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Chapter 2: Problem 31

Required lump sum payment You need \(\$ 10,000\) annually for 4 years to complete your education, starting next year. (One year from today you would withdraw the first S10,000.) Your uncle will deposit an amount today in a bank paying 5 percent annual interest, which would provide the needed \(\$ 10,000\) payments. a. How large must the deposit be? b. How much will be in the account immediately after you make the first withdrawal?

Short Answer

Expert verified
Uncle must deposit approximately \$35,450. Balance after first withdrawal is \$27,222.5.

Step by step solution

01

Identify the Required Future Payments

You require \\(10,000 annually for 4 years. These are the withdrawals at the end of each year. Let's denote these withdrawals as an annuity, \(PMT = \\)10,000\), for \(n = 4\) periods.
02

Calculate the Present Value of the Annuity

To find the required deposit, calculate the present value of an annuity using the formula: \[PV = PMT \times \left(1 - \left(1 + r\right)^{-n}\right) / r\] where \(r = 5\% = 0.05\). Substituting the values: \[ PV = 10,000 \times \left(1 - (1 + 0.05)^{-4}\right) / 0.05 \] Calculate:\[ PV = 10,000 \times (1 - 0.8227)/0.05 = 10,000 \times 3.545 \] Thus, \(PV \approx \\(35,450 \). Your uncle needs to deposit approximately \\)35,450 today.
03

Determine the Account Balance After First Withdrawal

After the first \\(10,000 withdrawal at the end of the first year, the remaining amount will accumulate interest. First, find the future value after one year's interest: \[FV_{after1yr} = PV \times (1 + r) = 35,450 \times 1.05 = 37,222.5\]The account balance immediately after the \\)10,000 withdrawal:\[\text{New Balance} = FV_{after1yr} - 10,000 = 37,222.5 - 10,000 = 27,222.5\]. So, the balance after the first withdrawal is \$27,222.5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Annuity Payments
An annuity refers to a series of equal payments made at regular intervals over a specified period. In our scenario, you need $10,000 at the end of each year for four years. These payments are examples of an annuity, specifically an ordinary annuity, where payments occur at the end of each period.

Understanding annuities is crucial for financial planning. They help you figure out how much you need to save today to meet future financial goals. In the context of education funding, annuities ensure that necessary funds are available when tuition and other educational expenses are due.
Interest Rate Impact
Interest rates significantly affect how much money needs to be deposited today to meet future expenses. When calculating the present value of an annuity, like in this exercise, the interest rate plays a vital role. This rate, also called the discount rate, determines how the value of future annuity payments is perceived in today's terms.

For instance, a 5% interest rate means that the money deposited today will grow each year by 5%. This allows smaller initial deposits to grow into larger amounts. Hence, even a slightly increased interest rate can reduce the amount you need to deposit initially, easing the burden of current-year savings. This showcases the impact of interest rates on financial planning decisions, allowing for optimized educational funding strategies.
Educational Funding
Funding education requires careful planning to ensure that students can afford all necessary costs over the duration of their studies. With increasing costs of tuition, planning for these educational expenses early can make a significant difference in financial outcomes.

The concept of annuity payments becomes vital here. When future cash flow requirements are known, such as $10,000 annually, one can calculate the present value required to fund those expenses. This calculation helps determine how much needs to be set aside today, considering any interest that the funds would earn leading up to future payments.

By understanding and calculating these values, students and their families can better manage their finances and New Balance = 27,222.5." target a specific goal for educational costs without unnecessary strain.
Financial Planning for Education
Financial planning for education combines various financial concepts to ensure that there are sufficient funds for all educational costs. This not only includes strategies for saving and investing but also requires applying concepts like the time value of money and annuities.

By engaging in financial planning, one can make informed decisions about how much to save and where to invest. It's about balancing current financial capabilities with future financial needs. In our example, calculating the necessary $35,450 deposit today ensures that there's $10,000 ready each year for four years. This takes into account growing productively over time due to the bank's interest.

Effective financial planning for education also considers how financial tools can be used most efficiently, providing peace of mind that funds will be accessible when needed. This can significantly reduce financial stress and allow one to focus on academic pursuits.

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Most popular questions from this chapter

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Future value: annuity versus annuity due What's the future value of a 7 percent, 5 -year ordinary annuity that pays \(\$ 300\) each year? If this were an annuity due, what would its future value be?

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Future value for various compounding periods Find the amount to which \(\$ 500\) will grow under each of these conditions: a. 12 percent compounded annually for 5 years. b. 12 percent compounded semiannually for 5 years. c. 12 percent compounded quarterly for 5 years. d. 12 percent compounded monthly for 5 years. e. 12 percent compounded daily for 5 years. f. Why does the observed pattern of FVs occur?

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