Compound intereset calculators - different results.

What do you mean by "flat rate"? Do you mean the mortgage rate rather than the APR?

The above sounds right if you repay the loan monthly, and your lender gives you immediate interest benefit for every overpayment (I think most do now, but some may still operate on an "annual rest" basis which means overpayments don't reduce the interest you pay till the end of the year).

No, the above assumes the whole lump sum is repaid at the end of the 5 years. If you borrowed 8000 at 5.19% and made no repayments/overpayments at all, the

In message , listerofsmeg writes

Mr R Raygun will be along in a minute. Whilst we wait whilst he oils his slide rule and cleans his new varifocals, I would point out that IF the rate is 5.19% FLAT then the payment would be £167.93, and compound interest doesnt come into it. FLAT means ((5.19 x the number of years x the amount borrowed /100) + the amount borrowed)/ the number of months.

Just compare the mortgage APR with the alternative loan APR.

Well, £151.67 is the payment which corresponds to an *actual* (as opposed to *flat*) rate of 5.19% per year: 5.19%pa = 0.4325%pm, P = Ar/(1-(1+r)^-n) = £8k*0.004325/(1-1.004325^-60) = £151.67.

But 5.19% flat for 5 years means 25.95% in all, and so the total repayable including principal is 125.95%. Over 60 months that's a bit under 2.1% per month, or £167.93 for £8000.

What do you mean *un*fortunately? Flat rate advertising shouldn't be allowed any longer. In fact, it isn't. Is it? Well, it certainly isn't allowed *not* to advertise APR.

Well, that just goes to show what to expect when using other people's calculators, especially web-based ones, especially if you just type in numbers without understanding what they bloody well mean. There's no substitute for doing the arithmetic oneself. It's terrible that people don't find out why they went to school until many years after they leave.

You scurrilous rogue, you. I'll have you know that I'm nowhere near old enough to succumb to the marketing blurb extolling the virtues of varifocals. I'm still not fully convinced that going for bifocals was a good idea, and there are times when I feel more comfortable with my old unifocals. Varifocals? No Way! Not Never, Not No-How!

Oops, yes, that's what I meant. I obviously have my terminology wrong. Have pity on the newbie!

Aha! I see now. Many thanks for clearing that up.

ps. I hate working with APR 'cus I don't understand what it is! As far as I am aware it takes into account other costs such as "arrangement fees" etc to allow a direct like for like comparison. This is great for comparisons, but not so great for working out actual repayment figures. Unless I've got it wrong that is! (More than likely)

Pretty much, but it also takes into account the frequency of interest payments. The APR equals the rate if there are no fees and interest is paid once a year at the end of the year.

With a mortgage of course the interest is paid monthly, so a 5.19% mortgage rate is really a 0.4325% monthly rate, which when compounded gives an annual rate of about 5.32%...this is the APR if there are no fees.

Except the mortgage APR will probably be next to useless, as it'll include all sorts of fees which may not be relevant to the extra borrowing, and if he's on a discounted rate it'll reflect the rate over the whole term rather than just the current rate.

In message , Andy Pandy writes

Not quite. It is that rate of discount which, if applied to a discounted cash flow comprising all of the payments made under the loan agreement, gives a net present value equivalent to the amount borrowed.

Its the *if interest paid once a year* bit that is wrong. That is the definition of AER.

Only with some loans.

I'm sure you're right, but doesn't it work out to the same thing when there are no fees?

If I took out a loan where interest was charged once a year at the end of the year and there were no fees, wouldn't the APR always be equal to the interest rate charged?

But isn't AER equal to APR if (as I specified) there are no fees?

... "Andy Pandy" wrote

Hmmmm. Using the method in the first quote above, the monthly rate would be

Which do you prefer? ;-)

Yes, quote 1 was the result of some online compound interest calculator so it must have assumed 5.19% APR (or should that be AER), rather than a mortgage type loan where the quoted annual rate is really the monthly rate times 12.

No, because APR takes into accountwhen the actual payments are made, not when interest is charged. For example, many loan repayments are calculated on the basis that interest is charged once a year, and borrower payments are applied to the loan account once a year, but nevertheless payments are

But if you could find a loan deal where you *really* only have to make annual payments, and not monthly ones, then yes, I think you'd be right. The nominal rate would equal the APR.

