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Chapter 7: Problem 44

The First City Bank pays 7 percent interest, compounded annually, on time deposits. The Second City Bank pays 6 percent interest, compounded quarterly. a. Based on effective, or equivalent, interest rates, in which bank would you prefer to deposit your money? 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? In answering this question, assume that funds must be left on deposit during the entire compounding period in order for you to receive any interest.

Short Answer

Expert verified
a. Choose First City Bank for higher EAR. b. Second City Bank is better if needing quarterly access.

Step by step solution

01

Understanding Compounded Interest

Compounded interest means that interest earned in each compounding period is added to the initial principal, so in the next period, interest is earned on a larger principal amount. Each bank applies a different compounding frequency: yearly for First City Bank and quarterly for Second City Bank.
02

Calculate the Effective Annual Rate (EAR) for First City Bank

For the First City Bank with 7% annual interest compounded yearly, the effective annual rate is the same as the nominal rate because there is only one compounding period: \( \text{EAR}_1 = (1 + 0.07)^1 - 1 = 0.07 \) or 7%.
03

Calculate the Effective Annual Rate (EAR) for Second City Bank

For Second City Bank with 6% annual interest compounded quarterly, use the EAR formula: \( \text{EAR}_2 = \left(1 + \frac{0.06}{4}\right)^4 - 1 \). This calculates to \( \left(1 + 0.015\right)^4 - 1 = 0.061364\) or 6.1364%.
04

Compare Effective Annual Rates (EARs)

Compare the EARs: First City Bank offers an EAR of 7%, while Second City Bank offers an EAR of about 6.1364%. The First City Bank offers a higher effective annual rate.
05

Consider Partial Year Withdrawals

If you wish to withdraw funds during the year, you must consider the compounding periods. With First City Bank, interest is only given at the end of the year. With Second City Bank, interest is added quarterly, so you might earn interest if withdrawing at quarter-end.

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Key Concepts

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

Compounded Interest
Compounded interest involves the interest earned in each period being added to the principal. This means in the next period, you earn interest on a larger amount due to the included interest. It's like a snowball effect, where your investment grows faster over time. The power of compounded interest lies in the frequency and the rate at which these additions to the principal occur.
  • The more frequently the interest is compounded, the more interest you earn on your deposited money.
  • Even a small difference in the interest rates can lead to significant changes over a long period due to compounding.
Different banks might offer different compounding periods, which can highly influence which bank gives a better return on your investment.
Compounding Frequency
The compounding frequency refers to how often the interest is calculated and added to the principal. The two common compounding frequencies are annually and quarterly, but there can also be monthly, weekly, or even daily options.
  • Annual compounding means the interest is added once a year, leading to one opportunity for reinvestment each year.
  • Quarterly compounding adds interest four times a year, effectively allowing your investment to grow faster than annual compounding at the same nominal rate.
Understanding how compounding frequency impacts your earnings is crucial for making informed decisions. Higher compounding frequencies result in higher effective rates of return, making them generally more advantageous if you plan to keep your money invested for longer periods.
Interest Rates Comparison
When comparing interest rates, it's important to look beyond just the nominal rate and consider the Effective Annual Rate (EAR). The EAR provides a truer picture of your potential earnings.
  • The EAR accounts for the effect of compounding within the year, providing a more accurate reflection of your returns.
  • For instance, a bank could advertise a lower nominal interest rate but with more frequent compounding, the EAR might be higher than a bank with a higher nominal rate but less frequent compounding.
Using EAR as a comparison tool can help determine which bank offers a better return on your investment over the same period.
Withdrawal Implications
Withdrawal implications refer to how your decision to withdraw funds might affect the interest earned, based on the bank's compounding policy. It's crucial to understand the rules related to withdrawal, especially if you think you might need your money before the maturity of the deposit.
  • For annually compounding banks, withdrawing before the end of the year may mean you receive no interest at all.
  • With quarterly compounding, if you withdraw at the end of a quarter, you might still earn some interest.
It's important to align your withdrawal needs with the compounding schedule of the bank to maximize returns while maintaining access to your funds when needed.

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Most popular questions from this chapter

As the manager of Oaks Mall Jewelry, you want to sell on credit, giving customers 3 months in which to pay. However, you will have to borrow from the bank to carry the accounts receivable. The bank will charge a nominal 15 percent, but with monthly compounding. You want to quote a nominal rate to your customers (all of whom are expected to pay on time) that will exactly cover your financing costs. What nominal annual rate should you quote to your credit customers?

Washington-Atlantic invests \(\$ 4\) million to clear a tract of land and to set out some young pine trees. The trees will mature in 10 years, at which time Washington-Atlantic plans to sell the forest at an expected price of \(\$ 8\) million. What is Washington-Atlantic's expected rate of return?

An investment pays \(\$ 20\) semiannully for the next 2 years. The investment has a 7 percent nominal interest rate, and interest is compounded quarterly. What is the future value of the investment?

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?

Your client is 40 years old and wants to begin saving for retirement. You advise the client to put \(\$ 5,000\) a year into the stock market. You estimate that the market's return will be, on average, 12 percent a year. Assume the investment will be made at the end of the year. a. If the client follows your advice, how much money will she have by age \(65 ?\) b. How much will she have by age 70 ?
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