A prospective property buyer is considering purchasing a home for $$850000.$ The buyer approaches a bank for a loan to cover the cost of the property.

The bank agrees to monthly repayments, in advance, of interest only for a term of $30$ years. The interest charge is $6\%$ per annum convertible monthly. The capital value of the loan must be repaid at the end of the $30$th year.

a$)$ Show that each monthly interest instalment is $$4250.[1]$

b$)$ Calculate the present value, at time $0,$ of the interest repayments only. $[5]$

The buyer can afford to pay $$5000$ per month in total and decides to invest the remaining $$750$ into a unit trust account at the beginning of each month.

c$)$ Calculate the minimum effective annual rate of interest that the unit trust account will need to earn so that the buyer can use the proceeds to pay the capital value of the loan at the end of $30$ years.