In my opinion, National Australia Bank Ltd. (ASX: NAB) shares edge out Australia and New Zealand Banking Group (ASX: ANZ) shares at today's prices.
Who is National Australia Bank?
National Australia Bank is Australia's fourth-largest bank, worth around $89.5 billion. NAB is known for its presence in the Australian business banking market, with more than 21% share of all business banking. It also controls around 14.6% of household lending (think: mortgages, personal loans, etc.).
A troubled past
In my opinion, you'd be crazy to look at a share price chart and make an 'investment' based on what you see. However, one look at the following chart tells us something about NAB's history:

It has a troubled past. NAB's former management expanded the business into the UK and USA, which ultimately severely hampered shareholder returns for more than a decade. As can be seen in the chart above, NAB shares greatly underperformed ANZ and the broader market, or S&P/ASX 200 (Index: ^AXJO) (ASX: XJO).
Looking ahead
Since the global financial crisis (GFC) of 2008/2009, which really hurt NAB's overseas businesses, the bank has been on a mission to divest its foreign operations. It sold down its bad commercial debts, divested both its USA and UK banking subsidiaries, and most of its insurance business.
With its focus now solely on its 'core' business banking and household lending franchises in Australia and New Zealand, the market has rewarded its efforts. From a valuation perspective, the market had 'discounted' the value of NAB shares because of its struggling overseas businesses. By removing them, management unlocked the true value of its local operations.
Why NAB over ANZ?
ANZ has performed very well for shareholders. Under the leadership of former CEO, Mike Smith, ANZ expanded its reach into Asia. However, analysts held concerns for the 'Super Regional Strategy' and in 2015 those fears were realised when Chinese financial markets went into a selling frenzy and ANZ shares plunged. In fact, the ANZ share price fell 38% within 12 months from its high in April 2015.
I would buy NAB shares before ANZ shares because I think NAB shares are a lower risk proposition. What's more, NAB shares are paying a 5.9% fully franked dividend while ANZ shares are offering 5% fully franked.
Buy, Hold or Sell
I think NAB shares could be a welcome addition to income-seeking investors' portfolios at the right price. Indeed, I would buy NAB before ANZ.
However, I do not believe NAB shares are a bargain at today's prices, so I'm choosing to be patient and wait for a better entry price.