Question

Laquita deposits $5500 in her retirement account every year. If her account pays an average 6% interest and she makes 38 deposits before she retires, how much monet can she withdraw in 20 equal payments beginning one year after her last deposit? please show cash flow diagram and solve NOT with excel but with the interest rate formula equations!!!

Answer 1

The total value that will be saved at the end of 38 deposits will be given by finding the future value of annuity.

FV of annuity = P*[((1+r)^n - 1)/r]
P - Periodic payment = 5500
r - rate per period = 6%
n - number of periods = 38
FV of annuity = 5500*(((1+0.06)^38 - 1)/0.06) = $747473.1318

This becomes the present value of annuity for future withdrawals.

Present value of annuity is given by,

PV of annuity = P*[(1-(1+r)^(-n)) / r]

P - Periodic payment = ?

r - rate per period = 0.06

n - number of periods = 20

747473.1318 = P*((1-(1+0.06)^(-20)) / 0.06)

747473.1318 = P*11.46992

P = 747473.1318/11.46992 = $65168.12

She can withdraw an amount of $65168.12 every year for 20 years.

Cash flows:

Year	Deposit amount	Amount at the end of the year = Deposit amount*(1+0.06)^year + Previous end of year amount
0	5500	5500
1	5500	11330
2	5500	17509.8
3	5500	24060.388
4	5500	31004.01128
5	5500	38364.25196
6	5500	46166.10707
7	5500	54436.0735
8	5500	63202.23791
9	5500	72494.37218
10	5500	82344.03451
11	5500	92784.67658
12	5500	103851.7572
13	5500	115582.8626
14	5500	128017.8344
15	5500	141198.9044
16	5500	155170.8387
17	5500	169981.089
18	5500	185679.9544
19	5500	202320.7516
20	5500	219959.9967
21	5500	238657.5965
22	5500	258477.0523
23	5500	279485.6754
24	5500	301754.816
25	5500	325360.1049
26	5500	350381.7112
27	5500	376904.6139
28	5500	405018.8907
29	5500	434820.0242
30	5500	466409.2256
31	5500	499893.7792
32	5500	535387.4059
33	5500	573010.6503
34	5500	612891.2893
35	5500	655164.7667
36	5500	699974.6527
37	5500	747473.1318

Withdrawal cashflows:

No. of Withdrawal	Amount at the beginning of withdrawal (A)	Amount in the account after withdrawal (B) = A - 65168.12	Amount at the end of the year = B*(1+0.06)
0	747473.13	747473.13	792321.52
1	792321.52	727153.40	770782.60
2	770782.60	705614.48	747951.35
3	747951.35	682783.23	723750.23
4	723750.23	658582.11	698097.03
5	698097.03	632928.91	670904.65
6	670904.65	605736.53	642080.72
7	642080.72	576912.60	611527.36
8	611527.36	546359.24	579140.79
9	579140.79	513972.67	544811.03
10	544811.03	479642.91	508421.48
11	508421.48	443253.36	469848.57
12	469848.57	404680.45	428961.27
13	428961.27	363793.15	385620.74
14	385620.74	320452.62	339679.78
15	339679.78	274511.66	290982.36
16	290982.36	225814.24	239363.09
17	239363.09	174194.97	184646.67
18	184646.67	119478.55	126647.26
19	126647.26	61479.14	65167.89
20	65167.89	-0.23	-0.24