Two Plans
Consider two health plans with deductibles, coinsurance rate, and maximum allowable outofpocket limit as presented in the table below.
Characteristics of two health plans
 
Health Plan One

Health Plan Two
 
Deductible

$5,000

$1,000

Coinsurance Rate

0.2

0.5

Maximum Allowable
OutofPocket Limit

$10,000

$15,000

Questions:
At what level of total health expenditures does the insurance company start paying 100 percent of approved claims?
Write a piecewise linear function where outofpocket 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 outofpocket expense function and the piecewise insurance payout function equals total health expenses in each part of the domain of the functions.
Definitions:
Deductible: A specified amount of money that a person must pay before an insurance company will pay a claim.
Coinsurance Rate: The percent of cost on a claim the insured person pays after the deductible is met and prior to the customer meeting the maximum allowable outofpocket limit.
Maximumallowableoutofpocket limit. The most an insured person is required to pay for claims during a calendar year. Once this limit is reached the insurance company pays 100% of all approved claims.
Piecewise Lines Answers
At what level of total health expenditures does the insurance company start paying 100 percent of approved claims?
The level of total health expenses triggering the maximum allowable outofpocket limit can be solved by solving the following equation
OPM = D + r*(TEMD)
Where
OPM is outofpocket maximum,
r is the coinsurance rate,
D is the deductible,
TEM is the level of TE triggering the maximum allowable outofpocket expense limit. (Once TEM is hit the insurance company pays 100 percent of all claims)
For equation one plug D=$5,000, r=0.2, and OPM into the equation. Solve for TEM and get TEM = $30,000
Confirm that when TEM=$30,000 the outofpocket expense level is indeed $10,000 by plugging $30,000 into the OPM equation and obtaining outofpocket expenses equal to $10,000.
For equation two plug D=$1,000, r=0.5 and OPM=$15,000 into the equation. Rearrange for TE. I get TEM=$29,000.
I check my answer by plugging TEM=$29,000 into the outofpocket equation with D and r and gat outofpocket expenses equal to $15,000.
The piecewise lines for outofpocket expenses:
Write a piecewise linear function where outofpocket health care expenses is the Y variable and total health expenses is the X variable for the two health plans.
There are three sections of the piecewise outofpocket expense function. The first section is for total expenses from 0 to the deductible. The second section is from the deductible to the value of total expenses triggering the outofpocket expenses. (We just calculated the trigger points.) The third section is for total expenses over the level triggering the outofpocket expense limit.
The piecewise outofpocket line is presented below.
Piecewise Functions for OutofPocket Expenses
 
Health Plan One
 
TE

OPE

$0 to $5,000

TE

$5,000 to $30,000

$5.000+0.20*(TE$5,000)

>$30,000

$10,000

Health Plan Two
 
$0 to $1,000

TE

$1,000 to $29,000

$1,000+o.5*(TE$1,000)

>$29,000

$15,000

Write a piecewise linear function where insurance company payment is the dependent variable and total health expenses is the X variable?
The insurance company pays $0 when total expenses are under the deductible, r*(TED) once the deductible is met and until the outofpocket limit is met.
At outofpocket maximum, the insurance company pays (1r)*(TEMD).
Above the outofpocketmaximum the insurance company pays
(1r)*(TEMD)+(TETEM).
(The term (1r)*(TEMD) is a constant. For the first health plan this constant is .8*(300005000) or $20,000.) Confirm the constant is $14,000 for the second health plan.
The piecewise Insurance Payment Lines are below
Piecewise Insurance Payout Lines
 
Health Plan One
 
TE

IP

$0 to $5,000

$0

$5,000 to $30,000

0.2*(TE$5,000)

>$30,000

20,000+(TE30000)

Health Plan Two
 
TE

IP

$0 to $1,000

$0

$1,000 to $29,000

0.5*(TED)

>$29,000

14,000+(TE29000)

Analyzing the sum of the two Piecewise Functions:
I will analyze the sum of the outofpocket and insurance payout piecewise functions for the first health plans and encourage the reader to do so for the second health plan.
For total health expenses between $0 and $5,000 the outofpocket expenses are $TE and the insurance payout is $0. The sum of the two is $TE.
For total health expenses between $5,000 and $30,000 outofpocket expenses are $5,000 +0.2*(TE5000) and insurance payouts are 0.8*(TE5000). The sum of two expressions is TE.
For total health expenses greater than $30,000, outofpocket expenses are $10,000 (the maximum) and insurance payouts are 20,000+TE30000. The sum of the two is again TE.
So the sum of total outofpocket expenses and insurance paid expenses is equal to total expenses.
Go here for other problems on insurance math:
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