When General Motors said it had repaid its debt to the U.S. government in April, I called the company out. Now GM is back on the public stage, letting us down once again.

You see, unlike fellow TARP recipients Bank of America (NYSE: BAC) and Citigroup (NYSE: C), who in fact repaid their debts, GM anted up just $7 billion to settle its government loans. Problem was, we lent these guys $50 billion and converted the bulk of the debt into common stock when GM turned out its wallets, revealing nothing but pocket lint.

"Never fear!" responded then-CEO Ed Whitacre. "We're good for the rest, too!" Or words to that effect. What Whitacre actually said was: "I think the stock could be worth a lot and the taxpayers could get all their money, plus."

"Plus" what?
Here's how the math works. Begin with the 60% stake the U.S. government took in GM in lieu of repayment of $43 billion worth of debt. For our 60% stake in GM to be worth $43 billion, the company needs to reach about a $71.7 billion market cap.

Following me so far? Great. So here's the second step: According to the S-1 filing GM made with the SEC in August, the company intends to go public with a grand total of 500 million shares outstanding. Now, the S-1 is still subject to revision and full of I's that need dotting, T's that need crossing, and blanks that need filling in. But earlier this week, the company tentatively filled in the most important blank: the per-share price.

This is part three of today's math problem. In a public statement Wednesday evening, former-CEO and now-Chairman Whitacre averred that "it's a little too early to say, but [the IPO price] is going to be somewhere in the $20 range … $20, $25, something like that would be my guess."

The IPO stress test
Now let's bring all these numbers together, and see whether GM's math passes muster -- and whether its IPO stands a chance of making good with U.S. taxpayers. Based on the numbers we're working with today, 500 million shares x $25 a share (optimistically) must equal at least $71.7 billion. But of course, it doesn't. It equals only $12.5 billion -- missing the mark by a good $59 billion and change. There's no way GM can raise enough money on its IPO at this price to repay what it owes U.S. taxpayers.

From all appearances, this should make GM's IPO dead on arrival.

Stress fractures
How, then, will GM make the math work? To reach the magic number of $71.7 billion on 500 million shares, it would need to set its IPO at $143.40 a share, a price that might scare away potential buyers.

Last year, Berkshire Hathaway (NYSE: BRK-B) faced the same problem when it bought Burlington Northern, and its target company worried that retail investors wouldn't want to own it at the then-current "Berkshire-B" share price of $3,000-plus. The solution Berkshire adopted, and the solution GM management appears to want to copy, was (and is) to split the shares. According to multiple sources, GM intends to take the shares it has and split each of 'em of four ways, resulting in a total share count at IPO of 2 billion shares.

GM = Goofy Math
Now here's the problem: 2 billion shares x $25 a share still equals only $50 billion. That would make the U.S. stake in the company (60%) worth $30 billion at the IPO -- and leave GM $13 billion short of what it needs to pay off taxpayers.

Splitting the shares may be part of GM's solution, but it's not the whole answer. Over the past 12 months, GM had about $124 billion in sales. To reach a market cap of $71.7 billion, therefore, the company somehow needs to convince investors that its shares are worth 0.59 times annual sales. On one hand, that's certainly possible -- Honda Motor (NYSE: HMC), for example, commands a 0.61 price-to-sales ratio.

But on the other hand, GM is no Honda. These days, it might be as good as a scandal-tarred Toyota (NYSE: TM), whose shares sell for 0.46 times sales. But even if investors can be convinced that GM is as good a company as Ford (NYSE: F), Ford shares fetch only 0.36 times sales!

Foolish takeaway
Seems to me that GM has several problems in making a success of its IPO and fulfilling its promise to repay U.S. taxpayers "all their money, plus":

  • First, it must convince investors that its stock is worth nearly twice the P/S ratio of Ford -- a company that, unlike GM, did not just finish running itself into a bankruptcy ditch.
  • Then it has to price its shares at the tippity-top of their projected $20-to-$25 range.
  • And if all that goes well, the company still must find a way to increase the resulting $50 billion market cap by 43% to reach a valuation necessary to repay U.S. taxpayers in full -- let alone the "plus" that Whitacre promised us.

Now, granted, the government doesn't intend to unload its entire stake in the company all in one lump on IPO day. It will sell its shares off periodically and piecemeal, hoping to maximize shareholder-slash-taxpayer value. As a result, GM has a little time to work with -- time to grow that market cap to a size great enough to give it a chance of getting us our money back.

Let's just hope GM uses that time wisely.

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Fool contributor Rich Smith owns no shares of any company named above. Berkshire Hathaway is a Motley Fool Inside Value recommendation. Berkshire Hathaway and Ford Motor are Motley Fool Stock Advisor picks. The Fool owns shares of Berkshire Hathaway. Try any of our Foolish newsletter services free for 30 days.

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