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The fallacy of time diversification

Financial planners, journalists and investment pundits regularly state that as time increases the standard deviation of the annualised return decreases. While this is true (as will be demonstrated below) the use to which it is put is often fundamentally misleading. For instance, the Vanguard group in the US said in one of its publications that"the volatility of stock market returns diminishes markedly over time...Clearly, over time, stock market risk hardly seems excessive even for the most cautious investor" ( Spring 1990 issue of Vanguard's US publication "In The Vanguard"). Financial planners and others argue from the decrease in the standard deviation of the annualised return (an average), that the uncertainty (which they equate either explicitly or implicitly with the standard deviation of the annualised return) of investing in volatile assets like stocks reduces as the time horizon increases.