  We introduce new mathematical methods to study the optimal portfolio size of
investment portfolios over time, considering investors with varying skill
levels. First, we explore the benefit of portfolio diversification on an annual
basis for poor, average and strong investors defined by the 10th, 50th and 90th
percentiles of risk-adjusted returns, respectively. Second, we conduct a
thorough regression experiment examining quantiles of risk-adjusted returns as
a function of portfolio size across investor ability, testing for trends and
curvature within these functions. Finally, we study the optimal portfolio size
for poor, average and strong investors in a continuously temporal manner using
more than 20 years of data. We show that strong investors should hold
concentrated portfolios, poor investors should hold diversified portfolios;
average investors have a less obvious distribution with the optimal number
varying materially over time.
