This article has been adapted from Fool UK, our sister site across the pond.

It's that time of year again -- when the banks tell us how well they are doing. It's almost as if the past three years' near-total financial meltdown didn't happen.

Today, it's the Royal Bank of Scotland's (NYSE: RBS) turn to let the world know just how far it has turned things around.

RBS' generally upbeat results follow similar announcements this week from Barclays, HSBC, and Lloyds Banking Group.

The headlines shout 1.1billion pounds in pre-tax profits for the first half, for the bank that we all had to bail out to the max. So is normal service now resumed, and are the shares about to climb steadily? If only it were that simple.

More art than science
The last time I wrote about RBS, in January, I thought the shares were a contrarian's buy at 35 pence. This proved to be a timely call, though I was, admittedly, short on reasoning and was justifiably criticized for this. But in some ways this is appropriate. RBS is a very difficult animal to get an accurate valuation on, and gut feeling can help one's judgment. This is more art than science, in other words.

Big banks aren't easy to value at the best of times, and the now-shriveled RBS, which is still more than 80%-owned by the taxpayer, is possibly the most difficult of the lot to get an accurate feel for.

Still, we have to try. And on this basis, Friday's half-year results show a once-great bank dragging itself back to the party with not a little chagrin after its former excesses.

Back to big profits
The bank saw its pre-tax profit increase to 1.1 billion pounds in the first half, from 15 million pounds a year earlier. It also reported an operating profit of 1.6 billion pounds compared with an operating loss of 3.4 billion pounds in 2009. 

Overall, RBS managed a small net profit for the half-year, assisted by falling bad debt charges as impairment losses fell to 2.5 billion pounds from 4.7 billion pounds a year ago -- though revenues from its investment banking business fell.

The boss, Stephen Hester, reckons the bank will meet all the targets it has set for the next three years, but he remains cautious over the near term.

As part of the grand plan to put things on a firmer footing and climb back to steady profitability, RBS has had to shed 23,000 jobs worldwide since October 2008, including more than 17,000 in the U.K. -- and earlier this week, it agreed to sell 318 branches to Santander (NYSE: STD).

The main concerns are those of a double-dipping nature. To some extent, the reckless lending of the past that caused the implosion has caused the pendulum to swing too far in the other direction as the banks now face criticism for their lending reticence which is said to be stifling economic recovery. There just isn't the appetite for risk on which they gorged themselves a few years ago.

Nevertheless, today's news clearly shows a bank on the mend. At the time of writing, the shares are ahead 1.25 pence to 53.25 pence, valuing the bank at 31 billion pounds. This looks too cheap by half if the recovery in earnings is to continue. The brokers see 3.8 pence in earnings next year, but it's the far longer term investors need to consider. At its current price, RBS is still valued at less than its earnings per share of a couple of years ago.

"So what?" you may reasonably ask. But in a few years' time, long after this article is a metaphorical fish-and-chip wrapper, how will the once venerable old bank's earnings be looking, what will HM Government interest be, and, more to the point, where will the share price be?

In my opinion, the answers will be higher, zero and much higher, respectively.

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Brian Richards prepared this article for publication. Neither Brian nor David owns shares of any companies mentioned. The Fool has a disclosure policy.