# Question about Compound Interest Formula

• B
• opus
In summary, the formula for compounding an investment at a rate of ##r##, ##n## times per year is $$A(t)=P\left(1+\frac{r}{n}\right)^{nt}$$ The 1 in the formula ensures that the new compounded value remains above the initial principal value. Removing the 1 would result in a value less than the initial principal. This formula can also be expressed in terms of capital R returns, which is the accumulated rate of return, and lower case r, which is the cumulative rate of return. The parentheses in the formula indicate that each iteration gives interest on the accumulated interest as well as on the principal, hence the term "compound".f

#### opus

Gold Member
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?

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?

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

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?
Yes. You can keep your investment. That is the 1 in the formula.

opus
Thanks guys.

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.

opus