Renter - page 29
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Just don't use the language of the ACS, please. The simpler the better.
Why do you dislike this language so much...
By the way, this language allows you to expand the model almost infinitely, introducing additional relationships, conditions, constraints...
For example, you can consider five different accounts, with different accruals, and eight pockets, with different fillings, and even stipulated ratios between pockets.
For this, my model needs to be augmented with a few blocks and connections. And for your model, Alexey, it is an impossible task.
Thank you Alexey for pointing out the error in my attempts to get an analytical expression for the amount of withdrawals over time t. I did indeed derive the percentage k BEFORE the interest q was credited to the account.
Given the above, I propose again to obtain an analytical (corrected) value for the withdrawal and compare it with the iterated form of the entry. For the discrete case, the deposit will grow according to the formula:
here the index will successively run through all values from 1 to t and
The error used to be in the last summand, where i-1 stood instead of the index i .
For the withdrawable funds we can write:
For this iterative representation, an analytical notation can be obtained:
Alexei, this expression should match the one you obtained by induction. If there are no errors, let us now compare the values for the iterative formula and for the analytic representation:
Here, red dots show values of all derived means by iteration formula, as a function of the relative value of k/q - I think such a representation is more clear (thanks to Oleg - he inspired me). Blue is the analytical analogue. It is seen, that the coincidence is exact and for the specified t and q there is a pronounced maximum of kOpt for the derived means.
Actually, after the corrections made, it is proposed to find an analytical expression for kOpt . Find the derivativeof k:
We equate it to zero:
We check that we have not made a mistake and the zero of this expression coincides with the maximum of the withdrawn funds:
Well, all is well! It remains to find an acceptable solution for the zero of this beast-like derivative in analytic form.
P.S. It's a mess.
actually, it seems logical that k is a fraction of q
since
"to withdraw a certain percentage k from the account each month which does not exceed the value of q"
that's not the point... But...
This is important as the formula is different.
We have in January month B=100
At B = 100 is charged (30% ie q = 0.3) - we have in February (1 + 0.3)*B = 1.3 * 100 = 130 = (1 + q)*B
i.e., a surcharge of 0.3*B = 30 = q*B
So far it's the same as mine.
We remove a part of this surcharge (50%, i.e. k=0.5) k*q*B = 0.5*0.3*100 = 15
As a result for calculation of charges for February we have B=130-15=115
and then
In February we have B=115
You seem to have it at 0.15, i.e. 15%. That's what we're going to base it on.
But that's where our paths diverge. Generally speaking, I no longer operate with fractions, but only with percents.
15% of all accumulated deposit we withdraw: k*(1+q)*B = 0.15*(1+0.3)*100 = 19.5As a result for calculation of charges for February we have B=130-19.5=110.5
and further
As a result, in the month of February we have B=110.5.
As a problem solver, letSergey think about which option is better for him.
P.S. I see your answer, Sergey. Well I have already written the solution before. My formula did not coincide with yours :(
This language is quite adequate to describe linear dynamical systems. Oleg, your reasoning about lattice functions, frankly, just killed me. There were no such complexities in the original problem.
I agree about the flexibility.
This is important as the formula is different.
So far it's the same as mine.
Oleg, you are incorrigible :) k is a percentage, not a fraction!!!You seem to have it equal to 0.15, i.e. 15%. That's what we're going to base it on.
But that's where our paths diverge. Generally speaking, I no longer operate with fractions, but only with percents.
15% of all accumulated deposit we withdraw: k*(1+q)*B = 0.15*(1+0.3)*100 = 19.5As a result, to calculate the charges for February we have B=130-19.5=110.5
and then
We have B=110.5 in February.
Sergey , as a problem-solver, let him think about which option suits him better.
P.S. I see your answer, Sergey. Well I have already written the solution before. My formula did not coincide with yours :(
It seems that indeed - everyone solves "his or her" problem...
To get into the deposit is kind of unacceptable at withdrawal, this will be another problem.
;)
And the joke with interest and shares - the bomb!
Crying
P.S. I see the answer, Sergey. Well, I have already written the solution before. My formula did not coincide with yours :(
Uh-huh... Let's take it one step at a time.
An iterative form:
It shows the growth of the deposit. The first term on the right-hand side of the equation shows how much money was there when interest was charged q. The second term shows how much money will be added after the accrual and the third term shows how much will be deducted from what it was after the withdrawal of interest k.
Do you have any comments?
I think I see the error in myself, Alexei! - In the iterative formula for withdrawal
I'm essentially withdrawing a percentage k from a deposit already "withdrawn" (see the formula above). The correct way to write it is like this:
Then the analytical form will be like this:
Probably matches yours now. Going to see...
Mathemat:
В конце t-го месяца на счете (по индукции) останется D((1+q)(1-k))^t.
Let's see what's left in my account at the end of period t:
So there will be some left over:
You have: D((1+q)(1-k))^t.
We don't have the same denominators.
My formula:
Removed = k(1+q) * ( 1-r^t ) / (1-r)
r = (1+q)(1-k)
Output of the formula: https://www.mql5.com/ru/forum/131914/page27, my post at 23:21.
It's hard to call it pretty.
Try it directly, without iterations. Iterations can always be screwed on afterwards.
Try it directly, without iterations. Iterations can always be screwed on afterwards.
You're right!
It has to be like this:
Now it's a match. Phew...
Here we go, starting all over again :) The tale of the white bull...
Oleg, join in, if it's clear.
FreeLance: Залазить в депозит вроде при снятии низзя - это будет другая задачка.
And who gets in there? It's OK so far: with the figures shown, there's more in the account than there was at the beginning of the month.
Shit, a postnumerando annuity, for fuck's sake...