Question III: Ratings $(30$ points $\backslash 6$ points each$)$ Suppose there are two ratings categories: A and B$,$ along with default. The ratings$-$migration probabilities look like this for a B$-$rated loan: The yield on A rated loans is $5\%$; the yield on B rated loans is $10\%.$ All term structures are flat $($i$.$e$.$ forward rates equal spot rates$).$ A loan in default pays off $50\%.$ a$.$ You have two loans in your portfolio, both are B$-$rated, $3-$year, $10\%$ coupon bonds $($paid annually$),$ each with $$100$ face value. Compute the possible prices of the loans next year in each ratings bucket $($just before the first coupon is paid$).$ b$.$ Compute next year$$s mean value for each loan. c$.$ Using the mean from part b as the benchmark, compute the $1-$year VaR with $99\%$ confidence interval for each loan $($based on the actual distribution$).$ d$.$ Suppose the returns on the two B$-$rated loans in your portfolio have zero correlation $($they are independent$).$ This means, for example, that the probability that one loan remains B$-$rated and the other defaults equals the product of the marginal probabilities, or $0\mathrm{.}90*0\mathrm{.}05=0\mathrm{.}045.$ Construct the probabilities and values for each possible outcome of your portfolio $($there are $6$ outcomes$).$ e$.$ Based on part d and using the mean of your portfolio as the benchmark, what is the $1-$year VaR for the whole portfolio with $99\%$ confidence interval $($based on the actual distribution$)?$