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Yahoo today reported fourth-quarter profits that met Wall Street’s relatively modest expectations, but the numbers also show how Yahoo’s 2009 search agreement with Microsoft is changing the financial picture for the Sunnyvale, Calif., company — and how uncertain the long-term outcome remains.

The background: Under the agreement, Microsoft operates the underlying search and search advertising technology for both companies, and gets 12 percent of the search revenue generated by Yahoo properties and affiliate sites in places where the partnership has taken effect, after accounting for the cost of acquiring traffic.

The Yahoo chart above, taken from its presentation to investors today, shows how that revenue sharing has impacted the company financially over the past year. Yahoo’s 2011 annual revenue was $4.38 billion, down 5 percent, and Yahoo said today that the decrease was “primarily due to the revenue share related to the Search Agreement with Microsoft.”

Why would Yahoo give up such a big slice of its business? The long-term bet, placed by former Yahoo CEO Carol Bartz, is that the company will be able to reduce its expenses by handing over the underlying search technology to Microsoft, while still innovating on the user experience and attracting more users to its search engine.

Reflecting that goal, Microsoft is reimbursing Yahoo for search operating expenses during the transition. Those reimbursements to Yahoo totaled $48 million in the fourth quarter, but they will decline over time.

Together, Microsoft and Yahoo are betting that they will be able to compete more effectively against Google and thereby grow their overall revenue more than either could on its own.

So how’s it going? It’s still early in the 10-year deal, but as one indication, Yahoo’s search revenue was down 3 percent, to $376 million, in the fourth quarter.

On the positive side, the decline wasn’t as much as the double-digit decreases in Yahoo’s search revenue in quarters past. And Yahoo was able to reduce its overall product development expenses by about $77 million in 2011, to just over $1 billion.

However, the latest search stats aren’t promising for Yahoo, showing the company slipping to the No. 3 spot in U.S. market share, with 14.5 percent, for the first time coming in behind Microsoft Bing, which is now at 15.1 percent.

Meanwhile, Google continues to grow, with nearly 66 percent of the U.S. market.

Another complication: Microsoft’s adCenter technology hasn’t been producing as much revenue for Yahoo as expected. That previously prompted Microsoft to extend, for an extra year, the revenue guarantees it provided Yahoo under the deal. Those guarantees are now slated to expire in March 2013 in the U.S. and Canada.

On the other side of the partnership, Microsoft’s Online Services Division posted a quarterly loss of $458 million last week — still deeply in the red, but also an 18 percent improvement over the $559 million loss posted in the same quarter the previous year. Microsoft credited factors including the growth of Bing and the rollout of its Yahoo partnership.

New Yahoo CEO Scott Thompson has a lot on his plate, trying to reverse the fortunes of an aging Internet icon, and it’s hard to imagine this Microsoft deal being easy for him to swallow.

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