Meg's pension plan is an annuity with a guaranteed return of 3% per year (compounded quarterly). She would like to retire with a pension of $50,000 per quarter for 10 years. If she works 26 years before retiring, how much money must she and her employer deposit each quarter? HINT [See Example 5.] (Round your answer to the nearest cent.)
To determine the amount of money that Meg and her employer need to deposit each quarter, we can use the formula for the present value of an annuity.
The formula for the present value of an annuity is:
PV = PMT × (1 - (1 + r)^(-n)) / r
Where:
PV = present value (amount to be deposited each quarter)
PMT = pension amount per quarter ($50,000)
r = interest rate per compounding period (quarterly rate = annual rate / compounding periods per year)
n = number of compounding periods (number of quarters Meg will receive the pension, 10 years × 4 quarters/year = 40 quarters)
Let's calculate the quarterly interest rate first:
Annual rate = 3%
Quarterly rate = Annual rate / compounding periods per year = 3% / 4 = 0.75%
Now we can plug in the values into the formula:
PV = $50,000 × (1 - (1 + 0.0075)^(-40)) / 0.0075
Using a calculator, we can calculate the present value:
PV ≈ $50,000 × (1 - 0.314593) / 0.0075
PV ≈ $50,000 × 0.685407 / 0.0075
PV ≈ $4,076,433.92
So, Meg and her employer would need to deposit approximately $4,076,433.92 each quarter for her to retire with a pension of $50,000 per quarter for 10 years.
To calculate how much money Meg and her employer must deposit each quarter, we need to use the formula for the present value of an ordinary annuity:
\[ PV = \dfrac{PMT \times (1 - (1+r)^{-n})}{r} \]
Where:
PV = Present Value
PMT = Payment per period
r = Interest rate per period
n = Number of periods
The payment per period (PMT) is given as $50,000. The interest rate per period (r) is 3% divided by 4 (since it is compounded quarterly), which is 0.75%. The number of periods (n) is 10 years multiplied by 4 (since there are 4 quarters in a year), which is 40.
Now, substituting the values into the formula:
\[ PV = \dfrac{50,000 \times (1 - (1+0.0075)^{-40})}{0.0075} \]
Solving this equation will give us the present value (PV), which represents the total amount of money that Meg and her employer must deposit each quarter.
Calculating this equation will give us the desired result.