#### Multi-Year Volatility

We just saw that volatility decreases with time; now for the reasons why. To make things clearer let's use something simpler than the stock market. Pretend that you're playing a coin-flipping game, where you win a dollar every time you get heads, and lose a dollar every time you get tails. If you just play for one flip, there are two possible outcomes (heads = +1, and tails = -1), their average is zero, and the standard deviation is 1.0 (remember the informal definition of standard deviation, that it's how far from the average the data points are likely to lie).

 1 Flip 2 Flips 3 Flips 4 Flips 5 Flips Results

Now suppose you play for two flips. This time there are four possible outcomes. The average is still zero (of course), but something interesting happens to volatility: the standard deviation of the outcomes grows by a factor of the square root of two (1.41) and the standard deviation of the outcome-per-flips shrinks by the same factor (1 / 1.41  =  0.71).

This pattern holds for any number of flips... and it's also the main reason why volatility decreases with time in stock investments: so for example the standard deviation of annualized five year returns is approximately equal to the standard deviation of the one year returns divided by the square root of five:

 Date range: through Time horizon: years Adjust for Inflation

Results
 S&P 500 returns (dividends included): Robert Shiller and Yahoo! Finance

#### The Market is More Interesting than the Coin!

You can see at least two other interesting patterns here, beyond the (dominant) "square root" one:

• With a two year time horizon, volatility shrinks less quickly than the square root rule predicts. This is apparently evidence of price momentum: when the market decides to get greedy or fearful, it stays that way for awhile.

• With a longer time horizon, volatility shrinks more quickly than the prediction. This is "reversion to the mean", the tendency for the market to get dragged away from short term fads and back to economic reality.

This page is saying that the market isn't really like a fair coin; it's more like a coin ridden by a tiny gymnast, who keeps track of the ratio of heads to tails, and keeps making an effort to bring it back to where it belongs (that's mean reversion). Except that the gymnast gets a little dizzy sometimes, and presses for heads or tails a couple-three times too many (that's momentum).

Now if you want to be a short-term stock trader, beating the market ought to be as easy as reading the mind of a tiny, dizzy gymnast... which actually sounds impossible. So we're left with the usual moral: sensible investing is a long-term thing.

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