Question

1. Find the accumulated value of an annuity due of $500 payable at the beginning of every month for nine years at 8% compounded monthly. 2. Lily purchased a boat valued at $19 000 on an installment plan requiring equal monthly payment for four years. If the first payment is due on the date of purchase and interest is 6.2% compounded monthly, what is the size of the monthly payment? 3. A car can be purchased by paying $27 000 now or can be leased by paying $725 per month for the next four years, with the first payment due on the day of signing lease. What nominal rate of interest is charged on the lease? 4. Raj contributes $150 at the beginning of each month into an RRSP paying interest at 5.5% compounded semi-annually. What will be the accumulated balance in the RRSP at the end of 20 years?

Answer 1

First 2 questions are being answered here:

1. Here we will use the future value of annuity due formula as per below:

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

where, FVAD is future value of annuity due, P is the periodical amount = $500, r is the rate of interest = 8% compounded monthly So, monthly rate = 8% / 12 = 0.6667% and n is the time period = 9 * 12 = 108 months.

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

FVAD = (1 + 0.6667%) * $500 * ((1 + 0.6667%)¹⁰⁸ - 1 / 0.6667%)

FVAD = (1 + 0.006667) * $500 * ((1 + 0.006667)¹⁰⁸ - 1 / 0.006667)

FVAD = (1.006667) * $500 * ((1.006667)¹⁰⁸ - 1 / 0.006667)

FVAD = (1.006667) * $500 * ((2.04953756525- 1) / 0.006667)

FVAD = (1.006667) * $500 * (1.04953756525 / 0.006667)

FVAD = (1.006667) * $500 * 157.430556072

FVAD = $79240.05

So accumulated value of annuity due is $79240.05

2. Here since the monthly payments occur at the beginning of each month, so it is annuity due. We will use the future value of annuity due formula as per below:

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

where, FVAD is future value of annuity due = $19000, P is the periodical / monthly amount, r is the rate of interest = 6.2% compounded monthly So, monthly rate = 6.2% / 12 = 0.51667% and n is the time period = 4 * 12 = 48 months.

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

$19000 = (1 + 0.51667%) * P * ((1 + 0.51667%)⁴⁸ - 1 / 0.51667%)

$19000 = (1 + 0.0051667) * P * ((1 + 0.0051667)⁴⁸ - 1 / 0.0051667)

$19000 = (1.0051667) * P * ((1.0051667)⁴⁸ - 1 / 0.0051667)

$19000 = (1.0051667) * P * ((1.28064202364 - 1) / 0.0051667)

$19000 = (1.0051667) * P * (0.128064202364 / 0.00516667)

$19000 = (1.00516667) * P * 54.3177759832

$19000 = P * 54.598180068

P = $348

So, monthly payments will be of $348.