This article was an interesting read. Thanks for posting it.
In my opinion the author is confusing readers by calculating improperly.
I'm not a financial professional so I could be wrong but...
If I want the P/E of a basket of 3 stocks, the proper way to do it is to add the prices, add the earnings and then divide the two sums.
The author in that article is calculating individual ratios and then averaging them which doesn't make any sense.
The author is then going "see we need to do this oddball this thing to the ratio" (invert it)
The reason he's getting the answer he wants by flipping the ratio is because he has chosen his hypothetical ratios to all have equal prices/weighting and only vary the earnings. (price 100)
If he tested with some prices other than 100, it would be evident that his calculation is wrong.
To see how screwed up his math is replace the third stock in his example with something with a price of 0.1 and earnings of 0.1. It will massively shift the average, despite being only a tiny contributor to the return. (his math: mean(3/100,8/100,0.1/0.1)=37% my math: (3+8+0.1)/(100+100+0.1)=5.5%)
It's funny because he does the dividend yield correction correctly, adding the dividend payouts for the numerator and the prices for the denominator, but then he immediately switches his method for PE.
He even talks about "verification math" in which he calculates the right way.
He also mentions capping losses at zero when doing his calculation which seems like a irrationally bullish way to do it. If you're claiming to calculate the P/E for a certain set of companies, I don't see how you can just exclude losses. He should be summing all the positive and negative numbers. I know that if I buy an index fund I'm going to be talking those losses if they happen.
He also further muddies the waters by saying that you should use future earnings. He never uses the proper term "forward PE" for this, instead acting like this change is some innovation.
IMO he does this deliberately because he's trying to get you to compare current forward PE to historical regular or "trailing" PE. (Doing so will create a bullish bias) He even goes as far as to say that using trailing PE is a "flaw".
He also uses the forward earnings to show coverage for the current dividend. (As if using the current earnings to determine the dividend coverage for the current dividend is somehow pessimistic.)
He claims that by fixing "bad math" the PE of the DJIA goes from 29 to 25.1. Since at any instant, the dow implies a certain number of shares of a certain set of stocks, inverting the fraction should not change the result of a proper calculation. (If you sum up all the appropriately weighted earnings and sum up the all the appropriately weighted prices, then you should get the right answer.) Since his example math breaks down if you have unequal weights on your stocks, it doesn't seem like what he's doing would give the right answer for an index like the dow.