Limitations of the Gaussian model
1. Limitations of the Gaussian model
The Gaussian model is popular in finance because it is simple to understand and straightforward to estimate. Also, the parameters have simple interpretations: the location "m" is the average return and the dispersion "s" is the volatility of the returns. Despite these attractive features, academic research has shown that, in reality, financial returns tend to deviate from the Gaussian model mainly for two reasons.2. Deviations from the Gaussian model
Firstly, returns are not symmetrically distributed. Indeed, we often observe small and positive returns followed by a couple of large and negative values. The empirical distribution of the returns is thus left-skewed.3. Deviations from the Gaussian model
Secondly, empirical data sometimes exhibit very large returns, either positive or negative, which cannot be captured by the Gaussian model. You can notice this with stock ABC. When you compare the empirical distribution with the calibrated Gaussian model, you notice the presence of large negative returns for which the associated probability given by the model, in red, is very close to zero.4. Metrics for asymmetry and extremes
Two metrics can be used to assess the presence of asymmetry and extremes in historical returns: the skewness and the kurtosis coefficients.5. Skewness coefficient
The skewness is a measure of the asymmetry of the distribution. The skewness value can be positive, negative, or zero. For symmetric distributions, the skewness coefficient is zero. This is the case for the Gaussian model. When the tail is on the left side of the distribution, the distribution is left-skewed and the skewness coefficient is negative. When the tail is on the right side of the distribution, the distribution is right-skewed, and the skewness coefficient is positive.6. Kurtosis coefficient
The kurtosis coefficient is a measure of tailedness or extremes in the distribution of returns. The kurtosis is 3 for the Gaussian model. A value larger than 3 indicates that extreme returns are more present in the data set than under the Gaussian model. When the value is below 3, this indicates the contrary. As 3 is a reference value, people rather use the so-called "excess kurtosis", which is the difference between the estimated kurtosis and 3.7. Skewness and kurtosis with Google Sheets
Computing skewness and kurtosis coefficients with spreadsheets is straightforward. Given a series of historical returns, to compute the skewness of the series you can use the SKEW() function. It expects one argument, the range of returns, and outputs the value of the skewness.8. Skewness and kurtosis with Google Sheets
For the kurtosis, you can use the function KURT(). It works exactly like SKEW(). Despite its name, it is important to note that the function computes the excess kurtosis, not the kurtosis. The input is the series of returns and the output is the excess kurtosis. In this case, the excess kurtosis is 0.63. This indicates that the tails of the empirical distribution are fatter than the tails of the Gaussian model.9. Pros and cons
Overall, the Gaussian model is great. It is simple to estimate and easy to interpret. But you must be careful when relying on it and understand its limitations. In real life, returns are not symmetrically distributed and exhibit extremes not captured well by the bell-shaped curve of the Gaussian model.10. It's time to practice!
In the next few exercises, you'll dig into these concepts. Time to practice!Create Your Free Account
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