Insurance Math

Insurance Math

Problem One:  Consider two health plans with deductibles, coinsurance rate, and maximum allowable out-of-pocket limit as presented in the table below.

Characteristics of two health plans
Health Plan One
Health Plan Two
Coinsurance Rate
Maximum Allowable
 Out-of-Pocket Limit


At what level of total health expenditures does the insurance company start paying 100 percent of approved claims?

Write a piecewise linear function where out-of-pocket health care expenses is the Y variable and total health expenses is the X variable for the two health plans.   

Write a piecewise linear function where insurance company payment is the dependent variable and total health expenses is the X variable?

Demonstrate that the sum of the piecewise linear out-of-pocket expense function and the piecewise insurance payout function equals total health expenses in each part of the domain of the functions.

Answer to Problem One:

Problem Two:   A person is considering two insurance plans.   The first plan has a $5,000 deductible and a $5,000 maximum allowable out-of-pocket expense limit.   The second plan has a  $3,000 deductible, a 20% coinsurance rate, and a $9,000 out-of-pocket expense limit.  

Write a function describing insurance payments received by the insured as a function of total health expenses incurred under the two health plans.   

When are actual out-of-pocket expenses under the first insurance plan greater than out-of-pocket expenses under the second insurance plan?

Answer to Problem Two:

Problem Three:   The table below lists characteristics (deductible, coinsurance rate, and out-of-pocket expense limits) for three different health plans

Three Health Insurance Plans
Plan Number One
Plan Number Two
Plan Number Three
Coinsurance Rate
Out of-pocket Expense  Limit

The table below describes total health expenditures (the amount paid by the insured and the amount paid by the insurance company) for a sample of 35,000 people.

Health Expenditures
Number of People


What is the mean and the total health expenditure paid by the insurance firm for the three health insurance plans and this group of individuals?

A company offers all three insurance policies to its employees.  Discuss why average health expenditures will differ across these three plans?

Answer to Problem Three:


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