Question

Yumi's grandparents presented her with a gift of $21,000 when she was 12 years old to be used for her college education. Over the next 5 years, until she turned 17, Yumi's parents had invested her money in a tax-free account that had yielded interest at the rate of 2.5%/year compounded monthly. Upon turning 17, Yumi now plans to withdraw her funds in equal annual installments over the next 4 years, starting at age 18. If the college fund is expected to earn interest at the rate of 3%/year, compounded annually, what will be the size of each installment? (Assume no interest is accrued from the point she turns 17 until she makes the first withdrawal. Round your answer to the nearest cent.)
$

Answer 1

First of all, we have to calculate the value of investment of $21000 after 5 years @ interest 2.5%pa  compounded monthly.

Compound interest is calculated by multiplying the initial principal amount by one plus the annual interest rate raised to the number of compound periods minus one. Interest can be compounded on any given frequency schedule.

Formula of compounded interest= (P*(1+r/n)^nt-p)

where P= Principal , R=Annual rate of interest, n=number of times interest applied per time period, nt= total period

Interest = (21000*(1+2.5%/12)^60)-21000

= $2793.024

Total value of investment at the end of 5 years when Yumi turn to 17 years= $21000+$2793.024= $23793.024

below are the equation of monthly interest of 60 months

months	Principal at beginning	half year interest rate	Interest	Balance at the end of month
A	B	C =2.5%/12	D= (B*C)	F = B+D
1	21000.00	0.21%	43.75	21,043.75
2	21043.75	0.21%	43.84	21,087.59
3	21087.59	0.21%	43.93	21,131.52
4	21131.52	0.21%	44.02	21,175.55
5	21175.55	0.21%	44.12	21,219.66
6	21219.66	0.21%	44.21	21,263.87
7	21263.87	0.21%	44.30	21,308.17
8	21308.17	0.21%	44.39	21,352.56
9	21352.56	0.21%	44.48	21,397.05
10	21397.05	0.21%	44.58	21,441.62
11	21441.62	0.21%	44.67	21,486.29
12	21486.29	0.21%	44.76	21,531.06
13	21531.06	0.21%	44.86	21,575.91
14	21575.91	0.21%	44.95	21,620.86
15	21620.86	0.21%	45.04	21,665.91
16	21665.91	0.21%	45.14	21,711.04
17	21711.04	0.21%	45.23	21,756.28
18	21756.28	0.21%	45.33	21,801.60
19	21801.60	0.21%	45.42	21,847.02
20	21847.02	0.21%	45.51	21,892.54
21	21892.54	0.21%	45.61	21,938.15
22	21938.15	0.21%	45.70	21,983.85
23	21983.85	0.21%	45.80	22,029.65
24	22029.65	0.21%	45.90	22,075.54
25	22075.54	0.21%	45.99	22,121.54
26	22121.54	0.21%	46.09	22,167.62
27	22167.62	0.21%	46.18	22,213.80
28	22213.80	0.21%	46.28	22,260.08
29	22260.08	0.21%	46.38	22,306.46
30	22306.46	0.21%	46.47	22,352.93
31	22352.93	0.21%	46.57	22,399.50
32	22399.50	0.21%	46.67	22,446.16
33	22446.16	0.21%	46.76	22,492.93
34	22492.93	0.21%	46.86	22,539.79
35	22539.79	0.21%	46.96	22,586.75
36	22586.75	0.21%	47.06	22,633.80
37	22633.80	0.21%	47.15	22,680.96
38	22680.96	0.21%	47.25	22,728.21
39	22728.21	0.21%	47.35	22,775.56
40	22775.56	0.21%	47.45	22,823.01
41	22823.01	0.21%	47.55	22,870.55
42	22870.55	0.21%	47.65	22,918.20
43	22918.20	0.21%	47.75	22,965.95
44	22965.95	0.21%	47.85	23,013.79
45	23013.79	0.21%	47.95	23,061.74
46	23061.74	0.21%	48.05	23,109.78
47	23109.78	0.21%	48.15	23,157.93
48	23157.93	0.21%	48.25	23,206.18
49	23206.18	0.21%	48.35	23,254.52
50	23254.52	0.21%	48.45	23,302.97
51	23302.97	0.21%	48.55	23,351.52
52	23351.52	0.21%	48.65	23,400.17
53	23400.17	0.21%	48.75	23,448.92
54	23448.92	0.21%	48.85	23,497.77
55	23497.77	0.21%	48.95	23,546.72
56	23546.72	0.21%	49.06	23,595.78
57	23595.78	0.21%	49.16	23,644.93
58	23644.93	0.21%	49.26	23,694.19
59	23694.19	0.21%	49.36	23,743.56
60	23743.56	0.21%	49.47	23,793.02

Yumi wants to withdraw above generated fund of $ 23,793.02 in four years starting 18 years of age in equal instalment and fund will earn interest @ 3% annually compounding.

To find out equal withdrawal amount, we can apply EMI formula= 'PMT(Rate,NPER,PV,FV,type)

   =PMT(3.85%,9,11000,0,0)

= -6400.97

It can be proved from below Table

year	Principal at beginning	interest Rate annually	Interest	withdrawal	Balance at the end
A	B	C	D= (B*C)	E (EMI)	F = B+D-E
1	23,793.02	3%	713.79	6400.97	18,105.84
2	18,105.84	3%	543.18	6400.97	12,248.05
3	12,248.05	3%	367.44	6400.97	6,214.52
4	6,214.52	3%	186.44	6400.97	(0.01)

So equal instalment of withdrawal will be $ 6400.97 for four years