Insurance Premium that covers only the loss and not the profit For example: The utility function is ln(W), where W refers to the Wealth level. The initial wealth is $10,000 $ and you have a equal chance of winning and losing $1000. What if the insurance policy only covers the loss, how are you willing to pay for the insurance premium? Need some guidance on solving this question.

So you end up with either $10000-p$ or $11000-p$ instead of either $9000$ or $11000$. You must solve for $p$ in $$\frac 12\ln(10000-p)+\frac12\ln(11000-p) = \frac12\ln 9000+\frac12\ln 11000$$ or equivalently $$ (10000-p)(11000-p)=9000\cdot 11000$$ whihc is a quadratic in $p$ where one of the roots is between $0$ and $1000$ (and the other makes both factors on the left negative and can be ignored). It turns out that the premium will eat more than half of your hoped-for profit ...