# How to calculate annually compounded interest?

I am having some troubles understanding this topic of interest rates.

For example, if i invest 1 dollar at a continuous compounding rate of 11% , then my end of year value is equal to $$e^{0.11}=1.116$$ dollars. From here it says that investing at 11% a year continuously compounded is the same as investing 11.6 a year annually compounded.

Now where is this 11.6 coming from ? Is it coming from the 1.(116) dollars value? So i just take the decimal 0.116 and transform it into a percentage?

Another question I have regards this problem:

Suppose the annually compounded rate is 18.5%. The present value of a $$100$$ perpetuity, with each cash flow received at the end of the year, is $$100/.185 =\540.54.$$ If the cash flow is received continuously, we must divide \\$100 by 17%, because 17% continuously compounded is equivalent to 18.5% annually compounded (with the explanation that $$e^{0.17}=1.185).$$

What is the relationship between the annually compounded rate and the continuously compounded rate ? How do I use one to calculate the other ? I am very confused.

$$11\%$$ per year, compounded continuously is (approximately) equivalent to $$11.6\%$$ per year, compounded annually. If $$i$$ is the annual interest rate, the equivalent continuous rate is $$\ln(1+i)$$. The reason is simply that $$e^{\ln(1+i)}=1+i,$$ so that if you invest a dollar at a continuous rate of $$\ln(1+i),$$ at the end of a year, you have exactly what you would have had you invested the dollar at a rate of $$i,$$ compounded annually.
Of course, if you know the continuous rate $$\delta$$ and you want the equivalent annual rate $$i,$$ it's just $$i=e^\delta-1$$