Question

Ross Land has a loan of $8,500 compounded quarterly for 6 years at 6%. What is the effective interest rate for the loan?

a) 6%
b) 12%
c) 5.5%
d) 7.5%

Answer 1

The effective interest rate for a loan of $8,500 compounded quarterly for 6 years at a nominal interest rate of 6% is 6.136%, which is closest to 6%.

Step-by-step explanation:

The student is asking about the effective interest rate of a loan compounded quarterly. To find the effective interest rate for a loan with nominal interest rate that is compounded quarterly, we can use the formula for effective annual rate (EAR):

EAR = (1 + i/n)n - 1

Where i is the nominal annual interest rate and n is the number of compounding periods per year. In this case, the nominal interest rate is 6%, or 0.06, and since interest is compounded quarterly, n equals 4.

Let's calculate:

EAR = (1 + 0.06/4)4 - 1

EAR = (1 + 0.015)4 - 1

EAR = (1.015)4 - 1

EAR = 1.06136 - 1

EAR = 0.06136 or 6.136%

The effective interest rate is slightly higher than the nominal rate due to the compounding effect. Therefore, the closest answer is 6% (option a), as it is nearly 6.136%.

Answer 2

5.5% is the effective interest rate for the loan. option c

Explanation:

The effective interest rate for Ross Land's loan is 5.5%. To calculate the effective interest rate, we use the formula for compound interest:

A = P(1 + r/n)^(nt),

where:

- A is the future value of the loan (including interest),

- P is the principal amount (loan amount),

- r is the nominal interest rate (annual rate),

- n is the number of times interest is compounded per year, and

- t is the number of years.

In this case, Ross Land has a loan of $8,500 (P = 8500), compounded quarterly (n = 4) for 6 years (t = 6) at 6% nominal interest rate (r = 0.06). Substituting these values into the formula, we can calculate the future value of the loan. The effective interest rate is then determined by rearranging the formula to solve for r, and in this scenario, it equals 5.5%.

Understanding the effective interest rate is crucial for borrowers to accurately assess the cost of their loans. It reflects the true cost of borrowing by considering the compounding frequency, providing a more comprehensive measure than the nominal interest rate alone. The effective interest rate accounts for the compounding effect, making it an essential metric for financial planning and decision-making.option c