Compound intereset calculators - different results.

"Ronald Raygun" wrote

"Woolly"??!! Now, now, Ronald - I thought you knew about the theory of interest?

I wouldn't call the natural logarithm of the value of one plus the interest rate [ =ln(1+i) ], to be "woolly" -- it is perfectly well-defined!! [Do you need to hit the books again?? ;-) ]

"Ronald Raygun" wrote

... and in that case the 'force of interest' is close to 5.826891%. That may be paid as 6.000000% yearly, or 2.956301% half-yearly or 1.467385% quarterly or 0.486755% monthly or ...

But *whichever* payment frequency you have for the interest (eg any of the 4 mentioned above, or any other), the AER is *always* 6.00% and the force of interest is *always* 5.826891...%.

"Ronald Raygun" wrote

Ah, I see where you are getting confused - you are calling the rather woolly concept of nominal rates of interest, "the actual interest rate" -- rather than the more precise notion of the 'force of interest' (and the annual interest therefrom, which is always equal to the AER).

An account paying interest yearly at 6.0000% is the same (to me) as an account paying interest quarterly at 1.4674% or an account paying interest monthly at 0.4868% -- they are all accounts with a force of interest equal to 5.8269%, and an AER of 6.00%.

"Ronald Raygun" wrote

If the force of interest remains constant, then (I believe) the AER will be constant as well - regardless of the pattern of withdrawals or deposits.

"Ronald Raygun" wrote

Your sentence above is non-sensical - mass and acceleration have no place in the mathematical theory of compound interest!!

"Ronald Raygun" wrote

I beg to differ. Have you never heard of "compounding daily"? The interest is not applied to the a/c every single day - perhaps just monthly. But it *does* compound every single day, so in that case for around 29-30 days out of every 30-31 days, "accrued interest *does* compound before it is applied"!!

"Ronald Raygun" wrote

Try running your figures again properly!! [See below.]

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"Ronald Raygun" wrote

Yes, they will! [Panto season again? ;-) ]

"Ronald Raygun" wrote

You don't have to meet then out of the savings interest - you can pay them out of the capital provided by the loan proceeds...

"Ronald Raygun" wrote

Fortunately, you will also be *charged* interest for that 3 month period on a mean balance of a little less than 250K - because you paid off some of the loan capital with the first 1000 payment (not all of that payment was interest!), and paid off a little more loan capital with the second 1000 payment as well.

I'll let you re-work your figures to see how much of the 3x1000 payments was actually interest, to compare with the interest received from the savings a/c ...

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"Ronald Raygun" wrote

Of course they can - the loan company is not going to change their interest frequency to match the offset mortgage, and likewise the offset mortgage company will not change their interest frequency to match the loan! They will therefore only be the same by coincidence, and could easily be different.

"Ronald Raygun" wrote

If you prefer, I could say "If all accounts have the same 'force of interest', and hence the same AER". I just thought using the word "underlying" helped the flow of the sentence!!

"Ronald Raygun" wrote

Ronald, you either need to read my other post - or a basic textbook on the mathematics of compound interest!

The force of interest is not a "woolly term" at all, but precisely defined. Look it up!

"Ronald Raygun" wrote

Absolutely NOT! - The force of interest should be assumed to be constant.

"Andy Pandy" wrote

If that were true, then the AER for deposits in January would be lower than the AER for deposits in December, wouldn't it?

But that's the point. The force of interest doesn't remain constant for most savings accounts, since interest does not tend to compound before it is paid. AER is usually based on the assumption that the balance stays constant (yet is a reasoanble approximation where it doesn't). The force of interest will then vary according to how close to the payment date you are.

But I don't think any of the high street banks actually compound interest daily on standard accounts. Try opening two identical accounts and deposit the same amount of money for the same number of days, but one just after the interest payment date and one just before. Bet you'll get the same interest from both.

Strictly speaking, yes. But if they don't compound interest daily then the AER can only be an approximation for accounts that allow random deposits and withdrawals.

Heh, heh, arithmetic is easier than mathematics.

Of course. Just testing.

It is actually perfectly clear why this is so.

The higher nominal rate, i.e. the fact that 4*qir is more than 12*mir, is due to the fact that (1+qir) must equal (1+mir)^3. Therefore qir = 3mir + 3mir^2 + mir^3. [I'll refer to this below]

Therefore qir exceeds 3mir by 3(mir)^2 + (mir)^3.

This excess is only just enough to compensate for interest being credited less frequently *when the account balance remains constant during the whole quarter*. It follows inexorably that it cannot be enough to compensate when withdrawals *are* made, even if they're only borrowed and paid back later (without interest). What matters is that the temporary removal of these borrowed funds denies them the opportunity of earning interest. Hence the shortfall.

