91Ó°ÊÓ

Required annuity payments 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 percent 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 percent annually. The father will make 6 equal annual deposits into her account; the ist deposit today and the 6 th on the day she starts college. How large must each of the 6 payments be? [Hint: Calculate the cost (inflated at 5 percent) for each year of college, then find the total present value of those costs, discounted at 6 percent, 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 \(\mathrm{PV}\) costs and the amount that would be in the savings account must be made up by the father's deposits, so find the 6 equal payments (starting immediately) that will compound to the required amount.

Step by step solution

01

Calculate Future Costs of College

The annual cost is currently $15,000 and is expected to increase by 5% annually. We will calculate these yearly costs:\[ C_n = 15000 \times (1.05)^n \] where \( C_n \) is the cost in year \( n \) after the initial year. For simplicity, we'll calculate the costs for her four years in college (years 5 to 8 from now).- Year 5 cost: \( 15000 \times (1.05)^5 = 19144.12 \)- Year 6 cost: \( 15000 \times (1.05)^6 = 20101.32 \)- Year 7 cost: \( 15000 \times (1.05)^7 = 21106.38 \)- Year 8 cost: \( 15000 \times (1.05)^8 = 22161.70 \)These are the costs at the time she enters each year of college.

02

Calculate Present Value of Future Costs

To find out how much to have when college starts to cover these costs in present value terms, discount each year's cost by 6%:\[ PV_n = \frac{C_n}{(1.06)^n} \]- Year 5: \( PV_5 = \frac{19144.12}{1.06^5} = 14302.91 \)- Year 6: \( PV_6 = \frac{20101.32}{1.06^6} = 14185.69 \)- Year 7: \( PV_7 = \frac{21106.38}{1.06^7} = 14071.82 \)- Year 8: \( PV_8 = \frac{22161.70}{1.06^8} = 13960.14 \)Sum these to find the total present value of the costs at the start:\[ 14302.91 + 14185.69 + 14071.82 + 13960.14 = 56520.56 \]

03

Calculate Future Value of Current Savings

The current amount in the saving account, \(\$7,500\), will compound over the next 5 years at 6%. Use the compound interest formula:\[ FV = 7500 \times (1.06)^5 = 10039.73 \]This is how much the current saving will grow by the start of college.

04

Determine Required Contribution from Deposits

The total present value of college costs that needs to be covered by deposits is the difference between the discounted cost of college and future value of current savings:\[ Required Amount = 56520.56 - 10039.73 = 46480.83 \]This is the amount needed from the future deposits.

05

Calculate Each Annuity Payment

We will solve for \( P \) in the annuity formula:\[ 46480.83 = P \times \left( \frac{(1.06)^5 - 1}{0.06} \times 1.06 \right) \]First, calculate the annuity factor:\[ \frac{(1.06)^5 - 1}{0.06} \times 1.06 = 6.9753 \]Rearrange the equation to solve for \( P \):\[ P = \frac{46480.83}{6.9753} = 6663.48 \]Therefore, each payment should be approximately \( \$6,663.48 \).

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Key Concepts

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

Annuity Payments

An annuity represents a series of payments made at regular intervals. They are crucial in financial planning, especially when saving for significant future expenses like college tuition. For instance, in our college example, the father plans to make organized savings through equal yearly payments, or annuities, to meet future needs.

In solving such a problem, it's essential to determine how much money will be required in the future and then calculate how these annuities can accumulate over time to reach that requirement. This involves using an annuity formula that factors in the interest rate and the number of periods. The key goal here is to ensure that the series of annuity payments grow, with interest, to cover projected costs.

• First, calculate the total future needs, including all expected increases in costs.
• Then, determine the present value to see how much needs to be saved now.
• Lastly, use it to calculate how much each annuity payment should be.

Annuity payments allow for steady savings, ensuring that the financial burden is spread over several years rather than having to come up with a large sum all at once.

Present Value of Future Costs

Understanding the present value (PV) of future costs is fundamental in financial planning. PV helps in determining how much needs to be saved today to cover future expenses, adjusted for the time value of money. This concept is crucial when dealing with increasing costs, like college tuition over several years.

In our example, costs are inflated annually by 5%. These amounts must be discounted back to what they'd be worth today, using a discount rate (6% here) representing the rate of return on saved money. Each year's future cost is translated back to present value that reflects its value at the start of the savings period.

• Calculate future costs for each year considering inflation.
• Use the formula: \( PV = \frac{C_n}{(1 + r)^n} \) where \( C_n \) is the future cost and \( r \) is the discount rate.
• Add up each present value amount to get the total needed today.

This gives a semblance of reality to future financial obligations, enabling more accurate planning and savings strategies.

Compound Interest

Compound interest plays a vital role in growing savings and understanding how money can multiply over time. If you are saving for future expenses, like in our case, maximizing the effect of compound interest can significantly boost your savings.

When your initial savings (here, $7,500) are invested at a compound interest rate of 6%, it means the interest earned each year gets added to the principal amount. This new total becomes the base for calculating the next year's interest. Over several years, this exponential growth can mean a substantial increase in funds just from initial savings and reinvested interest.

• Use the formula: \( FV = PV \times (1 + r)^n \), where \( FV \) is the future value.
• Calculate how your initial savings grow over time with this formula.
• A substantial portion of college costs can be covered using compounded savings.

This growth through compound interest alleviates part of the financial pressure, reducing the amount needed from annuity payments and making long-term goals more attainable.