To illustrate the increase in variance, let's trade S&P 500 index "shares" and cash from 1/3/1950 to 5/27/2015 (16455 closings = 16454 daily gains). Holding cash for the entire period would result in a nominal gain of 1.0 (eg. no gain). Holding S&P 500 index "shares" for the entire period would result in a nominal gain of 127.6. Holding shares half the time would result in an expected nominal gain of sqrt(127.6) = 11.28.
What happens if we were to hold shares for 8227 consecutive days and cash for 8227 consecutive days (period start chosen at random), as compared to holding shares during randomly chosen 8227 days? See graph.
Again, but only holding shares for one quarter of the days (4113 days shares, 12341 days cash):
Again, but only holding shares for one sixth of the days (2742 days shares, 13712 days cash):
Though the median gain remains similar, the variance of outcomes is higher under daily, monthly and yearly trading regimes. As the second and third plots show, mean reversion is a very long term effect: the increase in variance is only evident at the edges in the last two plots. This demonstrates that markets have a (very) long term tendency to revert to the mean, which is lost when trading often. In practice, this means one should not trade often and should not invest in actively managed funds. It also means holding shares for a very long time is the only way to "fully" compensate for the increased volatility (no surprise here).