For the last 19 years, Mary has been depositing $500 in her savings account , which has earned 5% per year, compounded annually and is expected to continue paying that amount. Mary will make one more $500 deposit one year from today. If Mary closes the account right after she makes the last deposit, how much will this account be worth at that time?
Mary has been working at the university for 25 years, with an excellent record of service. As a result, the board wants to reward her with a bonus to her retirement package. They are offering her $75,000 a year for 20 years, starting one year from her retirement date and each year for 19 years after that date. Mary would prefer a one-time payment the day after she retires. What would this amount be if the appropriate interest rate is 7%?Mary’s replacement is unexpectedly hired away by another school, and Mary is asked to stay in her position for another three years. The board assumes the bonus should stay the same, but Mary knows the present value of her bonus will change. What would be the present value of her deferred annuity?
Mary wants to help pay for her granddaughter Beth’s education. She has decided to pay for half of the tuition costs at State University, which are now $11,000 per year. Tuition is expected to increase at a rate of 7% per year into the foreseeable future. Beth just had her 12th birthday. Beth plans to start college on her 18th birthday and finish in four years. Mary will make a deposit today and continue making deposits each year until Beth starts college. The account will earn 4% interest, compounded annually. How much must Mary’s deposits be each year in order to pay half of Beth’s tuition at the beginning of each school each year?
To answer these financial questions, we will need to use different financial formulas and calculations. Let me explain how to find the solutions to each question.
1. For the first question about Mary's savings account:
To determine the future value of the account, we can use the formula for compound interest:
FV = PV * (1 + r)^n
Where:
- FV is the future value of the account
- PV is the present value (initial deposit)
- r is the interest rate
- n is the number of years
Given:
- PV = $500 (annual deposit)
- r = 5% = 0.05 (annual interest rate)
- n = 19 (number of years)
To calculate the future value, we need to calculate the future value of the annual deposits and add it to the future value of the initial deposit:
Future value of annual deposits = (Annual deposit * ((1 + r)^n - 1)) / r
Future value of the initial deposit = Initial deposit * (1 + r)^n
So, the future value of the account at the time of the last deposit will be the sum of these two values.
2. For the second question about Mary's retirement package:
To find the one-time payment amount, we need to calculate the present value of the future cash flows using the present value of an annuity formula:
PV = CF / (1 + r)^n
Where:
- PV is the present value
- CF is the cash flow amount
- r is the interest rate
- n is the number of years
Given:
- CF = $75,000 (annual cash flow)
- r = 7% = 0.07 (annual interest rate)
- n = 20 (number of years)
To calculate the present value, we need to calculate the present value of each annual cash flow and sum them up.
3. For the third question about the present value of Mary's deferred annuity:
Since Mary will work for an additional three years, the future cash flows will start one year later. We need to recalculate the present value using the same formula for the new time period.
4. For the fourth question about Mary's deposits for Beth's education:
To calculate the annual deposits, we need to determine the present value of the future tuition payments using the present value of an annuity formula:
PV = (CF * (1 - (1 + r)^(-n))) / r
Where:
- PV is the present value
- CF is the cash flow amount
- r is the interest rate
- n is the number of years
Given:
- CF = $11,000 (annual tuition payment)
- r = 4% = 0.04 (annual interest rate)
- n = 4 (number of years)
To calculate the annual deposits, we need to divide the present value by the number of years.
Using these formulas and calculations, we can find the solutions to each of the financial questions.