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You are currently 42 years old and you intend to retire at age 60
You are currently 42 years old and you intend to retire at age 60. You are planning to start a retirement account. At the beggining of each of your year's work, you intend to make a deposit into the retirement account. You think that the account will earn 8% per year. After retirement at age 60 you expect to live 23 more years. At the beggining of each of these years you intend to withdraw 80,000 dollars from your retirement account. Your account balances will continue to earn 8%. How much should you deposit annually into the account? Use 2 alternative methods Year Age Total, Payment at Withdrawal at beginning of beginning of beginning of year year year Total end of year
Expert Solution
| First, we need to find the accumulation at the end of 60 yrs./beginning of 61 st yr.---from which you start beginning-of-yr. drawings of $ 80000 annually---at 8% interest rate |
| so, we use the present value of annuity due(beginning-of-yr.) formula, |
| PVAdue=Pmt.*(1-(1+r)^-n)/r*(1+r) |
| where, |
| PVADue-- is the to-be accumulated retirement savings |
| Pmt.--is the beginning-of-yr. annual drawings, $ 80000 |
| r= rate of interest, 8% p.a. |
| n= beginning of yr.61 to beg-of yr. 83---23 no.of withdrawals |
| so, plugging in these values, in the formula, |
| PVAdue=80000*(1-1.08^-23)/0.08*1.08= |
| 896059.49 |
| So, the above is the amount to be accumulated by end yr.60 |
| It means that this is the Future value of annuity of savings from beg.-of-yr. 43 to beg.-of-yr. 60---ie. 18 no.of annual pmts. |
| here,we need touse the future value of annuity due(beginning-of-yr.) formula, |
| FVAdue=Pmt.*((1+r)^n-1)/r*(1+r) |
| where, |
| FVADue-- is the to-be accumulated retirement savings --- found out above as $ 896059.49 |
| Pmt.--is the beginning-of-yr. annual savings---- to be found out----?? |
| r= rate of interest, same 8% p.a. |
| n= beginning of yr.43 to beg-of yr. 60---18 no.of payments into the retirement a/c |
| so, plugging in these values, in the formula, |
| 896059.49=Pmt.*((1+0.08)^18-1)/0.08*(1+0.08) |
| Pmt.=896059.49/(((1+0.08)^18-1)/0.08*(1+0.08))= |
| 22154.32 |
| SO, the amt. you should d eposit annually, for 18 yrs.= 22154.32 |
| Method-2 |
| Instead of using two-steps as above, we can also calculate straight-away, |
| ie. Divide the PV of accumulation(as in Part-1) by the FVADue factor(as in part-2) |
| ie. (80000*(1-1.08^-23)/0.08*1.08)/(((1+0.08)^18-1)/0.08*(1+0.08))= |
| 22154.32 |
| (Answer) |
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