Suppose we have constructed an MBS CDO which has four primary tranches: an AAA$-$rated tranche $(78\%$ of the principal$),$ an A$-$rated tranche $(10\%$ of the principal$),$ a BBB$-$rated tranche $(10\%$ of the principal$),$ and a not$-$rated tranche $(2\%$ of the principal$).$ The BBB tranche is further divided into three sub$-$tranches: an AAA$-$rated sub$-$tranche $(75\%),$ a BBB$-$rated sub$-$tranche $(20\%$ of the principal$),$ and a not$-$rated tranche $(5\%$ of the principal$).$ The total value of the principal is $$100$ million. a$)$ Depict the structure CDO $($split into tranches and sub$-$tranches$)$ visually.b$)$ Calculate the total losses to each tranche and sub$-$tranche if the bad debt ratio is $($i$)5\%,($ii$)10\%,($iii$)15\%,($iv$)30\%.$