One of the things that I keep coming back to is probability estimates, and quantifying risk. (This is one motivation for reading Mandelbrot, as he forces me to think again about these things.) Conventional finance has such measures (the volatility of stock prices of a stock, or index for example), but I'm not sure how reliable it is.
Part of the things we studied in the theory seminars I've attended is the study of random walks. There are a lot of flavors, apparently, and it all boils down to the probability distributions that you use for each step. Conventional finance calculations seem to rely on Gaussian distributions, and for these, experimental estimates of probability can be obtained by taking the mean and standard deviation. From these, the sophisticated machinery of confidence intervals and parametric tests emerge.
The world, however, is not so nice. If, for example, the underlying distribution for the steps in a random walk are Cauchy distributions (an example of a fat-tailed distribution), then the probabilities of very large steps are more likely than an experimental measurement of mean and variance will indicate. In finance, these large steps either produce great fortunes or ruin.
And yet, I keep procrastinating on the studies of probability and statistics that I need.
Still, I need to dust off those finance books and random walk books, so that I can start reading and rereading, as well as calculating. Some of the finance material has applications after all. But I'm not sure about rereading; sometime, it's more fun to read enough to understand the problem, and then work out the solution, than it is to read both.
I do have lots of questions to think about: what determines the risk premium for stocks over bonds? How do we measure the performance of an index fund relative to the index it's supposed to follow? (I want quantitative measures, and a suitable motivation for the definition of the measure.) How do we know if a fund consistently beats the index? (The problem is finding the right definition of "beat the index")