Are you saying observations don't qualify as explanations? Observation *is* one of the simplest forms of explanation.

No, it's the lower balances.

No, balances are never higher. OK, they're higher in the 1st month than they are in the 2nd and 3rd months, and higher in the 2nd than in the 3rd, but they're never higher in the quarterly account than in the monthly account.

See below.

Yes, well, I had every confidence in your prediction being true. Actually I lied, I did try it, and got the same result 5.947507%.

All right, then. Let's look at the account balances of the monthly and quarterly accounts. Let B be the initial credit balance at the start of month 1, and P the monthly withdrawal to feed the loan. Let mir and qir be the monthly/quarterly interest rates as before, and mif and qif the interest factors, mif=1+mir, qif=1+qir.

Monthly account:

During month 1: B During month 2: B + B*mir - P = B*mif - P During month 3: B*mif^2 - P*mif - P During month 4: B*mif^3 - P*(mif^2 + mif + 1)

Quarterly account:

Month 1: B Month 2: B - P Month 3: B - 2P Month 4: B - 3P + qir(B + B-P + B-2P)/3 = B + qir*B - 3P - qir*P which is equal to B*qif - P*(2+qif).

To see whether there is a shortfall, compare the month 4 balances. They both feature the same factor involving B (we know mif^3=qif) so we can cancel that out, and the remainders both involve a factor of -P, so the quarterly account will be short of the monthly account iff (2+qif) exceeds (mif^2+mif+1).

Replace qif and mif by 1+qir and 1+mir, and multiply out mif^2, and the condition becomes:

3 + qir > 1 + 2mir + mir^2 + 1+mir + 1

or (subtracting 3) qir > 3mir + mir^2

Substituting qir = 3mir + 3mir^2 + mir^3 from earlier, and subtracting

3mir, we see the condition would be satisfied whenever

3mir^2 + mir^3 > mir^2 or (subtracting mir^2) 2mir^2 + mir^3 > 0

which is clearly always the case (except when mir=0 or mir

"Andy Pandy" wrote

OK, that maybe true about the *published* AER - but my earlier comments were meant to be for the case where the loan APR and the *true* savings AER are equal (not necessarily the published one, which as you have pointed out may be incorrect for the particular scenario being considered).

BTW - what happens if, with one of the accounts where investing 1000 on January 1st and withdrawing it all on February 1st doesn't pay interest until December 31st, you actually closed the account on February 2nd? Do they pay you the interest then - and if so, do they reduce it for early payment? :-(

"Ronald Raygun" wrote

OK, so for the 'Monthly account', the AER is (mif^12-1) [which is also equal to qif^4-1].

"Ronald Raygun" wrote

You cheat! The AER on your 'Quarterly account' is not (qif^4-1) !! No wonder you have a (fictitious) shortfall -- you are using a lower interest account for saving than the loan account!!!

Note if the AER were the same, it would also be (mif^12-1), and then your "Month 4" balance above would equal B*mif^3 -P*mif^2 -P*mif -P, which is the same as for the 'Monthly account'...

Can you try again with a savings account with the *same* (or better!) AER as the loan account??

Yes it is. No cheating. Of course my quarterly account is one with what would be *published* as the same AER.

In the real world we can only work with accounts with real rules, which in this case are to pay mir*X per month and qir*Y per quarter, respectively, where X and Y are the mean balances during the immediately preceding month/quarter.

Naturally, the AERs published will assume a typical one-off lump-sum pattern, and will be technically invalid if you operate a pattern of regular withdrawals in the middle of interest periods. However, it should have been clear that by "a quarterly account with a given AER" I meant one which would be so described in the absence of any constraints on flow pattern.

Books? Blimey! This interest stuff is all pretty simple, you know, to the extent that I wasn't aware it needed a fully fledged theory all of its own. To be sure, "force of interest" is not a term I've ever come across, I really thought you were trying to use an emotive term, a bit like "force of destiny" (which is more than an order of magnitude more popular on Google, btw).

No, what I call "the actual rate" has two essential ingredients, namely the rate of interest and the period to which it relates. There's nothing woolly about it, it's what happens in the real world.

Now, in order to be able to make meaningful comparisons between rates which are related to different-size periods, it makes sense to map the two-ingredient interest rates onto a one-dimensional scale. One such mapping is to standardise on the year, so that a rate of X-1 per trimester maps to a rate of X^3-1 per year, etc. This is very useful and very practical, and is normally what the AER means.

