But you can't get away with calling this the "true" FoI. It isn't according to any text book I've read.
But you cheated and didn't use the "true" FoI!
As per previous up until the final year.
Then after the last normal interest payment date *before* closure for each account (ie 31 Dec 2029, 31 Mar 2030, 30 June 2030), you need to account for closure, and set FoI#2 to the nominal rate on 17 July.
The spreadsheet, which calculates backwards, will now have the correct FoI#2 for all dates in the final year.
In fact, in the old days interest was calculated by hand. Everytime there was hand posted transaction the previous balance was multiplied by the number of days that balance had existed and the 'product' were recorded next to the balance column. Every time there was base rate change, or it got to the end of the quarter, the 'products' were added up and a book of tables with a page for every interest rate expressed as £sd % was used to convert the products to an accrued amount of interest. The first computer posting didnt calculate interest either, that still had to be done by hand. (All of this, of course, was told to me by somebody FAR older than myself ......
Ha! If only. AP said " 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. "
Just as, "in the old days" - mortality tables & life assurance premiums/valuations were calculated by hand. The mortality rate for each 'year-of-age' was computed from the number of deaths and sizes of living populations .... etc etc
Nowadays, we have computers - so we calculate the **force of mortality** instead. All other mortality factors are computed from the graduated formula for the force of mortality of a particular standard mortality table...
Exactly! All the accounts were already existing at the start ("... you *have* one account which ..., one which ..., and a third account which ...").
Because the 2nd account pays interest on 31Mar, and we don't know of any accounts which pay their *annual* interest part-way through the 'account-year', we deduce that A/c2 was opened on 1Apr the previous (or an earlier) year.
Of course the FoI is valid. But the *true* FoI(t) function is different for different "value" functions, and the 'non-closeable' account has a different value-function than the 'closeable' account.
"Andy Pandy" wrote
For a particular discount rate, OK, yes. But what *is* that discount rate? [See below.]
"Andy Pandy" wrote
Have you tried integrating your FoI#2 function, to get back to the "value" function?
I think that the V(t) function related to your FoI#2(t) function (for A/c1) is something like: V(t) = V(0) / [ 1 - i.t/(1+i) ]
Then consider that this must also be equal to: V(0) x [ 1 + i.t/(1+r)^(1-t) ] where r is the discount rate being used at time t.
From these two formulae (for the same V(t)), we can see that the discount rate which you are using (r) actually varies from 6.00% at the start of the year, to around 6.18% just before the end of the year.
So your FoI#2(t) function is the *incorrect* one for having the discount rate equal to the FoI....
(2) - As I said in an earlier post, you should discount at whatever rate at which you decide to value the future payment. [This could even be something unrelated to the AER / nominal rate / FoI(t) !]
So, I agree that using discount rate = FoI at each instant is an interesting possibility. I wonder what the FoI(t) function *would* look like under that premise?? ......
------------
"Andy Pandy" wrote
That's probably because the textbooks weren't working with the "constraint" of not being able to close the account.
In other words, the textbooks will all have been valuing the account as: V(t) = V(0) x [ 1 + (i x t) ] {t But you cheated and didn't use the "true" FoI!
I *did* use the "true" FoI - the FoI for the new value function, based on an account that can't (won't) be closed.
The accounts are "closeable". But we can chose not to close them. This should not affect the "true" FoI.
That is the interesting question....
No. I'm fairly sure I'm right and I prefer practical proof....see below...
I've already worked out the formula, in my last reply to Ronald, as being:
FoI#2 = 1/(1+1/i -t) for an annual interest payment account.
For interest payment at any other frequency, use the same formula with i as the nominal *periodic* rate (not annual rate), then multiply by the number of periods in a year.
For practical proof, it's best to use an extreme comparison. Using small interest rates and comparing annual/monthly accounts won't show up much difference.
Say you have two accounts, both of which pay 100% AER (conceivable in countries like Turkey!).
One pays interest annually on 31 Dec at 100% nominal, and the other compounds daily at a nominal rate of 2^(1/365) -1.
So if you invested 10,000 in each on 1 Jan, then on 1 Jan the next year both would be worth 20,000 (assume that you earn interest on the date of deposit but not date of withdrawal).
The question is, how could you maximise the interest by switching between these accounts.
Using FoI#2 as above - the FoI#2 for the daily compounding account is obviously going to be the same every day, and will be ln(2), ie 69.3147% on average (theoretically it'll vary through the day, but by very little).
FoI#2 is going to start lower than this on the annual account and rise throughout the year. So to maximise interest, we should be investing in the daily compounding account until the annual account's FoI#2 exceeds 69.3147%.
So we can use the formula above to find when this happens:
FoI#2 = 1/(1+1/i -t)
0.693147 = 1/(1+1/1 -t)
So 2 - t = 1/0.693147,
t = 2 - 1/0.693147 = 0.5573046.
Multiply by 365 and this tells us that 203.42 days into the year we should switch, so on the 204th day, ie the 23rd July.
