Question
Calculate the effective annual interest rate for a loan with a 0.5% monthly interest rate and a 3% deposit requirement in a non-interest-bearing account.
Answers
GPT-4o mini
To calculate the effective annual interest rate (EAR) for a loan with a monthly interest rate of 0.5% and a 3% deposit requirement, we need to consider two main aspects: the effect of the monthly interest and the effect of the deposit requirement.
### Step 1: Calculate the Effective Annual Rate (from monthly interest rate)
1. The monthly interest rate is given as 0.5% or 0.005 in decimal form.
2. The formula to calculate EAR from the monthly interest rate is:
\[
\text{EAR} = (1 + r)^n - 1
\]
where \( r \) is the monthly interest rate and \( n \) is the number of compounding periods per year (12 months).
\[
\text{EAR} = (1 + 0.005)^{12} - 1
\]
Calculating \( (1 + 0.005)^{12} \):
\[
(1.005)^{12} \approx 1.061677812
\]
Now, subtracting 1:
\[
\text{EAR} \approx 1.061677812 - 1 \approx 0.061677812
\]
Converting to a percentage:
\[
\text{EAR} \approx 6.1677812\%
\]
### Step 2: Consider the effect of the deposit requirement
Since there is a 3% deposit requirement that is held in a non-interest-bearing account, this effectively reduces the amount you have available for investment or spending.
When you borrow, you lose access to that 3% of the principal that you have to deposit. Therefore, the effective interest rate should also reflect this deposit. When you borrow, the loan is effectively against a smaller amount (97% of the principal, because you cannot use the 3% that is held in the deposit).
To incorporate the effect of the 3% deposit, you can adjust the interest you pay on the loan:
1. If \( P \) is the total amount of the loan, after depositing 3%, you effectively can use \( 0.97P \).
2. The loan interest on \( P \) is what you are charged, but you only have effective access to \( 0.97P \).
The effective interest rate that you are paying on the money you can use (considering the non-accessible deposit) can be shown using the formula:
\[
\text{Effective interest rate} = \frac{\text{Nominal interest rate}}{1 - \text{Deposit requirement}}
\]
Thus:
\[
\text{Effective interest rate} = \frac{0.061677812}{1 - 0.03} = \frac{0.061677812}{0.97} \approx 0.0636 \text{ or } 6.36\%
\]
### Conclusion:
The effective annual interest rate you would actually be facing, considering both the monthly interest charge and the impact of the deposit requirement, is approximately:
\[
\text{Effective Annual Interest Rate} \approx 6.36\%
\]
### Step 1: Calculate the Effective Annual Rate (from monthly interest rate)
1. The monthly interest rate is given as 0.5% or 0.005 in decimal form.
2. The formula to calculate EAR from the monthly interest rate is:
\[
\text{EAR} = (1 + r)^n - 1
\]
where \( r \) is the monthly interest rate and \( n \) is the number of compounding periods per year (12 months).
\[
\text{EAR} = (1 + 0.005)^{12} - 1
\]
Calculating \( (1 + 0.005)^{12} \):
\[
(1.005)^{12} \approx 1.061677812
\]
Now, subtracting 1:
\[
\text{EAR} \approx 1.061677812 - 1 \approx 0.061677812
\]
Converting to a percentage:
\[
\text{EAR} \approx 6.1677812\%
\]
### Step 2: Consider the effect of the deposit requirement
Since there is a 3% deposit requirement that is held in a non-interest-bearing account, this effectively reduces the amount you have available for investment or spending.
When you borrow, you lose access to that 3% of the principal that you have to deposit. Therefore, the effective interest rate should also reflect this deposit. When you borrow, the loan is effectively against a smaller amount (97% of the principal, because you cannot use the 3% that is held in the deposit).
To incorporate the effect of the 3% deposit, you can adjust the interest you pay on the loan:
1. If \( P \) is the total amount of the loan, after depositing 3%, you effectively can use \( 0.97P \).
2. The loan interest on \( P \) is what you are charged, but you only have effective access to \( 0.97P \).
The effective interest rate that you are paying on the money you can use (considering the non-accessible deposit) can be shown using the formula:
\[
\text{Effective interest rate} = \frac{\text{Nominal interest rate}}{1 - \text{Deposit requirement}}
\]
Thus:
\[
\text{Effective interest rate} = \frac{0.061677812}{1 - 0.03} = \frac{0.061677812}{0.97} \approx 0.0636 \text{ or } 6.36\%
\]
### Conclusion:
The effective annual interest rate you would actually be facing, considering both the monthly interest charge and the impact of the deposit requirement, is approximately:
\[
\text{Effective Annual Interest Rate} \approx 6.36\%
\]