Bayes Formula

CFA Level I Quant
#1

ok, so i found the link below and im sure this has been explained many times, how can someone try to reexplain the baye formula? I understand its looking at probability with additional information.

http://www.analystforum.com/forums/cfa-forums/cfa-level-i-forum/91307380

Bonds rated B have a 25% chance of default in 5 years. ABonds rated CCC have a 40% chance of defauly in 5 years. A portoflio consists of 30% B and 70% CCC rated bonds. if randomly selected bond defaults in 5 years, what is the probability that it was a B rated bond?

The final step is 0.075/0.355 = .211.

How is your mind processing to get to this point? And you just divide one by another, but the formula below, you divide and then multiply.

Bayes = (probabilty of new info for a given event / unconditional probabilty of a new informaiton) * Prior probability of the event?

THANKS!

#2

Bayes’ Formula is nothing more than writing P(XY) two ways:

P(X|Y)P(Y) = P(XY) = P(Y|X)P(X)

P(X|Y) = P(Y|X)P(X) / P(Y)

The events are:

  • X is having a B-rated bond
  • Y is having a bond that defaults

P(X) = 0.3 (30% of the portfolio is B-rated bonds)

P(Y|X) = 25% (there’s a 25% chance of a B-rated bond defaulting)

P(Y) = (0.3 × 25%) + (0.7 × 40%) = 0.355 (30% of the bonds have a 25% chance of defaulting, 70% of the bonds have a 40% chance of defaulting)

P(X|Y) = P(Y|X)P(X) / P(Y)

= 0.25 × 0.3 / 0.355

=0.211

If you draw a binomial tree, Bayes’ Formula problems become trivial.