Question 350487
If he wants to buy the annuity today, and he needs to know how much money should be invested in the annuity, and the annuity needs to last for 20 years, at which time it will be depleted, and the annuity earns 6% per year, and he will begin withdrawing from the annuity 1 year after it is bought, and he will be withdrawing on a yearly basis, then the present value of an annuity formula should do the trick.


That formula will tell him that he needs to invest $573,496.0609 in the annuity.


He will be able to withdraw $50,000 at the end of each year for a period of 20 years, at which time there will be no money left in the account.


A yearly cash flow, shown below, gives you the details about how the account will be depleted.


<pre>


YEAR	REVENUE	        BALANCE
0		        $573,496.06
1	$50,000.00	$557,905.82
2	$50,000.00	$541,380.17
3	$50,000.00	$523,862.98
4	$50,000.00	$505,294.76
5	$50,000.00	$485,612.45
6	$50,000.00	$464,749.20
7	$50,000.00	$442,634.15
8	$50,000.00	$419,192.20
9	$50,000.00	$394,343.73
10	$50,000.00	$368,004.35
11	$50,000.00	$340,084.61
12	$50,000.00	$310,489.69
13	$50,000.00	$279,119.07
14	$50,000.00	$245,866.22
15	$50,000.00	$210,618.19
16	$50,000.00	$173,255.28
17	$50,000.00	$133,650.60
18	$50,000.00	$91,669.63
19	$50,000.00	$47,169.81
20	$50,000.00	$0.00


</pre>


At year 0, he starts with a balance of $573,496.0609.
At year 1, that balance is multiplied by 1.06 and then has $50,000 subtracted from it.
This happens year over year until the 20th year at which time the account goes to 0.


The formula that is used to calculate the present value of that annuity is shown below.


PRESENT VALUE OF A PAYMENT


{{{ PV(PMT) = (PMT * (1 - (1 / (1+i)^n))/i) }}}


PV is the present value of the annuity.
PMT is the annuity he expects to receive each year.
i is the annual interest rate per time period
n is the number of time periods.


The same formula is used to calculate the present value of a loan.


If he expects to receive 1/12th of the annual amount at the end of every month and the interest on the account is calculated monthly with compounding, then the formula is adjusted as follows:


$50,000 / 12 = $4,1666.67 per monmth is received by him.
Annual Interest Rate of .06 is divided by 12 to get a monthly interest rate of .06 / 12 = .0005 per time period (time periods are now in months rather than years).


Number of time periods is determined by multiplying the number of years by 12 to get 20 * 12 = 240 time periods.


Payments of $4,1666.67 per month for 240 months are equivalent to a present value of $581,586.5487.


With monthly withdrawals from the account, and monthly compounding, he would need to invest $581,586.5487 in the account.


Once he does that, he will be able to withdraw $4,1666.67 per momth for 240 months, at which time the account will be depleted.


The monthly cash flows are shown below to provide you with the details of how the money in the account is depleted.


