Quick question: Is BASF’s day-to-day rate of stock returns (shown below) distributed normally?
Yes? In theory it isn’t! In theory rates of stock returns follow a lognormal distribution as I have shown in “On the distribution of stock return”. Unfortunately, most people don’t take the lognormal distribution serious, although it is very often at work! This article shows how the lognormal distribution arises and why its shape sometimes mistaken for a normal distribution.
Relationship between normal and log-normal distribution
A log-normal distribution arises when several random variables are multiplied. Let and be random variables. Then is normally distributed and is logarithmically normal distributed.
In case we can easily show the relationship between both normal and log-normal distribution. Just apply the logarithm laws.
As you’ve seen, exponentiating a normal distribution leads to the log-normal distribution. The conclusion is also that each log-normal distribution has its underlying normal distribution! For those of you who like graphics, I illustrated the transformation below.
Higher values of the underyling normal distribution get more spread out when exponentiated. Hence, the log-normal distribution is positively skewed (as show in the picture above).
Risk of confusion
Now, how could it be that both distributions can be confused with each other? To show that, I produced a normal distribution with mean zero and an relatively small standard deviation. See the transformation in this case:
Small sections of a function graph can reasonably be approximated by a straight line. In this special case the derivative of the e-function at equals 1. That is why the underlying normal distribution and the resulting log-normal distribution look almost identical. An example from real life? Well, daily rates of returns are closely centered around a mean of 1. That is, the mean of the underlying normal distribution lies around zero. Hence, return rates might be distributed log-normally according to their distribution’s shape!
You should bear this effect in mind when working with small values. But you are not much safer with high values either. As the mean of the underlying normal distribution increases, its standard deviation just has do decline to still achieve this effect.