A simple basic program confirms that switching on the 204th day does indeed maximise interest.
I can point you at several examples of accounts advertised online where the AER is quoted as being exactly the same as the nominal annual rate, even though the interest payment date is not in exactly 12 months time from now.
Agreed - with a nominal rate of 10% (not less) when compounded annually.
As an aside, do you know of any banks which actually compound interest "annually" after *3* months, *15* months, *27* months etc? And if so, is it really *legal* to quote AER% when nominal rate is only
9.819% ??
Because in that case, if someone invested 10,000 for exactly ten-and-a-quarter years (ie up to an interest application date), they'd only get 26,140 at the end. That's an equivalent annual rate of only 9.83% (ie much lower than the 10% quoted) ....
Of course you can - but that changes the required "value" function, and hence the (true) FoI(t) function.
"Andy Pandy" wrote
Rubbish. If you choose to close the account, you *gain* - by receiving the accrued interest earlier. That gain is shown in a higher FoI(t) !!
You've already noticed this when you said how you would use FoI#2 to determine which account(s) are best when you know you're going to close them on 17 July 2030 :-
For the same reason as you adjusted your FoI#2 when you know when closure occured above, the *true* FoI also varies depending on actual closure date.
"Andy Pandy" wrote
Right about what? Your function might produce a correct solution to maximising interest, but it is only a "true" FoI function if your discount rate for valuing accrued interest is above the nominal rate - not equal to the FoI, as you specified we "ought" to be using!
"Andy Pandy" wrote
No, I was asking what the *true* FoI(t) function would look like if the discount rate used (to value accrued interest) were equal to the FoI. Your FoI#2 function (as I showed earlier) has a discount rate starting at the nominal rate, and ending the year somewhat higher than the nominal rate!
"Andy Pandy" wrote
This is also confirmed by using the *true* FoI(t) function.
No, this only happens for 'Fixed Rate & Period' deposits. The interest is usually credited on a date not determined by when the account was opened or the deposit made. Typical annual dates are 31 December and 31 March.
No, the "true" FoI is unaffected by whether you leave the account open or not. That's the point of it.
FoI#2 *is* affected by closing the account earlier. The *true* FoI isn't.
The true FoI would show a value on a closure date equal to the worth of the account, ie what you'd get if you closed it. So it doesn't need adjusting. FoI#2 does. But the true FoI is pretty useless for the purposes I've been using FoI#2 for.
Yes, that's the whole point of it. If you have your money in the account with the highest FoI#2 at every point in time, you'll maximise the interest you earn.
FoI#2 is not the true FoI.
Looked again at this - I think you are looking at the "effective rate" over a period rather than the FoI.
FoI#2 is the rate at which the interest would grow if continually compounded. So I have checked the result by compounding the interest every day by having a column setting the value of the account V(t) to the the previous days value multiplied by e^(FoI#2(t)/365) (to convert the FoI to an effective daily rate).
It looks good on this basis - you get the correct amount at year end - which it wouldn't if it were incorrect in the way you describe. It also get the "switch" date right for accounts for extreme situations - so I'm happy that it is correct for the purposes I've used it for.
Yes, that is what I mean. The nominal rate you quoted would only be correct for an account opened on 1 Jan with AER% and withdrawal on 31 Dec - meaning the interest is credited after 3 months. We know A/c2 has interest paid annually, because that was part of its definition - so it would be annually thereafter (after the first
3 months after opening).
"john boyle" wrote
In that case, I'm glad I never trust any AER quotes!
"john boyle" wrote
The interest is credited at the end of the third month (31 March after opening 1 January) :-
Just after the first interest, they'd have 10K x (1 + 0.25 x 9.819%) 10,245. Then after 1.25 years (from the start), they'd have: 10,245 x 1.09819 11,251. Then after a further 9 years, they'd have: 11,251 x 1.09819^9 = 26,140.
This initial premise is highly suspect. Even if the force of interest were *constant*, you could still say "we actually lose interest on the interest earned during dt" - and yet z(t-dt) will be no lower than z(t) and your equation above will *not* follow!!
What interests me though, is that - even using this initial dodgy assumption - your FoI#2 function can actually still determine the correct "day to switch": day = 365 x [ (1+i) - i/ln(1+i) ] / i Very interesting!!
Just after the first interest, they'd have £10K x (1 + 0.25 x 10%) £10,250. Then after 1.25 years (from the start), they'd have: £10,250 x 1.10 £11275 Then after a further 9 years, they'd have: £11,275 x 1.1^9 = £26,585.86
FoI#2 is based on what the interest rate would be if interest immediately compounded at every instant in time. At the instant in time before interest payment, it *does* immediately compound.
FoI#2 cannot be constant unless you have an account where interest *does* actually compound immediately at every instant in time.
That's because it does what it says on the tin.
I believe it's correct as I defined it, and my experimental test results would appear to back this up.
BeanSmart website is not affiliated with any of the manufacturers or service providers discussed here.
All logos and trade names are the property of their respective owners.