<pre>

YEAR	REVENUE	        BALANCE
0		        $581,586.55
1	$4,166.67	$580,327.81
2	$4,166.67	$579,062.79
3	$4,166.67	$577,791.43
4	$4,166.67	$576,513.72
5	$4,166.67	$575,229.63
6	$4,166.67	$573,939.11
7	$4,166.67	$572,642.14
8	$4,166.67	$571,338.68
9	$4,166.67	$570,028.71
10	$4,166.67	$568,712.18
11	$4,166.67	$567,389.08
12	$4,166.67	$566,059.36
13	$4,166.67	$564,722.99
14	$4,166.67	$563,379.94
15	$4,166.67	$562,030.17
16	$4,166.67	$560,673.65
17	$4,166.67	$559,310.36
18	$4,166.67	$557,940.24
19	$4,166.67	$556,563.27
20	$4,166.67	$555,179.42
21	$4,166.67	$553,788.65
22	$4,166.67	$552,390.93
23	$4,166.67	$550,986.22
24	$4,166.67	$549,574.48
25	$4,166.67	$548,155.69
26	$4,166.67	$546,729.80
27	$4,166.67	$545,296.78
28	$4,166.67	$543,856.60
29	$4,166.67	$542,409.22
30	$4,166.67	$540,954.60
31	$4,166.67	$539,492.70
32	$4,166.67	$538,023.50
33	$4,166.67	$536,546.95
34	$4,166.67	$535,063.02
35	$4,166.67	$533,571.67
36	$4,166.67	$532,072.86
37	$4,166.67	$530,566.56
38	$4,166.67	$529,052.72
39	$4,166.67	$527,531.32
40	$4,166.67	$526,002.31
41	$4,166.67	$524,465.65
42	$4,166.67	$522,921.32
43	$4,166.67	$521,369.26
44	$4,166.67	$519,809.44
45	$4,166.67	$518,241.82
46	$4,166.67	$516,666.36
47	$4,166.67	$515,083.02
48	$4,166.67	$513,491.77
49	$4,166.67	$511,892.56
50	$4,166.67	$510,285.36
51	$4,166.67	$508,670.12
52	$4,166.67	$507,046.80
53	$4,166.67	$505,415.37
54	$4,166.67	$503,775.78
55	$4,166.67	$502,127.99
56	$4,166.67	$500,471.97
57	$4,166.67	$498,807.66
58	$4,166.67	$497,135.03
59	$4,166.67	$495,454.04
60	$4,166.67	$493,764.64
61	$4,166.67	$492,066.80
62	$4,166.67	$490,360.47
63	$4,166.67	$488,645.60
64	$4,166.67	$486,922.17
65	$4,166.67	$485,190.11
66	$4,166.67	$483,449.39
67	$4,166.67	$481,699.97
68	$4,166.67	$479,941.81
69	$4,166.67	$478,174.85
70	$4,166.67	$476,399.06
71	$4,166.67	$474,614.39
72	$4,166.67	$472,820.79
73	$4,166.67	$471,018.23
74	$4,166.67	$469,206.65
75	$4,166.67	$467,386.02
76	$4,166.67	$465,556.28
77	$4,166.67	$463,717.40
78	$4,166.67	$461,869.32
79	$4,166.67	$460,012.00
80	$4,166.67	$458,145.39
81	$4,166.67	$456,269.45
82	$4,166.67	$454,384.13
83	$4,166.67	$452,489.39
84	$4,166.67	$450,585.17
85	$4,166.67	$448,671.43
86	$4,166.67	$446,748.12
87	$4,166.67	$444,815.19
88	$4,166.67	$442,872.60
89	$4,166.67	$440,920.30
90	$4,166.67	$438,958.23
91	$4,166.67	$436,986.35
92	$4,166.67	$435,004.62
93	$4,166.67	$433,012.98
94	$4,166.67	$431,011.37
95	$4,166.67	$428,999.76
96	$4,166.67	$426,978.10
97	$4,166.67	$424,946.32
98	$4,166.67	$422,904.39
99	$4,166.67	$420,852.24
100	$4,166.67	$418,789.84
101	$4,166.67	$416,717.12
102	$4,166.67	$414,634.04
103	$4,166.67	$412,540.54
104	$4,166.67	$410,436.58
105	$4,166.67	$408,322.09
106	$4,166.67	$406,197.04
107	$4,166.67	$404,061.35
108	$4,166.67	$401,914.99
109	$4,166.67	$399,757.90
110	$4,166.67	$397,590.03
111	$4,166.67	$395,411.31
112	$4,166.67	$393,221.70
113	$4,166.67	$391,021.14
114	$4,166.67	$388,809.58
115	$4,166.67	$386,586.96
116	$4,166.67	$384,353.23
117	$4,166.67	$382,108.33
118	$4,166.67	$379,852.20
119	$4,166.67	$377,584.80
120	$4,166.67	$375,306.06
121	$4,166.67	$373,015.92
122	$4,166.67	$370,714.33
123	$4,166.67	$368,401.24
124	$4,166.67	$366,076.58
125	$4,166.67	$363,740.29
126	$4,166.67	$361,392.33
127	$4,166.67	$359,032.62
128	$4,166.67	$356,661.12
129	$4,166.67	$354,277.76
