Final answer:
The nominal annual interest rate for the loan is 18%, while the effective annual interest rate is 19.5618%. To find how much one would pay at the end of two years, we use either the period interest rate approach or the effective interest rate approach to determine the future value of the loan and then subtract the original loan amount to calculate the interest amount.
Step-by-step explanation:
Nominal and Effective Interest Rates
To answer your questions regarding loans from Gougos, let's deal with each part one by one:
a. Nominal Annual Interest Rate
The nominal annual interest rate is simply the monthly rate multiplied by the number of months in a year. Since the loan has an interest rate of 1.5% per month, the nominal annual rate would be 1.5% * 12 = 18%.
b. Effective Annual Interest Rate
The effective annual interest rate (EAR) takes compounding into account and is calculated using the formula:
(1 + monthly rate)^12 - 1. Substituting the monthly rate of 1.5% (or 0.015), the EAR would be (1 + 0.015)^12 - 1 = 0.195618, or 19.5618%.
c. Future Value of the Loan Using Period Interest Rate
To calculate the future value of the loan using the period interest rate, we use the compound interest formula: P(1 + i)^n, where P is the principal, i is the monthly interest rate, and n is the number of periods. In this case, P = $3,000, i = 0.015, and n = 24. The future value will be $3,000 * (1 + 0.015)^24.
d. Future Value of the Loan Using Effective Interest Rate
Using the effective annual interest rate, we calculate the future value by compounding the loan only once a year. The formula in this case simplifies to P(1 + EAR)^t, where EAR is the effective annual interest rate and t is the number of years. So, we'll have $3,000 * (1 + 0.195618)^2.
e. Interest Amount
The interest amount is calculated by subtracting the original loan amount from the future value amount calculated in either part (c) or (d).