Question

You are planning to make monthly deposits of $100 into a retirement account that pays 9 percent interest compounded monthly. If your first deposit will be made one month from now, how large will your retirement account be in 21 years?

Answer 1

Here, the deposits will be same every year, so it is an annuity. We have to find the future value of annuity. Here we will use the future value of annuity formula as per below:

FVA = P * ((1 + r)ⁿ - 1 / r)

where, FVA is future value of annuity, P is the periodical amount = $100 r is the rate of interest = 9% compounded monthly. Monthly rate = 9% / 12 = 0.75% and n is the time period = 21 * 12 = 252 months

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

FVA = $100 * ((1 + 0.75%)²⁵² - 1 / 0.75%)

FVA = $100 * ((1 + 0.0075)²⁵² - 1 / 0.0075)

FVA = $100 * ((1.0075)²⁵²- 1 / 0.0075)

FVA = $100 * ((6.57285138662 - 1)/ 0.0075)

FVA = $100 * (5.57285138662 / 0.0075)

FVA = $100 * 743.046851549

P = $74304.69

So, the amount in retirement account after 21 years will be $74304.69.

Answer 2

SOLUTION :

Monthly compounding.

Monthly deposit, A = 100 ($) (starting from the next month)

Monthly interest rate, r = 9/12 % = 9/1200 = 3/400 in fractions.

=> 1 + r = 403/400 

Periods, n = 21 * 12 = 252 months

So, 

FV after 21 years (252 months)

= A((1 + r)^n - 1) / r

= 100 * ((403/400)^252 - 1) / (3/400)

= 74304.69 ($)   (ANSWER).