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A father is now planning a savings program to put his daughter through college. She is \(13,\) she plans to enroll at the university in 5 years, and she should graduate in 4 years. Currently, the annual cost (for everything- food, clothing, tuition, books, transportation, and so forth) is \(\$ 15,000,\) but these costs are expected to increase by \(5 \%\) annually. The college requires that this amount be paid at the start of the year. She now has \(\$ 7,500\) in a college savings account that pays \(6 \%\) annually. Her father will make six equal annual deposits into her account; the first deposit today and the sixth on the day she starts college. How large must each of the six payments be? [Hint: Calculate the cost (inflated at \(5 \%\) ) for each year of college and find the total present value of those costs, discounted at \(6 \%\), as of the day she enters college. Then find the compounded value of her initial \(\$ 7,500\) on that same day. The difference between the \(P V\) costs and the amount that would be in the savings account must be made up by the father's deposits, so find the six equal payments (starting immediately) that will compound to the required amount.]

Step by step solution

01

Project Her College Costs

The cost for the first year of college is currently $15,000 and increases by 5% annually. Calculate the projected costs for each of the four years of college:- Year 1 Cost: \( C_1 = 15000 \)- Year 2 Cost: \( C_2 = 15000 \times 1.05 \)- Year 3 Cost: \( C_3 = 15000 \times (1.05)^2 \)- Year 4 Cost: \( C_4 = 15000 \times (1.05)^3 \)

02

Calculate Present Values for Each Year

Discount each future college cost to the present value at the time she starts college (in 5 years), using a 6% discount rate:\[ PV(C_1) = \frac{C_1}{(1+0.06)^1}, \quad PV(C_2) = \frac{C_2}{(1+0.06)^2}, \quad PV(C_3) = \frac{C_3}{(1+0.06)^3}, \quad PV(C_4) = \frac{C_4}{(1+0.06)^4} \]Sum these present values to find the total present value of all college costs at the start of college.

03

Future Value of Current Savings

Calculate the future value of her current savings ($7,500) in 5 years, using the 6% interest rate:\[ FV_{7500} = 7500 \times (1+0.06)^5 \]

04

Calculate Remaining Amount Needed

Subtract the future value of the current savings from the total present value of projected college costs to determine the total additional amount needed from the father's savings plan:\[ R = \text{Total PV College Costs} - FV_{7500} \]

05

Calculate Required Deposit Amounts

The father will make six equal deposits from today to the day she starts college. Use the annuity future value formula to find the equal annual deposit amount \(P\):\[ R = P \times \left(\frac{(1+0.06)^6 - 1}{0.06} \right) \]Solve for \(P\) to find the required deposit amount.

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Key Concepts

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

Present Value Calculation

When planning college expenses, understanding present value calculations is critical. The present value (PV) helps us figure out how much a future sum of money is worth today. In this scenario, we use it to compare the cost of attending college in the future to today’s dollars. We need to discount the future costs back to the present time using a specific interest or inflation rate.
This process helps account for how money's value changes over time due to inflation or investment opportunities. The formula used to calculate PV is: \[PV = \frac{C}{(1 + r)^n}\]
where \(C\)is the future cost, \(r\)is the discount rate, and \(n\)is the number of periods.
This allows for a structured, logical approach to assessing the full future costs of college, ensuring that financial planning is thorough and realistic.

Future Value of Savings

Understanding the future value (FV) of savings is significant when planning for future educational expenses. It helps us estimate how much savings today will grow over time, given a particular interest rate. With the college savings account earning 6% annually, the FV formula provides a glimpse into the future:
\[FV = P \times (1 + r)^n\]
where \(P\)is the initial amount, \(r\)is the annual interest rate, and \(n\)is the number of years.
By calculating FV, the father can predict how much the current savings will grow by the time his daughter starts college. This step is crucial as it helps determine how much more money needs to be saved or invested to cover future expenses.

Tuition Inflation

Tuition inflation refers to the annual rise in the cost of attending college. In this case, it’s reported as a 5% increase per year. Understanding this concept is crucial as it directly affects financial planning. Tuition costs can grow significantly over a few years, drastically changing how much money families need to put aside.
The formula to calculate tuition inflation for each year is simple:
\[C_n = C_0 \times (1 + i)^n\]
where:

• \(C_0\)is the current cost
• \(i\)is the inflation rate
• \(n\)is the number of years

By understanding the impact of tuition inflation, families can prepare better and adjust their savings plans to meet these escalating costs.

Annuity Payments

When saving for college, annuity payments are periodic contributions made to reach a financial goal, like paying for tuition. In this exercise, these payments occur yearly until the child starts college. Their size can be calculated using the annuity future value formula, which will determine the amount needed to cover any shortfall between present savings and future costs.
The formula is:
\[FV = P \times \left(\frac{(1 + r)^n - 1}{r}\right)\]
where:

• \(FV\)is the future value or goal to achieve
• \(P\)is the periodic payment amount
• \(r\)is the annual interest rate
• \(n\)is the number of payments

This approach helps break down the daunting task of saving for education into manageable, regular contributions, ensuring that the savings goal can be met without unnecessary financial strain.