Is there an investor on the planet that has not been enjoying 2013? With gains across the board, this feels like a real bull market.

Since the end of 2012, the FTSE 100 (UKX) is up 2.9%. Some shares have done far better, most notably the banks. My research suggests that big gains are still to come from bank shares.

I've taken a look at the three most interesting opportunities in the UK bank sector: Barclays (LSE: BARC) (NYSE: BCS.US), Lloyds Banking Group (LSE: LLOY) and Royal Bank of Scotland (LSE: RBS) (NYSE: RBS.US). First of all, the fundamentals:

Company Price (p) 2012 P/E (forecast) 2013 P/E (forecast) Market cap (£m)
Barclays 294 8.1 7.7 35,160
Lloyds Banking 53.5 20.4 13.3 35,820
Royal Bank of Scotland 350 19.5 12.2 20,280

Data from Stockopedia

Barclays

So far in 2013, shares in Barclays are up 12.7%. This has taken the shares to their highest price since April 2011.

If my analysis is right, then Barclays shares could have a lot further to rise in 2013. Despite recent rises, Barclays still trades at a large discount to its peers in both the banking sector and the FTSE 100.

In February, Barclays will report results for the full year 2012. Consensus expectations are for earnings per share (EPS) of 35.3p per share. This puts the shares on a price-to-earnings (P/E) ratio today of 8.1. Considering that Barclays is expected to grow its earnings in 2013, this seems cheap.

Before the financial crisis struck, Barclays shares typically traded on a P/E around 12. If Barclays win back the market's confidence, then 44% of further upside could be enjoyed.

If you are prepared to wait, the gains could be even larger. The consensus 2013 EPS estimate for Barclays is 37.2p for 2013. Applying a P/E of 12 to that estimate and adding the forecast dividends of 7.2p gives a possible return on Barclays shares of 54%.

Lloyds Banking Group (Lloyds)

Shares in Lloyds are up 9.2% so far this year. That only tells part of Lloyds' recovery story. In the last 12 months, the shares are up 102.2%. That's a massive gain in just one year from a company the size of Lloyds.

In my opinion, there were two triggers for those gains. The first was the political progress against the problems in the eurozone. The second was the large reduction in impairments that Lloyds reported throughout 2012. I think that the full effects of these two themes is yet to be reflected in the Lloyds share price.

The strain of the eurozone crisis that infected markets so badly in 2011 first started to worry investors around April 2010. At that time, shares in Lloyds were trading as high as 68.5p. That's 29% ahead of today's price. A recovery to that level would also mean that the government's investment in the bailout of Lloyds was into profit.

If Lloyds can maintain the rate of reduction of impairments then the shares could go even higher. The post-bailout high for the shares is 77p. There remains plenty of room for further share price appreciation if Lloyds can win back the market's confidence.

Royal Bank of Scotland (RBS)

RBS is the bank investors most like to speculate about. This makes the share price vulnerable to large swings. When economic news is good, investors start to guess how much money RBS could make. If things turn bad, people soon start suggesting that RBS may have to raise more capital.

So, I am surprised that RBS' rise so far this year is the least of the three. Since the turn of the year, RBS shares are up 10.5%. In the last 12 months, they have risen 70.6%. That's a large rise but it is still significantly less than Lloyds shares managed.

In recent years, RBS been detailing its net asset value in results announcements. With the 2011 results, RBS' net asset value per share was 501p. Market consensus is for RBS to report EPS of 19.5p for 2012. I estimate that at the end of 2012, RBS could have had a net asset value per share of around 520p.

One of my stock-market rules of thumb is that a profitable company should not trade at a discount to its net asset value. RBS is forecast to report profits for 2012 and 2013. I therefore expect to see the gap between the share price and the bank's book value to narrow.

A 51% rise in RBS share price would be needed for the bank to trade on a par with its net asset value. I will continue to hold my shares and wait.

Sticking with my shareholdings in the banks helped me make back all of my credit-crunch losses and more. If would like to learn how the power of the stock market can help you build your wealth then get the free Motley Fool report, "10 Steps To Making A Million In The Market". The report is 100% free and will be delivered to your inbox immediately. Just click here to start learning now.