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Chapter 5: Problem 33

You want to buy a house within 3 years, and you are currently saving for the down payment. You plan to save \(\$ 5,000\) at the end of the first year, and you anticipate that your annual savings will increase by \(10 \%\) annually thereafter. Your expected annual return is \(7 \% .\) How much will you have for a down payment at the end of Year \(3 ?\)

Short Answer

Expert verified
You will have $17,659.50 available for a down payment at the end of Year 3.

Step by step solution

01

Determine Savings at End of Year 1

You save $5,000 at the end of Year 1. There is no interest accumulated since it's the initial saving.
02

Calculate Savings for Year 2

Your savings increase by 10% annually. Therefore, for Year 2, your savings will be \(5,000 + 10% of \)5,000 = \(5,000 \times 1.10 = \)5,500.
03

Calculate Accumulated Savings and Interest for Year 2

The \(5,500 saved at the end of Year 2 earns a 7% return by the end of Year 3: \)5,500 \times 1.07 = $5,885.
04

Calculate Savings for Year 3

Your savings for Year 3 also increase by 10%. So, savings for Year 3 are \(5,500 \times 1.10 = \)6,050.
05

Calculate Interest for Year 1 Savings

The \(5,000 saved at the end of Year 1 earns interest for two years at a rate of 7% per year: \)5,000 \times 1.07^2 = 5,000 \times 1.1449 \approx 5,724.50.
06

Sum Up Total Savings for Down Payment

Add up the interest-accumulating savings from Year 1, the savings and interest from Year 2, and the savings from Year 3 (which do not have enough time to gain interest) to find the total amount available for a down payment: $5,724.50 + $5,885 + $6,050 = $17,659.50.

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Key Concepts

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

Future Value
The concept of Future Value is crucial when planning savings or investments. It's the value of a current asset at a future date based on an assumed rate of growth. In this exercise, you are calculating the future value of multiple savings contributions over three years. Each year, your savings grow not just because you add more funds but also due to interest on previous amounts.
  • Your initial savings of $5,000 has a future value because it gains interest over two years.
  • The savings in Year 2, which start at $5,500, grow through interest over one year.
  • The Year 3 savings, however, remain $6,050 as they do not accumulate interest yet.
Calculating these correctly shows how much your current savings will be worth in the future, a practical application of future value in financial planning.
Savings Plan
A Savings Plan is a detailed strategy to set aside money for future needs. In this exercise, your savings plan involves depositing a fixed amount annually with a consistent increase each year. This systematic approach helps in disciplined savings and showcases the effect of regular contributions.
  • Initially, you save $5,000 in Year 1.
  • In Year 2, you increase your savings by 10%, leading to a total of $5,500.
  • Continuing the 10% increase in Year 3, your savings become $6,050.
Changing savings each year demonstrates how structured contributions can enhance your financial stability and help build a substantial fund over time.
Compound Interest
Compound Interest is a method where the interest earned over time is added to the principal amount, causing the total value to increase faster. This exercise utilizes compound interest to grow your savings annually.
  • For Year 1's $5,000, interest is compounded over two years, transforming into $5,724.50.
  • Year 2's $5,500 grows at a 7% annual rate to become $5,885.
  • While Year 3's total of $6,050 doesn't yet compound, it stands ready to do so if saved longer.
The power of compound interest lies in its ability to magnify your savings, exponentially increasing your total funds.
Financial Planning
Financial Planning involves outlining your future goal, such as a down payment for a house, and strategically working towards it. This means organizing your savings, anticipating returns, and factoring in financial growth such as interest. By setting a target amount in three years as in this exercise, you can leverage both systematic savings and compound interest to accumulate sufficient funds.
  • Identify your target, like saving for a house.
  • Establish regular, and potentially increasing, savings contributions.
  • Utilize compound interest to maximize growth.
  • Regularly monitor progress and adjust strategy if needed.
This thorough planning ensures that you reach your financial goals efficiently and securely.

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Most popular questions from this chapter

You want to buy a house that \(\operatorname{costs} \$ 100,000 .\) You have \(\$ 10,000\) for a down payment, but your credit is such that mortgage companies will not lend you the required \(\$ 90,000\). However, the realtor persuades the seller to take a \(\$ 90,000\) mortgage (called a seller take-back mortgage) at a rate of \(7 \%\), provided the loan is paid off in full in 3 years. You expect to inherit \(\$ 100,000\) in 3 years; but right now all you have is \(\$ 10,000,\) and you can afford to make payments of no more than \(\$ 7,500\) per year given your salary. (The loan would call for monthly payments, but assume end-of-year annual payments to simplify things.) a. If the loan was amortized over 3 years, how large would each annual payment be? Could you afford those payments? b. If the loan was amortized over 30 years, what would each payment be? Could you afford those payments? c. To satisfy the seller, the 30 -year mortgage loan would be written as a balloon note, which means that at the end of the third year, you would have to make the regular payment plus the remaining balance on the loan. What would the loan balance be at the end of Year \(3,\) and what would the balloon payment be?

As a jewelry store manager, you want to offer credit, with interest on outstanding balances paid monthly. To carry receivables. you must borrow funds from your bank at a nominal \(6 \%\), monthly compounding. To offset your overhead, you want to charge your customers an EAR (or EFF\%) that is \(2 \%\) more than the bank is charging you. What APR rate should you charge your customers?

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.]

Bank A pays 4\% interest compounded annually on deposits, while Bank \(B\) pays \(3.5 \%\) compounded daily. a. Based on the EAR (or EFF\%), which bank should you use? b. Could your choice of banks be influenced by the fact that you might want to withdraw your funds during the year as opposed to at the end of the year? Assume that your funds must be left on deposit during an entire compounding period in order to receive any interest.

How long will it take \(\$ 200\) to double if it earns the following rates? Compounding occurs once a year. a. \(7 \%\) b. \(10 \%\) c. \(18 \%\) d. \(100 \%\)
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