# MATH275 Brock University Time Value of Money Problems Worksheet

Department of Mathematics and Statistics

1. In March 2016, Yves decided to save for a new truck. He deposited $500 at the end of every three months in a bank account earning interest at 5% compounded quarterly. He made his first deposit on June 1, 2016. On June 1, 2018, Yves decided that he needed the money to go to college, so on September 1, 2018, rather than making deposits, he started withdrawing $300 at the end of each quarter until December 1, 2019. How much is left in his account after the last withdrawal if his bank account interest rate changed to 6.5% compounded quarterly on March 1, 2019?

2. Nicole has just turned 41 and has accumulated $24 500 in her RRSP. She makes month-end contributions of $400 to the plan and intends to do so until she retires at the age of 60. The RRSP will be allowed to continue to accumulate until she reaches the age of 65. If the RRSP earns 6% compounded monthly for the next 24 years, how much will her RRSP contain when she turns 65?

3. Suzanne had a summer job working in the business office of Blast-It TV and Stereo, a local chain of home electronics stores. When Michael Jacobssen, the owner of the chain, heard she had completed one year of business courses, he asked Suzanne to calculate the profitability of two new large-screen televisions. He plans to offer a special payment plan for the two new models to attract customers to his stores. He wants to heavily promote the more profitable TV.

When Michael gave Suzanne the information about the two TVs, he told her to ignore all taxes when making her calculations. The cost of television A to the company is $1950 and the cost of television B to the company is $2160, after all trade discounts have been applied. The company plans to sell television A for a $500 down payment and $230 per month for 12 months, beginning 1 month from the date of the purchase. The company plans to sell television B for a $100 down payment and $260 per month for 18 months, beginning 1 month from the date of purchase. The monthly payments for both TVs reflect an interest rate of 15.5% compounded monthly.

Michael wants Suzanne to calculate the profit of television A and television B as a percent of the TV’s cost to the company. To calculate profit, Michael deducts overhead (which he calculates as 15% of cost) and the cost of the item from the selling price of the item. When he sells items that are paid for at a later time, he calculates the selling price as the *cash value *of the item. (Remember that cash value equals the down payment plus the present value of the periodic payments.)

Suzanne realized that she could calculate the profitability of each television by using her knowledge of ordinary annuities. She went to work on her assignment to provide Michael with the information he requested.

**Questions: **

a. What is the cash value of television A? Round your answer to the nearest dollar.

b. What is the cash value of television B? Round your answer to the nearest dollar.

c. Given Michael’s system of calculations, how much overhead should be assigned to television A?

d. How much overhead should be assigned to television B?

e. According to Michael’s system of calculations, what is the profit of television A as a percent of its cost?

f. What is the profit of television B as a percent of its cost?

g. Which TV should Suzanne recommend be more heavily promoted?

4. Three months later, due to Blast-It’s successful sales of television A and television B, the suppliers of each model gave the company new volume discounts. For television A, Blast-It received a discount of 9% off its current cost, and for television B one of 6%. The special payment plans for television A and television B will stay the same. Under these new conditions, which TV should Suzanne recommend be more heavily promoted?

5. After winning some money at a casino, Tony is considering purchasing an annuity that promises to pay him $300 at the end of each month for 12 months, then $350 at the end of each month for 24 months, and then $375 at the end of each month for 36 months. If the first payment is due at the end of the first month and interest is 7.5% compounded annually over the life of the annuity, find Tony’s purchase price.

6. A loan of $5600 is to be repaid at 9% compounded annually by making payments at the end of the next 10 quarters. Each of the last six payments is two times the amount of each of the first four payments. What is the size of each payment?

Karim Soltan is shopping for a new vehicle, and has noticed that many vehicle manufacturers are offering special deals to sell off the current year’s vehicles before the new models arrive. Karim’s local Ford dealership is advertising 3.9% financing for a full 48 months (i.e., 3.9% compounded monthly) or up to $4000 cash back on selected vehicles.

