The insurance policy pays \(\$ 100,000\) if one of them dies and \(\$ 150,000\) if both die during this time. What is a fair cost for this policy?

Question

A 65 year old couple are considering a joint life insurance policy. The man has a probability of \(.90\) of living at least 5 more years, \(.95\) for the woman (and assume the event of either person dying is independent of the other). The insurance policy pays \(\$ 100,000\) if one of them dies and \(\$ 150,000\) if both die during this time. What is a fair cost for this policy?

Answer 1

Since their deaths are assumed to be independent,
\(P\) (neither dies \()=(.9)(.95)=.855\)
\(P(\) one dies \()=P(\) only husband dies \()+P(\) only wife dies \()=(.1)(.95)+(.9)(.05)=\) \(.14\)
\(P(\) both die \()=(.1)(.05)=.095\)
The payout is \(0,100000,150000\) respectively in the three cases. The payout is the value the random variable takes. So its expected value is
\[
E(X)=0(.855)+100000(.14)+150000(.095)=28250
\]
So a fair price for the policy would be exactly \(\$ 28,250\), since that is the expected payout.