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Chapter 7: Problem 12

Which amount is worth more at 14 percent, compounded annually: \(\$ 1,000\) in hand today or \(\$ 2,000\) due in 6 years?

Short Answer

Expert verified
The $1,000 in hand today is worth more as it grows to $2,195.62, exceeding the $2,000 due in 6 years.

Step by step solution

01

Understanding the Problem

We have to compare the future value of $1,000 invested at an annual compound interest rate of 14% with a $2,000 sum due in 6 years. The goal is to determine which option has a greater value.
02

Calculate the Future Value of $1,000

To find the future value of $1,000 compounded annually at 14% over 6 years, we use the compound interest formula: \[ FV = PV \times (1 + r)^n \]where \(PV = 1,000\), \(r = 0.14\), and \(n = 6\).Calculate:\[ FV = 1,000 \times (1 + 0.14)^6 \]
03

Perform the Calculation

Calculate the inside of the parenthesis first: \[(1 + 0.14)^6 = (1.14)^6\]. Then multiply by $1,000.
04

Compare with $2,000 due in 6 years

Calculate the future value:\[ 1,000 \times 1.14^6 \approx 1,000 \times 2.195618 \approx 2,195.62 \]Now compare \(2,195.62 (future value of \)1,000) with $2,000.
05

Conclusion

The future value of $1,000 at 14% compounded annually for 6 years is approximately $2,195.62, which is more than $2,000 due in 6 years. Therefore, $1,000 today is worth more than $2,000 due in 6 years at the given interest rate.

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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 (FV) is essential in determining how much an investment made today will grow to in the future. The future value is calculated using the compound interest formula: \[ FV = PV \times (1 + r)^n \] where:
  • \(PV\) is the present value or the initial amount of money invested or loaned.
  • \(r\) is the annual interest rate expressed as a decimal.
  • \(n\) is the number of periods the money is invested for.
In our exercise, we calculated the future value of \(\\(1,000\) invested at a 14% annual interest rate over 6 years. This tells us how much the \(\\)1,000\) would be worth at the end of the investment period. By calculating the future value, you can make informed decisions about where to put your money for the highest return.
Annual Interest Rate
The Annual Interest Rate is a critical factor in compound interest calculations. It determines how much extra money you will earn on your investment for each year the money is invested. In our example, the annual interest rate was 14%, which informed us that the investment would grow by 14% each year, compounded annually.When dealing with compound interest, remember that your investment grows based not only on the initial amount but also on the interest already accrued. The formula used reflects this growth:\[ (1 + r)^n \]Where each year's interest compounds the previous years' accumulated interest, creating exponential growth over time. This means even a small change in the annual interest rate can significantly impact the future value of an investment over long periods.
Financial Decision Making
Making sound financial decisions involves comparing different future values to determine the most beneficial option. When faced with choices, such as in our exercise, understanding the future value of different sums lets you see which provides a better return.The decision often depends on:
  • Your time horizon: The length of time you can leave money invested.
  • The annual interest rate: Higher rates mean quicker growth, making earlier investments more valuable.
  • The risk level you are willing to accept.
In the example, by calculating that \(\\(1,000\) today would grow to approximately \(\\)2,195.62\) in 6 years, you avoid idle capital and make informed choices. Selecting the more valuable future amount ensures you maximize potential earnings, an essential practice for sound financial decision-making.

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Most popular questions from this chapter

John Roberts has \(\$ 42,180.53\) in a brokerage account, and he plans to contribute an additional \(\$ 5,000\) to the account at the end of every year. The brokerage account has an expected annual return of 12 percent. If John's goal is to accumulate \(\$ 250,000\) in the account, how many years will it take for John to reach his goal?

Find the interest rates, or rates of return, on each of the following: a. You borrow \(\$ 700\) and promise to pay back \(\$ 749\) at the end of 1 year. b. You lend \(\$ 700\) and receive a promise to be paid \(\$ 749\) at the end of 1 year. c. You borrow \(\$ 85,000\) and promise to pay back \(\$ 201,229\) at the end of 10 years. d. You borrow \(\$ 9,000\) and promise to make payments of \(\$ 2,684.80\) per year for 5 years.

If you deposit money today into an account that pays 6.5 percent interest, how long will it take for you to double your money?

The Jackson family is interested in buying a home. The family is applying for a \(\$ 125,000,30\) -year mortgage. Under the terms of the mortgage, they will receive \(\$ 125,000\) today to help purchase their home. The loan will be fully amortized over the next 30 years. Current mortgage rates are 8 percent. Interest is compounded monthly and all payments are due at the end of the month. a. What is the monthly mortgage payment? b. What portion of the mortgage payments made during the first year will go toward interest? c. What will be the remaining balance on the mortgage after 5 years? d. How much could the Jacksons borrow today if they were willing to have a \(\$ 1,200\) monthly mortgage payment? (Assume that the interest rate and the length of the loan remain the same.)

Find the following values, using the equations, and then work the problems using a financial calculator or the tables to check your answers. Disregard rounding differences. (Hint: If you are using a financial calculator, you can enter the known values, and then press the appropriate key to find the unknown variable. Then, without clearing the TVM register, you can "override" the variable that changes by simply entering a new value for it and then pressing the key for the unknown variable to obtain the second answer. This procedure can be used in parts b and \(\mathrm{d}\), and in many other situations, to see how changes in input variables affect the output variable.) Assume that compounding/discounting occurs once a year. a. An initial \(\$ 500\) compounded for 1 year at 6 percent. b. An initial \(\$ 500\) compounded for 2 years at 6 percent. c. The present value of \(\$ 500\) due in 1 year at a discount rate of 6 percent. d. The present value of \(\$ 500\) due in 2 years at a discount rate of 6 percent.
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