Correlation is not "arbitraged away" because there's no inherent arbitrage in correlation. I think in your first example, you had in mind pairs trading where there is lagged correlation, or negative autocorrelation of the spread. In the second example, mortgages, what you're saying's just wrong.
An important aspect that you also omit to mention is that efficient market means the risk free expectation is the risk free rate. That does not mean that up and down are equally likely. A junk bond is much more likely to go up than down, only it'll go up a... (read more)
Reading this I can't help but think of a character, The Sphinx from Mystery Men, who illustrates your topic. He only signals wisdom:
The Shoveller: I know this guy. Big crime-fighter from down South. Big-league hitter down there.
Mr. Furious: What's his power?
The Blue Raja: Well, he's terribly mysterious.
Mr. Furious: [dismissively] That's it? That's his power? He's mysterious?
The Blue Raja: Well, TERRIBLY mysterious.
The Sphinx: Your temper is very quick, my friend. But until you learn to master your rage...
Mr. Furious: ...your rage will become your master? That's what you were going to say. Right? Right?
The Sphinx: Not necessarily.
(now that wasn't signaling wisdom nor maturity)