Interest continuously compounding with a variable prinicple

In summary, the conversation discusses a problem involving continuously compounded interest and the goal of finding the time at which the interest earned in a year equals the principle invested. The speaker provides their own solution to the problem and asks for confirmation or a correct answer. They also mention trying various approaches and debunking other answers they have received.
  • #1
venik
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0

Homework Statement


This is a question I *might* have already got the answer to, I'd just like for someone very good with calculus and algebra to verify/answer the question themselves.

To be more exact with the problem, we are putting 1 dollar (a variable principle) into a continuously compounded interest account at rate r. At what moment does one year's interest (e^r-1), equal our principle of 1 dollar per year. Or at what time can we stop putting the dollar in completely.

I solved it as the most complicated substitution problem I, personally, have ever done. Feel free to do it any way you please, but I'm looking for both answers and/or possible mistakes in my math.

Homework Equations


F=Pe^rt

The Attempt at a Solution



Let F = Final, P = principle, r = interest rate, y = years.
Given that:
F = Pe^(ry)

And

P1 = ($)1 x y
(P1 because I'm going to have to use another P later)

Then replacing 1y for P we get

F = ye^(ry)

This gives us F for any time y, and rate r.

But we want a specific F, to get that I first defined what P2 is required for the next year's interest to be equal to the 1 dollar we are putting in every year.

P2(e^r-1) = 1

P2=1/(e^r-1)

In order to substitute this into our original equation we must substitute P2 into a separate F = Pe^(rt2)

We know that t2 = 1 because in the question we asked when does (e^r-1) of the *last* year equal $1.

We get

F = e^r/(e^r-1)

Then substitute this final into our F = ye^ry

we get

e^r/(e^r-1) = ye^ry

0 = ye^ry - e^r/(e^r-1)

at 15% interest I get y = 3.96 years. Graphing on my calculator because as far as I know that is unsolvable algebraically

at 8% interest I get y = 7.27 years.

Sounds too good to be true. Put $20k in a savings account for 4-7 years and you will raise your wage $20k/year for the rest of your life? Start investing! It will be free soon. lol.

This is only my most recent approach to this problem. Other answers which I have debunked are (for 8% interest) 13.8 years, 20 years, and 7.5 years. I'd like confirmation, or a correct answer please.
 
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  • #2
I think you haven't gotten any responses because it's not at all clear exactly what the set-up of the problem is.
 

FAQ: Interest continuously compounding with a variable prinicple

What is continuous compounding?

Continuous compounding is a method of calculating interest on a loan or investment where the interest is calculated and added to the principal continuously, rather than at specific intervals. This results in a higher total amount of interest earned over time compared to simple or compound interest.

What is the difference between continuous compounding and other types of compounding?

The main difference between continuous compounding and other types of compounding is the frequency at which interest is added to the principal. In continuous compounding, interest is added continuously, while in other types of compounding, interest is added at specific intervals such as monthly, quarterly, or annually.

How is the interest rate determined in continuous compounding with a variable principal?

In continuous compounding with a variable principal, the interest rate is typically determined by the market rate or the rate agreed upon in the loan or investment contract. This rate is then applied continuously to the changing principal amount, resulting in a continuously changing interest amount.

What are the benefits of continuous compounding with a variable principal?

The main benefit of continuous compounding with a variable principal is that it can result in a higher total amount of interest earned compared to other types of compounding. This is because the interest is added continuously, allowing it to compound more frequently and at a faster rate.

Are there any drawbacks to continuous compounding with a variable principal?

One potential drawback of continuous compounding with a variable principal is that it may be more difficult to calculate and understand compared to other types of compounding. Additionally, if the variable principal decreases, the interest earned may also decrease, which could impact overall earnings. It is important to carefully consider the terms and conditions of the loan or investment before choosing continuous compounding with a variable principal.

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