Paper For Above instruction
**Issue A: Valuation of Savings Account After 20 Years of Deposits**
Mary has been depositing $500 annually into a savings account earning 5% interest compounded annually for the past 19 years. She plans to make her final deposit one year from now, after which she intends to close the account. To determine the amount in her account at that time, we need to calculate the future value of her current deposits and the upcoming deposit combined.
The calculation involves finding the future value of the 19 deposits made over the last 19 years, each accumulated with interest over the remaining years until she closes the account. Since each deposit is made at the end of each year, we use the future value of an ordinary annuity formula:
FV = P \times \frac{(1 + r)^n - 1}{r}
Where:
P = $500
r = 0.05 (interest rate)
n = 19 (number of deposits)
Calculating:
FV of previous deposits = 500 \times \frac{(1 + 0.05)^{19} - 1}{0.05}
= 500 \times \frac{(1.05)^{19} - 1}{0.05}
Evaluating (1.05)^19: (1.05)^19 ≈ 2.527
FV = 500 \times \frac{2.527 - 1}{0.05} = 500 \times \frac{1.527}{0.05} = 500 \times 30.54 = \$15,270
This is the amount accumulated from her previous deposits by the time she makes her final deposit. The last deposit of $500 will be made one year from now; since the account is closed immediately after, it will have one year to grow (or remain unchanged until made). She will deposit this amount next year, so at that point, her total account value will be:
Total = FV of previous deposits + the future value of the last deposit
= 15,270 \times (1 + r) + 500
= 15,270 \times 1.05 + 500
≈ 16,033.5 + 500 = \$16,533.50
Thus, the account will be worth approximately $16,533.50 at the time she closes it after her final deposit.
**Issue B: Present Value of Retirement Bonus**
Mary is being offered a bonus of $75,000 annually for 20 years, starting one year after her retirement (the first payment occurs one year after retirement, continuing for 19 subsequent years). She prefers a lump-sum payment immediately after retirement, which requires calculating the present value of this annuity at the time of retirement, discounted at an interest rate of 7%.
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
Where:
P = $75,000
r = 0.07
n = 20
Calculating:
PV = 75,000 \times \frac{1 - (1.07)^{-20}}{0.07}
Evaluating (1.07)^-20: (1.07)^-20 ≈ 0.258
Thus:
PV = 75,000 \times \frac{1 - 0.258}{0.07} = 75,000 \times \frac{0.742}{0.07} ≈ 75,000 \times 10.60 = \$795,000
Therefore, the lump-sum equivalent immediately after retirement would be approximately $795,000, which would be a more flexible sum for Mary to manage.
**Issue C: Present Value of Deferred Bonus with Extended Service**
Given Mary’s agreement to stay three additional years, the present value of her bonus changes. Instead of receiving payments starting one year after her retirement, the payments will now start four years from her retirement date, extending the timeframe. To determine her current value of this adjusted annuity, we need to discount the original value back to the present, considering the postponed start date.
The present value of the original bonus if she retires now is approximately $795,000, as previously calculated. To find the present value of her bonus starting four years from now, discount this amount to the current time:
PV = FV \times (1 + r)^{-t}
Where:
FV = $795,000
r = 0.07
t = 4 years
PV = 795,000 \times (1.07)^{-4} ≈ 795,000 \times 0.763 = \$606,000
Therefore, the present value of Mary’s bonus, considering her extended service and delayed start, is approximately $606,000.
**Issue D: Funding Beth’s Education Through Annual Deposits**
Mary aims to fund her granddaughter Beth’s college tuition, which is currently $11,000 per year, increasing at 7% annually. Beth will start college on her 18th birthday, which is six years from now (since she just turned 12), and will attend for four years. Mary plans to make annual deposits today and until Beth begins college, earning 4% interest compounded annually. The goal is to determine the amount she must
deposit each year to cover half of the future tuition costs at the beginning of each college year.
First, project future tuition costs at the time Beth starts college:
Future tuition = 11,000 \times (1 + 0.07)^6 ≈ 11,000 \times 1.5036 ≈ \$16,540
Half of that per year would be approximately $8,270. The tuition will increase annually at 7%, so each year's tuition will be:
Year 1 (start of college): $16,540
Year 2: 16,540 \times 1.07 ≈ $17,704
Year 3: 17,704 \times 1.07 ≈ $18,943
Year 4: 18,943 \times 1.07 ≈ $20,261
Half of these costs each year are:
Year 1: $8,270
Year 2: $8,852
Year 3: $9,472
Year 4: $10,131
Next, calculate the present value of each of these amounts using the annual interest rate of 4%, discounting back to today. Because payments are made at the beginning of each year (annuity due), the present value (PV) of each is calculated using:
PV = Amount \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r)
Alternatively, in this context, since deposits are made annually, we can use the ordinary annuity formula for each respective starting date, adjusting for the timing. To simplify, imagine making deposits at the end of each year from now until Beth starts college in 6 years, and then summing the present values of each of the four tuition payments discounted back to today.
For each year, the deposit must grow to the required amount by Beth's college start date, considering the 4% growth rate, which suggests using the future value of an ordinary annuity for deposits. For clarity, the calculation involves ensuring the sum of the future value of deposits equals the required tuition costs.
By solving these equations, Mary can determine the annual deposit amount necessary to accumulate the required funds by Beth’s 18th birthday. After detailed calculations, it turns out that she must deposit approximately \$2,400 each year, starting today, to meet half of the projected tuition costs, considering the compounding interest and tuition growth.
Conclusion
Mary’s financial landscape involves planning for her retirement benefits, current savings, and her granddaughter’s education expenses. The calculations show that her savings account will grow substantially with consistent deposits, and a lump-sum payment can substitute ongoing bonuses for more flexible financial management. Additionally, understanding the present value of her bonus and careful planning for Beth’s tuition costs will help her make informed decisions. These financial strategies, rooted in fundamental principles of the time value of money, equip Mary with the knowledge to manage her resources effectively before and after retirement.
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