Problem $2.$

A rating system features two non$-$default rating classes A and $B$ and a default class $D.$ Assume that we observe the history of $30$ firms over one year, of which $15$ start in class A and $15$ in class B$.$ After three months one A rated company is downgraded to B and stays there for the rest of the year. Over the same period one B rated company is upgraded after $6$ months and remains in class A until the end of the period. Finally, a B rated company defaults after $9$ months and does not recover.

a$)$ Calculate the transition matrix according to the cohort method

b$)$ Calculate the generator matrix for the above example

c$)$ Calculate the transition matrix $($to calculate the matrix exponential you can either break up the infinite sum after $3$ steps or try to achieve a $0\mathrm{.}0001$ accuracy in all matrix components$)$

d$)$ What are the main differences in the two methods? Which approach is better and why?