>> "Tim" wrote
> >> > ... 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
> >> The accounts are "closeable". But we can chose not to close them. > > > "Tim" wrote
> > Of course you can - but that changes the required
> > "value" function, and hence the (true) FoI(t) function.
>
"Andy Pandy" wrote
No, the "true" FoI is unaffected by
> whether you leave the account open or not.
> That's the point of it.
That's not the point of it at all. If you think that, perhaps you need to hit the books again!
> "Andy Pandy" wrote
> > > This should not affect the "true" FoI.
> >
> "Tim" 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 :-
>
"Andy Pandy" wrote
FoI#2 *is* affected by closing the account earlier.
I'll accept that, because it is your creation.
"Andy Pandy" wrote
The *true* FoI isn't.
If you think that, perhaps you need to hit the books again!
"Tim" wrote
> >
> > > > "Andy Pandy" wrote
> > > > > 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. > > > >
> > 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
The true FoI would show a value on a closure
> date equal to the worth of the account, ...
... agreed (altho' strictly it's the value-function which shows this, not the FoI) ...
"Andy Pandy" wrote
... ie what you'd get if you closed
> it. So it doesn't need adjusting.
But if you *don't* close it, it is "worth" less! Hence, from your last comment ("The true FoI would show a value ... equal to the worth of the account"), coupled with the fact that the account **is worth less** than the "closure" value if you don't close it, then the true FoI is lower if you don't close it.
"Andy Pandy" wrote
But the true FoI is pretty useless for
> the purposes I've been using FoI#2 for.
Then how did I manage to confirm your solution for the switching day, using the true FoI??!
> > > "Andy Pandy" wrote
> > > > > But as in my reply to Ronald, I think we
> > > > > ought to be discounting by the *current*
> > > > > FoI at every instant in time, not the full rate i.
> > >
> > > "Tim" wrote
> > > > Have you tried integrating your FoI#2
> > > > function, to get back to the "value" function?
> >
> > "Andy Pandy" wrote
> > > No. I'm fairly sure I'm right and I
> > > prefer practical proof....see below...
> >
> "Tim" wrote
> > Right about what? Your function might produce
> > a correct solution to maximising interest,
>
"Andy Pandy" wrote
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.
OK. But you said "we ought to be discounting by the
*current* FoI at every instant in time", and I thought that you were suggesting that FoI#2 did this - which it doesn't.
"Tim" wrote
> > 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
FoI#2 is not the true FoI.
Well, with a particular discount rate for valuing accrued interest, the true FoI *would* equal your FoI#2 at each point in time. I thought you were suggesting that this discount rate were equal to the FoI at each point in time, which it is not for FoI#2.
> > "Tim" wrote
> > > > 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
> > > 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. > > > "Tim" 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
Looked again at this - I think you are looking at the
> "effective rate" over a period rather than the FoI.
No, I'm definitely looking at the *force* of interest as a function of time - not specially the "effective rate" over a period.
Although, as you should know, the exponent of the integral of the FoI over any period *is* the "effective rate" for that period.
"Andy Pandy" wrote
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).
That just describes the formula I've stated a number of times previously, for the value function if your FoI#2 were indeed the true FoI :- V(t) = V(0) / [ 1 - i.t/(1+i) ]
If this value is actually equal to V(0) plus "the accrued interest ( which is (V(0) x i x t) ) discounted at r", then r is as I stated before.
"Andy Pandy" wrote
It looks good on this basis - you get
> the correct amount at year end - ...
There are *lots* of functions which will get the amount at end-year correct!
"Andy Pandy" wrote
... which it wouldn't if it were incorrect in the way you describe.
I haven't described it as "incorrect", simply that it is only equal to the true FoI(t) with a particular (varying) discount rate for valuing accrued interest - and that discount rate is above the nominal rate across the year. In other words, it is the true FoI for a discount rate (on accrued interest) which *doesn't* do as you suggested: "we ought to be discounting by the *current* FoI at every instant in time".