  Given a geometric Brownian motion wealth process, a log-Normal lower bound is
constructed for the returns of a regular investing schedule. The distribution
parameters of this bound are computed recursively. For dollar cost averaging
(equal amounts in equal time intervals), parameters are computed in closed
form. A lump sum (single amount at time 0) investing schedule is described
which achieves a terminal wealth distribution that matches the wealth
distribution indicated by the lower bound. Results are applied to annual
returns of the S&P Composite Index from the last 150 years. Among data analysis
results, the probability of negative returns is less than 2.5% when annual
dollar cost averaging lasts over 40 years.
