Question

An investment manager has all her bonds divided in three portfolios: P1, P2 and P3. The rst
portfolio (P1) collects the 20 percent of the bonds, the second (P2) 50 percent, and the third one (P3)
the remaining 30 percent. However, 5 percent of the bonds in the rst portfolio are defaulted, 2 percent
in the second are defaulted and, nally, the percentage of defaulted bonds in the third portfolio is 10
percent. If you buy one bond from the above manager, and it turns out to be defaulted, what is the
probability that it belongs to the rst portfolio P1? And what is the probability that belongs to any
of the other two portfolios (P2 and P3)?

Answer 1

$P(1st)=20\%$ ;

$P(2nd)=50\%$ ;

$P(3rd)=30\%$ ;

$5\% \cdot 20\%=1\%$ of all bonds are defaulted in 1st portfolio;

$2\% \cdot 50\%=1 \%$ of all bonds are defaulted in 2nd portfolio;

$10\% \cdot 30\%=3 \%$ of all bonds are defaulted in 3rd portfolio;

$P(a=1st | \,a \, is\, defaulted)=\frac{1}{1+1+3}=\frac{1}{5}$ ;

$P(a=2nd\, or\,3rd|\,a\,is\,defaulted)=1-\frac{1}{5}=\frac{4}{5}$ .