Are Insurance and Risk premium totally different?

I've been studying various aspects of utility function and I came across the definition of risk premium and insurance, which are mathematically very different from each other.

In the book "Theory of Asset pricing", page 17, risk premium $\pi$ is defined as the amount that would satisfy $E[U(W+X)]=U(W-\pi)$ where $W$ is the investor's wealth and $X$ is a a zero-mean risk. If we consider the case of small risks, we then get that $\pi=-\frac{1}{2}E[X^2]\frac{U"(W)}{U'(W)}$.

For me, this means that risk premium is an amount of money that we could be ready to pay to get rid of a risk / loss. Isn't that the core of insurance ?

What does each of these two really mean ?

Thanks !

• Your formula is missing something, like the stdev of $X$. – MJ73550 Feb 13 '17 at 7:13
• You're right, I've corrected it. – mlx Feb 14 '17 at 23:06

Apart from that, I think you are correct, in that you can see $\pi$ in your formula as the amount an insurance customer is willing to pay to insure herself against the downside risk X.