# 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 $X$. Feb 13, 2017 at 7:13
• You're right, I've corrected it.
– mlx
Feb 14, 2017 at 23:06

## 1 Answer

An Insurance premium typically focuses solely on the downside of your Risk. An Insurance pays if you suffered some damage, but you do not give them some share of your profit if things are good.

That means you have to get rid of the positive part of X, which has than of course a non-zero mean.

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.

The insurer on the other hand has of course a totally different view and utility, when calculating the premium. For example she can't calculate each contract separatly because she has to factor in correlations.

• A good answer. But also insurance is simply not available for many situations (for eg. divorce insurance when you get married) that are non-insurable for various reasons. So risk-premium is more general, it also exists when insurance does not. Feb 15, 2017 at 0:45