interest rate maths help....

"NC" wrote

monthly interest rate ?

0.585% ?

It could be anywhere between about 0.577% and 0.589%, I guess.

Provided the 36 payments are equal and are the only payments being made, the APR is worked out by compounding the actual monthly rate (which is normally a twelfth of the nominal annual rate - the one that's usually quoted), and then rounding.

The rules used to permit rounding down, so a compounded rate of

7.299% could be called an APR of 7.2, but if the rules have changed (have they or is that just wishful thinking?) to require rounding to the nearest tenth, then the compound rate must be between 7.15% and 7.25%, corresponding to monthly rates between 0.5772% and 0.5850%. If they haven't changed, then the CR must be between 7.2% and 7.3%, corresponding to monthlies between 0.5811% and 0.5889%.

The calculation you want to perform is to add 1, take the 12th root, and take away 1 again. In simpler words, raise 1.072 to the power of 1/12, which gives 1.005811.

Uh...does this help?

Possibly not...

In message , NC writes

Despite the arithmetic accuracy of Ronald's post, the real answer is that you cant tell unless you know the interest accrual period. I think Ronald is assuming monthly compounding, but it could be quarterly half yearly or annually.

Most places calculate interest daily, don't they? I know most credit cards do. 5000 plus one year's interest at 7.2% is 5360. If the interest is calculated monthly, then the closest I got to match (5360.01) was a rate of

0.5810809% per month. If calculated daily, at a rate of 0.01905% per day (5359.999). Multiply by the number of days in a month (30.41) and you get 0.579438%.

Take three decimal places or you're talking about pennies, +/- 0.0001% per day being about +/- 20p in total over the course of one year. +/- 0.001% per month is about +/- 60p per year.

Correct me if I'm wrong please, just a few minutes with Excel.

Marcus

In message , Marcus Fox writes

Yes, that wasnt my point though.

Here you are assuming that the accrued daily interest for the month is being capitalised (i.e. compounding) every month. From the info supplied we dont know that. It could be capitalised monthly, quarterly, half yearly or annually. For a door to door loan shark it even be weekly!

However, many banks and building societies, only having old-fashioned steam powered clockwork computers, simply divide the yearly rate by

1. But they don't tell thir staff this. They make them log onto company web sites to do the arithmetic for customer queries, so when the network is down, they can't even divide by 12, even though they have both a computer and a financial calculator on their desk.

I don't think kids are taught how to divide by 12 at school any more. Something to do with decimalisation and having calculators. In the old days before calculators even schoolkids were taught compound interest and 12th rooting. Taking the 12th root of a number and doing compound interest calculations is now regarded as university level maths.

Hmmmm - you are assuming that the instrinsic annual rate is 7.200%. What if it is (eg) 7.248% ?!

Which is completely correct if you start with the 'nominal' rate instead of an APR. Banks & Bsoc start with the nominal rate from which the APR is calculated, not the other way round. The age of their computers is irrelevant.It only appears to be in this group where there is a belief that monthly payments are derived form APRs.

And run the risk of giving a non compliant quote. APR calcs are obligatory and can not be left to a chance input error on a calculator and if there are any charges involved then I defy you to do the calc quickly ESPECIALLY if the interest is applied quarterly on a day which is not the day the monthly payment is received. The an NPV calc may be required based on a cash flow of all the payments made.

Duh. Then you just tap 1.07248 into your calculator instead of

1.072 before doing the raising-to-the-power bit.

Do keep up. The main part of my post identified two possible ranges of APR and the ranges of monthly rates which, based on my usual assumptions, correspond to them.

The last part was only an illustration of how to do the calculation, as a guide for those who don't know. This guide used one sample value which, yes, happened to be exactly 7.2%. I thought repeating the illustration with different values would make me look a prat, so I desisted, forsooth. But no doubt some in our esteemed audience will be eternally grateful to you for pointing out the obvious, in case it wasn't.

My apologies. I didn't expect anyone to take my joke about clockwork computers seriously.

I think the problem derives from the way APRs are calculated in the first place. They were invented to provide a general figure which would be comparable across all company advertising literature. People naturally believed that a simplified easy to understand comparable yearly interest figure would behave according to the normal compound interest rules, e.g., take the 12th root to find your monthly percentage and therefore interest payment.

The problem is that the APR doesn't work like that. I've never understood why not, i.e., why they didn't decide on a definition of an APR which *did* behave like that.

