Meg's pension plan is an annuity with a guaranteed return of 7% per year (compounded quarterly). She would like to retire with a pension of $20,000 per quarter for 15 years. If she works 31 years before retiring, how much money must she and her employer deposit each quarter?
1.) PMT(1-(1+r/m)^-mt/(r/m)
20,000(1-(1+0.07/4)^(-4*15))/(0.07/4) =739279.71
2.) Plug back into FV
739279.71(0.07/4)/((1+0.07/4)^4*31-1)= 1703.31
Final answer= 1703.31 (rounded to the nearest cent)
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To determine the amount of money Meg and her employer must deposit each quarter, we need to calculate the present value of the annuity.
The formula to calculate the present value of an annuity is:
P = (C x (1 - (1 + r)^(-n))) / r
Where:
P = Present value of the annuity
C = Cash flow per period ($20,000 in this case)
r = Interest rate per period (7% per year compounded quarterly = 0.07/4 = 0.0175 per quarter)
n = Number of periods (15 years x 4 quarters per year = 60 quarters)
Let's substitute the values into the formula:
P = (20,000 x (1 - (1 + 0.0175)^(-60))) / 0.0175
Calculating this formula will give us the present value of the annuity.
To calculate the amount of money Meg and her employer must deposit each quarter, we can use the formula for the present value of an annuity. The present value of an annuity is the sum of the discounted future cash flows.
Here's how we can break down the calculation:
1. Determine the number of compounding periods. Since the guaranteed return is compounded quarterly and Meg will retire after 31 years of working, the number of compounding periods will be 31 * 4 = 124.
2. Calculate the interest rate per compounding period. The guaranteed return is 7% per year, compounded quarterly. To get the interest rate per compounding period, we divide the annual interest rate by the number of compounding periods per year: 7% / 4 = 1.75%.
3. Convert the number of years of retirement income into compounding periods. Meg wants to receive $20,000 per quarter for 15 years. Since there are 4 quarters in a year, the total number of compounding periods during her retirement will be 4 * 15 = 60.
4. Calculate the present value of the retirement cash flows. To calculate the present value, we use the formula:
PV = C * (1 - (1 + r)^(-n)) / r
where PV is the present value, C is the cash flow per period, r is the interest rate per period, and n is the number of periods.
Using the values we have:
C = $20,000 (retirement income per quarter)
r = 1.75% (interest rate per quarter)
n = 60 (number of quarters during retirement)
PV = 20000 * (1 - (1 + 0.0175)^(-60)) / 0.0175
Calculating this expression will give us the present value of the retirement cash flows.
5. Divide the present value by the total number of compounding periods during the working years. Since Meg works for 31 years, the total number of compounding periods during the working years is 31 * 4 = 124.
Determine the amount Meg and her employer must deposit each quarter by dividing the present value by the total number of compounding periods:
Deposit per quarter = PV / (31 * 4)
Calculating this expression will give us the desired amount that Meg and her employer must deposit each quarter.