Compounding interest formula going awry

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Discussion Overview

The discussion revolves around the application of the continuous compounding interest formula in the context of withdrawals from an account. Participants explore the implications of a withdrawal rate on the balance over time, particularly focusing on the conditions under which the account may become empty.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant presents a formula for the account balance after T years, incorporating continuous compounding and continuous withdrawals, but questions the behavior of the function when specific values are substituted.
  • Another participant points out that the total amount withdrawn over time should be calculated as 100t², suggesting a reconsideration of the withdrawal model.
  • A participant expresses confusion over the results obtained when substituting different values for the initial deposit and interest rate, indicating a discrepancy between expected and plotted outcomes.
  • Further clarification is provided regarding the rate of interest accumulation and the net rate of change in the principal, leading to a differential equation that describes the situation.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the correct interpretation of the withdrawal rate and its impact on the account balance. There is no consensus on the correct approach or resolution of the mathematical issues presented.

Contextual Notes

Participants note potential misunderstandings in the formulation of the problem, particularly regarding the nature of withdrawals and their mathematical representation. There are unresolved questions about the behavior of the function under different parameters.

Who May Find This Useful

This discussion may be useful for students or individuals interested in understanding continuous compounding interest, differential equations, and the implications of withdrawal rates on financial models.

Locoism
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If Po dollars are deposited in an account paying r percent compounded continuously and withdrawals are at a rate of 200t (continuously), what is the amount after T years?

I derived the formula by taking the limit as m -> ∞ of the compounding interest equation P(t) = Po(1+r/m)^mt which gives us P(t) = Po*e^rt. So including the withdrawal rate

- P(t) = Po * e^rt - 200t

And that would be the amount in the account after T years. But my question asks, if r = .1 and Po = $5000, when will the account be empty? My function doesn't cross zero, and I don't understand where I made the mistake.

Can anyone help me?
 
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If withdrawals are done at a rate of 200t, then isn't the total amount of money withdrawn between time 0 and t 100t2?
 
Sorry, the question says withdrawals are at an annual rate of 200t dollars.
I thought about that, tried it out and the question still doesn't work out. I'm really worried about this one...*edit* Actually, I am an idiot. For some reason I got no real roots the first time...
*edit* However the question asks me to consider the case Po = $20 000 and r=0.1. I don't see anything special about this. Intuitively I would say the amount stays constant, but plotting it gives me something completely different...

Thank you for your support :P
 
Last edited:
Locoism said:
Sorry, the question says withdrawals are at an annual rate of 200t dollars.
I thought about that, tried it out and the question still doesn't work out. I'm really worried about this one...


*edit* Actually, I am an idiot. For some reason I got no real roots the first time...
*edit* However the question asks me to consider the case Po = $20 000 and r=0.1. I don't see anything special about this. Intuitively I would say the amount stays constant, but plotting it gives me something completely different...

Thank you for your support :P

The rate at which interest is accumulating is Pr, where r is the yearly interest rate divided by 100. The rate at which withdrawals are being made is 200. Therefore, the net rate at which principal is increasing is given by the differential equation:

dP/dt = Pr - 200
 

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