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I'm hoping someone can clarify this for me.

The model/example is this:

We lend an amount of 1498.50 (loan amount). Other fees total 39.95. The term of the loan is for 12 months. There is no interest per month, per se, but a 10.00 fee is charged for each of the 12 months.

Using federally-approved software that calculates APR (to comply with Regulation Z, Appendix J), the calculation returns the APR to be 21.488%.

The monthly payment is calculated as follows:

(Loan Amount + (10.00 x 12 months)) / 12 = 134.87.

According to the amortization schedule, none of the interest for any particular month exceeds 14%. How can the APR for the entire term be ~ 21% when none of the effective monthly interest rates even come close to 21%?

My intuition tells me that there is some averaging going on, but to achieve an average of 21%, some months would have to exceed 21% by quite a bit, while being offset by smaller percentages on the fringes of the curve.

If you are familiar with Reg Z, Appendix J, and the given example above, would you say that the formula being used is not appropriate? (There are a few others that the feds provide; maybe we/the software applied the wrong one)

Additionally, playing around with the software, keeping all numbers the same, but changing the term (in months), I find that the APR also changes. I find this confusing and counter-intuitive.

For example: 3 months, APR is 39.126% 6 months, APR is 27.393% 12 months, APR is 21.488%.

Can someone please explain this?

*All numbers above are the input and output values of approved calculation software.

Huge thanks in advance!

PS: Apologies for the seemingly uninformative tag(s). I couldn't find one that was more meaningful.

  • 0
    I'm tr$y$ing to either understand the process of APR calculation or prove that the software is calculating incorrectl$y$ given the above model. According to a consultant with whom I'm working, the formula I should be using (and that the software supposedly is using) can be found at (http://fraser.stlouisfed.org/publications/frb/page/31359/3958/download/31359.pdf) ... Go to Page 56, formula (C)...first one on the left-hand column. That formula is way above my head.2011-05-14

2 Answers 2

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I make the annual interest rate about 21.532%, though there may be some rounding issue. Since $1.21532^{1/12}=1.01638$, this corresponds to a monthly interest rate of about 1.638%.

Initially the amount handed over less the fee is 1498.50-39.95 = 1458.55.

M   Inter-  Balance     Payment Balance       est    before              after                payment             payment     0           1498.500     39.950 1458.550 1   23.896  1482.446    134.875 1347.571 2   22.078  1369.649    134.875 1234.774 3   20.230  1255.004    134.875 1120.129 4   18.352  1138.481    134.875 1003.606 5   16.443  1020.048    134.875  885.173 6   14.502   899.676    134.875  764.801 7   12.530   777.331    134.875  642.456 8   10.526   652.982    134.875  518.107 9    8.488   526.595    134.875  391.720 10   6.418   398.138    134.875  263.263 11   4.313   267.576    134.875  132.701 12   2.174   134.875    134.875    0.000 

To calculate the monthly interest rate $i$, you need to find the solution to

$(L-f)(1+i)^m - p((1+i)^m-1)/i = 0$

where $L$ is the loan, $f$ is the fee, $m$ is the number of months, and $p$ is the monthly payment. Here, I get $i=0.01638$.

Then to get the annual interest rate

$(1+i)^{12}-1$

which I make $0.21532$. Multiply by 100 to get percentages.

  • 0
    The 23.896 is roughly (allowing for rounding) 1.638% of 1458.55. It may not be what the lender or the borrower think the monthly interest rate is, but it is what the formula thinks, and with 1.638% of the remaining balance applied each month leads to a nice 0.000 as the final balance after all the payments.2011-05-16
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First, if the other fees are paid as part of the loan, I find a payment of 138.1625. If they are paid in advance, I believe they still count as interest. Then the effective interest is 159.95. The average balance is about half of the amount borrowed, as you start off owing 1498.5 and end at 0. So the effective annual interest rate is about 159.95/(1498.5/2)=23.15%. I think they get a lower value because they do an amortization, which keeps the balance higher for longer, but it is not far off. For the shorter terms, the interest rate goes higher because the fixed 39.95 fee gets spread over fewer months.

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    If the fi$x$ed 39.95 is paid in front, I agree with the 134.875 you have for a payment. But the 39.95 still counts as part of the interest-any costs paid to the lender are part of it. Henry's calculation supports this.2011-05-16