Final answer:
To find out how long the annuity will last until the account balance is $0, we use the present value of an annuity formula and solve for the total number of payments. Since the annuity is compounded quarterly, we divide the interest rate by 4 and solve for 'n' which tells us the total number of quarters. This number can then be converted to years to get the duration of the annuity payout.
Step-by-step explanation:
The question involves calculating the duration of an annuity. An annuity is a financial product that pays out a fixed stream of payments to an individual, typically used as an income stream for retirees. In this case, the annuity pays $4500 at the end of each quarter and earns 5% interest, compounded quarterly. To find out how long it will take for the account balance to reach $0, we can use the formula for the present value of an annuity:
PV = Pmt \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)
Where:
PV is the present value of the annuity, which is $300,000 in this case
Pmt is the payment amount per period, which is $4500
r is the interest rate per period, which is 0.05/4 since the interest is compounded quarterly
n is the total number of payments
We need to solve for n, which requires rearranging the formula and then using numerical methods if necessary since there isn't an algebraic solution for n. A financial calculator or spreadsheet software can be very useful for this calculation. Once you solve for n, it will tell you the total number of quarters until the account balance is $0. To express that number in years, divide by 4 since there are 4 quarters in a year.