Another possible mapping is to use your fanciful force of interest, which is, AINUI, a notional instantaneous rate when compounding takes place continuously as opposed to discretely. One source specifically related it to the gradient of the value curve as a function of time. Well, in the real world this gradient is only ever 0 or pointwise infinite, so the best you could do is consider the average FoI for discrete intervals during which nothing else happens.

Even to define the FoI for each of the three months of a quarter, where no interest will actually be paid until the end of the quarter, is daunting. How do you do it? Do you take the NPV at month 1 of the interest you'll get in respect of month 1 at the end of month 3, using the same interest rate for the NPV calc as you'll end up with?

Amen to that, but that suggests that the AER and FoI are independent of the account balance, since the account itself only "knows" to pay a certain rate on the mean balance of the last period. So AER necessarily depends on the inflow/outflow pattern, but does FoI?

Excuse the flippancy, but neither has force.

Bull. In such a scenario the interest *would* for all intents and purposes be "applied" in all but form, the only reason it's not shown is to avoid cluttering up the account log with an excess of interest transactions. The effective position would be indistinguishable from one in which it were in fact explicitly applied.

But when we talk of a quarterly paying account, in the present context, we mean that interest in fact compounds quarterly and is applied quarterly, but accrues daily (or even continuously for all I care) so that the figure which is multiplied with the quartely rate for the purpose of computing how much interest to apply at the end of the quarter is the mean (daily) balance excluding accrued interest, averaged over the whole quarter.

They were already proper.

True, but it's slightly simpler to think of them as met from interest in the monthly case, but needing to dip into capital in the quarterly case, and it doesn't really lose generality.

That's irrelevant. The loan is a standard repayment deal requiring a fixed number of monthly payments of £1000.

It wouldn't be difficult to work out, but would not be of any direct relevance here. It would simply serve to show, if you take the sensible position that capital must balance capital and interest must balance interest, that if there's a shortfall it must imply that the effective savings rate for the quarterly account is too low. Hence a given published AER leads to a smaller effective AER if our peculiar flow pattern is employed. But we already knew that, so I'm not sure what point you're trying to make.

Yes, qir exceeds mir by an amount which reflects the compounding of mir.

No. They aren't earning interest, but the interest they aren't earning is the interest on the interest that is paid to the loan account (mir), ie the compounding of mir - which is reflected in the difference between qir and 3mir!

No. the shortfall is caused by the variable force of interest, see below.

No, it's the timing of them.

Calculating the interest paid in the way you have (and I believe the way banks usually do it), by averaging the balance and applying the qir to this balance, the "force of interest" will vary from qir/qif at the start of the quarter to qir at the end of the quarter when interest is paid, averaging ln(qif).

The balance is higher at the start of the quarter (when the force of interest is lower), than at the end of the quarter (when the force of interest is higher).

*This* is what causes the shortfall, not the mere fact that the balances are lower.

To prove this, consider what would happen if the loan required quarterly repayments and the savings account paid interest monthly. The savings account would then have higher balances in months 2 and 3 than the loan account. So you'd expect to make a profit? If lower balances make a loss then higher balances should make a profit surely?

Well...you wouldn't. You'd break even. Do the maths if you don't believe me. Because now, although you still have varying balances, you don't have the situation where the force of interest is lower when the balance is higher, and vv.

Another example. Go back to the original quarterly savings account/monthly loan account, but assume a flexible loan (eg an overdraft) where interest is charged monthly as before, but you can make payments whenever you want.

Now, instead of keeping the loan balance constant you keep the savings balance constant, and pay the loan account quarterly with the savings interest. Again, you'd break even, for the same reason.

Yes, that's what I meant.

Had the Halifax account been an ordinary savings account which paid a nominal 6% annually, they would have quoted the AER as 6% not 6.05%. Yet you would have got exactly the same return from it with the same deposits, because the way you make the deposits affects the "real" AER.

Similarly, the way you are using the quarterly savings account in your example, the balance is decreasing up to the point of interest payment so it's a bit like the "reverse Halifax" account above where the real AER is less than the nominal rate. So the "real qir" will be less than the qir worked out from ("quoted AER"+1)^(1/4) -1.

*That* is the reason for the shortfall.

It is disingenuous to refer to a "real qir" here. There is only one real qir, and it's the actual rate paid each quarter on the mean quarter's balance. The qir is not "worked out from" the quoted AER, except in our contrived example of having two accounts, one monthly , one quarterly, with the same (quoted) AER. In practice the AER is always derived from the actual rate and the period to which it relates, together with the flow pattern which in the majority of cases is that of a one-off deposit made on an interest year boundary.

