  We investigate the impact of big winner stocks on the performance of active
and passive investment strategies using a combination of numerical and
analytical techniques. Our analysis is based on historical stock price data
from 2006 to 2021 for a large variety of global indexes. We show that the
log-normal distribution provides a reasonable fit for total returns for the
majority of world stock indexes but highlight the limitations of this model.
Using an analytical expression for a finite sum of log-normal random variables,
we show that the typical return of a concentrated portfolio is less than that
of an equally weighted index. This finding indicates that active managers face
a significant risk of underperforming due to the potential for missing out on
the substantial returns generated by big winner stocks. Our results suggest
that passive investing strategies, that do not involve the selection of
individual stocks, are likely to be more effective in achieving long-term
financial goals.
