On May 6th, a group of heavyweight private equity firms began to circle the wagons. The aim: the largest private equity buyout since the financial crisis began in 2007, a deal estimated to be worth nearly $15 billion.

Discussions continued for almost two weeks, but a final agreement was never reached. The press hasn’t been able to fully explain why the deal was never sealed. Shares lost as much as -20% of their value in the aftermath.

So who were the players in this Wall Street game? Private equity giants Blackstone Group, TPG Capital and THL Partners. Their target? Fidelity National Information Services Inc. (NYSE: FIS), an undervalued payment processing firm. The deal would have would have valued the firm at about $32 a share, but in this case, individual investor’s can own the company for much less than that.

The reasons why the deal fell through are two-fold. For starters, Fidelity National was holding out for a higher price and balked at what it saw as an initial bid from the private equity consortium. Second, credit markets started to freeze up during the negotiations as a result of the debt crisis in Greece that quickly spread throughout Europe. This would have made raising the debt to close the transaction very difficult.

The fact that the buyout was never consummated represents a unique opportunity for individual investors to profit off the backs of the work private equity put into valuing Fidelity National.

The company bills itself as a leading provider of technology for processing banking payments across the globe. The firm’s software is used to allow deposit and lending systems to work across bank branches, the Internet, automatic teller machines and call centers as well as communicate with outside financial institutions.

The appeal for the private equity firms in this deal is clear. These transaction systems throw off stable, prodigious cash flow. Better yet, many of these functions are considered mission-critical and can’t be cut back or turned off during a business downturn. In Fidelity National’s case, the firm produced just more than $500 million in free cash flow off of $3.8 billion in total revenue last year, for a free cash flow margin of 13.2% — impressive in what was an extremely difficult economic environment. Fidelity National provides services to more than 14,000 financial firms, including 40 of the 50 largest global banks.

Significant profit upside exists with Fidelity National. Management has been focused on buying out industry rivals, which is important, as scale helps lower costs per transaction and wrings further profits out of the large and growing sales base. Most recently, it acquired archrival Metavante in October 2009. Sales are expected to exceed $5 billion this year as a result. Combined with cost cutting moves, this should push earnings to $2 per share. A commensurate amount of free cash flow is a conservative estimate as it assumes only flat year-over-year cash flow growth.

Disappointment when the deal fell through combined with a market correction sent Fidelity National’s shares quickly back to the mid-$20 range. In order to fend off another buyout attempt, management is planning to issue debt and repurchase a large amount of the common stock.

In other words, those hoping for upside from an acquisition are out of luck. However, reported earnings will see a boost with less shares outstanding after the recapitalization. This also means that cash flow will go to service the debt and to creditors instead of shareholders for a while.

Still, $32 a share is a very reasonable value to place on Fidelity National’s stock. The price is about +15% above current levels and, as mentioned above, management believes the firm’s value is even higher. Considering the current state of the existing business and upside potential from the integration of Metavante, they’re probably right.

Action to Take –> A combination of further acquisitions and organic growth has led to double digit sales growth during the past five years and steady, high single-digit cash flow growth. Coupled with multiple expansion opportunities, this could indicate double-digit stock returns for investors for at least the next three years.