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Chapter 5: Problem 14

First National Bank charges 12.4 percent compounded monthly on its business loans. First United Bank charges 12.7 percent compounded semiannually. As a potential borrower, which bank would you go to for a new \(\operatorname{loan} ?\)

Short Answer

Expert verified
The potential borrower should choose First National Bank, as its Effective Annual Rate (EAR) of 13.11% is lower than First United Bank's EAR of 13.14%.

Step by step solution

01

Understand the formula for Effective Annual Rate (EAR)

The formula for calculating the Effective Annual Rate (EAR) is given by: EAR = \((1 + i/n)^{n} - 1\) where i is the nominal annual interest rate, and n is the number of compounding periods per year.
02

Calculate the EAR for First National Bank

In this case, i = 12.4% and n = 12 (compounded monthly). So we have: EAR_FNB = \((1 + 0.124/12)^{12} - 1\) Calculate the expression in the brackets: \( (1 + 0.124/12) = 1.010333 \) Raise the result to the power of 12: \( (1.010333)^{12} = 1.1311 \) Subtract 1 to find the EAR: \( 1.1311 - 1 = 0.1311 \) The EAR for First National Bank is 0.1311 or 13.11%.
03

Calculate the EAR for First United Bank

In this case, i = 12.7% and n = 2 (compounded semiannually). So we have: EAR_FUB = \((1 + 0.127/2)^{2} - 1\) Calculate the expression in the brackets: \( (1 + 0.127/2) = 1.0635 \) Raise the result to the power of 2: \( (1.0635)^{2} = 1.1314 \) Subtract 1 to find the EAR: \( 1.1314 - 1 = 0.1314 \) The EAR for First United Bank is 0.1314, or 13.14%.
04

Compare the EARs and decide which bank to choose

The Effective Annual Rates for both banks are as follows: - First National Bank: 13.11% - First United Bank: 13.14% Since the EAR for First National Bank (13.11%) is less than the EAR for First United Bank (13.14%), the potential borrower should choose the First National Bank as it offers a better loan rate when considering the compounding effect.

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Key Concepts

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

Compounded Interest
Compounded interest is the process by which the interest earned on an investment or charged on a loan is reinvested or added to the principal, so that the interest also earns interest from that point on. This concept is central to understanding how investments grow over time and how loan costs can escalate.

For instance, if you have a business loan with a certain annual interest rate, compounded interest ensures that each month or year, interest is calculated not just on the original amount borrowed, but also on any interest accumulated up to that point. This compounding effect can significantly affect the total amount you will pay back over the life of the loan, making it a critical element to consider when comparing loan offers from financial institutions like First National Bank and First United Bank in our exercise.
Business Loans
Business loans are financial products that provide capital to businesses for various purposes, such as starting a new venture, expanding current operations, or bridging cash flow gaps. The cost of borrowing, reflected in the interest rate and how it compounds, plays a decisive role in the total repayment amount. Understanding the true cost of a loan is imperative for business owners to make informed decisions.

The type of interest rate鈥攚hether it's fixed or variable鈥攁s well as the frequency of compounding, as shown in the comparison between First National Bank and First United Bank, can lead to different Effective Annual Rates (EAR). Business owners must evaluate these rates to identify which loan option aligns best with their financial strategy and repayment capacity.
Interest Rate Calculation
Interest rate calculation is an integral skill when managing financial products. Calculating the Effective Annual Rate (EAR) allows individuals and businesses to understand the actual annual return on an investment or the actual annual cost of a loan after accounting for the compounding of interest.

The formula used for calculating EAR is an important tool that incorporates the nominal annual interest rate and the number of compounding periods within the year. By comparing the calculated EAR from different financial products, borrowers can make more accurate comparisons and decisions. It's evident in our exercise that even a seemingly small difference in nominal interest rates or compounding frequency can influence the cost of a loan, as the slight difference between the EAR of First National and First United Bank demonstrates.

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Most popular questions from this chapter

If you deposit \(\$ 3,000\) at the end of each of the next 20 years into an account paying 9.5 percent interest, how much money will you have in the account in 20 years? How much will you have if you make deposits for 40 years?

Beginning three months from now, you want to be able to withdraw \(\$ 2,200\) each quarter from your bank account to cover college expenses over the next four years. If the account pays .90 percent interest per quarter, how much do you need to have in your bank account today to meet your expense needs over the next four years?

This question illustrates what is known as discount interest. Imagine you are discussing a loan with a somewhat unscrupulous lender. You want to borrow \(\$ 12,000\) for one year. The interest rate is 13 percent. You and the lender agree that the interest on the loan will be \(.13 \times \$ 12,000=\$ 1,560 .\) So the lender deducts this interest amount from the loan up front and gives you \(\$ 10,440\). In this case, we say that the discount is \(\$ 1,560 .\) What's wrong here?

You are planning your retirement in 10 years. You currently have \(\$ 150,000\) in a bond account and \(\$ 450,000\) in a stock account. You plan to add \(\$ 9,000\) per year at the end of each of the next 10 years to your bond account. The stock account will earn an 11.5 percent return and the bond account will earn a 7.5 percent return. When you, retire, you plan to withdraw an equal amount for each of the next 25 years at the end of each year and have nothing left. Additionally, when you retire you will transfer your money to an account that earns 6.75 percent. How much can you withdraw each year?

You want to buy a new sports car from Muscle Motors for \(\$ 61,000\). The contract is in the form of a 60 -month annuity due at an 8.15 percent APR. What will your monthly payment be?
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