$2.$ Securitization provides a way to diversify idiosyncratic risk. You are given an example in the notes of a $ $100$ loan with a default rate of $5\%.$ For $100$ households, $95$ pay $$115$ and $5$ pay nothing. The mortgage$-$backed security created by pooling the loans guaranteed a payment of $$109\mathrm{.}25$ to each of the $100$ lenders. Now suppose that there is an unanticipated increase in the rate of default, from $5\%$ to $10\%.$ Determine the expected value of a loan and the standard deviation with the new default rate. The mortgagebacked security had guaranteed a payment of $$109\mathrm{.}25.$ Is that payment still possible with the higher default rate? Explain.

$3.$ We will use the example above and interpret this as health insurance. Suppose there are $100$ people: $95$ people will be healthy and $5$ will be sick, but at the time the insurance is purchased, no one knows who will be sick. Assume purchase of insurance is mandatory. If you are healthy, you incur no health costs. If you become sick, you incur costs of $$115.$ The total health expenses for the $5$ people will equal $$575($so $5$ times $115).$ If all $100$ people split the cost of the illness, what is the insurance premium paid by each person? Show that this payment is equal to the probability of illness times the cost per person, or the expected value of health costs for an individual. An important part of the statement of the problem was the assumption insurance purchase is mandatory. Suppose that insurance purchase is not mandatory and there is a no pre$-$condition clause, meaning you cannot be denied insurance coverage because of a pre$-$condition. Explain how the absence of mandatory insurance plus the no pre$-$condition clause leads to an adverse selection problem.