Effective interest calculation

[WallyMcCloud]

I am trying to figure out how to calculate the effective interest according to the EU-standards (as described here: https://en.wikipedia.org/wiki/Annual_percentage_rate), in other words extract APR from that formula and use that to calculate the effective interest based on, for example:

- Loan amount: 10.000
- Interest: 6%
- Period: 24 months
- Startup fee: 300
- Monthly fee: 25

According to an online calculator this should sum up to 1537 in fees and interest costs and an effective interest of 15.55%. I can't, for the love of Pythagoras, figure out how that interest is calculated! Anyone know how this is calculated?

Please note that there seems to be a difference of what is referred to as APR in this formula and what most Americans consider APR.

 

[tkhunny]

Well, your love of Pythagoras is not very well documented.

First glance, using your 468.21.

468.21 * 24 Covers Interest, All Monthly Fees, and Principal. It does NOT cover the 300.00 initiation fee which must be paid separately, up front. It is a little odd that you have this 300.00 an entire month prior to loan origination.

Thus, 468.21 * 24 - 10,000.00 + 300.00 = 1,537.04

Interestingly, the regular payment of 468.21 is calculated in a rather straightforward manner using ½% per month, which is 1/12 of the given 6% annually.

Okay, that's the loan structure. How would you like to calculate the total effective APR?

 

[WallyMcCloud]

Thanks, tkhunny.

Let me give you some more background then. In EU there is a law that everyone selling credit cards, loans etc must provide an "effective interest" which should include all interest costs and all relevant fees (excluding however fees for currency exchange, late fees, reminder fees etc). The example I provided is from an online Swedish calculator from an official consumer information website. Feel free to check it out: https://www.konsumenternas.se/lana/olika-lan/om-konsumtionslan/lanekalkyl# if you can Google translate it. I used the input given above to get the result in the screenshot I attached.

I can understand every bit in the loan structure but how they reach the conclusion of the "effective interest" beats me. It is supposed to be using the formula provided in the Wiki-article (scroll down to "European union") but I can't understand how to break out the "APR" in that formula. I am quite sure the answer must not be solved as in iteration by testing different APR.s As I mentioned before, when I have googled "APR" it seems like the APR in this formula doesn't perfectly equal what American APR calculations do. So that Wikipeda article formula is the documentation in that sense.

My thought here would be to calculate the interest by 1537/10000 which would sum up to 0,1537 but that does not take the period into account (and if you calculate this for, e.g. 12 months in that online calculator you get 928 as total costs for 10.000 and 18.64% as effective interest).

I am completely lost here. I have asked 5+ banks for clarification, the website and governmental service that provides the online calculator and many others but get no other replies than "Customer service standard replies" and used two freelancers and none have solved it. All this for a calculation that is demanded by EU law that you use when you provide a rather common financial product!

 

[tkhunny]

1) I did read your Wiki link. Why did you feel it necessary to re-quote it? Wasn't that why you posted the link?
2) Why didn't you mention that the effective interest calculation was the only part you didn't understand in the very first post.
3) I'm relatively sure that anyone required to disclose the effective interest is also required to provide a small document of explanation? Maybe they refer to the Wiki article for that? Can you walk in a brick bank and ask, "If I were to take a loan, what sort of disclosure concerning effective interest would you give me?"
4) Why not try some tiny examples in the calculator in an attempt to reverse engineer it?

So far, with various assumptions, I have managed 16.5767%, 18.0937%, 16.7466%, and 15.4364%. We're definitely in the neighborhood.

 

[WallyMcCloud]

1. Because there are more than one formula stated in it whereas the first one is very simple and the one I am trying to understand is not.
2. I mean, the whole formula is about calculate that effective interest, which my question was about but maybe I could have made that even more clear.
3. I have asked several banks and they can always give you a result as in an effective interest. They are, by law, required to show you the effective interest for a loan with a given loan sum, given nominal interest and given period of time (and the total cost of it). Having all the inputs and the results, even with that formula, does not provide the means however to calculate the effective interest. Every bank I have asked have just provided me with a standard wording "We are following the EU regulations according to this document https://eur-lex.europa.eu/legal-content/SV/TXT/PDF/?uri=CELEX:32008L0048&from=EN but cannot provide you with directions on HOW we calculate it.". It is highly frustrating.
4. I have used several examples of it but I don't see how I could reverse engineer a formula from the different results of it?

I am truly thankful for any help I can get on this matter. It is incredibly frustrating to have literally hundreds of companies online having this effective interest calculated, having a formula that doesn't provide any help and absolutely none of the banks are willing to assist you in explaining how they actually do it. In addition, the authority that is supposed to assert that the banks are following these rules will not help me. I don't really know where to address this question.

 

[tkhunny]

Sorry, I haven't had time to look at it for a couple of days. Keep having patience. We'll get it.

I did note that we have a problem in the U.S. that would be solved by this EU regulation. A furniture store says:
1) This is the price if you pay now.
2) If you want the 1-year, NO INTEREST payment plan, this HIGHER number is the price.

This, of course, makes no sense. The ONLY reason these prices would be different is a difference in the EFFECTIVE interest rate. However, they are not required to disclose the effective interest rate. Good work, EU! Maybe the US will catch up on this one.

 

[tkhunny]

Point #1 Without ANY fees, either startup or periodic administrative, it is a simple calculation of nominal vs effective.

Monthly: ( 1 + 0.06/12)^12 - 1 = 0.0616778
Quarterly: ( 1 + 0.06/4)^4 - 1 = 0.06136355

Point #2 Without a Startup fee, but with a level monthly admin fee, it is just the rate that reproduces the original loan value:

Loan 1,000.00
Interest 6%
Monthly
Admin Fee: 0.00
Payments: 24 @ 44.32

PV of 24 monthly payments of 44.32 @ effective annual 6.16778% is 1,000.00 <== As in Point #1

Admin Fee: 10.00
Payments: 24 @ 54.32

PV of 24 monthly payments of 54.32 @ effective annual 30.4533% is 1,000.00

Point #3 With a startup fee and NO monthly admin fee. This one doesn't quite make sense to me. Can you explain it? Maybe I'm missing something.

Loan 1,000.00
Interest 6%
Monthly
Startup Fee: 100.00
Payments: 24 @ 48.75

The new target for the date of the original loan is NOT 1,000.00. One might THINK it is 1,000.00 + 100.00 = 1,100, but that seems not to be the case. For some reason, the target is not quite that much. There is an additional discount of (100.00/1,000.00) * 100.00 = 10.00 (My first guess, anyway). Thus, we see:

PV of 24 monthly payments of 48.75 @ effective annual 17.9106% is 1,000.00 + 100.00 - 10.00 = 1,090.00

Point #4 Let's just see if we can guess with all we've learned:

Loan 1,000.00
Interest 6%
Monthly
Startup Fee: 50.00
Admin Fee 15.00
Payments: 24 @ 61.54

PV of 24 monthly payments of 61.54 @ effective annual 49.5514% is 1,000.00 + 50.00 - (50 * (50/1000)) = 1,047.50
Sadly, our Swedish friends say 51.56%!

In other words, we need to figure out that additional discount. Why isn't the PV target just the original loan value increased by the startup fee? What is the nature of that adjustment? Making the PC Target less increases the Effective Interest Rate.

We're almost there!