The Provincial Insurance Company has the choice of investing $$107,000$ in either a mortgage bond with annual payments based on a $10-$year amortization schedule with a maturity of five years at $10$ percent or a five$-$year corporate bond with annual interest payments and a final principal payment also yielding $10$ percent.

Required:

a$.$ Find the duration of each instrument if they are issued at par.

b$.$ If the market rate of interest on each bond fell from $10$ percent to $7$ percent and the durations found in part $($a$)$ remained constant, what would be the new price for each bond? Please show work.