The vehicle that Karim wants to purchase costs $24 600 including taxes, delivery, licence, and dealer preparation. This vehicle qualifies for $1800 cash back if Karim pays cash for the vehicle. Karim has a good credit rating and knows that he could arrange a vehicle loan at his bank for the full price of any vehicle he chooses. His other option is to take the dealer financing offered at 3.9% for 48 months.

Karim wants to know which option requires the lower monthly payment. He knows he can use annuity formulas to calculate the monthly payments.

**Questions **

a. Suppose Karim buys the vehicle on July 1. What monthly payment must Karim make if he chooses the dealer’s 3.9% financing option and pays off the loan over 48 months? (Assume he makes each monthly payment at the end of the month and his first payment is due on July 31.)

b. Suppose the bank offers Karim a 48-month loan with the interest compounded monthly and the payments due at the end of each month. If Karim accepts the bank loan, he can get $1800 cash back on this vehicle.

Help Karim work out a method to calculate the bank rate of interest required to make bank financing the same cost as dealer financing. First, calculate the monthly rate of interest that would make the monthly bank payments equal to the monthly dealer payments. Then calculate the effective rate of interest represented by the monthly compounded rate. If the financing from the bank is at a lower rate of interest compounded monthly, choose the bank financing. The reason is that the monthly payments for the bank’s financing would be lower than the monthly payments for the dealer’s 3.9% financing.

(i) How much money would Karim have to borrow from the bank to pay cash for this vehicle?

(ii) Using the method above, calculate the effective annual rate of interest and the nominal annual rate of interest required to make the monthly payments for bank financing exactly the same as for dealer financing.

c. Suppose Karim decides to explore the costs of financing a more expensive vehicle. The more expensive vehicle costs $34 900 in total and qualifies for the 3.9% dealer financing for 48 months or $2500 cash back. What is the highest effective annual rate of interest at which Karim should borrow from the bank instead of using the dealer’s 3.9% financing?

7. A regular deposit of $100 is made at the beginning of each year for 20 years. Simple interest is calculated at *i *% per year for the 20 years. At the end of the 20-year period, the total interest in the account is $840. Suppose that interest of *i *% compounded annually had been paid instead. How much interest would have been in the account at the end of the 20 years?

8. Herman has agreed to repay a debt by using the following repayment schedule. Starting today, he will make $100 payments at the beginning of each month for the next two-and-a-half years. He will then pay nothing for the next two years. Finally, after four-and-a-half years, he will make $200 payments at the beginning of each month for one year, which will pay off his debt completely. For the first four-and-a-half years, the interest on the debt is 9% compounded monthly. For the final year, the interest is lowered to 8.5% compounded monthly. Find the size of Herman’s debt. Round your answer to the nearest dollar.

9. Victor and Jasmine Gonzalez were discussing how to plan for their three young sons’ university education. Stephen turned 12-years old in April, Jack turned 9 in January, and Danny turned 7 in March. Although university was still a long way off for the boys, Victor and Jasmine wanted to ensure enough funds were available for their studies.

Victor and Jasmine decided to provide each son with a monthly allowance that would cover tuition and some living expenses. Because they were uncertain about the boys’ finding summer jobs in the future, Victor and Jasmine decided their sons would receive the allowance at the beginning of each month for four years. The parents also assumed that the costs of education would continue to increase.

Stephen would receive an allowance of $1000 per month starting September 1 of the year he turns 18.

Jack would receive an allowance that is 8% more than Stephen’s allowance. He would also receive it at the beginning of September 1 of the year he turns 18.

Danny would receive an allowance that is 10% more than Jack’s at the beginning of September of the year he turns 18.

Victor and Jasmine visited their local bank manager to fund the investment that would pay for the boys’ allowances for university. The bank manager suggested an investment paying interest of 4.0% compounded monthly, from now until the three boys had each completed their four years of education. Victor and Jasmine thought this sounded reasonable. So on June 1, a week after talking with the bank manager, they deposited the sum of money necessary to finance their sons’ post-secondary educations.

**Questions **

a. How much allowance will each of the boys receive per month based on their parents’ assumptions of price increases?

b.

(i) How much money must Victor and Jasmine invest for each son on June 1 to provide them the desired allowance?

(ii) Create a timeline of events for each of the sons.

(iii) What is the total amount invested on June 1?