Question about Compound Interest Formula

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  • #1
opus
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If I have an investment, that is compounded at some rate ##r##, ##n## times per year, it can be written as a function as such:

$$A(t)=P\left(1+\frac{r}{n}\right)^{nt}$$

My question is in regards to the 1 here. I think I have a general idea of what it's for, but I can't really put it into correct words.
What it seems to be doing, is keeping the new compounded value above ##P##. Where if the 1 wasn't there, we would be getting a value less than ##P##. But this seems wishy washy and I'd like to put it into more definitive terms so that I can understand it better. Can anyone help me out with this?
 

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  • #2
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If the compounding rate was say 4% per year compounded once per year then the expression would be 1.04 or 104%

Given a hundred dollar loan then with the 104% means after one year we’d need to pay back 104 dollars.
 
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  • #3
opus
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And under the same circumstances, and removing the 1 from inside the parentheses, that would be just $4. And when we do have the one inside the parentheses, we are adding that $4 to the initial principal investment?
 
  • #4
BWV
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Think about it in reverse. So in academic finance you often talk about capital R returns which is A(t+n) / A(t) so the R in the example above is 1.04. Lower case r is then used for the cumulative rate of return which is R-1.

The convenience of thinking about R is that you can then choose any units of time to subdivide you like, and most often you can forget about n and use log returns rather than compound as they are easier to work with so

r annualized = log(R)/t if t is in units of years, for example
 
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  • #5
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And under the same circumstances, and removing the 1 from inside the parentheses, that would be just $4. And when we do have the one inside the parentheses, we are adding that $4 to the initial principal investment?
Yes. You can keep your investment. That is the 1 in the formula.
 
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  • #6
opus
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Thanks guys.
 
  • #7
mathman
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The reason for the parentheses is to show that each iteration gives you interest on the accumulated interest as well as on the principal. That's why it is called compound.
 
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