Question 3 (7 marks)The residents’ association of a retirement village needs $275 000 to upgrade the village’s facilities.The association obtains a 5-year, interest-only loan for $275 000 that charges a flat interest rate of5.2% per annum.(a) Calculate the quarterly interest payment for the interest-only loan.(1 mark)In order to repay the loan, the association sets up a sinking fund and plans to make monthlypayments of $4116.44 over the 5 years. The sinking fund earns interest at a rate of 4.3% per annum,compounded monthly.(b) Calculate the total cost of the interest-only loan and the sinking fund.(2 marks)(c) (i) Calculate the value of the sinking fund at the end of the first year.(2 marks)(ii) Calculate how much interest the sinking fund will earn in the first year.(1 mark)

(a) To calculate the quarterly interest payment for the interest-only loan, we first need to find the annual interest.

The annual interest is calculated as follows: Principal amount * Interest rate = $275,000 * 5.2$14,300

Since the interest is paid quarterly, we divide the annual interest by 4: $14,300 / 4 =$3,575

So, the quarterly interest payment for the interest-only loan is $3,575.

(b) The total cost of the interest-only loan is the sum of the total interest paid over 5 years and the principal amount.

The total interest paid over 5 years is: Annual interest * Loan term = $14,300 * 5 =$71,500

So, the total cost of the loan is: Principal amount + Total interest = $275,000 +$71,500 = $346,500

The total cost of the sinking fund is the sum of all monthly payments made over 5 years.

The total monthly payments over 5 years is: Monthly payment * Number of months = $4116.44 * 60 =$247,586.4

So, the total cost of the sinking fund is $247,586.4.

(c) (i) To calculate the value of the sinking fund at the end of the first year, we use the formula for the future value of an ordinary annuity:

FV = P * [(1 + r/n)^(nt) - 1] / (r/n)

where: FV is the future value of the annuity, P is the monthly payment ($4116.44), r is the annual interest rate (4.3% or 0.043 as a decimal), n is the number of times the interest is compounded per year (12 for monthly compounding), and t is the number of years the money is invested for (1 year in this case).

Substituting the given values into the formula, we get:

FV = $4116.44 * [(1 + 0.043/12)^(12*1) - 1] / (0.043/12) =$50,330.96

So, the value of the sinking fund at the end of the first year is $50,330.96.

(c) (ii) The interest the sinking fund earns in the first year is the difference between the value of the sinking fund at the end of the first year and the total payments made into the fund during that year.

Total payments in the first year = Monthly payment * Number of months = $4116.44 * 12 =$49,397.28

So, the interest earned in the first year is: Value of the fund at the end of the year - Total payments = $50,330.96 -$49,397.28 = $933.68