# APR Calculation

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%.

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

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

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(1) Did you remember to factor in the "other fees"? They also go into the computation of the APR. (2) When you say the interest for a month "does not exceed 14%", does that mean 14% annualized, or 14% monthly? If you borrowed 100, and after one month you pay 14 dollars in interest, that 14% is monthly interest; the annualized interest would be much higher; even simple monthly interest would give you 14*12 = 168% annual interest doing it that way. –  Arturo Magidin May 13 '11 at 23:55
I entered the "other fees" as instructed by the software. It just doesn't seem that 39.95 would affect the APR that much, especially over 12 months, even if I were to leave it out. Regarding the 14%, yes, I meant 14% per month. EDIT: not 14% per month, but 14% at the highest when looking at each month's individual interest rates. –  Marco Brutini May 13 '11 at 23:59
Hmmm... I don't understand the "federally approved software calculation". Your total payments amount to 1658.45 (once the 39.95 in fees is added). That's 159.95 over the original loan amount, which is only about about 10.67%. How does the "federally approved software" make the computation? –  Arturo Magidin May 14 '11 at 0:05
@Marco: If it's a 14% monthly rate, then that would amount to well over a 150% APR. If I loan you 100, and charge you 14 interest every month, you end up paying me well more than 14 dollars in interest. –  Arturo Magidin May 14 '11 at 0:06
@Marco: E.g., see this table in Wikipedia. A 100 loan with 10 fees and a 5% monthly interest has an APR of 45%; the 5% is monthly, so the annual interest can be a much higher rate. –  Arturo Magidin May 14 '11 at 0:14

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.

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Close enough to 21.488% for me –  Ross Millikan May 14 '11 at 14:56
@Henry: That's great, thanks for the detailed breakdown. However, my understanding of interest is that it's the relationship of the surcharge (in this case, a constant \$10) w.r.t the monthly payment. Is that concept not applicable in this scenario? (EDIT: I'd vote you up if I could) –  Marco Brutini May 16 '11 at 18:54
@Marco: You can accept the answer if you wish. The point about the effective APR is that it takes all payments in each direction into account together with their timing. It doesn't care whether they are called principal, interest, fees, balloon payment or anything else. –  Henry May 16 '11 at 20:03
@Henry: the balance after the 39.95 fee should be 1498.50, though the borrower only receives 1458.55 net of fee. It will only change your numbers a bit. –  Ross Millikan May 16 '11 at 20:36
@Henry: So that means that the 23.896 interest rate in the first month is the result of this "amalgamation" of monies. It's not easily calculable looking at that one row to determine how that particular interest rate came about. I guess this is the main thing that has been throwing my mind for a spin. It almost seems useless to look at a month's interest rate as a consumer. –  Marco Brutini May 16 '11 at 21:13
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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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Even if the fixed 39.95 is paid up front in the first month? –  Marco Brutini May 16 '11 at 18:59
If the fixed 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. –  Ross Millikan May 16 '11 at 20:36