
Cost of Interest
Some students, because they do not have prior experience with debt and loan amortization, do not appreciate how much their loans will cost them. The concept of paying interest is not immediately transparent, in part because loan amortization calculations involve solving nonlinear equations. For example, if you ask most borrowers how much interest they will pay over the lifetime of a 10 year, $10,000 loan with 10% interest, very few give the correct answer of $5,858.15. Typical errors include assuming an interestonly loan, where the monthly payments do not include payments to reduce the principal balance, and either reporting just a single year's interest or the full term's interest, the student would either calculate $1,000 in interest (10% of $10,000) or $10,000 in interest (10% of $10,000 for 10 years). This either yields a value that is too low or a value that is too high. When estimating the monthly payments or the total amount repaid, including interest and principal, many borrowers omit either the interest or the original loan balance.
Even if the borrower understands the concept of interest, an interest rate does not feel very high on an emotional level. The interest rates are usually low single digit figures for federal education loans, and so don't feel like real money. For this reason, FinAid strongly encourages educators and borrowers to calculate the actual total interest paid over the lifetime of the loan, using calculators like FinAid's loan calculator. Seeing the actual cost of the loan, the total interest paid, and the monthly loan payment help borrowers make more realistic decisions concerning the amounts they borrow. In addition, here is a useful rule of thumb for calculating the interest on a loan that yields a total interest paid figure that is in the right ballpark: Total interest paid is slightly higher than the product of the amount borrowed multiplied by the interest rate and the length of the loan term in years and divided by 2. If B is the loan balance, I is the interest rate, and Y is the length of the loan term in years, this figure is BIY/2. In the 10year $10,000 at 10% interest example given above, this would yield $5,000 in interest over the lifetime of the loan, not far off from the actual value of $5,858.15. Multiply the BIY/2 figure by (1 + IY)/(1 + IY/2) to obtain an upper bound on the total interest paid. In the 10year $10,000 at 10% interest example, this would yield $6,666.67. The actual figure, $5,858.15, is midway between the two estimates. With level repayment, the monthly payment is the same for the life of the loan, and includes both interest payments and payments to reduce the principal balance. Initially the interest payments start off as a high percentage of the monthly loan payment  in the example given above, a little more than 3/5ths  and gradually drop as the monthly payments gradually pay off the loan principal. (The percentage of the monthly payment that is initially interest is usually at least the ratio x/(1 + x), where x is IY. A somewhat better approximation is (x + x*x/2)/(1 + x + x*x/2).) Calculating the size of the monthly payment necessary to fully pay off the loan within the loan term is called loan amortization. Since most of the monthly payment is initially interest, it takes time for the payments to principal to accumulate enough to reduce the interest portion appreciably. So the portion of the monthly payments that pay interest do not drop in a linear fashion, but rather in a slightly convex curve, with the degree of curvature related to the interest rate and the loan term. But a linear approximation based on the average of the first and last months interest payments represents a reasonably accurate approximation, yielding the rule of thumb highlighted in bold above.

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