In: Finance
Laquita deposits $5500 in her retirement account every year. If her account pays an average 6% interest and she makes 38 deposits before she retires, how much monet can she withdraw in 20 equal payments beginning one year after her last deposit? please show cash flow diagram and solve NOT with excel but with the interest rate formula equations!!!
The total value that will be saved at the end of 38 deposits will be given by finding the future value of annuity.
FV of annuity = P*[((1+r)^n - 1)/r]
P - Periodic payment = 5500
r - rate per period = 6%
n - number of periods = 38
FV of annuity = 5500*(((1+0.06)^38 - 1)/0.06) = $747473.1318
This becomes the present value of annuity for future withdrawals.
Present value of annuity is given by,
PV of annuity = P*[(1-(1+r)^(-n)) / r]
P - Periodic payment = ?
r - rate per period = 0.06
n - number of periods = 20
747473.1318 = P*((1-(1+0.06)^(-20)) / 0.06)
747473.1318 = P*11.46992
P = 747473.1318/11.46992 = $65168.12
She can withdraw an amount of $65168.12 every year for 20 years.
Cash flows:
Year | Deposit amount | Amount at the end of the year = Deposit amount*(1+0.06)^year + Previous end of year amount |
0 | 5500 | 5500 |
1 | 5500 | 11330 |
2 | 5500 | 17509.8 |
3 | 5500 | 24060.388 |
4 | 5500 | 31004.01128 |
5 | 5500 | 38364.25196 |
6 | 5500 | 46166.10707 |
7 | 5500 | 54436.0735 |
8 | 5500 | 63202.23791 |
9 | 5500 | 72494.37218 |
10 | 5500 | 82344.03451 |
11 | 5500 | 92784.67658 |
12 | 5500 | 103851.7572 |
13 | 5500 | 115582.8626 |
14 | 5500 | 128017.8344 |
15 | 5500 | 141198.9044 |
16 | 5500 | 155170.8387 |
17 | 5500 | 169981.089 |
18 | 5500 | 185679.9544 |
19 | 5500 | 202320.7516 |
20 | 5500 | 219959.9967 |
21 | 5500 | 238657.5965 |
22 | 5500 | 258477.0523 |
23 | 5500 | 279485.6754 |
24 | 5500 | 301754.816 |
25 | 5500 | 325360.1049 |
26 | 5500 | 350381.7112 |
27 | 5500 | 376904.6139 |
28 | 5500 | 405018.8907 |
29 | 5500 | 434820.0242 |
30 | 5500 | 466409.2256 |
31 | 5500 | 499893.7792 |
32 | 5500 | 535387.4059 |
33 | 5500 | 573010.6503 |
34 | 5500 | 612891.2893 |
35 | 5500 | 655164.7667 |
36 | 5500 | 699974.6527 |
37 | 5500 | 747473.1318 |
Withdrawal cashflows:
No. of Withdrawal | Amount at the beginning of withdrawal (A) | Amount in the account after withdrawal (B) = A - 65168.12 | Amount at the end of the year = B*(1+0.06) |
0 | 747473.13 | 747473.13 | 792321.52 |
1 | 792321.52 | 727153.40 | 770782.60 |
2 | 770782.60 | 705614.48 | 747951.35 |
3 | 747951.35 | 682783.23 | 723750.23 |
4 | 723750.23 | 658582.11 | 698097.03 |
5 | 698097.03 | 632928.91 | 670904.65 |
6 | 670904.65 | 605736.53 | 642080.72 |
7 | 642080.72 | 576912.60 | 611527.36 |
8 | 611527.36 | 546359.24 | 579140.79 |
9 | 579140.79 | 513972.67 | 544811.03 |
10 | 544811.03 | 479642.91 | 508421.48 |
11 | 508421.48 | 443253.36 | 469848.57 |
12 | 469848.57 | 404680.45 | 428961.27 |
13 | 428961.27 | 363793.15 | 385620.74 |
14 | 385620.74 | 320452.62 | 339679.78 |
15 | 339679.78 | 274511.66 | 290982.36 |
16 | 290982.36 | 225814.24 | 239363.09 |
17 | 239363.09 | 174194.97 | 184646.67 |
18 | 184646.67 | 119478.55 | 126647.26 |
19 | 126647.26 | 61479.14 | 65167.89 |
20 | 65167.89 | -0.23 | -0.24 |