Question

empirical vs. normal distribution of returns?

Answer 1

For understanding Empirical and Normal distribution of returns it is important to understand that the two concepts are not mutually exclusive of each other, rather empirical rules apply to the normal distribution concept and can be seen as the boundary conditions in the same. To better understand, let us break the answer and understand what each of the two mean:

Normal Distribution

It is a symmetric probability distribution describing how certain values or variables are distributed. In structure, the values are clusted around a central peak and the curve is bell shaped. The values that go further awar from the central peak (called mean), taper off equally in both the directions. So, if i draw a line right in the middle of the bell-curve, i obtain two equal and symmetric halves. The center or the mean is the most common/probable value in the distribution. For example, height of boys in a class can follow a normal distribution.

There are two key elements of a Normal Distribution: -

1. Mean

2. Standard Deviation (measures variation i.e. how far from the mean, do the values lie- in a way it is the distance between the average value of the curve i.e. the mean and the observed values)

Empirical Rule

This concept determines the percentage of data that lies within a specified standard deviation number within the distribution. This is a boundary rule established for Normal Distribution and can be seen as an extension of the concept.

Under Empirical rule, the following parameters have been established:

68% data lies within +/- 1 standard deviation of the mean

95% data lies within +/- 2 standard deviation of the mean

99.7% data lies within +/- 3 standard deviations of the mean

In a nutshell, Normal distribution of the returns represents the overall returns and the distribution along with average values of the returns and the correspondong data. While the Empirical rule helps us understand the percentage of data lying within specified standard deviations or variations from the average or mean.