Question

I want to buy a boat today today but have realised that if I take out a loan I can only repay $150 quarterly, with payments made at the beginning of each quarter, over the next 5 years. How much can I spend on my boat if the interest rate is 16% per annum compounded quarterly ?

Answer 1

Here, the payments will be same every quarter, so it is an annuity. And since the cash flows will start at the beginning of each quarter so it will be termed as an annuity due. We need to calculate the present value of annuity due here. For calculating the present value of annuity due, we will use the following formula:

PVAD = P * (1 - (1 / (1 + r)n / r) * (1 + r)

where, PVD is the present value of annuity due, P is the periodical amount = $150, r is the rate of interest = 16% compounded quartery, so quarterly arte = 16% / 4 = 4% and n is the time period = 5 * 4 = 20 quarters

Now, putting these values in the above formula, we get,

PVAD = $150 * (1 - (1 / (1 + 4%)20 / 4%) * (1 + 4%)

PVAD = $150 * (1 - (1 / (1 + 0.04)20 / 0.04) * (1 + 0.04)

PVAD = $150 * (1 - (1 / (1.04)20 / 0.04) * (1.04)

PVAD = $150 * (1 - (1 / 2.19112314303) / 0.04) * (1.04)

PVAD = $150 * ((1 - 0.45638694620) / 0.04) * (1.04)

PVAD = $150 * (0.5436130538 / 0.04) * (1.04)

PVAD = $150 * 13.59032645 * 1.04

PVAD = $2120.09

Amount that should be spent on boat is $2120.09