In a discount interest loan, you pay the interest payment up front. For example, if a 1-year loan is stated as $10,000 and the interest rate is 10 percent, the borrower “pays” 0.10 x $10,000 = $1,000 immediately, thereby receiving net funds of $9,000 and repaying $10,000 in a year.
a. What is the effective interest rate on this loan?
b. If you call the discount d (for example, d= 10% using our numbers), express the effective annual rate on the loan as a function of d.
c. Why is the effective annual rate always greater than the stated rate d?
10.38
a. To calculate the effective interest rate on this loan, we need to consider the net funds received and the amount repaid. In this case, the borrower receives $9,000 in net funds and repays $10,000 in a year. So, the effective interest rate can be calculated using the following formula:
Effective Interest Rate = (Amount Repaid - Net Funds Received) / Net Funds Received
Effective Interest Rate = ($10,000 - $9,000) / $9,000
Effective Interest Rate = $1,000 / $9,000
Effective Interest Rate = 0.1111 or 11.11%
b. To express the effective annual rate on the loan as a function of the discount d, we can use the following formula:
Effective Annual Rate = (1 + d) / (1 - d)
In this case, if d = 10% (or 0.1), we can substitute the value into the formula:
Effective Annual Rate = (1 + 0.1) / (1 - 0.1)
Effective Annual Rate = 1.1 / 0.9
Effective Annual Rate = 1.2222 or 12.22%
c. The effective annual rate is always greater than the stated rate because the borrower receives a net amount lower than the loan amount due to the upfront payment of interest. This lowers the actual funds received by the borrower, resulting in a higher effective interest rate. The effective annual rate captures the true cost of borrowing by considering this upfront payment and the subsequent repayment of the full loan amount.
a. The effective interest rate on the loan can be determined by calculating the interest paid relative to the net funds received. In this case, the borrower receives $9,000 and repays $10,000 in a year. Therefore, the interest paid is $10,000 - $9,000 = $1,000.
To calculate the effective interest rate, divide the interest paid by the net funds received and express it as a percentage:
Effective interest rate = (Interest paid / Net funds received) x 100%
= ($1,000 / $9,000) x 100%
≈ 11.11%
Therefore, the effective interest rate on this loan is approximately 11.11%.
b. The effective annual rate on the loan can be expressed as a function of the discount, denoted as "d". Since the discount represents the upfront interest payment, the net funds received can be calculated as the loan amount minus the discount:
Net funds received = Loan amount - Discount
In this case, the loan amount is $10,000 and the discount is 10%, which can be expressed as a decimal value of 0.10. Substituting these values, we get:
Net funds received = $10,000 - (0.10 x $10,000)
= $10,000 - $1,000
= $9,000
Now, to calculate the effective annual rate as a function of d, we can divide the interest paid by the net funds received (as calculated in part a) and express it as a percentage:
Effective annual rate = (Interest paid / Net funds received) x 100%
Since the interest paid is equal to the discount (d x Loan amount), and the net funds received is equal to the loan amount minus the discount, we can substitute these values:
Effective annual rate = (d x Loan amount / (Loan amount - d x Loan amount)) x 100%
= (d x Loan amount / Loan amount (1 - d)) x 100%
Simplifying further, we have:
Effective annual rate = (d / (1 - d)) x 100%
Therefore, the effective annual rate is expressed as a function of the discount d as (d / (1 - d)) x 100%.
c. The effective annual rate is always greater than the stated rate d because the effective rate takes into account the compounding effect of interest over time. In a discount interest loan, the borrower receives the net funds upfront, but the interest payment is calculated based on the full loan amount.
By paying the interest upfront, the borrower receives less money compared to a traditional loan where the interest is paid over time. This means that the interest payment is effectively higher, causing the effective annual rate to be greater than the stated rate. The difference between the two rates increases as the time period of the loan extends, as the compounding effect becomes more significant.
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