Question

1. You begin making contributions to a new retirement account on your thirtieth birthday. You make a contribution of $4000 at the beginning of each year through your sixty-fourth birthday. Starting at age sixty-five and continuing through your eightieth birthday, you made a level withdrawal on your birthday. Find the amount of these withdrawals if they completely exhaust the balance in your account and the annual effective rate is 6% until you are sixty-five and 5% thereafter.

Answer 1

First of all we will calculate Future Value of contributions on 65th Birthday (ie 36 years compounding)

4000 * FVAF 6%,36 = 4000*126.26 = 505072.47

Alternatively it can be calculated as follows

Year	Instalment	No of Year Compounding	Future Value
30	4000	36	32,589.01
31	4000	35	30,744.35
32	4000	34	29,004.10
33	4000	33	27,362.36
34	4000	32	25,813.55
35	4000	31	24,352.40
36	4000	30	22,973.96
37	4000	29	21,673.55
38	4000	28	20,446.75
39	4000	27	19,289.38
40	4000	26	18,197.53
41	4000	25	17,167.48
42	4000	24	16,195.74
43	4000	23	15,279.00
44	4000	22	14,414.15
45	4000	21	13,598.25
46	4000	20	12,828.54
47	4000	19	12,102.40
48	4000	18	11,417.36
49	4000	17	10,771.09
50	4000	16	10,161.41
51	4000	15	9,586.23
52	4000	14	9,043.62
53	4000	13	8,531.71
54	4000	12	8,048.79
55	4000	11	7,593.19
56	4000	10	7,163.39
57	4000	9	6,757.92
58	4000	8	6,375.39
59	4000	7	6,014.52
60	4000	6	5,674.08
61	4000	5	5,352.90
62	4000	4	5,049.91
63	4000	3	4,764.06
64	4000	2	4,494.40
65	4000	1	4,240.00
		FV at 65th Birthday	505,072.47

There is a withdrawl of 4000 on 65th birthday

Amount left in account on 65th Birthday = 501,072

(For understanding consider this amount as loan taken for 15 Years at 5% per annum interest rate, we have to calculate the Equal annual Instalment of this loan)

Let the annual withdrawl from the account be X

X * PVAF 5%, 15 Years = 501,072

10.3796 X = 501,072

X = 48,274.46

Amortization of this balance is shown below

Year	Opening Bal	5% Return	Withdrawl	Closing Balance
66	501,072	25,054	48,274	477,852
67	477,852	23,893	48,274	453,470
68	453,470	22,673	48,274	427,869
69	427,869	21,393	48,274	400,988
70	400,988	20,049	48,274	372,763
71	372,763	18,638	48,274	343,126
72	343,126	17,156	48,274	312,008
73	312,008	15,600	48,274	279,334
74	279,334	13,967	48,274	245,026
75	245,026	12,251	48,274	209,003
76	209,003	10,450	48,274	171,179
77	171,179	8,559	48,274	131,463
78	131,463	6,573	48,274	89,762
79	89,762	4,488	48,274	45,976
80	45,976	2,299	48,274	0