B Question about Compound Interest Formula

[opus]

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?

 

[jedishrfu]

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.

 

[opus]

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?

 

[BWV]

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

 

[mfb]

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.

 

[opus]

Thanks guys.

 

[mathman]

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.