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Question 1086548: An instructor plans to retire in exactly one year and want an account that will pay him
25,000.00 a year for the next 15 years. Assuming a 6% annual effective interest rate, what is the amount he would need to deposit now? (The fund will be depleted after 15 years.)
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Draw out a timeline. A simple numberline will do
The values alone the number line indicate the year number.
The timeline is broken down into two phases
Phase1: The period from year0 to year1. This is a period of 1 year. The money is not withdrawn during this phase (the instructor hasn't retired yet). Only one deposit is made at the beginning of the year. This value is unknown but we can use the present value formula to find it.
Phase2: The period from year1 to year16 which is a period of 16-1 = 15 years. During this phase, the professor has returned. The money is now withdrawn at a rate of $25,000 per year.
There are 15 years, so n = 15
The cashflow amounts, or payment amounts, are C = 25000
The interest rate is r = 0.06
Plug these values into the present value of an annuity formula
PV = P*( (1-(1+r)^(-n))/r )
PV = 25000*( (1-(1+0.06)^(-15))/0.06 )
PV = 242806.224693525
Rounding up to the nearest cent, we get PV = 242806.23
You must round up to clear the hurdle for that last year.
Answer: $242,806.23
This is what I get when I use a spreadsheet to calculate the various balances throughout the years. As you can see, the last balance is 0.01 dollars which is close enough to $0.
The balance when n = 0 indicates the initial deposit
The balance when n = 1 is the amount after 1 year (still in phase 1)
The balance when n = 2 is when phase 2 starts and the withdrawals start to occur.
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