Compound intereset calculators - different results.

OK, I think I can almost relate to that. Thanks.

The scenario you illustrate above is, however, not complicated enough to show the usefulness of the concept, since you can just get the interest over the time period by subtracting the balance at t=0 from that at t=1.

If we now look at a piecewise simple scenario such as the three months of the quarter, and assume that the FoI is constant over each of the 3 months but not over the whole quarter, then the values Q, F, S, T which I concocted in my previous message are valid, subject to the little detail of not calling them FoI but the value which the FoI function takes over the respective intervals. Right?

I meant "real qir" as in the case (like the Halifax account) where the AER is worked out (perhaps in retrospect) for the *actual* inflow/ouflow pattern rather that assuming a constant capital balance.

Yup.

As pointed out by Tim, the force of interest is not a wooly concept, it is quite simply the rate at which the account balance is increasing, if we include interest accrued but not yet paid. Just like a car can be travelling at 50mph at a particular instant in time, the FoI is the rate at which the balance is increasing at a particular instant in time.

It isn't! That's not what I said, see below..

Yes. It is not affected by the account balance at all. For a particular account with particular interest crediting rules, the force of interest at a particular point in time can be calculated. However the force of interest will vary according to the date, and even time to be pedantic.

For instance, for an account which pays 6% nominal, annually on 31 Dec at midnight. The force of interest on 31 Dec at 23:59 will be 6%, as interest will compound immediately. The force of interest on 1 Jan at 00:01 will be 6%/1.06 ie

The "true AER" is the result of mapping the flow pattern onto the FoI (which varies with time). If the balance is at its peak when the FoI is at its peak, then the "true AER" is higher than the "quoted AER" (if the quoted AER was based on no capital movement). And vv, as in the "quaterly loan/monthly interest" example.

Sounds right.

Sounds right, although F, S & T are also averages of the FoI over the particular month. Which is fine if the balance is not changing.

Sounds right.

All sounds OK, but I'm not going to check your maths.

Yes.

Yes.

Yes. And this is bit you need to remember (see below)....

I think you meant lower.

Yes.

Oh dear...

Yes.

Can you not see the flaw in this? You've lost out on interest in month 1 and gained it back in month 3. Remember the FoI is based on continuous compounding. Therefore you've lost out on the "compounding" of the interest you temporarily went without. That is why it makes no difference, despite you now having bigger balances when the FoI is greater.

Of course, since you are temporarily losing interest which is compounding at a higher FoI.

"Ronald Raygun" wrote

No problem!

"Ronald Raygun" wrote

OK...

"Ronald Raygun" wrote

Hold on right there - you can't just *assume* that something is constant, when it is *not*! The force of interest is already well-defined by the other features of your account&cashflow.

For instance, we already know that if the force of interest really *were* constant, then the balance as a function of time (effective balance, including interest accrued but not yet applied - ie the amount you'd receive if you closed the account at that time), would be something like :- B(t) = B(0) x (1+i)^t But your a/c doesn't work like that, does it?

Wouldn't your account's 'effective balance' (between deposits/withdrawals/interest) be more like :- B(t) = B(0) x (1 + (i x t)) [ie simple interest as opposed to compound] ?

So, the next question is: "How exactly does the force of interest in your case vary over time?" [Andy helped you with this earlier.]

Looking at the "text books" - it seems I got it the wrong way round. I said the FoI increases with time, the text books say the FoI *decreases* with time.

After much head scratching it seems the reason is the text books use the "effective balance", ie what the account is worth, what you'd get if you closed it. I used the actual (or perhaps "written") balance, ie the balance you'd see if you looked at your statement or logged onto internet banking.

The "effective balance" is given by the formula above. So if an account pays nominal 6% annually and you invest 100 then the "effective balance" after a month would be 105. This is what you'd get if you closed it. The "actual balance" would be 100.

So the "effective balance" for this account would be 105 after one month, 110 after 2 months, 115 after 3 months, etc. So it can be seen that the "effective interest rate" (and so the FoI) is actually *decreasing*.

This is because the interest earned is growing at 0% while the capital is growing at 6%, and as the ratio of interest to capital increases, the "effective interest rate" and so the FoI decreases.

