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I start with a value A. I decrease it by M each month m. Every year (ie, m mod 12 =0), I calculate the average of what A has been throughout the year, multiply that by t, and add it to A. This goes on until A reaches 0 (t and M are set in such a way that A decreases every year). How many months m would that take ?

Similarly, given A and M, if I want A to reach 0 after m months, what does the value of t need to be ?

In other words, I have a capital A that has been given to me by a bank B1. B1 asks me to return M dollars every month, until it reaches a value N that was predefined by B1 (B1 also calculates this in terms of a interest percentage, but I don't understand the definition of that interest value). I put the money A that B1 gave me into a bank B2. What does the yearly interest t in bank B2 need to be so that it is the threshold between winning and losing money ?

These may seem like 3 different questions, but I think that once put into equation, I think it's easy to answer all of these by just expressing one variable as a function of others.

concrete example:

I ask a bank B1 to lend me A=10,000 dollars. In exchange for that 'donation', B1 wants me to give M=100 dollars for 10 years (m=120). In the end, as you can see, I end up giving more (12000 dollars) than I was given. However, I do not plan to use these 10,000 dollars. Instead, I just put it into a bank B2 and wait for interest rates to inflate my money. If don't know how this is done in your country, but in mine, I think the interest rates are given at the end of the year, but calculated monthly (I think that one other way to say that is to say that it is calculated yearly, but with the average of what you had in your balance throughout the year, hence my explanation in the first paragraph). Interests over interests don't apply monthly, but yearly. This is to say that if you have 100 dollars the first year, with an interest rate of 2%, then the second year you have 100+2, but the third you have 100*1.02*1.02=1.0404 instead of just 100+2+2=104. So if I could have these 10,000 dollars for 10 years, I would have 10000*t^(m/12). But I don't have A=10000 dollars for 10 years, because they keep decreasing as I have to give M=100 dollars away every month. So first of all, I'm trying to find a formula (not a estimate by excel, but an exact mathematical formula) that says what is my balance after m months. Then I'm trying to find what value t (the interest rate from bank B2) needs to be (depending on the value of M) so that I end up winning money by asking B1 to give me money and placing this money into B2.

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You "decrease A by M..." What does this mean? Is M some quantity in the same units as A, is it some percentage of A...or what? Also, what does "decrease it by M each month m" means? –  DonAntonio Jul 19 '12 at 3:26
Yes M is in the same units as A, it's not a percentage. And by dicreasing each month, I mean that A_m=A_{m-1}-M. After m months, A is A-M*m. But when m is a multiple of 12, A_m=X*t (where X is the average of the values A_{m-12}...A_m). Maybe reading the last question will help you to understand what I meant in the first one, they are all related, and the first one was perhaps a failed try to formalize the issue. M is just the money I need to give to a bank B1 each month. The money is put into a bank B2, so each year, I get yearly interests that depend on how much money was on B2 on average. –  JLock Jul 19 '12 at 3:46
There remain ambiguities. Do we decrement $M$ immediately (at $t=0)$, or at the end of the first month? Does the bank consider the decremented value of the min. monthly balance (probably, they are greedy) for purposes of averaging? –  André Nicolas Jul 19 '12 at 4:10
I am more interested in a rough estimate of the result, and the explanation of how to get there, than the actual result to my question, which is indeed not well asked. You can explicitly mention in your solution the ambiguities you find, and use what you think is the closest to real life. –  JLock Jul 19 '12 at 8:11
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1 Answer

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Writing the equation is following what you have written, but there are a few details not perfectly specified. Let $B(m)$ be the balance owing to the first bank in month $m$. The first sentence tells us that $B(0)=A$. Then you say that $B(m)=B(m-1)-M$ unless $m$ is a multiple of $12$. It is not clear what values are averaged for the yearly increment, but it looks like we should average B(0) through B(11). Do you also decrement by M in month 12? It would seem that $B(12)=B(11)-M+\frac t{12}\sum_{i=0}^{11}B(i)$. Then we go back to $B(m)=B(m-1)-M$ until month $24$, when $B(24)=B(23)-M+\frac t{12}\sum_{i=12}^{23}B(i)$ and so on. Presumably you deposit the A in the second bank. You are using $t$ for a constant in the bank B1 equation as well as the interest paid by bank B2. Do you know that they will be the same? Probably not, so I will use $s$ as the annual interest paid by B2 and assume that it is compounded monthly. Then let $C(m)$ be the balance in B2. We have $C(0)=A, C(m)=C(m-1)(1+\frac s{12})-M$. Your net position at any month is $C(m)-B(m)$. This is made for a spreadsheet-make columns for $m, B(m), C(m), C(m)-B(m)$ with $t,s$ in cells you reference by name or by fixed reference (Excel uses dollar signs). Given $A, M, t$, you can look to find the month you pay off the loan. Then Excel will let you goal seek to make the balance zero by changing $s$. If you use a spreadsheet without that capability, you can just do it by hand, looking for the correct $s$.

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Maybe I didn't understand your answer, but I think you misunderstood my question. Sorry for not explaining myself clearly. The bank B1 just asks me to give monthly payments to refund the money it lent me. That's it, A(m)=A(m-1)-M. The second part of the equation, where I talk about "average", was an attempt to explain what happens when I put A into a bank B2, which gives interests every year, but calculated monthly (because I think that's how a bank calculates its interest rates). So basically, ask for A=10.000$ to B1, put it into B2, and see whether you end up winning or losing money. –  JLock Jul 19 '12 at 8:20
I added an example to my question to show you what I mean. I hope it's more understandable now. Feel free to ask for details. If it's just minor ambiguities, you can use your better judgment to take whatever solution is more appropriate. –  JLock Jul 19 '12 at 8:37
I'm sorry to say that after your explanation I still can't fully understand your question...or questions. One thing I know, though: don't mess with banks. Those guys aren't the nicest living things out there (poisonous snakes, ebola and HIV viruses included), so you'll hardly end up even with them, let alone earning money. Most likely you'll end up losing big time and crying. –  DonAntonio Jul 19 '12 at 12:48
lol. This is just a theoretical question, I'm sure I'm not the only one who has thought about this before, so yes it's not viable in real life, but I'd like to know how it works as far as the math is concerned. I'd like to know the threshold between winning and losing money, I think it would help to better understand why banks choose one interest rate over another. If you don't understand the question, feel free to ask for details. –  JLock Jul 19 '12 at 13:34
@JLock: roughly speaking, if you can borrow at interest rate $t$ and lend at $s$, you need $s \gt t$ to make money. The way I interpreted it, your lending is compounded monthly while your borrowing is compounded yearly, so you need $(1+\frac s{12})^{12} \gt 1+t$. There will be some "end effects", so a spreadsheet is your friend. There are financial functions that let you calculate the effective interest rate from a payment stream so you know what the first bank is charging you. –  Ross Millikan Jul 19 '12 at 13:52
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