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 5 years. If she works 21 years before retiring, how much money must she and her employer deposit each quarter?

To calculate how much money Meg and her employer must deposit each quarter, we need to determine the future value of the annuity. The future value is the accumulated amount that the annuity will grow to over time.

The formula to calculate the future value of an annuity is:

FV = PV * (1 + r/n)^(nt)

Where:
FV = future value
PV = present value (the amount of each deposit)
r = annual interest rate (3% in this case)
n = number of compounding periods per year (quarterly, so n = 4)
t = number of years

We know that Meg wants to retire with a pension of $50,000 per quarter for 5 years. So, the future value of the annuity is $50,000 * 5 = $250,000.

Let's substitute the values into the formula:

$250,000 = PV * (1 + 0.03/4)^(4*21)

Simplifying the equation:

$250,000 = PV * (1.0075)^(84)

To solve for PV (the present value), we need to isolate it on one side of the equation. Divide both sides of the equation by (1.0075)^(84):

PV = $250,000 / (1.0075)^(84)

Using a calculator, we can find the exact value of (1.0075)^(84) = 1.310798823.

So, PV = $250,000 / 1.310798823 = $190,590.53 (rounded to the nearest cent).

Since Meg and her employer are depositing the same amount each quarter, they will each need to deposit $190,590.53 / 2 = $95,295.27 (rounded to the nearest cent) every quarter.

Let's pick the date of her retirement as a reference point (focal point).

i = .03/4 = .0075
n for working period = 21(4) = 84
n for retirement period = 5(4) = 20
let her and her employer's contribution be x

So the "amount" of the contributions = Present Value of her annuities at that time.
x(1.0075^84 - 1)/.0075 = 50,000(1 - 1.0075^-20)/.0075
multiply both sides by .0075
x(1.0075^84 - 1) = 50,000(1 - 1.0075^-20)
x(.87320...) = 50000(.1388101...)
x = 7948.34

check my arithmetic

strange pension. So she is allowing her retirement to last only 5 years, with an annual income of appr. $200,000 ?
Looks like she is going to live it up for 5 years, and then .... ???