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Chapter 2: Problem 1

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

Short Answer

Expert verified
The future value after 5 years is \$16,105.10.

Step by step solution

01

Understand the Formula

The formula to calculate the future value with compound interest annually is given by \( FV = P (1 + r)^n \), where \( FV \) is the future value, \( P \) is the principal amount, \( r \) is the annual interest rate, and \( n \) is the number of years.
02

Identify the Given Values

From the problem, the principal amount \( P \) is \( \$10,000 \), the annual interest rate \( r \) is 10% or 0.10, and the number of years \( n \) is 5 years.
03

Plug in the Values

Substitute the given values into the formula: \( FV = 10,000(1 + 0.10)^5 \).
04

Calculate Inside the Parentheses

First, calculate \( 1 + 0.10 \), which equals 1.10.
05

Raise to the Power of Years

Next, calculate \( 1.10^5 \). This equals approximately 1.61051.
06

Multiply by the Principal

Finally, multiply the result by the principal: \( 10,000 \times 1.61051 \). This equals \( 16,105.10 \).

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Key Concepts

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

Compound Interest
Compound interest is a powerful concept in finance that allows your investments to grow faster compared to simple interest. Unlike simple interest, which only earns interest on the initial principal, compound interest earns interest on both the initial principal and the accumulated interest over time.
When you invest money, the interest you gain is added back to the principal amount, and future interest is calculated on this new principal. This is why the future value of an investment can grow significantly over a period of time.
In the given exercise, the compound interest helps us determine the total amount in a bank account after 5 years by using the principal and the interest rate given:
  • Principal (\( P \)) = $10,000
  • Annual interest rate (\( r \)) = 10% or 0.10
  • Time (\( n \)) = 5 years
By applying these values in the compound interest formula, you can easily calculate the future value of your investment.
Annual Interest Rate
The annual interest rate is a critical factor in calculating how much your investment will grow over a certain period. It refers to the percentage at which your account balance increases each year.
An interest rate of 10% means that each year, the investment grows by 10% of the starting balance. This increase includes the principal for the first year, and then compounds as the interest is recalculated on the new balance each following year.
It is important to note that the annual interest rate directly affects the outcome of the future value calculation. Higher interest rates will contribute to a faster-growing investment. In this example, the rate is set at 10%, which provides a good growth over the course of 5 years.
Investment Growth
Investment growth refers to how much your money increases over a given time through mechanisms like compound interest. The growth is measured in terms of the final amount you have compared to the initial amount you invested.
In the exercise, we calculated the future value to be approximately $16,105.10 after 5 years, starting from an initial investment of $10,000. This demonstrates a growth of $6,105.10, which is solely possible due to the compound interest effect.
This growth showcases the power of investing and the benefit of allowing time to enhance your investment. By understanding investment growth, you can make informed decisions on future investing, selecting the options and time frames that best meet your financial goals.

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Most popular questions from this chapter

Building credit cost into prices 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 bank at a nominal 12 percent, 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?

FV of uneven cash flow 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 percent annually thereafter. Your expected annual return is 7 percent. How much would you have for a down payment at the end of Year \(3 ?\)

Required lump sum payment You need \(\$ 10,000\) annually for 4 years to complete your education, starting next year. (One year from today you would withdraw the first S10,000.) Your uncle will deposit an amount today in a bank paying 5 percent annual interest, which would provide the needed \(\$ 10,000\) payments. a. How large must the deposit be? b. How much will be in the account immediately after you make the first withdrawal?

Present and future values for different interest rates Find the following values. Compounding/discounting occurs annually. a. An initial \(\$ 500\) compounded for 10 years at 6 percent. b. \(\quad\) An initial \(\$ 500\) compounded for 10 years at 12 percent. c. The present value of \(\$ 500\) due in 10 years at 6 percent. d. The present value of \(\$ 1,552.90\) due in 10 years at 12 percent and also at 6 percent. e. Define present value, and illustrate it using a time line with data from part d. How are present values affected by interest rates?

\(\mathrm{PV}\) and loan eligibility You have saved \(\$ 4,000\) for a down payment on a new car. The largest monthly payment you can afford is \(\$ 350\). The loan would have a 12 percent APR based on end-of-month payments. What is the most expensive car you could afford if you finance it for 48 months? For 60 months?
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