More laws of large numbers?

Question

My teacher went over this one in class but I don't jive well with his teaching method. This is one of our examples for the chapter... Please help!

 

Suppose a company charges an annual premium of $120 for an insurance policy for minor injuries. Actuarial studies show that in case of an injury claim, the company will pay out an average of $900 for outpatient care and an average of $3000 for an overnight stay in the hospital. They also determine that, on average, each year there are five claims made that result in outpatient care for every 1000 policies and three claims made that result in an overnight stay out of every 1000 policies. What is the expected annual profit of an insurance policy for the company?

Answer 1

Problem Summary

A company charges an annual premium of $120 for a minor‑injury insurance policy. Actuarial data show the following claim information per 1,000 policies:

• 5 outpatient claims, each averaging $900

• 3 overnight‑stay claims, each averaging $3000

The task is to compute the expected annual profit per policy by calculating the expected payout and subtracting it from the premium.

Step by Step answer

Convert claim frequencies to probabilities:

  • Outpatient claim probability = 5 / 1000 = 0.005

  • Overnight stay probability = 3 / 1000 = 0.003

Compute the expected payout per policy:

  Expected payout = (0.005 × 900) + (0.003 × 3000)

  = 4.50 + 9.00

  = $13.50

Compute the expected profit:

  Expected profit = Premium − Expected payout

  = 120 − 13.50

  = $106.50

The expected annual profit per policy for a 1000 policies if priced at $120 per policies is $106.50 or $106,500 on the portfolio of policies (106.50 x 1000) assuming the exact same experience.

However remember life never (or very rarely) follows the model precisely.