With the economy struggling after the worst recession on record, an ongoing debt crisis in Europe, a brewing trade war with China, a collapsing dollar, and yet another mortgage scandal, I'm going to make the case that stocks could be at all-time highs by this time next year.

My prediction doesn't require a particularly rosy view of the economy -- just simple arithmetic.

Let's start with this: $103 per share. That's the record high for expected operating earnings for the S&P 500. It was set almost exactly three years ago. Not coincidentally, the S&P 500 reached its all-time high (1565) around the same time.

Now another number: $63 per share. That's the low for expected earnings for the S&P 500, set in May 2009, a month before the bottom of the recession, but two months after the bottom for stocks.

The 39% drop from $103 to $63 is quite a tumble. It's not the worst earnings drop in history, but it is the worst since 1938. And it happened even though the gross domestic product fell only 1.1% over the same period (not adjusted for inflation).

Now, a third number: $92. That's where expected earnings per share are right now. That's a huge gain off the low -- an amazing 46%. That stellar gain took place while GDP grew only 5% (again, not adjusted for inflation, and assuming that third-quarter growth was the same as second-quarter growth).

As I discussed in this column a couple of months ago, this kind of earnings jump happens because of "operating leverage." When a recession hits, earnings are hit far worse than the overall economy. And when the recovery begins, even a modest recovery, earnings come surging back. It's as close to a law of nature as you're ever going to find in investing.

which, as I explained. If 5% GDP growth can trigger a 46% earnings recovery, it doesn't seem like that much of a stretch to think that we could get the further 12% earnings recovery we need, even if GDP growth is downright mediocre.

In fact, that 5% GDP growth from the recession bottom is, itself, downright mediocre. After the horrible recession we endured, it should have come bouncing back much harder. Yet earnings exploded higher, nevertheless. That tells me there's a lot more earnings growth potential ahead.

So here comes the punch-line. (Thanks for your patience.)

If earnings estimates get back to all-time highs, shouldn t stocks get back to all-time highs, too? After all, don't expected earnings drive stock prices?

The problem is they aren't the only thing that does. The other key driver of stock prices is investor sentiment. In other words, stock movement is not just a question of the objective value of expected earnings -- it's also a question of how investors subjectively feel about those earnings.

The simplest way to capture sentiment is through price-to-earnings ratios. As every investor knows, when P/Es are high, that means sentiment is good because investors are willing to pay up for earnings. When P/Es are low, that means sentiment is bad, because investors are not willing to pay much for the same earnings.

This may be a problem. Back in October 2007, when the stock market and expected earnings were both at all-time highs, P/Es were also high. On the same day that expected earnings topped out at 103, the P/E ratio for the S&P 500 was a little above 15.

At the bottom for stocks in March 2009, the P/E ratio fell all the way down to about 10. Now it's a little below 13.

This all means that for stocks to reach new all-time highs, two things have to happen at the same time. First, earnings have to return to 103. Second, the P/E ratio has to get back to 15. That, by the way, explains why earnings have to grow only 12% to get back to all-time highs. But stocks would have to rise 33%.

It's a cinch to me that earnings can make it all the way back. Can the P/E ratio? I'm not so sure. Pessimism is so deep that it wouldn't surprise me to see earnings at new highs but stocks still underwater.

But I wouldn't rule it out. Just last April, when hopes for a robust economic recovery were higher than they are now, the P/E ratio for the S&P 500 got to around 14.5. Why couldn't that happen again, based on nothing more than the obviously fantastic news that earnings will have completely recovered, even if the economy is still somewhat sluggish?

Look, I'm as gun-shy as the next investor. So my purpose here isn't to make a raging bull case for stocks. But I think the case I'm making is a pretty solid reason for some degree of optimism. As I travel around the country talking to my institutional investor clients, I walk through this arithmetic with them, and I am continually struck by how surprised they are by it. The numbers are easily available and very well known. And yet the implications are not well grasped, because it seems completely unthinkable that stocks could ever make new all-time highs, and certainly not within just a year.

But do the arithmetic. It is totally possible. And even if stocks don't make it all the way, you won't want to miss it even if they make it halfway.