Question

a personal account earmarked as a retirement supplement contains $292,200. suppose $250,000 is used to establish an annuity that earns 6%, compounded quarterly, and pays $6000 at the end of each quarter. how long will it be until the account balance is $0? (round your answer up to the nearest quarter.)

Answer 1

Suppose $250,000 is used to establish an annuity that earns 6%, compounded quarterly, and pays $6000 at the end of each quarter. It will take about 120 quarters until the account balance reaches $0.

Amount invested (Present value) = $250000

Quarterly payment (At the end of each quarter) (P) = $4500

Interest Rate (Quarterly) (r) =  6% /4

= 1.5% = 0.015

A number of quarters (n) = ?

Future value at the end  = 0

Present value of Annuity formula:

Present value  = P × $(1-(1+r))^{(-n)} / r$

250000 =  4500 × $(1-((1+0.015))^{(-n)} / 0.015$

250000 =  300000 × $(1-((1+0.015)}}^{(-n)}$

250000 / 300000 = $1-(1+0.015)^{(-n)}$

0.83333 = $1-(1.015)^{(-n)$

n = 120

Hence is shall take 120 Quarters until the account balance is $0.

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