First, to understand this rather geeky entry, you should go here first, then here. Jeremy Siegel argues that the price-earnings ratio for the S&P 500 Index is badly calculated and would be more accurate if weighted by the market capitalization of its members. He correctly notes that the large losses of a few firms get added to the returns of healthy firms, thus dragging down overall profits and inflating the overall P/E ratio. Because of this, Siegel says stocks look more expensive than they really are. Note: If you’re lost already, see the next paragraph.

(Market capitalization equals number of shares outstanding times the price of each. Price/earnings ratio refers to the price of the stock divided by the earnings per share.)

Now let’s dispense with the first obvious objection: even if the P/E ratios are high now, we’re calculating them using the same formula as always, so we’re comparing apples to apples. If you applied Siegel’s weighting method retroactively, there’s a chance that that “historical average of 15” that he cites (for the index’s P/E) could shift downward to say 12, so stocks may still be far from cheap. In any event, let’s shove this objection aside, because his math fails anyway.

In his Yahoo! article, he gives a concrete example that we can parse to see where the train comes off the tracks (bold mine):

Here is where the distortion comes in. Exxon Mobil has a market value of $350 billion, while AIG's value is now a mere $15 billion (and it was only $5 billion a month ago). That means that the average investor owns more than 20 times as much Exxon Mobil stock in their portfolio as AIG stock, so that for the average portfolio of those two stocks, the oil giant has over a 95 percent share and AIG has less than a 5 percent share.Now, if you’ve got a really quick mathematical mind (mine isn’t; it took me a while to realize this), you’ll wonder about a contradiction hidden in this example. The market cap, after all, is simply the sum of what all investors have paid for all the shares. So investors as a whole would be looking at an infinite P/E ratio for Exxon Mobil and AIG combined, because of AIG’s huge losses. But Siegel’s saying that your investment, even when it’s split to reflect the exact proportions of each company’s market cap, should have about a 9 P/E ratio.

S&P says that an investor holding 95 percent of his portfolio in Exxon Mobil and 5 percent in AIG has negative aggregate earnings and an infinite price-to-earnings ratio because the losses of AIG are greater than the profits of Exxon Mobil, no matter how much you hold in each. S&P would say this even though 95 percent of your portfolio is in Exxon Mobil, a stock that sells for less than 8 times its earnings.

My methodology would weight the $45 billion earned by Exxon Mobil by 95 percent and the $99 billion loss of AIG by 5 percent to obtain a weighted average earnings of $39 billion for the portfolio. With a weighted average market value of AIG and Exxon Mobil of $335 billion, this would lead to approximately a 9 P/E ratio for the portfolio, not the infinite P/E computed by Standard & Poor's.

Okay, that’s a bit confusing so let me try this analogy. You thoroughly mix the shares of Exxon Mobil and AIG in a large bowl. Anything you extract from the bowl (a teaspoonful, a cupful, whatever) is composed of about 95 percent Exxon shares, 5 percent AIG. Now the entire bowl has an infinite P/E ratio. But Siegel is saying that the proper P/E ratio for what you take from that bowl as an investor should be “weighted” to equal roughly 9.

So you may be scratching your pate, thinking “Where the heck did all that astronomical P/E go?” Answer: nowhere. Siegel’s math makes a subtle slip-up.

Just assign hypothetical numbers to his example. Let’s say you have $100 billion to put into Exxon and AIG shares and split the money according to their S&P weighting. Here’s where Siegel’s math crumbles: that amount of money buys proportionately much less of the larger market cap company (Exxon), so the need for readjusting the P/E's through a weighting process just washes out.

Try it with real math, assuming a mega-large investor, to keep the calculations easier: $95 billion would buy you about one-third of Exxon’s stock and the remaining $5 billion would buy about one-third of AIG’s stock. So (theoretically) you are entitled to one-third of Exxon’s $45 billion of earnings ($15 billion) but you also have to eat one-third of AIG’s losses of $99 billion ($33 billion).

So you have ... a P/E of infinity, just as the S&P calculation suggests.

But there’s a larger point where I think Siegel is correct: namely, that we have a large basically insolvent company at the bottom of the S&P that is behaving like a black hole, sucking in massive amounts of money. These are taxpayer funds, so investors are largely getting a free ride. The shareholders don’t absorb that fat loss; Uncle Sam does. So the S&P is cheaper than it looks thanks to the U.S. government.

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