An individual borrows $105,000$ from a bank using a $25-$year fixed$-$interest mortgage. The mortgage is repayable by monthly instalments paid in advance. The instalments are calculated using a rate of interest of $5\%$ per annum effective. After ten years, the borrower has the option to repay any outstanding loan and take out a new loan, equal to the amount of the outstanding balance

$$FM$2024$

Page $2$ of $3$

on the original loan, if interest rates on $15-$ year loans are less than $5\%$ per annum effective at that time. The new loan will also be repaid by monthly instalments in advance.

$($i$)$ Calculate the amount of the monthly instalments for the original loan. $[2]$

$($ii$)$ Calculate the interest and capital components of the $25$th instalment. $[3]$

$($iii$)$ The borrower takes advantage of the option to repay the loan after ten years. The rate of interest on mortgages of length $15$ years has fallen to $1\mathrm{.}5\%$ per annum effective at that time. The first loan is repaid and a new loan is taken out, repayable over a $15-$year period, for the same sum as the capital outstanding after ten years on the original loan.

$($a$)$ Calculate the revised monthly instalment for the new loan.

$($b$)$ Calculate the accumulated value of the reduction in instalments at a rate of interest of $1\mathrm{.}5\%$ per annum effective, if the borrower exercises the option.