The problem with the text book method is that it works fine where there's no capital movement, and the FoI can be defined by a formula. However it is useless where the capital balance is changing - as this will change the ratio of interest to capital and so the FoI.

My definition should work regardless of capital movement, since I simply pretend that the interest compounds immediately, and so define the FoI as the nominal rate reduced by the appropriate amount such that if compounded up to the interest payment date, it equals the nominal rate. This can be defined simply by reference to the nominal rate and how long till the interest payment date, capital movement doesn't affect it.

The one time this breaks down is if the account is closed early - here you'd get a bit more (as you have effectively brought the interest payment date forwards). In fact, this is a way you could actually profit from the "quartely saving, monthly loan, same APR/AER" accounts. Simply close the savings account after a month and pay off the loan! You'll be in profit by:

A(qir/3 - mir) ie A((mif^3-1)/3 - mir).

"Andy Pandy" wrote

Yes, of course.

"Andy Pandy" wrote

I haven't looked at my textbook on the subject for - ooooohhh - what must be over 15 years now -- but it's nice to know that I've not been talking bullshit here!

"Andy Pandy" wrote

But, when you then compute the amount of interest from a miniscule time period, divided by this balance and divided by the size of the miniscule time period - wouldn't that give a *constant* force of interest??!

"Andy Pandy" wrote

I'm hoping that your monetary figures above are all missing a '0' digit after the first '1'?? :-( [1000, 1005, 1000; 1005, 1010, 1015.]

"Andy Pandy" wrote

Yes, of course.

"Andy Pandy" wrote

The forces of interest applied to deposits made at different times, may be different in the particular type of account we're considering here (but only for those deposits made since the last interest application date).

This is why, eg if interest is applied on 31st December, a payment made on

"Andy Pandy" wrote

Errmmm - doesn't "interest compounds immediately" mean that under your method, the *actual* balance increases continuously using the formula B(t) = B(0) x (1+i)^t, between capital changes - and doesn't actually stay constant??

"Andy Pandy" wrote

The true (decreasing) force of interest can be defined simply from the nominal rate and the length of time *since* either the latest interest application or the time of the deposit (if no interest has yet been added for that deposit).

"Andy Pandy" wrote

The correct method doesn't have this problem!

No - because I apply a "discount" to the interest earned to reflect that it doesn't get paid until the next interest payment date.

Oops.

There's also withdrawals to consider. What if an amount is withdrawn before the interest payment date?

No, because the interest earned doesn't affect the *actual* balance. The actual balance is only affected by *capital* changes until the interest payment date. The interest earned to date is a separate virtual entity until the payment date.

But that means you have different FoI's acting on different bits of the balance depending on when the deposits were made. It also doesn't seem to be able to cope with withdrawals (other than closure) before the interest payment date.

For instance, how would the "true" FoI act on our "annual interest on 31 Dec" account, if you deposited 1000 on 1 Jan and withdrew 900 on 1 Feb?

No, but it has another that mine doesn't. If instead of closing the account, you simply empty it, the "correct" method will fail to account for the fact that you'll be waiting till the end of the year for your interest. Mine won't.

The "correct" method would seem to treat any withdrawal as a closure, whereupon the interest on the amount withdrawn is paid immediately.

"Andy Pandy" wrote

OK, so how do you then make use of this 'force of interest#2' which increases over time for this type of account? I mean, what do you use it for & how?

"Andy Pandy" wrote

That's not a problem - a withdrawal is simply a *negative* deposit - and so the force of interest is the same as for any other (positive or negative) deposit made at that time.

"Andy Pandy" wrote

Unfortunately, that is a feature of this type of bank account.

"Andy Pandy" wrote

Nope - it copes perfectly (see above & below).

"Andy Pandy" wrote

Well, assuming a nominal annual rate of 6%:- For the +1000 deposit on 1 Jan the force of interest starts at 6% on 1 Jan then decreases to around 5.66% on 31 Dec. For the -900 deposit (ie 900 withdrawal) on 1 Feb the force of interest again starts at 6%, but this time on 1 Feb, and then decreases to around

Note that, for the last day of the year, the interest earned on the 1000 deposit is 16.4p. This can be computed as either (1/365) of 6% of 1000, or something close to (1/365) of 5.66% (the FoI) of 1060 (the effective balance on that day). [I say "something close to" merely because the FoI applies to an *instant* in time, and not for the whole *day* (but a day is a sufficiently small proportion of the whole year as to get meaningful results without integration).]

