UNDERSTANDING
INTEREST RATES
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 interest-only 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.
Education loans are even more complicated
than this, adding to the confusion. For example, they often involve capitalized
interest, which increases the size of the loan. They also involve a variety of
loan discounts and loan fees. All of this combines to make it harder for
borrowers to appreciate just how much the loan will cost them to repay.
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 10-year $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 10-year
$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.