Question

Please show full work so that I can understand these types of problems, thank you!

Complete the following Complex Time Value of Money problem.

You are doing some long-range retirement planning. On the day you retire (23 years from now) you want to be able to withdraw $200,000. Then, you want to withdraw the following amounts at the end of each year after that (during your retirement period).

                           Years 1-4            $160,000

                           Years 5-9            $175,000

                           Years 10-15       $165,000

                           Years 16-26       $145,000

At the end of the 26^th year in retirement, you’d like to have $500,000 remaining in your retirement account available for withdraw. During your retirement years, you anticipate earning a 4.5% rate of return.

You currently have $275,000 that you are going to use to start your retirement savings today. In addition, you plan to save $700 at the end of each month for the next 8 years. At that point (8 years from today) you will add another $150,000 to your retirement fund. Then, over the remaining 15 years, how much must you save at the end of each month to reach your goal if you earn 8.9% as a rate of return during the first 8 years and 7.6% over the final 15 years in which you are saving for retirement?

Answer 1

Given you require to accumulate certain amount of wealth by the end of 23rd year from today,
		Calculation of amount to be accumulated:
		Given the future values of amounts required as below
Timeline terms used in Question	End of 23rd year	Each year (years 1-4)	Each year (years 5-9)	Each year (years 10-15)	Each year (years 16-26)	End of 26th year
Actual terms of timeline	(i.e. 23)	(i.e. 24 to 27)	(i.e. 28 to 32)	(i.e. 33 to 38)	(i.e. 39 to 49)	(i.e. 49)
Taking 23rd year as base year	Year 0	Year 1-4	Year 5-9	Year 10-15	Year 16-26	Year 26
Cashflows	$ 200000	$ 160000	$ 175000	$ 165000	$ 145000	$ 500000

Since we need amount from the end of 23rd year we should discount the given required future cashflows
using 4.5% rate of return as given in question as anticipated return to get amount required to be accumulated
by the end of 23rd year in order to receive required amounts on future dates.
Since we require present value of cashflows by the end of 23rd year - I am taking 23rd year as base year and
discounting the cashflows with 4.5% discounting rate in the below table

Actual terms of timeline	Taking 23rd year as base year	Cashflows required (A) in $	Working for B	Discount rate @ 4.5% by taking 23rd year as base year (B)	Present value of cashflow (C=A*B) in $
23	0	200000	1	1.00	200000.00
24	1	160000	1/1.045	0.96	153110.05
25	2	160000	1/(1.045^2)	0.92	146516.79
26	3	160000	1/(1.045^3)	0.88	140207.46
27	4	160000	1/(1.045^4)	0.84	134169.81
28	5	175000	1/(1.045^5)	0.80	140428.93
29	6	175000	1/(1.045^6)	0.77	134381.75
30	7	175000	1/(1.045^7)	0.73	128594.98
31	8	175000	1/(1.045^8)	0.70	123057.40
32	9	175000	1/(1.045^9)	0.67	117758.27
33	10	165000	1/(1.045^10)	0.64	106248.07
34	11	165000	1/(1.045^11)	0.62	101672.79
35	12	165000	1/(1.045^12)	0.59	97294.54
36	13	165000	1/(1.045^13)	0.56	93104.82
37	14	165000	1/(1.045^14)	0.54	89095.52
38	15	165000	1/(1.045^15)	0.52	85258.87
39	16	145000	1/(1.045^16)	0.49	71698.05
40	17	145000	1/(1.045^17)	0.47	68610.58
41	18	145000	1/(1.045^18)	0.45	65656.05
42	19	145000	1/(1.045^19)	0.43	62828.76
43	20	145000	1/(1.045^20)	0.41	60123.21
44	21	145000	1/(1.045^21)	0.40	57534.18
45	22	145000	1/(1.045^22)	0.38	55056.63
46	23	145000	1/(1.045^23)	0.36	52685.77
47	24	145000	1/(1.045^24)	0.35	50417.00
48	25	145000	1/(1.045^25)	0.33	48245.94
49	26	145000	1/(1.045^26)	0.32	46168.36
49	26	500000	1/(1.045^26)	0.32	160000.00	extra amount required
Total amount required at the end of 23rd year (Sum of Column C)	2789924.59

Given the cashflows of savings being made from year 0 to year 23

Also the rate of return expected from such savings is given

If we find the future value of these savings by the end of 23rd year we can

compare it with required accumulation by 23rd year and calculate the additional amount required if any

In the below table I am compounding the savings to find the future value of savings by 23rd year end

Compounding rate is 8.9% for first eight years, 7.6% for remaining 15 years

Year	Savings (Given)(A)	Working for B	Compounding rate for years 0-8)(B)	Future Value (C=A*B)
0	275000	[(1.076)^14]*[(1.089)^9]	6.01	1651792.20
1	8400	[(1.076)^14]*[(1.089)^8]	5.52	46331.26
2	8400	[(1.076)^14]*[(1.089)^7]	5.06	42544.78
3	8400	[(1.076)^14]*[(1.089)^6]	4.65	39067.75
4	8400	[(1.076)^14]*[(1.089)^5]	4.27	35874.88
5	8400	[(1.076)^14]*[(1.089)^4]	3.92	32942.96
6	8400	[(1.076)^14]*[(1.089)^3]	3.60	30250.65
7	8400	[(1.076)^14]*[(1.089)^2]	3.31	27778.38
8	8400	[(1.076)^14]*1.089	3.04	25508.15
8	150000	[(1.076)^14]*1.089	3.04	455502.68
9	X*12	(1.076)^14	2.79	33.46X
10	X*12	(1.076)^13	2.59	31.10X
11	X*12	(1.076)^12	2.41	28.90X
12	X*12	(1.076)^11	2.24	26.86X
13	X*12	(1.076)^10	2.08	24.96X
14	X*12	(1.076)^9	1.93	23.20X
15	X*12	(1.076)^8	1.80	21.56X
16	X*12	(1.076)^7	1.67	20.04X
17	X*12	(1.076)^6	1.55	18.62X
18	X*12	(1.076)^5	1.44	17.31X
19	X*12	(1.076)^4	1.34	16.09X
20	X*12	(1.076)^3	1.25	14.95X
21	X*12	(1.076)^2	1.16	13.89X
22	X*12	1.076	1.08	12.91X
23	X*12	1	1.00	12.00X
			Sum of Column C	2387593.68+315.86X

As per our calculations 2387593.68+315.86X = 2789924.59
315.86X=	402330.9
X=	1273.76
Therefore , from 9th year to 23rd year every month the person should save $ 1273.76
Notes:
1. Discounting rate of nth year = 1/[(1+rate)^n]
2. Compounding rate of nth year = (1+rate)^n