Similarly, again for the last day of the year, the interest lost on the 900 withdrawal is 14.8p. This can be computed as either (1/365) of 6% of 900, or something close to (1/365) of 5.69% (the FoI) of 949 (the effective balance on that day).

In other words, the interest applied, for the very last day of the year, on the combined balance of just 100, is 16.4p - 14.8p = 1.6p.

This could be computed as either (1/365) of 6% of 100, or as (1/365) of

Happy? ;-)

"Andy Pandy" wrote

I'm listening ...

"Andy Pandy" wrote

OK...

"Andy Pandy" wrote

The correct method doesn't pretend to help determine the wishy-washy balance figure which you call "actual balance". What it *does* help determine, is the actual *value* of the account at any particular point in time. That value being the sum of your "actual balance" plus the 'accrued interest not yet applied'.

"Andy Pandy" wrote

It continues to, accurately, determine the true value of the account.

To work out the best place for your money at any instant in time!

For instance if you have 10000 which you want to invest from Jan-Mar, and you have one account which pay interest annually on 31 Dec, one which pays annually on 31 Mar, and a third account which pay interest monthly.

FoI#2 varies from i/(i+1) just after the interest payment date to i just before - so using this is should be blatently obvious that, given the same AER's, of the 3 accounts above the last one above is the best place to put your money.

More usefully, if the accounts have *different* AER's as well as payment dates, as well as perhaps payment frequencies, and you have 10000 to invest permanently, you could use it to work out that on a particular date of the year you'd be best transferring the money from one to another. You could include flexible mortgage accounts like my Nationwide one where (I believe) interest compounds daily.

I'm not sure of the formula to work out the FoI#2 on any particular date, but there's not really much point since the easiest way to do it is to use a spreadsheet (or perhaps a program).

Eg put dates of the year in column 1 and the FoI's of each account in the other columns, setting the FoI to the nominal rate on the payment date and for all other dates, the FoI equals *(1-/365) where x is the cell containing the

Duh, obviously. I should have thought of that.

Which could get very complicated with an account that has lots of deposits and withdrawals (eg a current account)!

Yes, I understand it now. But I still prefer mine :-)

"Andy Pandy" wrote

Ermmm - but the best place (at any instant in time) is where the *true* force of interest is highest!

"Andy Pandy" wrote

Eh? Let's see if I've got this right (using AER=6%) :-

Your 2nd a/c (with interest annually on 31 Mar) has FoI#2 = 5.66% around Apr

Your 3rd a/c (with interest monthly) has FoI#2 varying between 5.66% on the

So won't your **2nd** a/c have a higher average FoI#2 across Jan-Mar (not the *last* one), being always over 5.9% -- whereas your last a/c will have an average FoI#2 somewhere closer to midway-between 5.66% and 6.0% (5.83%), ie *lower* than 5.9%?

Of course, I'd say that *either* of the 1st or the 2nd accounts are better than the last one - because the true FoI lies between 6.0% and 5.9% for the initial three months (Jan-Mar) immediately after the deposit - but that the (true) FoI varies between 6.0%pa & 5.66%pa each month for the last a/c.

OK - let's see how much interest you *do* get:

A/c1 & A/c2 : 10K x 6% x (3/12) = 150. [Which you can draw out with the original 10K capital on Apr 1, after your period Jan-Mar.]

A/c3 : Monthly rate is 1.06^(1/12) - 1 = 0.486755% [to give AER=6%] Month 1 - 10,000.00 x 0.486755% = 48.68. Month 2 - 10,048.68 x 0.486755% = 48.91. Month 3 - 10,097.59 x 0.486755% = 49.15. Total interest = 48.68 + 48.91 + 49.15 = 146.74.

Check: 10K x 1.06^(1/12) x 1.06^(1/12) x 1.06^(1/12) = 10,146.74.

Looks like your 1st & 2nd accounts are equivalent, with the last account being worse. Where have I mis-understood your method?

"Andy Pandy" wrote

No, you need the *true* FoI to do that!

"Andy Pandy" wrote

You can do it *either* way for the true FoI..

Of course it gets complicated - because **the account** is complicated.

A deposit into any account will always have the true FoI equal to the nominal rate to start with. If you're moving money around you would need to work out the effect of the FoI on the withdrawal from one account plus the deposit in another over a possibly unknown time period.

Oops, sorry, I meant the second one.

Yes.

Yes. I have worked out the formula to be FoI#2 = i/(it+1) where t is the proportion of the year *remaining* till interest payment (for annual interest).

Not quite. If the AER's are the same, the monthly rate paid would be 0.4867551%. The FoI#2 (or indeed the real FoI AIUI) can never be greater than the nominal rate.

It'll vary between about 5.81% and 5.84% (use the same formula but using the real nominal monthly rate, then multiply by 12 to annualise it).

Yes, that's what I meant...

But with a/c 1, unless you close it you'll have to wait till 31 Dec for your interest.

Because I'm assuming these are all existing accounts which perhaps have other money in, which you don't want to keep opening and closing. You are simply looking at the situation on 31 Mar and what each account is worth if you close it on that date. See below....

That would be a nightmare. Do you want a go? Same 3 accounts, you want to invest the 10,000 for 30 years and get the maximum possible interest. You don't want the hassle of continually opening and closing accounts.

Using my method:

You can write a simple spreadsheet to work out FoI#2 for every day of the year with a "flag" column telling you when to switch (when the account with the biggest FoI#2 changes).

This tells me to invest in account 2 from 1 Jan to 31 Mar, then move to account

Now, how would you work that out using the true FoI?

In message , Andy Pandy writes

I dont agree.

A = annually @ 31/12. B = annually @ 31/3, C = Compounds monthly.

If all have an AER of 10% (say) then the amount in the account plus the accrued interest as at 31/12 must be £11000 in each case.

Working backwards using an inelegant spreadsheet (please dont scream at me) shows that the nominal rate needed for each of these scenarios is :

A = 10% (because there is no compounding at all)

B = 9.819 (because there is one compounding of interest at the end of the investment period)

C = 9.569 (because there are twelve compoundings)

The AER must assume that you leave the dosh in the account for a full twelve months.

If you were to close each account on the 31/3 then it is the first account that would pay out the most A = £10,250

B = £10,245.48

C = £10,241.11

No it ACCRUES daily. 'Compounding' is when the accrued interest is added to capital and then, itself, accrues interest.

I meant the second is the best, ie B below.

The actual AER, yes (which will vary according to the deposit/withdrawal pattern).

But the quoted AER will probably be the same as the nominal rate for A & B as it will usually assume a constant capital balance.

Yes, but I was talking about 3 accounts with same "quoted AER", where the quoted AER was based on the usual assumption of constant capital balance. Which means the nominal rates for A and B will be the same as the AER, and the nominal rate for C would be (AER+1)^(1/12) -1 per month.

In this case, the best place for the money from Jan-Mar is B (however it is not then the best place to then keep the interest earned on 31 March).

I did think it compounded daily, but I could be wrong...

In message , Andy Pandy writes

Accepted, but... seem my below

No (well not quite). The definition of AER does not vary with the transaction pattern in the way you describe. In fact, the requirements of AER quotations are such that if the AER were to vary according to the date when the deposit was made, then the quoted AER must also change. So, for example, in the case of an account that compounds in, say, June annually, or an account that has a 'bonus' interest period that expires in June, then every advertisement that quotes an AER between now and June must be adjusted as time passes towards those dates to account of the changing AER.

No. The only balance changes that can effect the AER are when the capitalisation of interest occurs during the first annual period. Contrary to your other assertions, other capital transactions make no difference to the AER.

No. The AER calc measures the benefit received by the borrower in one years time, even if the accrued interest is not actually capitalised (or compounded) on day 365. Therefore, when comparing the differing AER and nominal rates of different accounts, ANY account which compounds (or capitalises) during the year MUST mean that :

a) in the case of a constant AER, there is a lower nominal rate OR

n) in the case of a constant nominal rate, there MUST be an increased AER.

No. I am not getting involved in the pure maths, just its application in the definition of AER. IF the AER is the same for A, B & C then C must be offering the lowest nominal rate and A offering the highest.

Do you really think any banker has a daily interest rate which it raises to the power of 365?

Yes, though as I think you've since corrected yourself elsewhere, this is back to front. The FoI decreases from 6% on 1st Jan to 6%/1.06 on 31st Dec.

There are 3 simple models of capital growth:

For all of these consider the time interval t=0 to t=1, with an initial deposit of C, using a nominal interest rate r per time unit (meaning the balance including interest at t=1 will be C*(1+r)), and use the interest factor f as shorthand for r+1. So in all cases the balance increases from C to C*f.

(1) You postulate continuous compounding.

The balance B(t) = C*f^t, and consequently the interest earned by time t is I(t) = B(t)-C = C*(f^t-1). The derivative of this, dI/dt = C*f^t*lnf. The FoI is this derivative related to the balance: FoI(t) = (dI/dt)/B(t). That's C*f^t*lnf/C*f^t = lnf which is constant.

(2) You postulate incremental compounding at time unit boundaries, but taking the (necessarily linearly) accrued interest into account in computing B(t).

Clearly the interest I(t) is C*r*t, and hence B(t) = C+I(t) = C(1+r*t). The derivative dI/dt is C*r, which is constant. FoI is thus r/(1+r*t), which decreases from r to r/f.

(3) You compound incrementally as above, but discount the accrued interest, by scaling it by multiplying by f^t and dividing by f.

I(t) = C*r*t*f*(t-1) B(t) = C*(1+r*t*f^(t-1)) dI/dt = [need to use product rule] C*r*(t*f^(t-1)*lnf + f^(t-1)) = C*r*f^(t-1)*(1 + t*lnf) Divide this by B(t) and the expression gets a little heavy, and I won't bother even to type it out, but suffice it to say that it's easy enough to evaluate for the cases t=0 and t=1, and to show that it *increases* from r/f to (r/f)*(1+lnf).

If you plot the balances against time for each of these methods, it transpires not only that the exponential (model 1) balance is always less than the linear (model 2) balance (except at the very start and end of the interval, when they're equal), but also that the discounted linear balance (model 3) always lies below even the exponential balance.

Which model to use? The first is unrealistic because no real account actually compounds continuously. The second is what you and Tim have been favouring but it suffers from the drawback that it's not 100% kosher to consider the accrued interest as genuinely part of the balance, because it is ineligible for withdrawal prior to the next time unit boundary. The third tries to compensate for this by discounting, but it's not clear whether that compensation is adequate.

A fourth model, also simple, is to consider the balance constant, i.e. to disregard the accrued interest for balance purposes. One might call this the "real life" method. :-) Here FoI is constant and equal to r, since B(t)=c, I(t) = C*r*t, dI/dt = C*r, FoI = C*r/C = r. But it's dodgy because of discontinuities at interval boundaries.

Looking at annually compounding accounts, it would be possible to boost the interest earned by closing and re-opening them every month (or indeed more frequently) if this forces the whole accrued interest to be paid out immediately (i.e. prematurely). This is essentially an exploitation of method 2. But Tim's rotating balance method involving 12 different annual accounts which each pay interest in a different month is a particularly cool way of avoiding the prohibitive administrative hassle associated with regular destruction and creation of accounts, and ironically this means that within each one of these accounts the balance sits in it in the month when the "true" (method 2) FoI is lowest, so perhaps this is more an exploitation of method 3, or is it method 4?

"john boyle" wrote

No - the computer does it! ;-)

"john boyle" wrote

John, the A/c2 (B above) was opened on 1st April of the previous year, with an AER of 10%. ;-)

"Andy Pandy" wrote

Ah, OK then!

"Andy Pandy" wrote

OK -

Well, taking the A/c1, the true FoI we had before was for a *closeable* account, where: Value is V(t) = V(0) x [ 1 + i.t ] { for t

"Ronald Raygun" wrote

Yes, good.

"Ronald Raygun" wrote

Yes, this (2) is the case for a "closeable" account, ie where the value of the account at any point in time can be considered to be equal to the "actual balance" plus the "accrued interest not yet applied" - the available amount if the account were closed.

"Ronald Raygun" wrote

Yes, this (3) is the case for a "non-closeable" account, ie where the value of the account at any point in time can be considered to be equal to the "actual balance" plus the **discounted value** of the "accrued interest not yet applied".

