Weight P/E by P*E ?

A while ago, Jeremy Siegel attacked P/E ratio calculated by S&P for overestimating the real value (here and here. HT Greg mankiw's Blog).

S&P are now using the following formula:


where V_i is the market value and E_i is the earning of issue i, and PER denotes P/E ratio.
Siegel asserts that S&P should weight both market value(numerator) and earning(divisor) by market value, such as:



Siegel says that this weighting will alleviate the problem of overestimating the overall P/E ratio, because the huge loss of the company like AIG will be downplayed by its relatively small market value, in comparison with S&P formula .


However, when you transform S&P formula as follows, you can see that it is also a kind of weighted sum:



That is, individual P/E ratio is weighted by earning.


When you look at Seigel's formula, it can be re-written as follows:



That is, individual P/E ratio is weighted by the product of value and earning.

It can also transformed as follows:


Individual P/E ratio is weighted by the product of P/E and squared earning.

From these transformations, you can see that the meaning of Siegel's formula is much harder to grasp than S&P one.


For Siegel's purpose, maybe it's better to weight individual earning by market value, instead of by earning in S&P formula, as follows: