Difference Equation Bank Account Problem

[Houeto]

Homework Statement

Homework 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

The Attempt at a Solution

Can someone shed some lights on it?

Thanks[/B]

 

[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 :)

 

[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.

 

[Stephen Tashi]

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).
.

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.

 

[Ray Vickson]

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, jusdt to check the expression: f(0), f(1), f(2) etc.

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.

 

[BvU]

Do bean counters work with z-transforms ?

 

[Ray Vickson]

Do bean counters work with z-transforms ?

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.

 

[TomHart]

The equation describing the balance will be f(n+1)=f(n)+R/12*Dm-Cf

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.