Question

HW17. Suppose that beginning on your 25th birthday, you put $1,000 into a savings account, and you make a new $1000 deposit every 3 months, up to and including your 45th birthday. The savings account pays 6% annual interest, compounded quarterly. \\

a. How much money will be in the account on your 45th birthday?

b. How much will your first $1000 deposit end up contributing to the amount in your bank account, on your 45th birthday?

c. The total account value in part (a) is really the result of computing a long sum (even though you used a formula as a shortcut, the formula was developed and explained in the text as a shortcut for a specific long sum of terms). Write out the first two terms (corresponding to your 25th birthday, and 3 months after your 25th birthday) and last two terms of that sum (corresponding to 3 months before your 45th birthday, and the day of your 45th birthday). How many terms are in the sum?

FV = PMT (1+?) ?− 1/ ? = PMT 1/ ? (1 − 1/ (1+?) ? ) . FV = PV(1 + i) n

Answer 1

a. Let the money in the account on 45th birthday be A

Recurring deposit amount = P = $ 1000

Compounding period = N = 21 years x 12 months = 252 months

Interest rate = r = 6% per annum = 6/4 % per quarter

A = P + interest earned

= P + P x [(n(n+1))/(2x12)] x r/100

= 1000 + 1000 x [(252 x (252+1))/(2 x 12)] x 6/4%

= 1000 + 1000 x [63756 / 24] x 1.5%

= 1000 + 39847.5

= 40847.5

On 45th birthday the amount in account = $ 40847.5

b. Amount contributed by initial deposit of $ 1000 = P (1+r/n) ^ n x t

where n = number of compounding in a year

therefore amount = 1000 ( 1+ 6/4)^ 21 x 4

= 1000(2.5)^84

= 1000*3.49

=3492.59

Amount contributed by initial deposit = $ 3492.59 - $ 1000 = $ 2492.59 interest.