Renter - page 18
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What I did: I decomposed (1+q-k)^t = (1+epsilon)^t to the third power binomial. Suppose q = 0.01 and hence epsilon <~ 0.01.
Suppose that t=50. Then on the calculator, (1+0.01)^50 = 1.645. Binomial approximation to the 3rd degree: (1+0.01)^50 ~ 1 + 50*0.01 + 50*49/2*0.01^2 + 50*49*48/6*0.01^3 = 1 + 0.5 + 0.1225 + 0.0196 = 1.6421. Well, yes, that's pretty accurate.
But here, say, at t=100 (just over 8 years) the exact result is 2.7048...(almost an e-number, by the way).
It is not by accident. The number e (or the Second Noble Limit) is defined exactly like that: e=lim(1+1/n)^n, at n->inf. In your example, n=100 and epsilon <~ 0.01, so you get 2.7...
Right, of course.
My ordeal seems to be coming to an end. If everything is clear to you in Mikhail Andreyevich's reasoning, I don't need to publish my decision (I'll just write an answer, perhaps) :) There is nothing beautiful there.
Sergey, by the way, I haven't asked you the main question: what's the order of q? Can it be equal to, say, 0.4 (40%) - or is it something about a bank's interest, i.e. 0.01?
Sergei!
Are you satisfied with the solution?
But Mikhail Andreyevich is wrong about the need to make contributions to the consumption fund - in the conditions of the problem they do not exist, as I understand it?
So the optimal strategy, in the correct sense of the word, would be the initial accumulation of the maximum possible amount in the account, and only after that - withdrawal of all accrued interest until the end of the deposit.
My ordeal seems to be coming to an end. If everything is clear to you in Mikhail Andreyevich's reasoning, I don't need to publish my decision (I'll just write an answer, perhaps) :) There's nothing beautiful there.
Sergey, by the way, I haven't asked you the main question: what is the order of q? Can it be equal to, say, 0.4 (40%) - or is it something like bank interest, i.e. 0.01?
Draw your answer with explanations. I still need time to understand it.
q lies within the range 0.1<q<0.3 (relevant to Forex).
q is in the range 0.1<q<0.3 (relevant for Forex).
Then, according to my conclusions, we have to assume that the period of using the deposit should be at least 30 months - this is for q=30% per annum.
For a 10% interest per year TT(q/12) from the previous page would already require 85 months ...
;)
If everything is clear to you in Michail Andreevich's reasoning, I don't have to publish my solution (I'll just write an answer maybe) :)
Sorento:
Sergey! Are you satisfied with the decision?
But Mikhail Andreyevich is wrong about the necessity of making deductions to the consumption fund - in the conditions of the problem there are none, as I understand it?
Is this a joke - " Mikhail Andreyevich's reasoning"?
What kind of decision is this? What in this solution follows from where? Some kind of formulae... trigonometric ones, too. You, Mikhail Andreyevich, could you at least give us a hint as to where your solution comes from.
This must be a spell of a shaman: "...First of all, the possibility of applying the technique of not withdrawing all accrued interest must be decided:
It's probably obvious to everyone but me where these logarithms come from! Well, this could not be mentioned at all:
every real kid in kindergarten understands that cosine is cool! (especially for our problem).
In short, Michael Andreevich, you with the same success can result here the proof of Fermat's Theorem, not troubling itself with superfluous comments.
Therefore the optimal strategy, in the correct sense of the word, would be a strategy of primary accumulation of maximum possible amount on the account, and only after that - withdrawal of all accrued interest until the end of the deposit use.
So why is that, Sorento? And what meaning do you give to "... the proper meaning of the word,..."?
Why suddenly (where does it come from) your statement: "... the optimal strategy would be to firstly accumulate the maximum possible amount in the account, and only after that - to withdraw all accrued interest...". "? We showed above many times with numerical solution, that there is an optimal withdrawal percentage kOpt and it is greater than zero and less than or equal to the accrued fixed interest q (it depends on the amount of accrued interest and usage time t) .
Is this a joke - " Mikhail Andreyevich's reasoning"?
What kind of decision is this? What in this solution follows from where? Some kind of formulae... trigonometric ones, too. You, Mikhail Andreyevich, could you at least give us a hint as to where your solution comes from.
This must be a spell of a shaman: "...First of all, the possibility of applying the technique of not withdrawing all accrued interest must be decided:
It's probably obvious to everyone but me where these logarithms come from! Well, this could not be mentioned at all:
,
every real kid in kindergarten understands that cosine is cool! (especially for our problem).
In short, Michael Andreevich, you may as well give a proof of the Great Theorem by Fermat, without bothering with unnecessary comments.
Reshetov kind of explained to us about hedgehogs.
Everything is easy and understandable for them. :)
As for the criterion to calculate the TT function, it's really simple - try to solve the problem of finding the time when 100 rubles deposited with accumulated interest will double.
The fact that if accrued interest is not to withdraw and reinvest, the terms of YOUR problem they can not withdraw, except in the form of accrued interest on them.
That's where the twos and logarithms come from...
As for sines and cosines, a mistake. The reasoning about the area of the circle is misleading. And the result as you can see is still better.
But the optimal strategy is described above.
I have not finished formulas, maybe I will do it next week.
Therefore - why is that, Sorento? And what is your meaning in "... the right sense of the word,..."?
Why all of a sudden (where does it come from) your statement: "... the optimal strategy would be to first accumulate the maximum possible amount in the account, and only after that - to withdraw all accrued interest...". "? We have shown above many times with numerical solution, that there is an optimal withdrawal percentage kOpt and it is greater than zero and less than or equal to accrued fixed interest q (it depends on interest accrual and time of deposit t) .
1) an extremum... ;)
2)first of all by the conditions of your problem, which I wrote about earlier - in the discussion of TT.
As for "repeatedly demonstrated with numerical solution, that there exists an optimal withdrawal percentage kOpt..." you should evaluate result with this shamanistic coefficient and with my method.
;)
Sorento:
Что касается критерия вычисления функции ТТ, то и вправду просто - попробуйте решить задачу нахождения времени при котором 100 рубле положенные на вклад с накоплением процентов удвоятся.
So, Sorento, then who is Mikhail Andreyevich. Are you for him or is everything clear to you?
I've got it right with cosines, but the time to double the count for compound interest is different: TT(q)=ln(2)/ln(1+q)
Sorento:
1) extremum... ;)
2)first of all by the conditions of your problem, which I wrote about earlier - in the reasoning of TT.
As for "repeatedly demonstrated with numerical solution, that there exists an optimal withdrawal percentage kOpt ..." you should evaluate the result with this shamanistic coefficient and with my method
Evaluate with your method and give the result.
So, Sorento, then who is Mikhail Andreyevich. Are you for him or is everything clear to you?
It is clear with cosines, but the time of doubling the count for compound interest is different for me: TT(q)=ln(2)/ln(1+q)
How is it different? Because it takes strictly more. :)
As Hodja-not Yusuf used to say: "There must be a profit"...
Otherwise the sense of reinvestment? Moreover, in real tasks there is always a discount - I also spoke about it.
;)