91Ó°ÊÓ

A father is planning a savings program to put his daughter through college. His daughter is now 13 years old. She plans to enroll at the university in 5 years, and it should take her 4 years to complete her education. Currently, the cost per year (for everything \(-\) food, clothing, tuition, books, transportation, and so forth) is \(\$ 12,500,\) but a 5 percent annual inflation rate in these costs is forecasted. The daughter recently received \(\$ 7,500\) from her grandfather's estate; this money, which is invested in a bank account paying 8 percent interest, compounded annually, will be used to help meet the costs of the daughter's education. The remaining costs will be met by money the father will deposit in the savings account. He will make 6 equal deposits to the account, one deposit in each year from now until his daughter starts college. These deposits will begin today and will also earn 8 percent interest, compounded annually. a. What will be the present value of the cost of 4 years of education at the time the daughter becomes 18 ? [Hint: Calculate the future value of the cost (at \(5 \%\) ) for each year of her education, then discount 3 of these costs back (at \(8 \%\) ) to the year in which she turns \(18,\) then sum the 4 costs. b. What will be the value of the \(\$ 7,500\) that the daughter received from her grandfather's estate when she starts college at age 18 ? (Hint: Compound for 5 years at an 8 percent annual rate. c. If the father is planning to make the first of 6 deposits today, how large must each deposit be for him to be able to put his daughter through college? (Hint: An annuity due assumes interest is earned on all deposits; however, the 6 th deposit earns no interest - therefore, the deposits are an ordinary annuity.)

Step by step solution

01

Calculate Future Costs of Education

First, compute the future cost for each year the daughter will be in college. Currently, the cost is $12,500. We need to calculate the cost for each year, factoring in a 5% inflation rate.1. Future cost for the 1st year of college in year 5: \[ FV_1 = 12500 \times (1+0.05)^5 \]2. Future cost for the 2nd year of college in year 6: \[ FV_2 = 12500 \times (1+0.05)^6 \]3. Future cost for the 3rd year of college in year 7: \[ FV_3 = 12500 \times (1+0.05)^7 \]4. Future cost for the 4th year of college in year 8: \[ FV_4 = 12500 \times (1+0.05)^8 \]

02

Discount Costs to Time 18

Now, we discount the future costs back to the time when the daughter begins college at age 18, using an 8% discount rate.1. Present value of cost during the first year of college (year 5): \[ PV_1 = FV_1 / (1+0.08)^0 \] 2. Present value of cost during the second year of college (year 6): \[ PV_2 = FV_2 / (1+0.08)^1 \] 3. Present value of cost during the third year of college (year 7): \[ PV_3 = FV_3 / (1+0.08)^2 \] 4. Present value of cost during the fourth year of college (year 8): \[ PV_4 = FV_4 / (1+0.08)^3 \] Sum these present values to find the total present value of all costs at the daughter's age of 18.

03

Future Value of Grandfather's Gift

To determine the value of the $7,500 gift at age 18, we compound the amount for 5 years at an 8% rate.\[ FV_{gift} = 7500 \times (1+0.08)^5 \]

04

Calculate Necessary Annual Deposits

First, determine the present value of the cost required to pay for college at the daughter's age of 18 by subtracting the future value of the gift from the total present value of college costs.Then calculate the required annual deposit:- Consider these deposits as a 6-year ordinary annuity (as beginning today means the first deposit does not earn interest, mimicking an annuity due without interest on the last deposit).- Solve for the annuity payment \( P \):\[ PV = P \times \left( \frac{(1-(1+0.08)^{-6})}{0.08} \right) (1+0.08) \]Calculate \( P \).

05

Solve and Summarize

Perform the calculations necessary to solve for the required payment per year based on the previous steps, revealing the amount the father will need to deposit annually.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Future Value Calculation

Calculating future value is essential when planning for education costs, especially as they may increase over time due to inflation. Future value (FV) calculations help us determine how much a current amount of money will be worth in the future. In our scenario, the father needs to predict his daughter's future college expenses, starting with today's costs of $12,500 per year. To account for a 5% inflation rate, we use the formula: \[ FV = PV \times (1 + r)^n \]Where:

• FV is the future value,
• PV is the present value (current cost),
• r is the inflation rate (5% or 0.05 in this case),
• n is the number of years until the cost is incurred.

This calculation provides a projection of how much money will be needed each year, factoring in expected cost increases.

Interest Compounding

Interest compounding refers to the process where interest is added to the principal sum, so that from that moment on, the interest that has been added also earns interest. This is a crucial concept in calculating how much money will grow over time when it is invested or saved. In our case, the daughter’s \(7,500 inheritance is invested and earns 8% interest, compounded annually. Each year, the interest earned the previous year gets added to the principal.The formula for compound interest is: \[ A = P \times (1 + r)^n \]Where:

• A is the amount of money accumulated after n years, including interest.
• P is the principal amount (initial investment).
• r is the annual interest rate (8% or 0.08 here).
• n is the number of years the money is invested.

By compounding the interest, the \)7,500 can grow significantly, aiding in covering future educational costs.

Annuity Payments

An annuity refers to a series of equal payments made at regular intervals. When it comes to saving for college, annuity payments help us determine how much should be saved periodically to reach a financial goal. The father plans to make six equal deposits into the investment account, starting today, to prepare for his daughter's education expenses.This type of saving involves an ordinary annuity, where deposits are made at the end of each period, and is adjusted to reflect an annuity due at the start. Mathematically, the present value of an ordinary annuity is: \[ PV_{annuity} = P \times \frac{(1-(1+r)^{-n})}{r} \times (1+r) \]Where:

• PV is the present value of the annuity,
• P is the annuity payment per period,
• r is the interest rate (8% here),
• n is the number of payments.

Calculating this helps the father determine the annual amount he needs to save to meet future college costs.

Inflation Impact on Costs

Inflation significantly affects the future costs of goods and services, including education. When planning for expenses several years ahead, it's important to consider how inflation might increase those costs. In the provided scenario, inflation is anticipated to be 5% annually. Inflation impacts future costs by decreasing the purchasing power of money over time. To mitigate this, we must adjust current prices to reflect future values using the aforementioned formula for future value. The result is a projection of how much will be needed per year for college education in terms of future dollars, allowing for a more accurate financial plan. Understanding inflation's impact is crucial for making informed decisions about saving and investment to ensure future needs are met. This helps avoid underestimating the necessary savings or investments required to match expected inflation-driven price increases.