a loan of $8,500 compounded quarterly for five years at 8% what is the effective interest rate for the loan
Did you mean the effective annual interest rate ?
if so, then
(1+i) = (1.02)^4
1+i = 1.0824..
i = .0824
or appr 8.24% per annum
To calculate the effective interest rate for a loan, you need to know the loan amount, the compounding frequency, the loan period, and the nominal interest rate. In this case, we have:
Loan amount (principal): $8,500
Compounding frequency: Quarterly
Loan period: 5 years
Nominal interest rate: 8%
Here's how you can calculate the effective interest rate for this loan:
Step 1: Convert the nominal interest rate to a decimal form.
Nominal interest rate = 8% = 0.08
Step 2: Determine the number of compounding periods.
Since the loan is compounded quarterly for five years, the total number of compounding periods will be:
Number of compounding periods = 4 (quarters per year) × 5 (years) = 20
Step 3: Calculate the periodic interest rate.
The periodic interest rate is determined by dividing the nominal interest rate by the number of compounding periods in a year:
Periodic interest rate = Nominal interest rate / Number of compounding periods per year
Periodic interest rate = 0.08 / 4 = 0.02
Step 4: Calculate the effective interest rate using the formula:
Effective interest rate = (1 + Periodic interest rate)^Number of compounding periods - 1
Plugging in the known values:
Effective interest rate = (1 + 0.02)^20 - 1
Now, let's do the calculations.
(1 + 0.02)^20 = 1.4859475
Effective interest rate = 1.4859475 - 1 = 0.4859475
This means that the effective interest rate for the loan is approximately 48.6%.