Your wealth is $€14400$. If you have no accident, your wealth remains unchanged, but if you do, you need to pay $€4400$ to cover medical costs. If you have an accident, the insurance reimburses the medical costs but you need to pay a premium $p=€2300$ whether you have an accident or not. Your utility over wealth is $U(w) = {\sqrt w}$, the probability of an accident is $q$.
(1) Suppose that you believe there is a probability $q=0.1$ of having an accident. What is the highest premium $p$ you would be willing to pay for the insurance?
In this case the expected utility without insurance would be:
$E [U(N)]=q*U(14400-4400)+(1-q)*U(14400)$
$E [U(N)]=q*U(10000)+(1-q)*U(14400)$
$E [U(N)]=q*{\sqrt 10000}+(1-q)*{\sqrt 14400}$
$E [U(N)]=q*100+(1-q)*120$
$E [U(N)]=0.1*100+0.9*120$
$E [U(N)]=118$
In order to buy the insurance, expected utility with insurance (minus premium price) must be grater or equal the expected utility without insurance. $E [U(I-premium)] ≥ E [U(N)]$.
So, $E [U(I-premium)] ≥ 118]$
Is it correct?
But I have also seen this this rule:
$E [U(I)]=q*U(14400-premium)+(1-q)*U(14400-premium)≥0$
$E [U(I)]=0.1*U(14400-premium)+0.9*U(14400-premium)≥0$
And when I calculate it, I obtain that, with a probability of accident of $q=0.1$, the risk highest risk premium that a person would be willing to pay is exactly $€14400$, which is her/his entire wealth. This is a nonsense.
How can I calculate the highest premium that a person would be willing to pay for the insurance?
Any help is appreciated.
(2) Suppose that you still believe $q=0.1$, but that your utility over wealth is now $U(w) = w$. What is the highest premium $p$ you would be willing to pay for the insurance? Explain intuitively how this premium compares with the one you found in the previous point.
Here is the difference that this person is risk neutral, because of utility function $U(w) = w$, and in the first case we know that the person is risk averse, because of $U(w) = {\sqrt w}$ is a concave function. But how does the premium change? How can I calculate it?
Any help is appreciated.
Thanks in advance.
Suppose the insurance premium is $p$, and your wealth is $I$. You’ve found that your expected utility with insurance is just $U(I-p) = \sqrt{I-p}$.
You prefer to purchase insurance whenever the above quantity is higher than your expected utility without insurance, which you’ve found to be $118$. Thus, you are looking for the largest $p$ such that $$ \sqrt{14400 -p} \ge 118.$$
You can follow similar steps when $U(w)=w$.