Python Tutorial : Mean, Variance, and Normal Distributions

Key Takeaways

since financial risk can simply be thought of as a measure of uncertainty of future returns it is good practice to examine the natural distribution of returns in order to better understand the risk of investments probability distributions have common properties known as moments which can be analyzed and compared to other distributions the first moment is the mean or mu which is essentially the average outcome of a random sample of the distribution the second moment is the variance which is simply the standard deviation or Sigma squared this is a measure of the variability in outcomes the third and fourth moments are less commonly used the third moment skewness is a measure of the tilt of a distribution while the fourth moment kurtosis is an important measure of the thickness of the tails of a distribution we'll get to those later there are many different types of common distributions you might be familiar with the Gaussian normal distribution if you've taken statistics class before the standard normal distribution is a special case of the normal distribution when Sigma equals 1 and mu equals 0 all normal distributions tend to have a skewness near 0 and a kurtosis near 3 financial returns on the other hand tend to have positive skewness and a kurtosis higher than 3 this means financial returns tend to have a higher probability of both outliers and of positive returns than a normal distribution we'll talk more about kurtosis and skewness and later here we compare the daily returns of IBM stock with a sample from the normal distribution with the same mean and standard deviation there are statistical tests for normality such as Shapiro Wilk and jock Berra but a high kurtosis or large skewness is a simple indicator of non normal returns you can use Python to estimate the moments of a return distribution to calculate the average daily return of an asset or mu assuming you have already calculated daily returns simply use numb PI's mean function to compute the average now if you want to analyze that number you'll first need to add 1 to the decimal before raising the quantity to the power of 252 which is the typical number of trading days in a year for example a daily return of just 0.03 percent works out to an annualized return of seven point eight five percent when it is compounded every day for 250 two trading days in a row the second moment of the distribution is variance which is this square of the standard deviation or volatility volatility is one of the most important concepts for risk management in finance some traders simply refer to it as volve for short the fundamental takeaway is simple an investment with higher volatility is viewed as a higher risk investment volatility is just another measure of the dispersion of returns just like variance you can easily calculate the standard deviation returns using the STD function if the returns are daily returns then this function will output the daily standard deviation in order to calculate the variance simply square the standard deviation it's very important to understand that volatility scales with the square root of time to properly scale the daily volatility of an asset simply multiply the volatility by the square root of the number of trading days in the year this is easy to accomplish using numpy by simply multiplying the standard deviation of daily returns by the square root of 252 time to put this into practice