Question

Derek will deposit $3,270.00 per year for 13.00 years into an account that earns 14.00%. Assuming the first deposit is made 6.00 years from today, how much will be in the account 30.00 years from today?

Answer 1

Part A

FV of Annuity :
Annuity is series of cash flows that are deposited at regular intervals for specific period of time.

FV of Annuity = CF [ (1+r)^n - 1 ] / r
r - Int rate per period
n - No. of periods

Particulars	Amount
Cash Flow	$      3,270.00
Int Rate	14.000%
Periods	13

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 3270 * [ [ ( 1 + 0.14 ) ^ 13 ] - 1 ] / 0.14
= $ 3270 * [ [ ( 1.14 ) ^ 13 ] - 1 ] / 0.14
= $ 3270 * [ [5.4924] - 1 ] / 0.14
= $ 3270 * [4.4924] /0.14
= $ 104929.9
first deposit made at 6 years from today then Annuity starts at end of 5 years

Balance in account at the end of 18 (13+5) years $ 104929.9

Part B

Balance in account after 30 years , remaining period is 12 (30-18) years

Future Value:
FV = PV (1+r)^n
Where r is Int rate per period
n - No. of periods

Particulars	Amount
Present Value @ 18 years	$ 104,929.9
Int Rate	14.0000%
Periods	12

Future Value = Present Value * ( 1 + r )^n
= $ 104929.9 ( 1 + 0.14) ^ 12
= $ 104929.9 ( 1.14 ^ 12)
= $ 104929.9 * 4.8179
= $ 505542.27

Balance in account after 30 years = $ 505542.27

Alternative Answer

First deposit made after 6 years so remaining years 24

Future value factor = (1+r) ^ n

r - rate

n - remaining period

Future value = Cash flow * Future value factor

Year	Cash flow	Balance period	Future value factor @ 14 %	Future value
6	3270	24	23.21220685	75903.92
7	3270	23	20.36158496	66582.38
8	3270	22	17.86103944	58405.60
9	3270	21	15.66757845	51232.98
10	3270	20	13.74348987	44941.21
11	3270	19	12.05569287	39422.12
12	3270	18	10.57516918	34580.80
13	3270	17	9.276464197	30334.04
14	3270	16	8.137249295	26608.81
15	3270	15	7.137937978	23341.06
16	3270	14	6.261349104	20474.61
17	3270	13	5.492411495	17960.19
18	3270	12	4.81790482	15754.55

Balance in account after 30 years $ 505542.27

please comment if any further assistance is required