The term AER is only used for saving, and APR is used only for borrowing, so to compare them is meaningless.

"Ronald Raygun" wrote

Surely not "meaningless"?

If you found a loan with an APR of 5.8% (say), and a savings a/c with an AER of over 6% - is it not worthwhile comparing the two, to deduce that you could make a little money there? [Assume non-taxpayer, or that the savings a/c is tax-free.]

Yes.

No. You imply that if the APR and AER were the same in this case, that this borrow-to-invest scheme would break even. Not so, unless the payment frequencies are the same. For instance, if you had to make monthly loan repayments (and they were applied monthly with interest charges applied monthly too), but savings interest were only paid quarterly (meaning that you had to fund two loan repayments each quarter solely by "temporarily" eating into savings capital), then even though APR®R, the scheme would make a loss, because the interest credited immediately after the third payment each quarter would not be quite enough to beef up the savings balance to lie level with the loan balance. Never take "equivalent" too literally.

Always nice to assume the unlikely! :-) Sadly, AERs do assume non-taxpayers, but alas for the majority of savers who do pay tax, it's not easy to compute a "net AER" from a gross AER without reference to payment frequency, which is a pity since it effectively renders the AER concept itself meaningless (exaggeration, read "less than ideal" -- rather like APR which is also less than helpful precisely *because* it includes charges etc and in many cases makes assumptions which are unlikely to be satisfied (not only the most obvious one that rates remain fixed for the term, but also, in the case of discount deals where the APR takes account of the total loan term, i.e. the discounted period and any lock-in period,

"Ronald Raygun" wrote

"Ronald Raygun" wrote

I believe so, yes.

"Ronald Raygun" wrote

Eh? Without any fees (as specified by Mr Pandy), the APR represents the annual interest rate for a particular (constant) underlying "force of interest" - does it not? And if the AER equals the APR, then the AER also represents the annual interest rate for the *same* underlying "force of interest" - does it not??

This same "force of interest" can be applied to cashflows at *any* frequency, or indeed even ad-hoc (irregular) payments... It doesn't matter when the interest is actually *applied* - what matters is the total amount of interest that has "accrued even if not yet been applied".

If the balance on the loan a/c and the balance on the savings a/c are equal at the start, with the force of interest applying to each a/c always being equal to each other (even if both vary in a similar manner), then the two balances (including all interest "earned even if not yet applied") will

"Ronald Raygun" wrote

By "savings capital", do you mean the money provided up-front from the loan? The loan co gives you a lump of cash, which you stash in an a/c (bearing interest at AER = APR of loan) and take money out of that a/c "as and when" needed to pay for loan repayments.

Imagine putting the loan proceeds into an a/c linked to an offset mortgage. All loan repayments come from one or other accounts linked to the same offset mortgage. If the AER and APR really *are* the same, then at the end of the loan period the offset mortgage balance will be the same as it would have been if no loan had been taken.

"Ronald Raygun" wrote

If all accounts have the same underlying interest rates, it should be perfectly the right amount!

"Ronald Raygun" wrote

Offset mortgages are, I understand, getting quite common nowadays. Any linked account works to the similar effect as receiving interest tax-free, at the mortgage rate.

"Ronald Raygun" wrote

Even tax payers can have tax-free accounts - offset mortgages are one type, as are ISA's ...

"Force of interest"? Cor. I like it. A nice woolly concept. But yes, I'd agree that it does. But don't forget that APR is based on the assumption of a rigid cashflow pattern.

Not quite. AER also assumes a cashflow pattern, the default assumption being that a lump sum is invested at the start of the interest year and no withdrawals or further deposits are made other than at year boundaries. In effect, this means that if you put £1000 in at an AER of 6.0%, then you will have exactly £1060.00 at the end of the year, no matter what the payment frequency is. It also means that if the payment frequency is N times a year, you can compute the actual interest rate, as opposed to any of the woolly ones, as (AER+1)^(1/N)-1, and that (1+qir) = (1+mir)^3, where qir and mir stand for quarterly and monthly interest rates, respectively.

But if you introduce a pattern of withdrawals or further deposits, everything changes *unless* the AER was calculated taking them into account, which generally is not the case.

