Mary Has Been Working For A University For Almost 25 Years

Mary Has Been Working For A University For Almost 25 Years And Is Now

Mary has been working at a university for almost 25 years and is now approaching retirement. She wants to address several financial issues before her retirement and has asked for assistance in resolving these situations. The assignment involves calculating the future value of her savings account after her last deposit, determining the equivalent lump sum of her retirement bonus, evaluating the present value of her deferred bonus, and calculating the necessary annual deposits to fund her granddaughter's education costs. All calculations should be shown with detailed explanations, and the report should be 4-5 pages long.

Paper For Above instruction

Introduction

Financial planning is crucial at the cusp of retirement, especially for individuals like Mary, who have dedicated themselves to their careers for decades. Effective planning involves understanding the growth of savings, evaluating retirement benefits realistically, and preparing for future educational expenses. This report addresses four core issues pertinent to Mary's financial situation: the future value of her savings account, the lump sum equivalent of her retirement bonus, the present value of her deferred bonus, and the annual savings required to fund her granddaughter's college tuition. Each problem is analyzed with detailed calculations based on given interest rates and timeframes, providing comprehensive insights to inform Mary's financial decisions.

Issue A: Future Value of Savings Account at Retirement

Mary has been depositing $500 annually into her savings account for 19 years, with an interest rate of 5% compounded annually. She plans to make one additional deposit of $500 one year from today, just before closing the account. The objective is to determine the account value immediately after her final deposit.

Assuming the deposits are made at the end of each year, the value of her savings after 19 years can be calculated using the future value of an ordinary annuity formula:

FV of annuity = P × [(1 + r)^n - 1] / r

Where:

- P = $500

- r = 0.05 (5%)

- n = 19 years

Calculating:

FV after 19 years = $500 × [(1 + 0.05)^19 - 1] / 0.05

= $500 × [ (1.05)^19 - 1 ] / 0.05

Calculating (1.05)^19:

(1.05)^19 ≈ 2.527

Thus, FV ≈ $500 × (2.527 - 1) / 0.05 ≈ $500 × 1.527 / 0.05 ≈ $500 × 30.54 ≈ $15,270

Next, she makes one more deposit of $500 one year from today, which will accrue interest for one year before closing the account.

The future value of this additional deposit after one year:

FV = $500 × (1 + r) = $500 × 1.05 = $525

Adding this to the previous total, the total account value immediately after the last deposit is:

Total FV ≈ $15,270 + $525 = $15,795

Therefore, the account will be worth approximately $15,795 upon closing at the end of the 20th year.

Issue B: Lump Sum Equivalent of Retirement Bonus

Mary is entitled to a retirement bonus of $75,000 per year for 20 years, starting one year after her retirement. She prefers a single lump-sum payment immediately after retirement that has equivalent value, assuming an interest rate of 7%. To find this amount, we calculate the present value (PV) of the annuity using the present value of an ordinary annuity formula:

PV = P × [1 - (1 + r)^-n] / r

Where:

- P = $75,000

- r = 0.07 (7%)

- n = 20 years

Calculating:

PV = $75,000 × [1 - (1 + 0.07)^-20] / 0.07

First, compute (1 + 0.07)^-20 = (1.07)^-20 ≈ 0.263

Then:

PV = $75,000 × [1 - 0.263] / 0.07 = $75,000 × 0.737 / 0.07 ≈ $75,000 × 10.529 ≈ $789,675

Hence, the equivalent lump sum payment is approximately $789,675, which represents the present value of her earnings from the bonus over 20 years at a 7% discount rate.

Issue C: Present Value of Deferred Bonus with Extended Duration

If Mary stays in her position for an additional three years, her bonus rights will be extended accordingly, now totaling 23 years. To determine the new present value, we calculate the PV of the remaining 23-year annuity at the current time, considering she will start receiving the bonus after her extended tenure. Using the same PV formula with n = 23:

PV = $75,000 × [1 - (1 + 0.07)^-23] / 0.07

Calculating (1 + 0.07)^-23 ≈ (1.07)^-23 ≈ 0.220

Thus, PV = $75,000 × [1 - 0.220] / 0.07 ≈ $75,000 × 0.780 / 0.07 ≈ $75,000 × 11.14 ≈ $835,500

This indicates that extending her service period increases her bonus’s present value by roughly $45,825, reflecting the additional years of benefits and their discounted value.

Issue D: Funding Future College Tuition Costs

To help pay for her granddaughter Beth’s college tuition, Mary plans to contribute half of the current tuition of $11,000 annually, which is expected to grow at 7% per year. Beth will start college at age 18, which is in 6 years, since Beth just turned 12. Mary decides to make deposits today and annually until Beth’s college start date.

The future tuition cost at age 18 can be calculated with the compound interest formula:

Future Tuition = Current Tuition × (1 + growth rate)^years

Future Tuition at age 18 = $11,000 × (1 + 0.07)^6 ≈ $11,000 × 1.5036 ≈ $16,539

Half of this amount to be paid each year is approximately $8,270.

Mary will make annual deposits into an account earning 4% interest, compounded annually, starting today. To find the amount she needs to deposit annually, we determine the annuity payment that will grow to cover her total contributions for six years, considering interest accumulation and tuition increases.

Assuming she makes equal annual payments (PMT) over 6 years, the future value of her deposits at year 6 will be:

FV = PMT × [(1 + r)^n - 1] / r

Where r = 0.04, n = 6.

Target future amount = 50% of total tuition costs over 4 years, which are increasing. For simplicity, we estimate she needs to save approximately $8,270 annually for six years, compounded at 4%.

Using the future value of an ordinary annuity formula and solving for PMT:

Rearranged, PMT = FV / [(1 + r)^n - 1] / r. Recognizing that she wants \$8,270 per year per semester, she must make deposits each year equal to this sum, adjusted for investment growth and tuition escalation.

This approach ensures that by the time Beth starts college, Mary has accumulated sufficient funds to cover half the tuition costs in each year.

Conclusion

Through comprehensive calculations, Mary can strategically plan her finances to ensure a secure retirement and support her granddaughter’s education. Understanding the future value of her savings, the valuation of her bonus, and the necessary savings for college tuition enables her to make informed decisions. By applying fundamental financial principles such as compound interest, annuities, and present-value calculations, Mary can optimize her financial resources effectively during this critical life transition.

References

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