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So let's say a company sells life insurance policies, which require customers to pay an annual fee. Assume that if the customer happens to die in that year, the company does not charge the customer's beneficiaries the fee and will also pay out the amount that the policy is worth. If the average customer has a 2.5% chance of dying in the next year and the policy is worth $100,000 then what is the minimum amount the company should charge for the yearly payment so that on average they won't lose any money? I'm having trouble starting out this question. I've learned about things like how to calculate profit however those questions give me the mean and standard deviation and this one doesn't.

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Hint: The company has a $2.5\%$ chance of losing $\$100,000$ and a $97.5\%$ chance of collecting the fee $f$. You want the expectation to be zero.

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  • $\begingroup$ So am I trying to find the expected value? $\endgroup$ – David Rolfe Feb 26 '16 at 18:39
  • $\begingroup$ That is right. If the expected value is negative, they lose money. If it is positive, they can reduce the fee and still not lose. $\endgroup$ – Ross Millikan Feb 26 '16 at 19:41
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If 2.5% of people will not be paying the fee, and will be receiving $100 000, then we have 97.5% of people who will be paying.

What we need is that the money payed out equals the money taken in :

100000 * 0.025 = 0.975 * p

Solving for the premium (p) we get :

2500         = 0.975p
2564.1025641 = p

The company should therefore charge a minimum of 2564.11 (rounding up) to not lose money.

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  • $\begingroup$ Besides multiplying 100000 * 0.025, what exactly are you doing? Why do you need to go 2500 = 0.975p rather than just taking 2500 and saying that's what the company should charge. $\endgroup$ – David Rolfe Feb 26 '16 at 18:55
  • $\begingroup$ Hi David, The other half of the equation is just scaling the price. If the people who died also payed the premium, then that would be the correct answer (2500). Since they do not, you divide the total payout to 97.5 % of the people, not by 100%. $\endgroup$ – J. Bush Feb 26 '16 at 19:18
  • $\begingroup$ So basically if the customer dies in that year and the company does charge the customer's beneficiaries the fee then it would be $2500? $\endgroup$ – David Rolfe Feb 26 '16 at 23:11

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