Bmary Has Been Working At The University For 25 Years

Issue Bmary Has Been Working At The University For 25 Years With An

Mary has been working at the university for 25 years, with an excellent record of service. 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%? A push in the right direction is helpful.

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Retirement planning often involves evaluating the value of periodic payments versus a lump sum payment, considering the time value of money. In this scenario, Mary has received an offer for an annuity—$75,000 annually for 20 years starting one year after her retirement—and she prefers a lump sum payment immediately upon retirement. The primary financial task is to determine the present value of these future payments at the time of her retirement, discounted at a 7% interest rate, which represents the appropriate rate of return or discount rate.

Understanding the concept of present value (PV) is crucial in this context. The present value of an annuity reflects the current worth of a series of future cash flows, discounted at a specific interest rate. The formula for the present value of an ordinary annuity (which assumes payments occur at the end of each period) is expressed as:

PV = P × [(1 - (1 + r)^-n) / r]

where P is the annual payment, r is the interest rate per period, and n is the total number of payments. Using this formula, we can calculate the present value of the 20-year annuity that Mary is being offered.

Given that P = $75,000, r = 7% or 0.07, and n = 20, the calculation proceeds as follows:

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

Calculating the denominator first:

(1 + 0.07)^-20 = (1.07)^-20 ≈ 0.25842

Then, the expression inside the brackets becomes:

1 - 0.25842 = 0.74158

Next, dividing by the interest rate:

0.74158 / 0.07 ≈ 10.594

Finally, multiplying by the annual payment:

PV ≈ 75,000 × 10.594 ≈ $794,550

This calculation provides the present value of the structured annuity payment schedule. Therefore, the lump sum equivalent that Mary would be willing to accept immediately, given a 7% discount rate, is approximately $794,550.

This lump sum translates her future series of payments into its equivalent today, enabling her to evaluate if the immediate payment is attractive compared to the annuity. It also serves as a critical decision-making tool for financial planning during retirement. Offering her this amount allows her to invest or use the funds according to her preferences, while the university benefits by consolidating the payout into a single, manageable sum rather than a long-term pension obligation.

It is essential to emphasize the significance of the discount rate in this calculation. A higher discount rate would lower the present value, while a lower rate increases it. In real-world applications, the chosen discount rate often reflects the risk-free rate or the expected return on alternative investments, underscoring the importance of considering broader economic conditions in such estimates.

In conclusion, the present value of Mary’s retirement bonus, structured as a 20-year annuity at a 7% interest rate, is approximately $794,550. This lump sum provides a comparable economic value to the series of future annual payments, offering her immediate liquidity and financial flexibility upon retirement. Such calculations are fundamental in retirement planning, ensuring fairness and clarity in long-term compensation arrangements.

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