Expected Utility of Wealth with Coinsurance and a Premium

Summary

The document works through expected utility for a person facing a uniformly distributed loss, with a specified wealth level, coinsurance rate, and gross premium. The key correction is to represent residual wealth as initial wealth minus the premium and the portion of the loss that remains uncovered. Expected utility is then the average of the utility of that residual wealth over the full loss distribution.

This setup applies the stated power utility function to wealth after insurance and integrates across possible losses. It corrects the questioner’s piecewise expression, which treats the premium as if it fully eliminates losses below a threshold. The answer gives a numerical expected utility matching the stated reference result. The example is narrow: it depends on the assumed utility function, loss distribution, contract terms, and interpretation of coinsurance, and it does not compare insurance choices or establish a general pricing rule.

Key ideas

• Expected utility is calculated by averaging utility over all possible loss outcomes.
• Residual wealth equals initial wealth less the premium and the uninsured share of the loss.
• Coinsurance leaves a portion of each loss with the policyholder under the interpretation used.
• A piecewise calculation can be wrong if it assumes coverage stops or changes at an unsupported loss threshold.
• The numerical answer depends on the specified utility, wealth, loss distribution, and insurance terms.

Tags

# How to compute the mean for utility function?


# How to compute the mean for utility function?












Let $u(x)=x^{2/3}$, $x>0$ be the utility function, $X \sim U(0, 100)$ is loss, wealth $w=\\\$150$.

Calculate $\mathbb{E}(u(w_r))$ if a coinsurance is $80\%$ and gross premium is $\\\$43$.

My attempt is:

$E(u(w_r))=\frac{1}{100}\int_0^{43}(150-43)^{2/3}dt + \int_{43}^{100} (150-0.8t)^{2/3}dt=21.3486$

But answer is $21.103$.

Question. How to compute the mean for utility function?

## Answer by Kermittfrog (score 3)

https://quant.stackexchange.com/a/70256

Maybe you misinterpreted the protection? Also, utility does not stand in $\\\$$. I'd come up with

$$ \frac{1}{100}\int_0^{100}(150-43-0.2t)^{2/3}\mathrm{d}t=21.103 $$

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