Question

Sally has a sum of $30000 that she invests at 8% compounded monthly. What equal monthly payments can she receive over a period of

a) 8 years? Answer = $

b) 17 years? Answer = $

Answer 1

(a) Here, the payments will be same every month, so it is an annuity. We will use the follwoing present value of annuity formula to calculate the monthly payments:

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

where, PVA = Present value of annuity = $30000, P is the periodical amount, r is the rate of interest = 8% compounded monthly, so monthly rate = 8% / 12 = 0.667% and n is the time period = 8 * 12 = 96 months

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

$30000 = P * (1 - (1 + 0.6667%)-96 / 0.667%)

$30000 = P * (1 - ( 1+ 0.00667)-96 / 0.00667)

$30000 = P * (1 - ( 1.00667)-96 / 0.00667)

$30000 = P * (1 - 0.52841185031) / 0.00667)

$30000 = P * (0.47158814969 / 0.00667)

$30000 = P * 70.734685719

P = $30000 / 70.734685719

P = $424.12

So, equal monthly payments are $424.12

(b) Here, the payments will be same every month, so it is an annuity. We will use the follwoing present value of annuity formula to calculate the monthly payments:

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

where, PVA = Present value of annuity = $30000, P is the periodical amount, r is the rate of interest = 8% compounded monthly, so monthly rate = 8% / 12 = 0.667% and n is the time period = 8 * 17 = 204 months

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

$30000 = P * (1 - (1 + 0.6667%)-204 / 0.667%)

$30000 = P * (1 - ( 1+ 0.00667)-204 / 0.00667)

$30000 = P * (1 - ( 1.00667)-204 / 0.00667)

$30000 = P * (1 - 0.25782003671) / 0.00667)

$30000 = P * (0.74217996329 / 0.00667)

$30000 = P * 111.3214284

P = $30000 / 111.3214284

P = $269.49

So, equal monthly payments are $269.49