Question

An insurance company is contemplating whether to sell insurance to a group of 121 hospitals for $730 million. For each bankruptcy, the insurance company would pay out $500 million for up to 6 bankruptcies (or $3 billion total). The probability for bankruptcy is given as 3.76%. Hospital bankruptcies are independent w the same probability. What is the overall net expected value of the investment? Now suppose p(bankruptcy) varies w the economy. There's a 41.7% chance that p is is.0376, a 50.45% chance that p is .0025 and a 7.85% chance the p is .2630. What is the overall net expected value of the investment?

Answer 1

Consider a random variable X denoting the number of bankruptcies...x=0,1,2,3,4,5,6

a) Then X is a binomial random variable with probabiillty of success p as .0376 and failure 1-p =0.9624

Since for each bankruptcy the insurance company should have to pary 500 million. So the expected amount the company has to pay for the expected value of bankrupcy= 500*0.2256=$112.8 million

So the net expected value of investment is $730-$112.8=$617.2 million

b) If the p varies with economy then the distribution of X will be a mixture of three binomial distributions with probabiliities 0.417,0.5045 and .0785 respectively

P(X=x) =P(x bankruptcy given that p=.0376).P(p=.0376)+ P(x bankruptcy given that

p=.0025).P(p=.0025)+P(x bankruptcy given that p=.2630).P(p=.2630)

Expected value that the company has to pay =$500*0.2255=$112.75 million

Overall net expected value of the investment=730-112.75=$617.25 million