FM for Actuaries

178 CHAPTER 5
5.33 A $200,000 real estate mortgage is to be repaid with monthly payments at
12% interest convertible monthly over 30 years. Calculate
(a) the principal portion in the 100th monthly payment,
(b) the total principal repaid in the 100th to 200th monthly payments,
(c) the sum of the present value of the interest payments made in the 100th
to 200th monthly payments.
Comment on the results in (b) and (c).
5.34 Stephen takes out a 15-year loan at 6% effective, to be repaid with annual
repayments at the end of each year. Each loan repayment is 10% more than
the previous year’s repayment. Let P be the first payment.
(a) Show that for t =0, 1, ··· , 15,
B t =25P
[ (1.1)
15
(1.06) 15−t − 1.1t ]
.
(b) Show that negative amortization takes place only in the first 2 years.
(c) It is given that the principal portion of the 9th repayment is $1,132.29.
Find the interest portion of the 5th repayment.
5.35 Betty borrows $1,000 from Ada for 2 years at a nominal rate of 12% compounded
quarterly. What is the difference in interest earned by Ada if Betty
repays with semiannual level installments instead of quarterly level installments?
5.36 June takes out a 20-year loan at an effective rate i to be repaid with annual
payments. The payments are 7 at the end of year 1 to 6, 13 at the end of year
7 to 13, and 9 at the end of year 14 to 20. If the interest portion of the 7th
payment is twice as much as that of the 14th payment, find i.
5.37 A man borrows a housing loan of $500,000 from Bank A to be repaid by
monthly installments over 20 years at nominal rate of interest of 5% per
year. After 5 years Bank B offers the man a loan at rate of interest of 4.5% to
be repaid over the same period. However, if the man wants to re-finance the
loan he has to pay Bank A a penalty equal to j of the outstanding balance. If
there are no other re-financing costs, what is the value of j such that the man
is indifferent to the two options?