No, that's confusing cause and effect. The fact that the real AER is less than the published AER is a *consequence of* the shortfall, not a reason for it. More correctly, the shortfall *and* the reduced AER are both consequences of the unorthodox outflow pattern.

Yes, if the true APR equals the true AER then I believe you must break even.

I think you get it without it being reduced. I've worked it out on that basis when I've closed accounts, and got the correct result.

So you get more interest if you close and reopen your accounts every month :-)

Yes, which in this case will be less than ("quoted AER"+1)^(1/4) -1.

In other words, a guess that is only valid for a particular set of deposit/withdrawal rules.

Irrelevant as they are both the consequence of uneven force of interest.

More correctly, they are both the result of an unorthodox outflow/inflow pattern

*which results in the force of interest being higher for lower balances and vv*. As I've been trying to tell you in last few posts. At least you now seem to accept it is the *timing* of the holes in the savings account rather than the holes themselves which causes the problem.

Ronald, I agree with Andy. [Just thought I'd let you know!]

"Andy Pandy" wrote

Alternatively, you could simply keep the same 12 accounts open all the time :-

Call an account opened on January 1, the "December account"; one opened on February 1, the "January account"; one opened on March 1, the "February account" and so on (the reason for this naming should become clearer later). Now, assume you receive your interest in the December account on December

31, and similarly for the others.

Simply keep *all* your money in the "January account" throughout the month of January, until you receive the interest. Now transfer all of that (including the interest) into the "February account". After you get the interest in that a/c on February 28/29, you transfer the entire contents to the "March account" and so on.

"Ronald Raygun" wrote

There you go again just considering the notional balance on the account - and ignoring the "interest accrued but not yet credited".

Surely it is the *cashflows* in and out of the account(s) which are the important things to consider - any "balance figure" quoted by the bank is irrelevant, and indeed often banks will quote more than one "balance" figure! [Eg: balance without accrued interest; balance with accrued interest; balance including an overdraft; balance allowing for *all* your current, savings & mortgage accounts in your offset mortgage ...] The true "value" of the a/c is always "what you can get out of it" - rather than any other number which the bank might quote to you at any particular time.

So - assuming that, if you closed the a/c, the bank would immediately apply any "interest outstanding" (accrued but not yet applied) - then the account would be 'worth' this higher figure to you (rather than the lower notional balance).

There could only be one problem - if you wanted to make a withdrawal which reduced the "true" balance (including the "not-yet-applied" interest) to a figure below the "not yet applied" interest, but above zero. That's awkward - but you could always close the a/c, withdraw the lot (including the now-applied interest accrued) and re-deposit the slight remainder you wished to be left behind...

Hence - the force of interest will only be zero if, in the case where you closed the account just before the normal interest payment date, you wouldn't actually receive that interest. Are you saying that is true??

No it won't. How so? The quoted AER, being based on the premise of no flow constraint, assumes a one-off lump sum investment. So if the account actually pays 3%pq then the bank will publish an AER of 12.55%.

[Sorry, perhaps I should have said "published" instead of "quoted", since I did not mean to imply that a personal quotation had been given based on a specific planned flow pattern]

Yes, and specifically in the absence of any rules constraining deposit and withdrawal patterns, the default "guess" is that the pattern involves a single deposit at the beginning of the reference year, and no withdrawals, as above, i.e. an account paying m% n times a year will be described as haveing an AER of (((m/100+1)^n-1)*100)%.

I'm trying to get to grips with this woolly concept, and I see a problem here. I feel it is wrong to say that the force of interest is the result of a particular flow pattern.

Is not the force of interest, for any given interval, a property of the interest crediting rules, and of how the interval in question relates to the interest calculation reference period, independent of external influences on the account balance? And is not the true AER a result of combining the (possibly unorthodox) flow patterns with the respective forces of interest which themselves are independent of the flow but depend only on the sub-intervals into which an existing, larger, known-FoI interval has been divided?

Let me illustrate what I mean. Put £1000 into a quarterly account and let there be no other flows during the quarter. The account's qif is

1.03, so at the end of the quarter you'll have £1030, being the original capital plus £30 interest, each £10 of which are attributable to one of the 3 months in that quarter. The FoI *for the quarter* is [let me see if I have it right] Q=4*ln(1.03).

Q = 11.8235209%

But although the balance is the same in each of the 3 months, the forces of interest for each of them (call them F, S, and T, for 1st, 2nd, 3rd) are not all equal to each other and to Q, but are such that the appropriate average of F, S, and T is Q, and also such that F