Question

You want to buy a car which will cost you $10,000. You do not have sufficient funds to purchase the car. You do not expect the price of the car to change in the foreseeable future. You can either save money or borrow money to buy the car.

• Plan 1: You decide to open a bank account and start saving money. You will purchase the car when you have sufficient savings. The nominal interest rate for the bank account is 6% per annum compounded monthly.

a) You will make regular deposits in your bank account at the start of each month for the next 2.5 years. Calculate the minimum required monthly savings to be deposited into the bank such that you would have sufficient funds to purchase the car in 2.5 years. (1 mark)

b) You will make regular deposits in your bank account at the start of each week for the next 2.5 years. Calculate the minimum required weekly savings to be deposited into the bank such that you would have sufficient funds to purchase the car in 2.5 years.

c) You will make regular deposits of $2,000 at the end of each year. Calculate how long will it take for you to have sufficient funds to purchase the car. (1 mark)

• Plan 2: You decide to borrow $13,000 from the bank and purchase the car now, as well as cover some other expenses. The bank offers two options for the structure of the repayments.

- Option 1: The first repayment will not start until you graduate from university. Therefore, no month-end-instalments will be made for the first 36 months. Then, commencing at the end of the 37^th month, a total of 30 month-end-instalments of $X will be made over the life of the loan. The nominal interest rate is 6% per annum compounded monthly.

d) Calculate X. (2 mark)

e) Your parents agree to help you repay the loan by contributing a lump sum of $1,800 when you successfully graduate from university. Calculate the new value of X. (1 mark)

- Option 2: For the first 36 months (while you are still studying), you will be making month-end-instalments of $Y. Then, commencing at the end of the 37^th month (when you graduate from university), you will double the amount of monthly repayment for the remaining 30 month-end-instalments. The nominal interest rate is 6% per annum compounded monthly.

f) Calculate the value of Y.

Answer 1

a). N (number of deposits) = 12*2.5 = 30; rate = annual rate/12 = 6%/12 = 0.5%; FV = 10,000; Type = 1 (beginning of the period deposits), solve for PMT.

Monthly deposit required = 308.25

b). N (number of deposits) = 52*2.5 = 130; rate (weekly) = [(1 + 6%/12)^(12/52)] -1 = 0.1152%; FV = 10,000; Type = 1 (beginning of the month payments), solve for PMT.

Weekly deposit required = 71.27

c). rate (annual) = [(1+6%/12)^(12)] -1 = 6.17%; FV = 10,000; PMT = -2,000; Type = 1 (end of the period deposits), solve for NPER.

Number of years for 10,000 to be collected = 4.49 years

d). If $X is paid for 30 months then PV of amount paid at the end of 36 months will be X*[1 - (1+0.5%)^-30]/0.5%

This amount has to be discounted back for 36 months so amount at graduation = X*[1 - (1+0.5%)^-30]/(0.5%*(1+0.5%)^36) (using PV of annuity formula)

This amount has to equal the loan amount of 13,000 so we have

X*[1 - (1+0.5%)^-30]/(0.5%*(1+0.5%)^36) = 13,000

X = 13,000/23.226 = 559.72

e). Total amount due after 36 months = amount borrowed*(1+0.5%)^36 = 15,556.85

Net amount due after 36 months = total amount due - lump sum of 1,800 = 15,556.85 - 1,800 = 13,756.85

PV of annuity of X = 13,756.85

X*[1 - (1+0.5%)^-30]/0.5% = 13,756.85

X = 494.96

f). PV of $Y paid for 36 months + PV of $(2Y) paid for 30 months = loan amount

Y/(1+0.5%)^36 + 2Y*[1 - (1+0.5%)^-30]/(0.5%*(1+0.5%)^36) = 13,000

Solving for Y, we get Y = 274.91