief valuator of Intrinsic Value. He is one of the three leading experts in Israel in the areas of quantitative finance, option valuation, risk management and financial engineering. He has consulted for accounting firms, financial advisory firms, investigative auditing firms and publicly-traded and privately-held companies in Israel on risk analysis, valuation, and real options, and has written numerous papers and articles on those topics. He is the founder and Chairman of the Israel Association of Financial Valuators and Actuaries. Actuary Polanitzer is also the founder and CEO of the Professional Data Scientists’ Israel Association and the owner and chief data scientist of Prediction consultants. He has an MBA in business administration and BA in economics. He is a certified Financial Risk Manager (FRM), a certified Chartered Risk Manager (CRM), a certified Quantitative Finance Valuator (QFV), a certified Financial and Economic Modeler (FEM), a certified Market Risk Actuary (MRA), a certified Credit Risk Actuary (CRA), a certified Python Data Analyst (PDA), and a certified Professional Data Scientist (PDS).