Question $1$: Mortgage Pass$-$throughs $(25$ points$)$

A $4-$year, $5\mathrm{.}5\%,$$$10,000,000$ mortgage pool has been established for the issue of $100$ units of pass$-$through security. The mortgages require annual mortgage payments. The pass$-$through will carry a pass$-$through rate of $5\%$ and a servicing fee of $0\mathrm{.}5\%.$ Consider the following two assumptions for mortgage borrowers' prepayment behaviour as a response to the movement in market interest rate.

$\backslash$table$[[$Pass$-$through A$,\backslash$table$[[$Pass$-$through B$]]],[\backslash$table$[[-$ No prepayment throughout the entire$],[$term of the mortgages regardless of$],[$the market interest rate level.$]],\backslash$table$[[-12\%$ of the outstanding pool balance is$],[$expected to be prepaid at the end of$],[$each year from year $2$ to $3$ if market$],[$interest rate is favourable for$],[$prepayment$.]]]]$

a$)$ Determine the market value of pass$-$through $A$ if the market interest rate for the passthrough is $4\mathrm{.}75\%.(10$ points$)$

b$)$ Determine the market value of pass$-$through $B$ if the market interest rate for the passthrough is $4\mathrm{.}75\%.(10$ points$)$

c$)$ If the market interest rate increases to $5\mathrm{.}75\%,$ what is the difference in price between $B$ and A$?(5$ points$)$