The easiest way to approach a question like this is to make up some numbers. So we will say that the loan amount is $100,000. It is a discounted note, which means that the interest is subtracted from the proceeds of the loan (the amount of money the borrower actually receives), and the borrower must pay back the entire $100,000. So 12% of $100,000, or $12,000, will be subtracted from the proceeds, and the borrower will receive $88,000. The borrower will be obligated to pay that amount back at the end of one year, plus the $12,000 interest, for a total of $100,000. Therefore, the effective interest rate will be $12,000 ÷ $88,000 = .1364 or 13.64%. Another way to calculate this answer is simply to divide .12 by (1 ? .12). .12 ÷ .88 = .1364 or 13.64%.
12% is the nominal interest rate on a discounted note. The question is asking for the effective interest rate. The effective annual interest rate is the amount of interest paid for a year divided by the proceeds of the loan (the amount the borrower actually receives). The amount the borrower actually receives will be the face amount of the note minus the discounted interest amount.
The easiest way to approach a question like this is to make up a gross amount for the loan and multiply it by the nominal rate to calculate the amount of the discounted interest. Then subtract the amount of the discounted interest from the gross amount of the loan to calculate the amount of the proceeds of the loan. Finally, divide the amount of the annual interest by the amount of the proceeds of the loan. Since in this question the loan is for one year, the amount of the annual interest is the same as the amount of the discounted interest. (If the loan were for a period other than one year, it would be necessary to annualize the amount of the discounted interest.) This answer results from adding the amount of the discounted interest to the gross amount for the loan to get the amount of the proceeds for the loan, instead of subtracting it. Or, it may result from dividing .12 by (1 + the nominal rate of.12).
This answer results from dividing .12 by the present value of $1 factor for 10% for 1 year. The easiest way to approach a question like this is to make up a gross amount for the loan and multiply it by the nominal rate to calculate the amount of the discounted interest. Then subtract the amount of the discounted interest from the gross amount of the loan to calculate the amount of the proceeds of the loan. Finally, divide the amount of the annual interest by the amount of the proceeds of the loan. Since in this question the loan is for one year, the amount of the annual interest is the same as the amount of the discounted interest. (If the loan were for a period other than one year, it would be necessary to annualize the amount of the discounted interest.)
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