When calculating the effective rate of a loan, which statement or statements must be true if n is equal to 1?
I. The nominal rate equals the effective rate.
II. The length of the loan is exactly one year.
III. The interest is compounded annually.
A. I and III
B. II and III
C. I only
D. III only
Answer (1)
The effective interest rate formula is r e ff = ( 1 + n r ) n − 1 .
Substitute n = 1 into the formula: r e ff = ( 1 + 1 r ) 1 − 1 = r .
Conclude that the nominal rate equals the effective rate when n = 1 .
Determine that interest is compounded annually when n = 1 , and the loan length is not necessarily one year, thus the final answer is I an d III .
Explanation
Understanding the Problem We are given that n = 1 when calculating the effective rate of a loan. We need to determine which of the following statements must be true:
I. The nominal rate equals the effective rate. II. The length of the loan is exactly one year. III. The interest is compounded annually.
Defining Variables and Formula Let's define the nominal rate as r and the effective rate as r e ff . The formula for the effective rate is:
r e ff = ( 1 + n r ) n − 1
Substituting n=1 Since n = 1 , we can substitute this value into the formula:
r e ff = ( 1 + 1 r ) 1 − 1 = ( 1 + r ) − 1 = r
Analyzing the Statements Therefore, when n = 1 , the effective rate equals the nominal rate. So, statement I is true.
If n = 1 , the interest is compounded once per year, which means it is compounded annually. So, statement III is true.
The length of the loan does not necessarily have to be one year. n = 1 simply means the number of compounding periods per year is 1. The loan could be for any number of years. For example, a loan could be for 5 years with interest compounded annually. So, statement II is not necessarily true.
Conclusion Therefore, statements I and III must be true.
Examples
Understanding effective interest rates is crucial in personal finance. For instance, when comparing two loan offers with the same nominal interest rate but different compounding frequencies, calculating the effective rate helps determine which loan is actually cheaper. If you're taking out a loan or making an investment, knowing the effective rate allows you to make informed decisions and accurately assess the true cost or return.
