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To buy more of a security in which one already has a long position after the price declines. For example, one may buy 500 shares in Company A at $50 per share, and then 500 more when the price declines to $35 per share. One doubles up on a security when one is exceedingly confident in its long term prospects. Doubling up carries relatively high risks.
To purchase an equal number of additional shares when the price of a stock declines. For example, an investor who purchases 500 shares of a stock at $40 per share would double up by purchasing an additional 500 shares if the price of the stock drops. This investment technique can also be applied to short sales. The risk of doubling up is that a bad decision on an initial trade will be compounded when additional shares of the same stock are purchased. Imagine doubling up on high-tech stocks during the dot-com bust.