130	$4,166.67	$351,882.48
131	$4,166.67	$349,475.23
132	$4,166.67	$347,055.94
133	$4,166.67	$344,624.55
134	$4,166.67	$342,181.00
135	$4,166.67	$339,725.24
136	$4,166.67	$337,257.20
137	$4,166.67	$334,776.82
138	$4,166.67	$332,284.04
139	$4,166.67	$329,778.79
140	$4,166.67	$327,261.02
141	$4,166.67	$324,730.66
142	$4,166.67	$322,187.64
143	$4,166.67	$319,631.92
144	$4,166.67	$317,063.41
145	$4,166.67	$314,482.06
146	$4,166.67	$311,887.80
147	$4,166.67	$309,280.58
148	$4,166.67	$306,660.31
149	$4,166.67	$304,026.95
150	$4,166.67	$301,380.41
151	$4,166.67	$298,720.65
152	$4,166.67	$296,047.59
153	$4,166.67	$293,361.16
154	$4,166.67	$290,661.30
155	$4,166.67	$287,947.94
156	$4,166.67	$285,221.01
157	$4,166.67	$282,480.45
158	$4,166.67	$279,726.18
159	$4,166.67	$276,958.15
160	$4,166.67	$274,176.27
161	$4,166.67	$271,380.49
162	$4,166.67	$268,570.72
163	$4,166.67	$265,746.91
164	$4,166.67	$262,908.98
165	$4,166.67	$260,056.86
166	$4,166.67	$257,190.47
167	$4,166.67	$254,309.76
168	$4,166.67	$251,414.64
169	$4,166.67	$248,505.05
170	$4,166.67	$245,580.91
171	$4,166.67	$242,642.14
172	$4,166.67	$239,688.69
173	$4,166.67	$236,720.47
174	$4,166.67	$233,737.40
175	$4,166.67	$230,739.42
176	$4,166.67	$227,726.45
177	$4,166.67	$224,698.42
178	$4,166.67	$221,655.24
179	$4,166.67	$218,596.85
180	$4,166.67	$215,523.17
181	$4,166.67	$212,434.12
182	$4,166.67	$209,329.62
183	$4,166.67	$206,209.60
184	$4,166.67	$203,073.99
185	$4,166.67	$199,922.69
186	$4,166.67	$196,755.64
187	$4,166.67	$193,572.75
188	$4,166.67	$190,373.94
189	$4,166.67	$187,159.15
190	$4,166.67	$183,928.28
191	$4,166.67	$180,681.25
192	$4,166.67	$177,417.99
193	$4,166.67	$174,138.41
194	$4,166.67	$170,842.44
195	$4,166.67	$167,529.99
196	$4,166.67	$164,200.97
197	$4,166.67	$160,855.31
198	$4,166.67	$157,492.92
199	$4,166.67	$154,113.71
200	$4,166.67	$150,717.62
201	$4,166.67	$147,304.54
202	$4,166.67	$143,874.39
203	$4,166.67	$140,427.10
204	$4,166.67	$136,962.57
205	$4,166.67	$133,480.71
206	$4,166.67	$129,981.45
207	$4,166.67	$126,464.69
208	$4,166.67	$122,930.35
209	$4,166.67	$119,378.33
210	$4,166.67	$115,808.56
211	$4,166.67	$112,220.93
212	$4,166.67	$108,615.37
213	$4,166.67	$104,991.78
214	$4,166.67	$101,350.07
215	$4,166.67	$97,690.16
216	$4,166.67	$94,011.94
217	$4,166.67	$90,315.34
218	$4,166.67	$86,600.25
219	$4,166.67	$82,866.58
220	$4,166.67	$79,114.25
221	$4,166.67	$75,343.15
222	$4,166.67	$71,553.20
223	$4,166.67	$67,744.30
224	$4,166.67	$63,916.35
225	$4,166.67	$60,069.27
226	$4,166.67	$56,202.95
227	$4,166.67	$52,317.30
228	$4,166.67	$48,412.22
229	$4,166.67	$44,487.61
230	$4,166.67	$40,543.38
231	$4,166.67	$36,579.43
232	$4,166.67	$32,595.66
233	$4,166.67	$28,591.98
234	$4,166.67	$24,568.27
235	$4,166.67	$20,524.44
236	$4,166.67	$16,460.40
237	$4,166.67	$12,376.03
238	$4,166.67	$8,271.25
239	$4,166.67	$4,145.94
240	$4,166.67	$0.00


</pre>


I confirmed the formulas will provide you with the same answer by reproducing the answers using them.


If the question is what I think you were asking, then these answers would be correct.


the question I think you were asking is how  much money needs to be invested today to provide $50,000 of annual income for the next 20 years.


At the end of the 20 year period, your father will have nothing left in that account.