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Chapter 5: Problem 1

If you deposit \(\$ 10,000\) in a bank account that pays \(10 \%\) interest annually, how much will be in your account after 5 years?

Short Answer

Expert verified
The account will have $16,105.10 after 5 years.

Step by step solution

01

Understand the Compound Interest Formula

To find the future value of an investment with annually compounded interest, we use the formula: \(A = P(1 + r)^t\), where \(A\) is the amount of money accumulated after \(t\) years, including interest. \(P\) is the principal amount (initial investment), \(r\) is the annual interest rate, and \(t\) is the number of years the money is invested for.
02

Identify Given Values

From the problem, we know that \(P = \$10,000\), \(r = 0.10\) (since 10% is 0.10 as a decimal), and \(t = 5\) years. We will plug these values into the compound interest formula.
03

Substitute Values into the Formula

Substitute the values into the formula: \[A = 10000(1 + 0.10)^5\]. This simplifies to \[A = 10000(1.10)^5\].
04

Calculate the Compound Interest Formula

First, calculate \((1.10)^5\). Using a calculator, \((1.10)^5 = 1.61051\). Then, multiply by 10000: \(A = 10000 \times 1.61051\). This gives \(A = 16105.1\).
05

Round to the Nearest Cent

Since money is usually expressed in two decimal places, round \(16105.1\) to \$16105.10.

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Key Concepts

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

Future Value Calculation
Calculating the future value of an investment allows you to understand how much your initial deposit will grow over a specific period. The future value is the amount you will have in your account after the interest has been applied over the years.

The formula to calculate the future value with compound interest is:
  • \( A = P(1 + r)^t \)
Where:
  • \( A \) is the future value of the investment/loan, including interest
  • \( P \) is the principal investment amount (the initial deposit)
  • \( r \) is the annual interest rate (as a decimal)
  • \( t \) is the number of years the money is invested for
Plugging these into the formula will tell you exactly how much your investment will grow. For example, depositing \(10,000 at a 10% annual interest rate for 5 years will result in a future value of \)16,105.10.

Understanding the future value calculation helps you plan and make informed financial decisions.
Interest Rate
The interest rate is the percentage at which your money grows over time. In terms of investment, it is the amount the bank pays you for keeping your money in their account. An annual interest rate reflects what you will earn on your investment each year.

Typically expressed as a percentage, you need to convert the interest rate to a decimal when using it in the compound interest formula.
  • 10% becomes 0.10
  • 5% becomes 0.05
A higher interest rate means more growth potential for your investment. Given the same initial investment and time, an account with a 10% interest rate will grow faster than one with a 5% interest rate.

Understanding how the interest rate affects your investment is crucial for maximizing your returns.
Investment Growth
Investment growth refers to the increase in your investment's value over time. When your investment earns interest, this amount gets added back into the principal, increasing the total amount that earns interest in the future. This is what we call compounding.

Every time compound interest is applied, your investment grows even more. This growth is exponential, not linear, meaning it accelerates over time. For example, your $10,000 deposit will be worth $16,105.10 after 5 years at a 10% interest rate.

Using compound interest helps your money grow faster with the principle of "earning on the earnings." This concept is essential in financial planning, aiding in meeting financial goals and ensuring long-term security.
Financial Mathematics
Financial mathematics is the field of mathematics used in analyzing the financial world, including calculations involving interest rates, investments, and the growth of money. It provides tools for making calculations like the future value of an investment.

By utilizing formulas and mathematical principles, we can predict and plan our financial futures. For instance, understanding how various interest rates affect a savings account can guide you in choosing the best bank or investment strategy.

In practice, financial mathematics involves various calculations:
  • Estimating future investment values (future value calculation)
  • Determining the required savings needed for future goals
  • Analyzing loans and mortgage payments
Mastery of financial mathematics empowers you to make educated financial decisions, helping to secure a stable financial future.

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Most popular questions from this chapter

Your client is 40 years old; and she wants to begin saving for retirement, with the first payment to come one year from now. She can save \(\$ 5,000\) per year; and you advise her to invest it in the stock market, which you expect to provide an average return of \(9 \%\) in the future. a. If she follows your advice, how much money will she have at \(65 ?\) b. How much will she have at \(70 ?\) c. She expects to live for 20 years if she retires at 65 and for 15 years if she retires at 70 . If her investments continue to earn the same rate, how much will she be able to withdraw at the end of each year after retirement at each retirement age?

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?

Your firm sells for cash only; but it is thinking of offering credit, allowing customers 90 days to pay. Customers understand the time value of money, so they would all wait and pay on the 90 th day. To carry these receivables, you would have to borrow funds from your bankat a nominal \(12 \%\), daily compounding based on a 360 -day year. You want to increase your base prices by exactly enough to offset your bank interest cost. To the closest whole percentage point, by how much should you raise your product prices?

An investment will pay \(\$ 100\) at the end of each of the next 3 years, \(\$ 200\) at the end of Year \(4, \$ 300\) at the end of Year \(5,\) and \(\$ 500\) at the end of Year \(6 .\) If other investments of equal risk earn \(8 \%\) annually, what is its present value? its future value?

Jan sold her house on December 31 and took a \(\$ 10,000\) mortgage as part of the payment. The 10 -year mortgage has a \(10 \%\) nominal interest rate, but it calls for semiannual payments beginning next June \(30 .\) Next year Jan must report on Schedule \(B\) of her IRS Form 1040 the amount of interest that was included in the two payments she received during the year. a. What is the dollar amount of each payment Jan receives? b. How much interest was included in the first payment? How much repayment of principal was included? How do these values change for the second payment? c. How much interest must Jan report on Schedule \(B\) for the first year? Will her interest income be the same next year? d. If the payments are constant, why does the amount of interest income change over time?
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