I'm reading ch. 1, and looking at figure 1-2. I'm confused, and I'm thinking there's a typo. The chart shows the probability of losing the bet. As Nate explains, the probability of losing the bet is higher if there is some connection or correlation between whether each homeowner is likely to default. But the chart shows higher probabilities in the column labeled "uncorrelated". Is that a typo, or am I confused about it?
Statistics major here. I'm pretty much positive it's a typo. "Perfectly Uncorrelated" means that they're completely independent, and "Perfectly Correlated" means that they're completely dependent. If they're completely independent, the probability of all failing is going to be (0.05)^5 (0.000003, as explained in the book). If they're completely dependent, it's going to be 0.05 no matter how many mortgages there are because if one fails, they all fail.
In addition to the columns being mislabeled, it looks like the calculations are off too. Or did I forget how to do math? If defaults are independent, the probability that the epsilon pool loses is the probability that at least one defaults: P(1+ defaults) = 1 - P(0 defaults) = 1- 0.95 ^ 5 = 22.6%, not 20.4%
Some of the other numbers don't look right either.
Haven't seen the book, but remember that "uncorrelated" means "there is no *linear* relationship," NOT they are independent. However, if they are independent, there is no relationship of any kind, so they are uncorrelated.
I'm thinking of buying this book as its seems to cover a lot of wide ranging interesting stuff and to be well thought of. one question/comment .. browsing the index I don't see a reference to shrinkage or Stein's paradox ... surely thats a key issue in prediction and isn't that advanced or technical (Efron and Morris wrote a brilliant general intro to the subject in the 70's, I think in Scientific American)
It's an obvious typo because if the underlying mortgages are perfectly correlated as to risk of default, it doesn't matter how the bet is structured for a pool. If all the mortgages have the same risk basis (correlated), it doesn't matter which pool you are invested in; same risk (5%).