问一道题：NO.PZ2016082406000078

问题如下：

Consider the following homogeneous reference portfolio in a synthetic CDO: number of reference entities, 100; CDS spread, $s=150bp$; recovery rate $f=50\%$. Assume that defaults are independent. On a single name the annual default probability is constant over five years and obeys the relation: $s={(1-f)}PD$. What is the expected number of defaulting entities over the next five years, and which of the following tranches would be entirely wiped out (lose 100% of the principal invested) by the expected number of defaulting entities?

选项：

There would likely be 14 defaults and a [3%—14%] tranche would be wiped out.

There would likely be 3 defaults and a [0%—l%] tranche would be wiped out.

There would likely be 7 defaults and a [2 %—3%] tranche would be wiped out.

There would likely be 14 defaults and a [6%—7%] tranche would be wiped out.

解释：

ANSWER: D

The annual marginal PD is $d=\frac{1.5\%}{1-0.50}=3.00\%$. Hence the cumulative PD for the five years is $d+S_1d+S_2d+S_3d+S_4d=3\%(1+0.970+0.941+0.913+0.885)=14.1\%$，where the survival rates are $S_1={(1-3\%)}=0.970$, $S_2=S_1{(1-3\%)}=0.941$, and so on. The expected number of defaults is therefore $100\times14.1\%$, or 14. With a recovery rate of 50%, the expected loss is 7% of the notional. So, all the tranches up to the 7% point are wiped out.