"Ronald Raygun" wrote

Yes, this is because the "closeable" account (your (2) ie 'linear' model) is good value, for any specific AER.

"Ronald Raygun" wrote

Yes, the "non-closeable" account (your (3) ie 'discounted linear' model) has a true FoI varying from 5.66% up to 5.99% (if AER=6%) whereas your 'exponential' account has a constant FoI equal to 5.83% (again for AER=6%).

So your model (1) 'exponential' beats your model (3) 'discounted linear' at the start of the year, but the overall levels of FoI across the year conspire to give the same end-year balance.

"Ronald Raygun" wrote

Well, if you consider the value of the account at any point-in-time to be the amount you'd get if you closed it, use model (2), 'linear'.

However, if you want to assume you'll *never* close the account, use model (3), 'discounted linear'.

"Ronald Raygun" wrote

You simply need to consider it as an account whose "value" at that point in time is (B+A), although the bank will not let you reduce the balance below A. You can still withdraw the rest (B), if it would be advantageous...

"Ronald Raygun" wrote

It solely depends on your valuation of the *value* of the future (accrued but not yet applied) interest.

You can value this any way you like, & come up with different V(t) functions, and hence FoI(t) functions. Above, we have discounted at the AER rate - which appears suitable. You could try other valuations of this accrued interest if you like!

"Ronald Raygun" wrote

No! The FoI(t) is equal to the derivative of B(t) wrt t, divided by B(t). Thus FoI(t) would be *zero* throughout a period where B(t) was constant.

If, instead, FoI *was* "constant and equal to r", we'd have a different B(t) function as follows: B(t) = C . exp(r.t)

Put r=ln(1+i) and you should see this from the situation before, when FoI(t)=ln(1+i) [constant], gave B(t)=C.(1+i)^t.

"Ronald Raygun" wrote

Indeed, FoI(t) would be pointwise infinite at those interval boundaries!

"Ronald Raygun" wrote

"Ronald Raygun" wrote

Actually, no. Only a deposit/withdrawal on/before the last interest application date, would have the low FoI.

For instance, deposit 10K on 1st Jan in the "Dec 31 interest" account and the interest on the last day of the year is 1.64 (as for any other day). The "value" (balance plus accrued interest) is around 10,600 - hence the Fo I is **5.66%**.

But, for a deposit of 10K on 1st Dec in the same "Dec 31 interest" account, the interest on the last day of the year is still 1.64 (as for any other day). But the "value" (balance plus accrued interest) is only around

Well, sort of. The above refers to the what I had assumed was the text book definition of the FoI. Where in fact the text book definition is the other way around. However, I still think my definition of the FoI has some validity and practical use (see the 30 year investment example in my previous post), and we have started referring to it as "FoI#2".

The "true" FoI decreases with time. FoI#2 increases with time.

Yes.

Sounds right.

Yes. This is the "true" FoI.

But this doesn't really make sense, because you are discounting the accrued interest by the full nominal rate, not the *current* FoI. The interest should be discounted such that if it plus the capital balance both continually grow at the current FoI at every instant in time between t=0 and t=1, you end up with the correct result. What you (I think) are doing above ends up with the accrued interest growing at a different rate to the capital balance.

The way I worked it out was:

Let z be the force of interest, so z is the rate at which the balance grows at an instant in time dt..

At the instant in time immediately before interest is paid (ie t=1) z must be equal to the nominal rate r.

Going back dt in time, ie at t=1-dt, z is slightly lower because we actually lose interest on the interest earned during dt. And similarly for all previous instants in time.

So we can say that z(t-dt) = z(t) - z(t)^2dt

so z(t)-z(t-dt) = z(t)^2dt

ie dz = z^2dt

Which gives:

dt/dz = z^-2

Therefore knowing the differential dt/dz, we can integrate to get the formula:

t = -z^-1 + k where k is a constant.

t = k - 1/z

We know that when t=1, z=r.

So k = 1 + 1/r

So the formula is:

t = 1 + 1/r - 1/z

So z = 1/(1+1/r -t)

or z = r/(r+1-rt)

or z = r/(f-rt)

So z varies from r/f at t=0 to r at t=1.