Appendix One: Derivation of Median Returns
Let Xi=1+ri, where ri is the return in period i.
Assume that returns are distributed lognormally and are serially independent (which are standard assumptions supported by the Central Limit Theorem and market efficiency, respectively).
(A1) Therefore, 
(A2) Geometric return over N years is defined as: 
(A3) So its log is: 
The expected value of this is:
(A4)
ln(1+gN) is normally distributed (because it is the sum of ln(Xi), which are normally distributed). Therefore it has a symmetrical distribution and so its median equals its mean. So:
(A5)
That is, half of the distribution of ln(1+gN) is below μ. Therefore, half of the distribution of (1+gN) is below eμ: Hence:
(A6)
and 
Therefore, although the expected geometric return declines as the time horizon increases, the median geometric return is a constant, invariant to the time horizon. It is the same as the median arithmetic return (because g1=r1), and it is less than both the expected geometric and expected arithmetic return.
It was noted above that the expected size of a stock over time (E[Sn]) is calculated by compounding the expected annual arithmetic return over the time horizon: E[Sn]=(1+E[r])n. A corresponding derivation to that used in this appendix can be used to show that the median stock also grows exponentially: M[Sn]=enμ.
