# Difference Equation Bank Account Problem

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1. Jul 25, 2016

### Houeto

1. The problem statement, all variables and given/known data

2. Relevant equations
The equation describing the balance will be f(n+1)=f(n)+R/12*Dm-Cf
with f(n)=initial deposit
R=Annual Rate
Dm=Each mouth Deposit 150
Cf= each month fee

3. The attempt at a solution

Can someone shed some lights on it?

Thanks

2. Jul 26, 2016

### Replusz

Your initial equation seems wrong to me. The interest comes over the whole amount of money, not just the new input. But maybe I'm wrong, pls double-check :)

3. Jul 26, 2016

### BvU

Not only that, the exercise clearly states that the 6% is an annual rate. That means one of two things: either the 6% is applied in intervals of 12 months or it is converted to a monthly interest rate that is not equal to 0.5% (but less -- compound interest and all that). There is a third possibility: the exercise composer overlooked these two things altogether and wants you to use the 0.5% nevertheless. You decide.

In your formula you don't mean f(n) = initial deposit (I hope?), but balance after n months.
In your formula I don't understand why you do not add Dm itself every month.

Why not build up a little table, just to check the expression: f(0), f(1), f(2) etc.

Last edited: Jul 26, 2016
4. Jul 26, 2016

### Stephen Tashi

One definition of the "nominal APR" is : ( number of payment periods per year) ( interest rate per payment period), so the person who composed the problem may be knowledgeable about that terminology and expect students to use R/12.

5. Jul 26, 2016

### Ray Vickson

In financial calculations it is common to use a monthly interest equal to 1/12 the annual interest, even under monthly compounding. Of course, that is incorrect mathematically, but that is how it is done by financial institutions, for the most part. If the OP's problem occurred in a "finance" or engineering economic type course, it is likely the intended monthly interest rate be taken as 0.5% exactly, in accordance with common institutional standards.; if it is in a "math" course, the intention is less clear.

Another criticism of the OP's contribution is his/her failure to explain the "timing" aspects clearly. Is the balance at month $n$ the amount in the account at the start of a month, before any deposits or payments or after any deposits or payments? Is it the balance at the end of the month, before (or after) any payments and/or interest income? Different answers will lead to slightly different recurrence relations.

Last edited: Jul 26, 2016
6. Jul 26, 2016

### BvU

Do bean counters work with z-transforms ?

7. Jul 26, 2016

### Ray Vickson

No, but sometimes "advanced" courses in finance or economics may introduce z-transforms.

Anyway, even if this was in a "math" course, the issue is still valid: should the OP take the monthly interest rate as 0.5% (because the instructor may explain that in financial problems, that is a common standard), or should it be $100 \times (1.06^{1/12}-1)$?

I hope the OP realizes there is an issue here, and if so, will likely know which way to proceed.

8. Jul 27, 2016

### TomHart

It looks like the equation is wrong. Should it possibly be:
After thinking about it, maybe I'm not supposed to correct the equation. I almost did it anyway because I figure for someone doing Z transforms, coming up with the equation is the simple part of the problem. So I will just say that the equation is definitely incorrect. As far as Z-transforms, for me that was long ago and far away.