Question

During your working career you make equal monthly deposits of $500 into an investment fund paying

5% interest per year (nominal rate). The fund compounds interest continuously. The first deposit will be made one month from today. How much money can you withdraw from this account immediately when you retire after working for 25 years? (Assume you make a payment into the fund at the end of your working career.)

Answer 1

Given,

Monthly deposit=$500

Interest rate=5%

Deposit is made every month for 25 years.

The fund compounds continously(i.e., every day)

Formula for compund interest is A=P(1+r/n)^(n*t)

where A is maturity amount

P is Principal

r is interest rate

n is the no. of times interest is compounded per unit 't'

Sol: Since it is monthly payment it is better to convert the interest rate to monthy interest rate = 5%/12=0.4167%(Rounded off)

So the r= 0.4167%

Since the fund compounds interest continously(i.e., per day), the no. of times interest is compunded in a month is 30.

So n=30

P=$500

Applying these values in the above given formula:

Month	No. of remaining months(t)	Maturtiy amount
1	299	1737.937903
2	298	1730.711485
3	297	1723.515114
4	296	1716.348666
5	295	1709.212016
6	294	1702.105041
7	293	1695.027617
8	292	1687.979621
9	291	1680.960931
10	290	1673.971425
11	289	1667.010981
12	288	1660.07948
13	287	1653.176799
14	286	1646.302821
15	285	1639.457425
16	284	1632.640492
17	283	1625.851905
18	282	1619.091544
19	281	1612.359294
20	280	1605.655036
21	279	1598.978655
22	278	1592.330035
23	277	1585.70906
24	276	1579.115615
25	275	1572.549586
26	274	1566.010859
27	273	1559.49932
28	272	1553.014856
29	271	1546.557356
30	270	1540.126705
31	269	1533.722794
32	268	1527.34551
33	267	1520.994743
34	266	1514.670384
35	265	1508.372321
36	264	1502.100445
37	263	1495.854649
38	262	1489.634822
39	261	1483.440858
40	260	1477.272649
41	259	1471.130087
42	258	1465.013067
43	257	1458.921481
44	256	1452.855224
45	255	1446.814191
46	254	1440.798277
47	253	1434.807377
48	252	1428.841388
49	251	1422.900206
50	250	1416.983727
51	249	1411.091849
52	248	1405.22447
53	247	1399.381488
54	246	1393.562801
55	245	1387.768309
56	244	1381.99791
57	243	1376.251504
58	242	1370.528993
59	241	1364.830276
60	240	1359.155254
61	239	1353.50383
62	238	1347.875904
63	237	1342.27138
64	236	1336.690159
65	235	1331.132145
66	234	1325.597242
67	233	1320.085353
68	232	1314.596383
69	231	1309.130236
70	230	1303.686817
71	229	1298.266033
72	228	1292.867788
73	227	1287.49199
74	226	1282.138544
75	225	1276.807358
76	224	1271.49834
77	223	1266.211396
78	222	1260.946436
79	221	1255.703368
80	220	1250.482101
81	219	1245.282544
82	218	1240.104607
83	217	1234.9482
84	216	1229.813234
85	215	1224.699619
86	214	1219.607266
87	213	1214.536088
88	212	1209.485996
89	211	1204.456903
90	210	1199.44872
91	209	1194.461362
92	208	1189.494742
93	207	1184.548773
94	206	1179.623369
95	205	1174.718446
96	204	1169.833917
97	203	1164.969699
98	202	1160.125706
99	201	1155.301855
100	200	1150.498061
101	199	1145.714242
102	198	1140.950314
103	197	1136.206195
104	196	1131.481802
105	195	1126.777053
106	194	1122.091867
107	193	1117.426162
108	192	1112.779857
109	191	1108.152872
110	190	1103.545125
111	189	1098.956539
112	188	1094.387031
113	187	1089.836524
114	186	1085.304938
115	185	1080.792195
116	184	1076.298215
117	183	1071.822922
118	182	1067.366238
119	181	1062.928084
120	180	1058.508384
121	179	1054.107062
122	178	1049.724041
123	177	1045.359244
124	176	1041.012597
125	175	1036.684023
126	174	1032.373447
127	173	1028.080795
128	172	1023.805992
129	171	1019.548964
130	170	1015.309636
131	169	1011.087936
132	168	1006.88379
133	167	1002.697125
134	166	998.5278684
135	165	994.3759477
