Theory of Interest Assignment

Faculty of Science Department of Mathematics and Statistics

Theory of Interest – MATH 2600 / STAT 2600 Fall 2017

Assignment 4 Due: Thursday, November 16, 2017 at 08:35 a.m.

This assignment has a total of 8 questions (total of 24 marks). The maximum mark is 20.

It is recommended that before you solve these questions, you attempt Assignments 3 and 4 from Fall 2016 (available on Brightspace) and compare your results with the posted solutions.

1. A loan of $65,000 is to be repaid in 8 bi-monthly payments starting two months from the time the loan is made. The lender charges a nominal annual interest rate of 15% compounded monthly. Construct an amortization schedule in each of the following cases:

(a)1/2 the borrower makes interest-only payments, with a lump sum principal payment at the end;

(b)1/2 the borrower makes level payments of principal;

(c)1 the borrower makes equal payments;

(d)1 the borrower makes payments that form an arithmetic progression: K, 2K, . . . , 8K;

(e)11/2 the borrower makes payments that form an arithmetic progression: 2000 + 7K, 2000 + 6K, . . . , 2000 + K, 2000.

2. Jenn lends Joe $24,000 at a nominal annual interest rate of 18%, compounded monthly. Joe is to repay the loan by making monthly payments over the course of two years starting one month after the loan was made. Find the amount of payment, interest, principal repayment, and outstanding balance for Joe’s 13th payment in each of the following cases:

(a)1/2 Joe repays the loan by making interest-only payments, and a lump sum payment of prin- cipal at the end.

(b)1/2 Joe makes level payments of principal (plus interest accrued).

(c)1 Joe makes equal monthly payments.

3.2 Jenn is looking for a loan for her business. Bank A offers Jenn a 5-year loan at an annual effective interest rate of 3.7% but with a 5% administrative fee (payable when the loan is made). Bank B offers Jenn a 10-year loan at an effective annual interest rate of 5% but with a 3% administrative fee (payable when the loan is made). Which loan is more favourable for Jenn? [Hint: APR.]

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THEORY OF INTEREST MATH 2600 / STAT 2600

FALL 2017 ASSIGNMENT 4

4. A loan of $250,000 is to be repaid in seven annual installments, starting one year after the loan was made. Construct an amortization schedule for the loan in each of the following cases:

(a)11/2 The loan is repaid in equal annual payments, and the effective annual interest rate for the first two years is 4% and then it drops to 2%.

(b)11/2 The first three payments are equal and the last four payments are equal. The first payment is four times the last payment. The effective annual interest rate is 3% throughout the loan. [Hint: The payments are 4K, 4K, 4K, K, K, K, K.]

(c)11/2 The amount of annual payment increases by 5% every year, and the effective annual interest rate is 3%.

5. A loan is repaid in 20 equal annual payments starting one year from the time the loan was made. Suppose that the first interest payment (I1) is $7,877.50 and the 11th interest payment (I11) is $5,011.97.

(a)1 Find the effective annual interest rate.

(b)1/2 Find the amount of the annual payment.

(c)1/2 Find the amount of the loan.

6. Joe lends Jenn money for four years at a nominal annual interest rate of 7%, compounded semi-annually. Jenn is to make semi-annual payments of interest starting 6 months after the loan was made, and a lump sum payment of principal at the end. Jenn accumulates the principal to be repaid in a sinking fund that earns a nominal annual interest of 4% compounded monthly. (Jenn deposits a certain amount into the sinking fund every 6 months.)

(a)1 If Jenn is able to spend at most $5,000 per 6 months on repaying the loan, what is the greatest amount that she is able to repay to Joe using this method? [Hint: First, find the effective semi-annual rate of the loan and the sinking fund.]

(b)1 Suppose that Jenn takes a loan in the amount you found in part (a). Construct a schedule showing for each period of 6 months the following items: Jenn’s outlay, payment to Joe, interest due, balance owed to Joe, deposit to the sinking fund, interest earned, balance of the sinking fund, net interest, and net loan (outstanding balance).

7. Edward has a savings account earning an annual interest of 6%. On January 30, 2016, he deposited $90,000 into his account. On March 23, 2016, he withdrew $26,000. On April 13, 2016, he withdrew $16,000 from his account. On May 12, 2016, he deposited $24,000. Find the balance of Edward’s account on May 31, 2016 using:

(a)1 the Merchant’s Rule;

(b)1 the US Rule.

[Hint: 2016 was a leap year.]

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THEORY OF INTEREST MATH 2600 / STAT 2600

FALL 2017 ASSIGNMENT 4

8. A 12-year loan of $144,000 at an effective annual interest rate of 8% is sold after four years to an investor. If the investor values the loan at an effective annual interest rate of 13%, find how much the investor should pay for the loan, if:

(a)1 the borrower makes equal annual payments, starting one year after the loan was made;

(b)11/2 the borrower makes equal annual payments, starting three year after the loan was made;

(c)1 the borrower makes interest-only payments annually, and a lump sum principal payment at the end;

(d)11/2 the borrower makes level payments of principal annually, starting one year after the loan was made. [Hint: Break down the loan as a sum of loans and use Makeham’s Formula.]

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THEORY OF INTEREST MATH 2600 / STAT 2600

FALL 2017 ASSIGNMENT 4

REMINDER:

No programmable or financial calculators are permitted on the midterm / final exam.

You may use only a scientific calculator.

If in doubt, please contact the instructor.

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