Meg's pension plan is an annuity with a guaranteed return of 4% per year (compounded quarterly). She would like to retire with a pension of $20,000 per quarter for 10 years. If she works 30 years before retiring, how much money must she and her employer deposit each quarter?
To calculate the amount of money Meg and her employer need to deposit each quarter into her pension plan, we need to use the formula for the future value of an annuity.
The formula for the future value of an annuity is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = future value of the annuity
P = payment per period (quarter in this case)
r = interest rate per period (compounded quarterly, so we divide the annual interest rate by 4)
n = number of periods (30 years, so 120 quarters)
In this case, Meg wants to receive $20,000 per quarter for 10 years, which is equivalent to 40 quarters (10 years x 4 quarters/year).
Substituting the given values into the formula:
FV = $20,000 * [(1 + 0.04/4)^120 - 1] / (0.04/4)
FV = $20,000 * [1.01^120 - 1] / 0.01
Now we can calculate the future value of the annuity:
FV = $20,000 * (2.10306291006644 - 1) / 0.01
FV = $20,000 * 1.10306291006644 / 0.01
FV = $20,000 * 110.306291006644
FV = $2,206,125.82
So, the total amount needed in the pension plan after 30 years of contributions is approximately $2,206,125.82.
To determine how much money Meg and her employer need to deposit each quarter, we can use the formula for the present value of an annuity:
PV = P * [1 - (1 + r)^(-n)] / r
In this case, we need to solve for P:
$2,206,125.82 = P * [1 - (1 + 0.04/4)^(-120)] / (0.04/4)
Simplifying the equation:
$2,206,125.82 = P * [1 - (1.01)^(-120)] / 0.01
Now solve for P:
P = $2,206,125.82 * 0.01 / [1 - (1.01)^(-120)]
P ≈ $9,760.55
Therefore, Meg and her employer need to deposit approximately $9,760.55 each quarter into her pension plan.