Question

Imagine you have some rich cousin who looks down at you. She thinks you are incapable of managing your own affairs and talks you into supplementing some form of future retirement income through some individual retirement plan. You are supposed to retire in 15 years. You have to make 15 annual deposits into your account until you retire. The first deposit is $10K, and each subsequent deposit will increase at a 4% rate. Then, good times come and you can have 10 equal annual withdrawals. The last withdrawal is at the end of 25^th year. Calculate (A), the amount of the equal withdrawals that you will get in the last 10 years. Your cousin has told you that the interest rate will always remain at 8%, compounded annually. Also, draw the cash flow diagram.

Answer 1

Given:

At the beginning of the first year, $10,000 is deposited for 15 years. Future Value of 1st $10,000 invested for 15 years at 8% compound rate, FV1 = $10000 * (1+0.08)^15 = $31,721.69

Subsequent years, deposit increases at 4%.

So for the second year, amount deposited = $10,000 *(1+0.04) = $10,400. This amount is deposited for 14 years, that grows at 8%p.a compounded annually. Future Value of $10,400 invested for 14 years FV2 = $10400 * (1+0.08)^14 = $30,546.81

For 3rd year, amount invested = $10,000 * (1+0.04)^2 = $10,816 which is invested for 13 years. Its future value, FV3 = $10,816 * (1+0.08)^13 = $29,415.45

and so on for remaining 12 years....

Similar calculation for the last deposit, that is made at the starting of 14th year is [$10,000 * (1+0.04)^14 = $17,316.76 which will be invested for 1 year. Future value of the last deposit $17,316.76, invested for 1 year is FV15 = $17,316.76 * (1+0.08) = $18,702.11

Future value of the 15 deposits made is the sum of all the future values of each deposit. V(15) = FV1 + FV2 + ... + FV15 = $31,721.69 + $30,546.81 + $29,415.45 + ...+ $18,702.11 = $370,230.91

This value V15 = $370,230.91 is at the end of 14th year. (or start of 15th year)

With V15 as present value, n = 10 years, rate = 8%, Equated annual amount is calculated as:

$370,230.91 = [A / (1+0.08)] + [A/(1+0.08)^2] + .... + [A/(1+0.08)^10]

Solving for A, we get A = $55,175.32

However, deposit in the account grows at 8% p.a. So $370,230.91 grows to $399,849.38 after 1 year. [i.e; V16 = V15 * (1+0.08) = $370,230.91 * (1+0.08) = $399,849.38]

At the beginning of 2nd year of post-retirement, amount left in the account will be $344,674.06[i.e; $399,849.38 - $55,175.32] That is, after the first withdrawal.

Now, $344,674.06 will grow at 8% for 1 year to $372,247.98, from which again $55,175.32 will be withdrawn for the 2nd time. Ending balance after 2 years of retirement = $317,072.66 [i.e; $372,247.98 - $55,175.32]

This calculation is shown below as tabular form:

Year	Beginning Balance	Investment growth after 1 year at 8%	Amt withdrew at year end	Ending balance
16	370,230.91	399,849.38	55,175.32	344,674.06
17	344,674.06	372,247.98	55,175.32	317,072.66
18	317,072.66	342,438.47	55,175.32	287,263.15
19	287,263.15	310,244.20	55,175.32	255,068.88
20	255,068.88	275,474.39	55,175.32	220,299.06
21	220,299.06	237,922.99	55,175.32	182,747.66
22	182,747.66	197,367.48	55,175.32	142,192.15
23	142,192.15	153,567.53	55,175.32	98,392.20
24	98,392.20	106,263.58	55,175.32	51,088.25
25	51,088.25	55,175.31	55,175.32	(0.01)

Cash Flow diagram as below: