Interesting read. While I cannot comment on the value of a CFA Charter in a quant career the field appeals to me personally too. Given my modest progress with the CFA Program (up for the level 1 exam this December) I can only comment from my computer science background. And this tells me that quite some fun can be had with one or the other algorithm. During my phd a colleague derived a very interesting prediction model using little more than modified Hidden Markov Model. Such models may be useful in forecast models. Strictly speaking, programming may not be required, in most cases. Excel will do, learning a few good tools such as SPSS, SAS or MathLab will be enough to explore interesting algorithms. Given the enormous learning curve while studying the CFA Curriculum and the modest knowledge I picked up during my phd I tend to not recommend schools for learning at the moment. Others already suggested studying a few courses up to the level to asked qualified questions. I agree. You may find it interesting to look up a few of these topics: Hidden Markov Models, Bayesian Networks, Neural Networks, Structured Equation Modelling, Kernel Machines. A book I can safely recomend on these topics is “Artificial Intelligence - A Modern Approach” by Stuart Russel and Peter Norvig. Peter makes most of our lives easier at his current employer. The book comes with a comprehensive companion site (http://aima.cs.berkeley.edu/), which may help you to get started with a few hand on exercises. I hope this (rather lengthy) post is of some value to you.
I think it would be interesting to talk about the uses of those thinbgs in finance (my opinion is not as useful as egal thinks), but I am sure that it is not nearly as useful as lots of other quant things the OP hasn’t studied like: a) Probability b) Statistics c) Linear programming d) Fancier OR for portfolio management e) Stochastic processes f) Intro stoch calc (and whatever else they teach in an MFE program)…
The field is certainly rather broad and requires the understanding of more than a few particular concepts. I could only try to mention a few topics I took some personal interest in, given my particular background, without any claim of completeness 
I’ glad you are interested in those things - feel free to post anything interesting about what you can do with Hidden Markov Models or whatever.