Yeh. But if the mass is different, the same force will give a different acceleration. :-)

Not so. Accrued interest does not compound before it is applied, and this creates a shortfall. See below.

No, they will not.

If you have monthly loan repayments of (say) £1000 to meet, and if you

This is because after month 1 you withdraw £1k, and after month 2 you do the same, and so at the end of month 3 you will be paid interest which has been accruing on a mean balance of £249k: £250k for the first month, £249k for the second, and £248k for the third. It will transpire that £(250k/3 + 249k/3 + 248k/3)*qir is some 3p short of £3000.

Yes.

Not a good counterexample, since in such scenarios the credit and debit interest frequencies cannot be different.

Now, now. "Underlying"? That's one woolly term too many!

In message , Andy Pandy writes

No the method of application of interest also matters

Likekly, it would be so, so long as the interest was applied on the last day of the accrual period and any changes of balance were refelcted in the daily interest that accrues.

No, see above, but I take your point.

I like it too. I've been grappling with this mathematically and you are of course correct...the payment frequency does matter. Not much, but it does matter (proportional to the interest rate cubed).

But I've been puzzling over why, conceptually, when the "force of interest" is the same.

But then I realised that although overall the "force of interest" is the same, it does actually vary for each account according to how close to the payment date you are.

Consider a savings account which pays interest annually on 31 Dec. If you invest

So, if you take out a loan which has to paid monthly and put the cash into a saving account which pays quarterly (APR®R), and fund the loan repayments from the saving account, the saving account balance will vary and will be lower when you are closer to the saving account interest payment date (when the "force of interest" is at it's highest), and higher just after the payment date (when the "force of interest" is at its lowest). So you'll lose out.

But sometimes is - and this explains an issue ISTR we were discussing a year ago. The Halifax were offering a regular saver account at a nominal interest rate of 6% but a quoted AER of 6.05%, interest being paid annually at the end of the year. How can the AER differ from the nominal rate when interest is paid annually was the cry (from me amongst others).

But it now becomes clear. Based on the "force of interest" being greater the closer to the payment date you are, with a regular saver account where interest is paid annually, you will be better off than if interest was paid monthly at the rate of(1+i)^(1/12) -1, because the "force of interest" is greatest when the balance is the greatest.

And it ties up - I cheated and used a spreadsheet - using a monthly interest rate of 0.4907% (1.0605^(1/12)-1) you get the same result as using an annual rate of just 6%.

That's not the problem. The compounding is taken care of, the problem is uneven "force of interest".

Sorry, in what units do you measure "matter"? :-) What precisely do you claim is proportional to the cube of the interest rate?

I don't know whether it would be fair to say that. The *effect*, in terms of actual interest earned, is greater when you invest in December than when you invest in January. But this effect is a consequence of timing your investment in an environment of *equal* "force of interest". Unless someone defines what exactly this woolly term is taken to mean, we're condemned to wander aimlessly around the Tower of Babel.

Well, the real reason you lose out is that, were the application frequencies the same (both monthly), then for AER=APR the two monthly actual rates would also be the same, *and* the average balance on which interest is earned would be the same as the average balance on which it is charged, and therefore the interest earned would equal the interest charged and so the savings balance would always exactly track the loan balance.

But when the savings schedule changes to monthly, the average balance on which interest is earned is less, and the fact that the quarterly interest factor is the cube of the monthly factor (because the AER is unchanged) is not enough to compensate for this. I think this is a simpler explanation than yours. It doesn't matter where the "holes" (the lower balances) are so much as the fact that they exist at all.

Exactly, and I almost mentioned this particular account in my post. The reason they calculate it the way they did is because the terms of the account make that particular cashflow pattern mandatory.

By that reasoning, would you say that if the terms of the account were that you must have a decreasing rather than increasing balance (weird, I know, but just suppose you had to invest a lump sum and withdraw a twelfth of it each month), then the AER would work out less than the nominal rate? Do try it - I haven't and won't - I await your report.

For shame!! :-)

Yes it is the problem. Woolly concept meets cold hard fact. That accrued interest is not compounded before it is applied is the real reason behind the airy-fairy perceived unevenness in your "force of interest".