FM for Actuaries

Loans and Costs of Borrowing 173
5.5 A loan of amount $3,000 is to be repaid with bi-monthly payments of $269.36
for 2 years.
(a) Find the nominal rate of interest and the outstanding loan balance immediately
after the 6th payment is made.
(b) Explain why the answer in (a) is larger than one half of the initial loan
balance.
5.6 A loan of amount $5,000 is to be repaid with P , 0.95P , 0.95 2 P and 0.95 3 P
at the end of the first 4 years. The annual rate of interest is 7%.
(a) Find P .
(b) Find the outstanding loan balance after 2 years.
5.7 A debt of $10,000 is to be repaid with quarterly installments of $4,000,
$3,000, $2,000, $1,000 plus a final payment which pays off the loan after
5 quarters. Interest rate is 12% compounded quarterly. Find the outstanding
loan balance after the second payment.
5.8 A loan is to be repaid with annual installments P t = n − t +1for t =
1, 2, ··· ,n. Show that the prospective method and the retrospective method
give the same outstanding balance.
5.9 Construct an amortization schedule for the loan in Exercise 5.6.
5.10 The following shows an amortization schedule for a loan which calls for
level semi-annual payments over 2 years. Fill in the missing values in the
amortization schedule and calculate the effective rate of interest.
Year Installment Interest paid Principal repaid Outstanding balance
0
0.5 9,088.51
1 2,969.71
1.5 122.38
2 3,151.49 0
5.11 In the amortization schedule in Table 5.1, show that for t =1, 2, ··· ,n− 1
and i>0,
B t > 1 2 (B t−1 + B t+1 ).
Hence, describe the shape of the plot of B t versus t.