A firm borrows $500,000 from a bank. Interest is charged on the loan at a rate of 12% per annum compounded quarterly.

(a) Suppose the firm agrees to pay back the loan by making 40 quarterly payments over 10 years. Calculate the required quarterly payment if the first payment is made one quarter after the loan amount is received by the firm?

(b) Suppose instead that the firm agrees to pay back the loan by making 40 equal quarterly payments with the first payment made one year after the loan amount is received by the firm. Calculate the quarterly payment in this case.

A company is running a competition in which the first prize is a perpetuity paying $100 each month. Supposing an interest rate of 9% per annum compounded monthly, how much money will the company need to fund the prize now if the first payment is made

(a) in one month's time?

(b) immediately?

(c) a year from now?

In order to replace a large freezer system in the future, an abattoir plans to deposit 8 equal amounts into a fund at yearly intervals, with the first deposit made immediately, so that the amount in the fund will be $50,000 in 8 years' time.

(a) Assuming the fund earns 7% per annum compounded annually, calculate the annual deposit amount.

(b) Suppose now that although the fund initially earns 7% per annum compounded annually, after 3 years, i.e. just after the fourthdeposit is made, the interest rate unexpectedly falls to 6% per annum compounded annually. If the firm changes its annual deposit to take account of the lower interest rate, what will be the new annual deposit amount for the last 4 deposits?