the amount t a person would need to deposit today to be able to withdraw $6000 each year for 10 years from an account earning 6 percent

Question

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Answer 1

Hello, thank you for taking the time to post your question!

The underlying formula that you want to use here is the present value of an annuity formula, meaning

PV = PMT * [(1 – (1+r)^(-n))/r]

For the parameters given in this equation that becomes

PV = 6000 * [(1 – (1 + 0.06)^(-10))/0.06)

PV = 6000 * 7.36008717

PV = $44,160.52

So the amount that you need to deposit today to meet these criteria is $44,160.52

I hope that helps get you moving in the right direction! Feel free to reach out if you have any additional questions beyond that :)

Answer 2

Hi Kelli!

 

Ok this problem involves something called the Annuity Due Payment, and there is a standard formula for solving this type of problem:

 

Pmt=PV * ( r / ( 1-(1+r)^-n ) ) * 1/(1+r)

 

where Pmt is the annual payment ($6,000.00 in this case), PV is the present value, or the amount deposited today, r is the interest rate per period (6%), and n is the number of periods (10). So, we have to plug a few things in:

 

$6,000 = PV * ( 0.06 / (1-1.06^-10) ) * 1/1.06

 

Multiplying out the right hand side gives us

 

$6,000 = PV * 0.128177319

 

Dividing both sides by 0.128177319 yields

 

PV = $46,810.15

 

To illustrate what happens to this money, I will write a quick table for you:

 

 

n          PV           PV-6,000

1     46,810.15     40,810.15

2     43,258.76     37,258.76
3     39,494.28     33,494.28
4     35,503.94     29,503.94
5     31,274.18     25,274.18
6     26,790.63     20,790.63
7     22,038.07     16,038.07
8     17,000.35     11,000.35
9     11,660.37       5,660.37
10     5,999.99             0.00

 

In the table above, for n = 2 and greater, PV was calculated by adding interest of 1.06X from the remaining balance from the previous period after the withdraw of $6,000.00 was made.