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To use a moving average to buy troughs in a trend, what period should an investor use for the moving average?

You are posing a very good question. It goes to the heart of the problem with moving averages and the reason I find them hard to use for this purpose personally.

Different trends move at different speeds. Also the one trend can change speeds as it develops - either to get steeper or to get shallower.

Essentially, you have to fit the moving average to the trend you are looking at. One trend might work with a 150 day moving average and another with a 100 day moving average.

One problem is that the earlier you are trying to enter a trend, the less history you have to work with and this method is hard to make work. However, if you have a trend that has been around for a year or more, then if it is fairly regular in its speed, you can fit a moving average to it. So, it is really a technique for entering an established trend, not a new one.