Risk Premiums and Insurance Premiums Measure Different Exposures

Summary

The document compares a utility-based risk premium with the price of insurance. It defines the risk premium as the certain reduction in wealth that makes an investor indifferent to facing a zero-mean risk, and gives a small-risk approximation tied to the risk’s variance and the investor’s local risk aversion. This frames the premium as a willingness to pay to remove uncertainty.

Insurance usually covers specified losses while leaving the policyholder with gains when outcomes are favorable. Since that protection removes only the downside, it changes the risk’s distribution and generally its mean; it is not the same exposure as eliminating a zero-mean risk. The response says the utility-based amount can still represent a customer’s willingness to pay for downside protection, but an insurer must price from its own perspective and account for dependence across contracts. The exchange is conceptual: it provides no numerical example or detailed insurance pricing model, and the approximation applies to small risks.

Key ideas

• A utility-based risk premium is the certain wealth reduction that compensates for bearing a zero-mean risk.
• For small risks, the premium depends on risk variance and the investor’s local risk aversion.
• Insurance commonly removes downside losses while allowing the policyholder to retain favorable outcomes.
• An insurer’s premium calculation must account for its own risk and correlations among policies.

Tags

# Are Insurance and Risk premium totally different?


# 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 !

## Answer by Ami44 (score 2)

https://quant.stackexchange.com/a/32367

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.