expected value probability

Question

You are selling security insurance services to a small business, and wish to know how to price your policy. In particular, you will reimburse the client for the costs of any attack that results in more than $10,000 of damage, up to a limit of $1,000,000. You know based on your data that attacks that result in more than $1,000,000 occur during a given year no more than 5% of the time, and attacks costing more than $250,000 but less than $1,000,000 occur no more than 15% of the time. How much should you charge the client on a yearly basis to ensure that you do not lose money on the insurance policy in expectation? Note that your information is incomplete, so you should make pessimistic assumptions where necessary to guarantee your profits. 

Answer 1

To break even, you charge your client the EV for the costs. ( Obviously, you have to charge more than this to make money and as a buffer for bad luck. You can't calculate the EV without knowing how often the lower cases occur, so you could make a different assumption that makes sense to you.

EV = .05*(1,000,000) + .15*(625,000) + .8(24/25*(130,000) )

top group: you only pay 1M

middle group: EV from 250K to 1M is 625K

Bottom group: I took the rest of the cases to be claims below 250K including cases that were below 10K at a rate equal to 1/25 of the rest of the under 250K cases. This is ludicrous because there may be 50% no claims making this exercise a bit futile. If you assumed 50% no claims (completely arbitrary), the the last term would be .3(130,000). Note that the last term is the largest contributor as is, but with 50% no claim, it becomes the smallest contributor to the EV. Good luck.