If I have current earnings for the SP500 of $160, should trailing or forward P/E be used to calculate the SP500's price?
What is the difference in using one over the other?
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The answer depends on what the future price for the Index will be used for.
Trailing Earnings, or Trailing P/E is used as a consistent and quite reliable metric for Index performance relative to its long-term historical trend, and as such, it is the measure to use when a more quantitative metric is required.
Forward Earnings or Forward P/E is a forecast, an educated and informed estimation. As such, it is useful when attempting to asses Risk. However, Forward P/E depends on factors that are generally not under anyone's control, such as future interest rates, future commodity prices, regulatory norms, geo-political forces, etc. As Master Yoda would say: "The future, easy to see it is; but the future is always changing."
For the purposes of investment strategy, both must be used. Historical P/E will be used to ascertain where current Index price is relative to its historical trend, and Forward P/E will be used to ascertain Risk -- at the same time.
There is no "answer".
Some traders happen to like to look at the forward P/E.
Some traders happen to like to look at the trailing P/E.
Many traders have utterly no interest in the P/E, and have never looked at it for any reason.
Most basically, your sentence says something like "can I calculate the price using..."
You can't, in any way, "calculate" the price of a stock. It's just what someone happens to pay last in a trade.