Yes, I've looked up the horrible arcane computations by which APR is derived. I don't know the motivation behind the details of the formula, but being familiar with the kinds of compound interest approximations that banks etc. often used to use in the old days of clockwork calculators which couldn't do roots, it looked to me as though what they had done was produce an APR formula which was compatible with the old clockwork algorithms which were at that time enshrined in banking bureaucratic procedures.

For all I know they still are. In fact, the relatively recent arrival of such financial devices things as offset mortgages and interest calculated on daily balances suggests to me that since banks etc. have had the computational power to do that kind of thing since at least the 1970s, it wasn't lack of computational power that precluded such devices, but the fact that banking accounting procedures had enshrined and ossified the old algorithms of the clockwork days, and it's taken the intervening decades for the bureaucracy to adapt to the kind and frequency of calculations that computers long ago made possible.

Of course. But I've had bank managers (ok, customer liaison officers) unable even to give me approximate ball-park non-binding guesses while their computers were down, not because they couldn't calculate a secure APR (which I wasn't asking for), but because they actually didn't know how to do compound interest calculations. That this was the problem was made clear when I offered to do the calculations for them, and we proceeded informally and happily on that basis. In one case the manager insisted that I used his expensive Hewlett-Packard financial calculator, instead of my cheap Casio, because he'd been issued with the HP (which he didn't know how to use) while being warned that some cheap calculators didn't do the proper financial calculations.

You're perfectly correct. But I wasn't asking for APRs, nor calculations which were exact, or contractually binding. I was asking for quick approximate back-of-envelope calculations which would allow us to continue exploring the approximate (and considerably different) costs of various options while the computer was down, and I was taking full responsibility for being misled by these approximations. I knew that the degree of inaccuracy introduced by such approximation was at least an order of magnitude smaller than the differences between the figures of the various different options I was comparing.

"Ronald Raygun" wrote

I suspect it would have been nice to see your post - but my server didn't show it.

It's annoying when that happens, so I repeat it below for your edification. The bit I'm not sure about is whether the rounding rules have in fact changed.

**END**

That ought to be obvious. In fact it *does* work like that, but only in those cases where the payment and charging schedules both coincide and their actual timings coincide with the theoretical ones. In many cases they do, such as where payments are applied on the same day they are collected and interest is also charged on the same day.

The problem arises when that is not the case, and in particular when account needs to be taken of payments which do not strictly form part of the interest regime but nonetheless form part of the cost of the loan and need therefore to take part in whatever process is used to come up with a "score" for comparing different loan products. How else do you compare two deals where one involves a slightly higher interest rate but the other instead involves an up-front fee?

Unfortunately, the APR rules don't even address that type of situation satisfactorily, because they make the unrealistic assumption not only that the interest rate will remain at its present level for the life of the loan (but that's OK if you assume that lenders are subject to the same external forces and if one changes its rate, chances are the other will too, and by the same amount, and if two products' APRs are already neck and neck, then it's likely they will remain so even after a rate change), but also that the up-front fee is amortised over the full term, whereas in practice it's more likely -these days- that a borrower might pay £350 for a 3-year deal and then switch products, and hence the fee should be amortised over the projected deal period, not the term.

Perhaps there *is* no fair way to compare loans when the parameters are too vague, and perhaps, then, the best way is to go for a straight compound rate and deal with the appendages separately instead of trying to incorporate them into what is essentially a nonsense ficticious interest rate.

Two things which make the official APR method less useful than it should be are:

(1) This is the minor one which is potentially easily fixed. Rounding to a tenth of a percent is all very well when interest rates are high, but in these times of low rates, the measure is too coarse. Two loans for the same amount over the same term, with the same schedule, both scoring an APR of 4.0, can involve actual payments differing by of the order of 2.5%, because the real interest rate can differ by up to 0.1%, which is 2.5% of 4%.

(2) This one is the killer. You can't compare loans by informal window shopping, because in many cases lenders are unable to publish their APRs, they can only publish a "typical APR" based on typical paramaters, which leaves you at a disadvantage if your parameters are untypical. This is particularly the case where fixed "application" fees apply, which are not scaled to the size of the loan, and will therefore contribute disproportionately to the APR for smaller loans.

"Ronald Raygun" wrote

Many thanks.

"Ronald Raygun" wrote

My reply to OP of 5.85% was meant to be in the upper range, so at least this is consistent!