Question

Robert expects to make 24 monthly deposits of $330 into an investment account that earns 9.9% compounded quarterly. After a one year investment pause, he intends to take out 30 equal monthly amounts from his investment account, the first withdrawal being exactly one year from the date of his last deposit. What is the dollar amount of each withdrawal assuming after the last one he has drained his investment account to zero. Assume the account earns the same rate of interest stated above throughout the investment horizon.

Answer 1

Quarterly Interest = 9.9/4 = 2.475%

Month	Deposit/(Withdrawal) [beginning of period]	Opening Balance [Previous Closing+/-Deposit/Withdrawal]	Interest [Opening Balance*(0.02475)]	Closing Balance [Opening Balance+Interest]	Deposit/(Withdrawal) [end of period]
0				0	330
1		330		330	330
2		660		660	330
3		990	24.5025	1014.5025	330
4		1344.5025		1344.5025	330
5		1674.5025		1674.5025	330
6		2004.5025	49.61143688	2054.113937	330
7		2384.113937		2384.113937	330
8		2714.113937		2714.113937	330
9		3044.113937	75.34181994	3119.455757	330
10		3449.455757		3449.455757	330
11		3779.455757		3779.455757	330
12		4109.455757	101.70903	4211.164787	330
13		4541.164787		4541.164787	330
14		4871.164787		4871.164787	330
15		5201.164787	128.7288285	5329.893615	330
16		5659.893615		5659.893615	330
17		5989.893615		5989.893615	330
18		6319.893615	156.417367	6476.310982	330
19		6806.310982		6806.310982	330
20		7136.310982		7136.310982	330
21		7466.310982	184.7911968	7651.102179	330
22		7981.102179		7981.102179	330
23		8311.102179		8311.102179	330
24		8641.102179	213.8672789	8854.969458	330
25		9184.969458		9184.969458
26		9184.969458		9184.969458
27		9184.969458	227.3279941	9412.297452
28		9412.297452		9412.297452
29		9412.297452		9412.297452
30		9412.297452	232.9543619	9645.251814
31		9645.251814		9645.251814
32		9645.251814		9645.251814
33		9645.251814	238.7199824	9883.971796
34		9883.971796		9883.971796
35		9883.971796		9883.971796
36		9883.971796	244.628302	10128.6001

37		10128.6001		10128.6001	-379
38		9749.600098		9749.600098	-379
39		9370.600098	231.9223524	9602.522451	-379
40		9223.522451		9223.522451	-379
41		8844.522451		8844.522451	-379
42		8465.522451	209.5216807	8675.044131	-379
43		8296.044131		8296.044131	-379
44		7917.044131		7917.044131	-379
45		7538.044131	186.5665923	7724.610724	-379
46		7345.610724		7345.610724	-379
47		6966.610724		6966.610724	-379
48		6587.610724	163.0433654	6750.654089	-379
49		6371.654089		6371.654089	-379
50		5992.654089		5992.654089	-379
51		5613.654089	138.9379387	5752.592028	-379
52		5373.592028		5373.592028	-379
53		4994.592028		4994.592028	-379
54		4615.592028	114.2359027	4729.827931	-379
55		4350.827931		4350.827931	-379
56		3971.827931		3971.827931	-379
57		3592.827931	88.92249128	3681.750422	-379
58		3302.750422		3302.750422	-379
59		2923.750422		2923.750422	-379
60		2544.750422	62.98257294	2607.732995	-379
61		2228.732995		2228.732995	-379
62		1849.732995		1849.732995	-379
63		1470.732995	36.40064162	1507.133636	-379
64		1128.133636		1128.133636	-379
65		749.1336364		749.1336364	-379
66		370.1336364	9.1608075	379.2944439	-379

Account Balance 1 year from the last deposit = $10128.6

Therefore, Amount that will be withdrawn every QUARTER = EMI

EMI = P*i*(1+i)^n/[{(1+i)^n}-1]

Where,

P = Principal = 10128.6

i= Interest Rate = 0.02475

n= Number of periods = 30 Months/3 = 10 Quarters

Therefore, EMI = 10128.6*0.02475*(1+0.02475)^10/[{(1+0.02475)^10}-1]

= 250.68285*(1.276966)/[1.276966-1] = 320.113476/0.276966 = $1155.786

Monthly Withdrawal = Quarterly Withdrawal/3 = 1155.786/3 = $385.262

Approximation of above Monthly Withdrawal(because interest will not be considered for first 2 installments) = 385.262-[{(385.262*2)*2.475%}/3] = 378.905 = $379(approx)