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Chapter 14: Problem 3

Suppose that when you were one year old, your grandmother gave you a shiny silver dollar. Your parents put that silver dollar in a savings account with a guaranteed \(9 \%\) interest rate, and then promptly forgot about it. a. Use the Rule of 72 to estimate how much that account will grow to by the time you are \(65 .\) b. Calculate exactly how much you will have in that account using the formula for compound interest. c. How close are your answers to (a) and (b)?

Short Answer

Expert verified
a) Double every 8 years, 8.25x doubling b) \$512.74 c) The estimates are similar, with the exact amount being \$512.74.

Step by step solution

01

Understanding the Rule of 72

The Rule of 72 is a quick, useful formula that is used to estimate the number of years required to double the investment at an annual compounded interest rate. To use the Rule of 72, you divide 72 by the interest rate. In this case, with a 9% interest rate: \(\text{Years to double} = \frac{72}{9} = 8\).

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Key Concepts

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

Rule of 72
The "Rule of 72" is a simple yet powerful tool used in finance to estimate the time needed for an investment to double in value at a fixed annual interest rate. The rule states that you divide the number 72 by the annual interest rate to find the approximate number of years required for the investment to double. For example:
  • If you have an interest rate of 9%, you calculate: \( \frac{72}{9} = 8 \) years. This means in approximately 8 years, your investment will double in value.
The beauty of this rule lies in its simplicity. While it doesn鈥檛 give the exact answer, it provides a reasonable estimate without the need for a calculator. It鈥檚 especially useful for quick mental math assessments of investment growth.
Interest Rate
The interest rate is a crucial concept in finance, as it represents the cost of borrowing money or the reward for saving or investing money. It is typically expressed as a percentage of the principal amount. Here's how it works:
  • The interest rate on a savings account determines how much additional money the bank pays you for letting them hold your funds. For example, at a 9% interest rate, each year your savings earn 9% of the account balance.
  • There are two main types of interest rates: simple and compound. Simple interest is calculated solely on the principal, while compound interest is calculated on both the principal and the accumulated interest.
A higher interest rate results in more substantial gains on your savings over time, assuming the same principal.
Savings Account
A savings account is a type of bank account where you can store money securely and earn interest over time. It's often used for setting aside funds for future expenses or for emergency savings. Key features include:
  • Safety: Banks are generally insured, making them a secure place for your money.
  • Interest Earnings: Savings accounts earn interest over time, which increases your balance without additional deposits.
  • Liquidity: While savings accounts are accessible, transactions may be limited to encourage saving.
For a long-term goal or when planning for growth, selecting a savings account with a high-interest rate is advisable, as this maximizes the potential earnings on your stored funds.
Investment Growth
Investment growth refers to the increase in value of an investment over time. This growth is primarily driven by the interest rate and the compounding effect. Here's how it plays out:
  • The principal grows over time due to interest being added to the original amount.
  • Compound interest plays a significant role, as it allows interest to be calculated on accumulated interest as well as the principal.
  • The growth potential of an investment can be substantial over long periods, especially with higher interest rates.
Consider a savings account opened when you were one year old with an interest rate of 9%. By letting this account grow without withdrawals, the compounding effect will substantially increase the initial amount by the time you reach 65, demonstrating the power of investment growth over the long term.

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Most popular questions from this chapter

As a New Year's gift to yourself, you buy your roommate's 1976 Ford Pinto. She has given you the option of two payment plans. Under Plan A, you pay \(\$ 500\) now, plus \(\$ 500\) at the beginning of each of the next two years. Under Plan B, you would pay nothing down, but \(\$ 800\) at the beginning of each of the next two years. a. Calculate the present value of each plan's payments if interest rates are \(10 \%\). Should you choose Plan A or Plan B? b. Recalculate the present value of each plan's payments using a \(20 \%\) interest rate. Should you choose Plan A or Plan B? c. Explain why your answers to (a) and (b) differ.

You are considering the purchase of an old fire station, which you plan to convert to an indoor playground. The fire station can be purchased for \(\$ 200,000,\) and the playground will generate lifetime profits (excluding the cost of the building) of \(\$ 700,000\). (Assume that those profits are all realized one year after opening.) However, there is a \(20 \%\) chance that the city council will re-zone the district to exclude establishments such as yours; a hearing is scheduled for the coming year, and if your building is re- zoned, your profit will be zero. Assume that there is no other building currently under consideration. a. Assume an interest rate of \(10 \% .\) Calculate the net present value of opening the playground today. Note that the cost of purchasing the building today is certain, but the benefits are uncertain. b. Calculate the net present value today of opening the playground in one year, after the zoning issues have been decided. Note that the benefits of opening the playground are uncertain today, but will be certain in one year. c. Based on your answers to (a) and (b), should you open the playground today, or should you wait until the zoning commission reaches its decision?

You are currently driving a gas-guzzling Oldsmobuick that you expect to be able to drive for the next five years. A recent spike in gas prices to \(\$ 5\) per gallon has you considering a trade to a fuel-efficient hybrid Prius. Your Oldsmobuick has no resale value and gets 15 miles per gallon. A new Prius costs \(\$ 25,000\) and gets 45 miles per gallon. You drive 10,000 miles each year. a. Calculate your annual fuel expenditures for the Prius and the Oldsmobuick. b. Assume that the interest rate is \(7 \%\). Calculate the present value of your costs if you continue to drive the Oldsmobuick for another five years. Assume that you purchase a new Prius at the end of the fifth year, and that a Prius still costs \(\$ 25,000\). Also assume that fuel is paid for at the end of each year. (Carry out your cost calculations for only five years.) c. Calculate the present value of your costs if you purchase a new Prius today. Again, carry out your cost calculations for only five years. d. Based on your answers to (b) and (c), should you buy a Prius now, or should you wait for five years? e. Would your answer change if your Oldsmobuick got 30 miles per gallon instead of \(15 ?\)

You have \(\$ 832.66\) in a savings account that offers a \(5.25 \%\) interest rate. a. If you leave your money in that account for 20 years, how much will you have in the account? b. Suppose that inflation is expected to run at \(3.25 \%\) for the next 20 years. Use the real interest rate to calculate the inflation-adjusted amount your account will contain at the end of the 20 -year period. c. The amount you calculated in (b) is smaller than the amount you calculated in (a). Explain exactly what the amount you calculated in (b) tells you, and why the difference arises.

You are romantically interested in Chris, but have always wanted to date the president of the Economics Club. As it turns out, Chris is battling Pat for control of the Econ Club. That battle should be decided in a year, and you estimate the odds of Chris winning at \(60 \%\). Attracting Chris and kindling a relationship will involve \(\$ 1,000\) of effort on your part; if Chris wins the presidency, you will receive benefits worth \(\$ 2,200\) (assume you receive these benefits one year after beginning the relationship). If Chris loses the election, you receive nothing. a. Assume an interest rate of \(10 \% .\) Calculate the net present value of building a relationship with Chris today. Notice that the costs of kindling a relationship today are certain, but the benefits are uncertain. b. Considering only your answer to (a), should you initiate a relationship with Chris at this time? Assume you are risk-neutral in formulating your answer. c. Calculate the net present value of waiting until the presidency is decided to build a relationship with Chris. Note that both the costs and benefits of kindling a relationship are uncertain at this point, but that the two will be certain in one year. d. Based on your answers to both (a) and (c), should you initiate a relationship with Chris today, or should you wait to initiate the relationship until the presidency is determined?
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