# Intuitive way to expalin why mode goes down early when compounding lognormal returns

If you are compounding lognormal returns to model stock price movements, you get a somewhat odd result that makes sense mathematically, but is hard to explain.

Suppose you start with \$100, which we represent as having a zero (or very, very small) risk of being far away (say, the standard deviation is a penny, or even zero, though zero will blow up many software packages). Now, compound it by multiplying it by a lognormal RV that has mean 10% and stdev 15% or so. The issue is what happens right away. After the first multiplication (done by adding the underlying normals that the lognormals are built out of), you will find that the mean of the new lognormal is dead on at \$110. Indeed, it will match the deterministic solution exactly, next being \$121, then \$133.10, etc. So far, so good.

Here is the problem. In order to support the 'expanding' spread to the right in the shape of the distribution, and preserve the right value for the mean, there needs to be a 'mass concentration' below the mean. So, in the example above, while the mode (to start) is very close to \$100, after one cycle the mean is \$110, but the mode is around \$105. How can one explain this intuitively? Clearly, if you invest the \$100 (which is a certain number), and the mean is \$110, how can the most likely outcome be \$5,000 short of that?