Continuously Compounded Interest What exactly does it mean? By continuously compounded it makes me think it is almost like multiplied as time goes on. Could someone also explain what the constant e is and how it originated? Also how do they get the compounded interest formula from the constant e?
So lets look at the what we mean by compound interest. Let use $r$ as a rate for a year. If we want to define a semiannual compound rate we do the following
$$
\left(1+\frac{r}{2}\right)\left(1+\frac{r}{2}\right) = \left(1+\frac{r}{2}\right)^{2}
$$
for quarterly we have
$$
\left(1+\frac{r}{4}\right)^4
$$
since at the end of the year we have $4$ growths where we have invested the increased value.
in general we have for $n$-periods we have
$$
\left(1+\frac{r}{n}\right)^n
$$
now if we have a continuous growth then we are effectively saying that period becomes infinitesimally small i.e. $n\to\infty$
Mathematically speaking
$$
\lim_{n\to\infty}\left(1+\frac{r}{n}\right)^n\to e^r
$$ 
where $\mathrm{e}$ is the exponential constant. A really special function in mathematics is $\mathrm{e}^x$ where the gradient at any point of the graph $\mathrm{e}^x$ is proportional to itself, so ideal for growth formula.

Now we can extend this from one year to many years by using 
$$
\left(1+\frac{r}{n}\right)^{nt}
$$
where we can adjust the index by $nt=k$ we get
$$
\left(1+\frac{rt}{k}\right)^{k}
$$
if we have $r,t&lt;\infty$ then we have 
$$
\lim_{k\to\infty}\left(1+\frac{rt}{k}\right)^{k} \to \mathrm{e}^{rt}
$$
Let denote the interest per annum (year) as $i$. Therefore the compound interest is $1+i$. Now you can compound twice a year. But you take only the half of the interest rate, because the period between the compoundings are only half as long as before.

$\left(1+\frac{i}{2} \right) ^2$

Then you compound 4 times a year (every three months). Thus you divide i by 4 and exponentiate it by 4.

$\left(1+\frac{i}{4} \right) ^4$

Now imagine you do compound infinitely times in a year. The expression becomes 

$$\lim_{n \to \infty}\left(1+\frac{i}{n} \right)^n=e^i $$

This means that the interest per year is effectively $(e^i-1)\cdot 100\% \neq i$

$\color{blue}{\text{numerical example}: i=0.05}$

$(e^{0.05}-1)\cdot 100\% =5.127\%\neq 5\%$

You see, that this tansformation from yearly compounding to continuous compounding is not equivalent.