136	164	990.2412908
137	163	986.123826
138	162	982.0234818
139	161	977.940187
140	160	973.8738707
141	159	969.8244623
142	158	965.7918915
143	157	961.7760883
144	156	957.776983
145	155	953.7945061
146	154	949.8285886
147	153	945.8791615
148	152	941.9461563
149	151	938.0295046
150	150	934.1291386
151	149	930.2449905
152	148	926.3769928
153	147	922.5250784
154	146	918.6891805
155	145	914.8692323
156	144	911.0651677
157	143	907.2769205
158	142	903.504425
159	141	899.7476158
160	140	896.0064274
161	139	892.2807951
162	138	888.5706542
163	137	884.8759401
164	136	881.1965889
165	135	877.5325365
166	134	873.8837194
167	133	870.2500743
168	132	866.631538
169	131	863.0280477
170	130	859.4395409
171	129	855.8659552
172	128	852.3072287
173	127	848.7632995
174	126	845.2341061
175	125	841.7195872
176	124	838.2196819
177	123	834.7343293
178	122	831.263469
179	121	827.8070406
180	120	824.3649842
181	119	820.9372401
182	118	817.5237486
183	117	814.1244506
184	116	810.7392869
185	115	807.368199
186	114	804.0111282
187	113	800.6680162
188	112	797.338805
189	111	794.0234368
190	110	790.7218541
191	109	787.4339994
192	108	784.1598158
193	107	780.8992464
194	106	777.6522346
195	105	774.418724
196	104	771.1986585
197	103	767.9919821
198	102	764.7986393
199	101	761.6185745
200	100	758.4517325
201	99	755.2980584
202	98	752.1574974
203	97	749.029995
204	96	745.9154968
205	95	742.8139489
206	94	739.7252974
207	93	736.6494886
208	92	733.5864691
209	91	730.5361858
210	90	727.4985857
211	89	724.4736161
212	88	721.4612244
213	87	718.4613584
214	86	715.4739659
215	85	712.4989952
216	84	709.5363945
217	83	706.5861124
218	82	703.6480977
219	81	700.7222994
220	80	697.8086667
221	79	694.907149
222	78	692.0176959
223	77	689.1402573
224	76	686.2747833
225	75	683.4212239
226	74	680.5795298
227	73	677.7496516
228	72	674.9315401
229	71	672.1251465
230	70	669.3304219
231	69	666.547318
232	68	663.7757863
233	67	661.0157787
234	66	658.2672474
235	65	655.5301446
236	64	652.8044227
237	63	650.0900346
238	62	647.3869329
239	61	644.6950709
240	60	642.0144018
241	59	639.344879
242	58	636.6864562
243	57	634.0390872
244	56	631.4027261
245	55	628.7773271
246	54	626.1628446
247	53	623.5592332
248	52	620.9664478
249	51	618.3844432
250	50	615.8131748
251	49	613.2525978
252	48	610.7026678
253	47	608.1633405
254	46	605.6345718
255	45	603.1163178
256	44	600.6085349
257	43	598.1111794
258	42	595.6242081
259	41	593.1475776
260	40	590.6812451
261	39	588.2251677
262	38	585.7793028
263	37	583.3436079
264	36	580.9180407
265	35	578.5025592
266	34	576.0971213
267	33	573.7016853
268	32	571.3162096
269	31	568.9406529
270	30	566.5749738
271	29	564.2191313
272	28	561.8730845
273	27	559.5367926
274	26	557.2102152
275	25	554.8933117
276	24	552.5860421
277	23	550.2883661
278	22	548.000244
279	21	545.721636
280	20	543.4525026
281	19	541.1928043
282	18	538.9425019
283	17	536.7015564
284	16	534.4699288
285	15	532.2475804
286	14	530.0344727
287	13	527.8305671
288	12	525.6358255
289	11	523.4502097
290	10	521.2736818
291	9	519.1062039
292	8	516.9477386
293	7	514.7982482
294	6	512.6576955
295	5	510.5260433
296	4	508.4032546
297	3	506.2892925
298	2	504.1841204
299	1	502.0877017
300	0	500
Total	298221.36

The amount that can be withdrawn at the end of the 25th year is $298221.36

Points to be noted:

1) No. of remaining months(t) for the first month in the above given table is taken as 299 months because first deposit is made at the end of first month.

2) Formula used in excel for the first month is

=500*(1+0.4167%/30)^(299*30)

for the second month the formula is

=500*(1+0.4167%/30)^(298*30)

and so on....

If you notice the only change in the formula for every month is t(i.e., no. of remaining months).